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A uniformly but not normally convergent function series
Consider a functions series \(\displaystyle \sum f_n\) of functions defined on a set \(S\) to \(\mathbb R\) or \(\mathbb C\). It is known that if \(\displaystyle \sum f_n\) is normally convergent,
then \(\displaystyle \sum f_n\) is uniformly convergent.
The converse is not true and we provide two counterexamples.
Consider first the sequence of functions \((g_n)\) defined on \(\mathbb R\) by:
\[g_n(x) = \begin{cases}
\frac{\sin^2 x}{n} & \text{for } x \in (n \pi, (n+1) \pi)\\
0 & \text{else}
\end{cases}\] The series \(\displaystyle \sum \Vert g_n \Vert_\infty\) diverges as for all \(n \in \mathbb N\), \(\Vert g_n \Vert_\infty = \frac{1}{n}\) and the harmonic series \(\sum \frac{1}{n}\)
diverges. However the series \(\displaystyle \sum g_n\) converges uniformly as for \(x \in \mathbb R\) the sum \(\displaystyle \sum g_n(x)\) is having only one term and \[
\vert R_n(x) \vert = \left\vert \sum_{k=n+1}^\infty g_k(x) \right\vert \le \frac{1}{n+1}\]
For our second example, we consider the sequence of functions \((f_n)\) defined on \([0,1]\) by \(f_n(x) = (-1)^n \frac{x^n}{n}\). For \(x \in [0,1]\) \(\displaystyle \sum (-1)^n \frac{x^n}{n}\) is
an alternating series whose absolute value of the terms converge to \(0\) monotonically. According to Leibniz test, \(\displaystyle \sum (-1)^n \frac{x^n}{n}\) is well defined and we can apply the
classical inequality \[
\displaystyle \left\vert \sum_{k=1}^\infty (-1)^k \frac{x^k}{k} – \sum_{k=1}^m (-1)^k \frac{x^k}{k} \right\vert \le \frac{x^{m+1}}{m+1} \le \frac{1}{m+1}\] for \(m \ge 1\). Which proves that \(\
displaystyle \sum (-1)^n \frac{x^n}{n}\) converges uniformly on \([0,1]\).
However the convergence is not normal as \(\sup\limits_{x \in [0,1]} \frac{x^n}{n} = \frac{1}{n}\).
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purgeR tutorial
purgeR is a package for the estimation of inbreeding-purging genetic parameters in pedigreed populations. These parameters include the inbreeding coefficient (\(F\)), partial (\(F_{i(j)}\)),
ancestral (\(F_{a}\)) and purged (\(g\)) inbreeding coefficients, as well as the total and expressed opportunity of purging (\(O\) and \(O_{e}\), respectively). Only genealogical records are required
to estimate them, and all individual estimates will be stored in a dataframe. Thus, purgeR provides the raw material for subsequent analysis on inbreeding depression and genetic purging (see the ‘ip’
vignette for more detailed examples on this).
In addition, functions are also included for the pre-processing of pedigrees, and for the analysis of population diversity (e.g. effective population size), and the inference of time (e.g. number of
equivalent to complete generations), bottlenecks (e.g. effective number of founders and ancestors, and founder genome equivalents) and fitness (e.g. breeding success and reproductive value), among
others. All these functions are helpful to contextualize the demographic circumstances under which inbreeding and purging occur, as well as their consequences.
The next sections give a practical introduction to all functions contained in the purgeR package. The tidyverse R dialect is used throughout the tutorial, including the pipe operator %>%. Users
unfamiliar with it are encouraged to read the introductory book R for Data Science.
Basic input format
Most functions contain a mandatory argument ‘ped’, that will be used to input a dataframe with pedigree information. Pedigree dataframes need to follow some rules:
• Three columns are mandatory: individuals, mothers and fathers identities.
• Individuals must be sorted from older to younger.
To facilitate the usage of pedigrees and improve reproducibility, most functions will in addition require that columns for individuals, mother and fathers’ identities are named ‘id’, ‘dam’, and
‘sire’ respectively, all of type integer, with unknown parents named ‘0’. Individuals should in addition be named in order, from 1 to N.
There is no restriction to the addition of more columns, e.g. containing measures of individual genetic or environmental factors.
Sort and rename individuals
Example pedigrees in this package are given already sorted, which means that ancestors are always placed on top of descendants. This is a requirement for all functions in the package, except for
ped_sort, which is a function dedicated to sort individuals following Zhang et al. (2009) algorithm. See ?ped_sort for an example of use.
The function ped_rename is the most important pre-processing function in the package, and it will make sure that all input requirements are met, while making the changes needed for the remaining
functions to work properly. Consider the example below using the pedigree of the Darwin/Wedgwood family:
William Darwin I Unknown unknown
Mary Healey unk UNK
Gilbert Wedgwood 0 NA
Margaret Burslem 0 0
Anne Earle 0 0
William Darwin II Mary Healey William Darwin I
After using ped_rename, the pedigree is checked, and individuals are renamed in a proper format:
darwin <- purgeR::ped_rename(
ped = darwin,
id = "Individual",
dam = "Mother",
sire = "Father",
keep_names = TRUE
1 0 0 William Darwin I
2 0 0 Mary Healey
3 0 0 Gilbert Wedgwood
4 0 0 Margaret Burslem
5 0 0 Anne Earle
6 2 1 William Darwin II
Note that in the example only the first 6 rows are shown. In the renamed dataframe, Charles R. Darwin will appear with id = 52. Note as well the use of the option keep_names = TRUE. This will store
the original individual identities on a separate column ‘names’.
Reduce pedigree size
Downstream analyses may require at least one additional variable (column) containing some measurement of biological fitness (or any other value), meaning that individuals with no data available
(i.e. with NA value) can be filtered out, as long as they are not ancestors of any other individual with available data. This is the job of the function ped_clean, that will reduce the size of the
pedigree, and may improve the performance of inbreeding/purging functions in large pedigrees.
Taking as example the Dama gazelle pedigree (1316 individuals), ped_clean will reduce the pedigree size to 1176 individuals for the analyses of 15-days survival, and to 389 only when analyzing female
dama %>% nrow()
#> [1] 1316
dama %>%
purgeR::ped_clean(value_from = "survival15") %>%
#> [1] 1025
dama %>%
purgeR::ped_clean(value_from = "prod") %>%
#> [1] 375
Note that ped_clean will require a renamed input pedigree. After its filtering step, it will automatically rename again the output pedigree.
Inbreeding and Purging
Several measures of inbreeding and purging can be computed, based on the probability of allele identity by descent of individuals of the pedigree. All functions related to inbreeding and purging are
prefixed with ip_.
Wright’s inbreeding coefficient
The inbreeding coefficient (\(F\), Wright 1922), here also referred to as standard inbreeding, is defined as the probability that an individual inherits two alleles derived from the same ancestor
(i.e. identical by descent, IBD). In pedigreed populations, this can be calculated for an individual \(i\) as the kinship coefficient of its parents \(j\) and \(k\) (\(F_{i} = f_{j,k}\)), which can
be calculated as:
\[f_{j,k {} (j=k)}=\frac{1}{2}(1+F_{j})\]
\[f_{j,k { } (j\neq k)}=\frac{1}{2}(f_{j,k_{d}}+f_{j,k_{s}})\] Where \(k_{d}\) and \(k_{s}\) refer to \(k\)’s dam and sire (see Falconer & Mackay 1996).
The function ip_F computes the inbreeding coefficient, given an input pedigree. Note that the value of \(F\) will be saved in a new column of the dataframe, as it is usually convenient to save it
this way to simplify the computation of further inbreeding and purging parameters, as well as for later analyses.
The example below shows the inbreeding coefficient of William E. Darwin (son of Charles R. Darwin and Emma Wedgwood).
darwin <- darwin %>% purgeR::ip_F()
darwin %>% dplyr::filter(names == "William Erasmus Darwin")
#> id dam sire names Fi
#> 1 60 54 52 William Erasmus Darwin 0.06298828
\(F\) can also be estimated based on population estimates of the effective population size \(N_{e}\) and generation numbers, using the classical expression (Falconer and Mackay 1996):
\[F_{t} = 1 - (1-\frac{1}{2N})^{t}\] This can be achieved with the function exp_F (e.g. exp_F (Ne = 50, t = 50)).
Partial inbreeding coefficient
As mentioned above, IBD happens when alleles are inherited from the same ancestor and appear in homozygosis. Thus, \(F_{i}\) can be partitioned as the additive contribution of its ancestors to \(F_
{i}\). The partial inbreeding coefficient \(F_{i(j)}\) is defined as \(i\)’s probability of IBD for alleles coming from ancestor \(j\). It can be computed from partial kinship coefficients (\(f_{p_
{1},p_{2}(j)}\), where \(p_{1}\) and \(p_{2}\) refer to \(i\)’s parents), so that \(F_{i(j)}=f_{p_{1},p_{2}(j)}\), using the tabular method as described by Gulisija & Crow (2007). Given an ancestor \
• All \(f_{p_{1},p_{2}(j)}\) values are initialized to \(0\), except for the diagonal entry corresponding to founder \(j\), which takes value \(1/2\).
• Values for intermediate ancestors are computed as follows:
□ When \(p_{1}=p_{2}\): \(f_{p_{1},p_{2}(j)} = f_{p_{1},j}+\frac{1}{2}f_{p_{1d},p_{1s}}\), where \(p_{1d}\) and \(p_{1s}\) are \(p_{1}\)’s parents.
□ When \(p_{1}\neq p_{2}\): \(f_{p_{1},p_{2}(j)} = \frac{1}{2}(f_{p_{1d},p_{2}}+f_{p_{1s},p_{2}})\), where \(p_{2}\) is older than \(p_{1}\).
The function ip_Fij will return a matrix object with all possible values of the partial inbreeding coefficient. In that matrix, the value in row \(i\) and column \(j\) indicates the probability of
IBD of individual \(i\) for alleles coming from ancestor \(j\). Values in the upper diagonal of the matrix always take values of zero. Of course, the summation of \(F_{i(j)}\) over every column \(j\)
equals \(F_{i}\) when \(j\) are founder ancestors.
m <- ip_Fij(arrui, mode = "founders") # ancestors considered are founders (by default)
base::rowSums(m) # this equals ip_F(arrui) %>% .$Fi
By default, ip_Fij only considers partial inbreeding conditional to founders, but it can also be extended to any ancestor using the mode = "all" argument. A custom number of individuals can also be
used (see ?ip_Fij). Note however that for a large number of individuals, the computation of this matrix may require a substantial amount of time. In every case, columns of the returned matrix are
sorted by ancestor identity use.
Figure below shows the contribution of the two founders in the Barbary sheep pedigree to inbreeding values \(F>0.35\).
arrui <- arrui %>% purgeR::ip_F()
tibble::tibble(founder1 = m[, 1], founder2 = m[, 2], Fi = plyr::round_any(arrui$Fi, 0.025)) %>%
tidyr::pivot_longer(cols = c(founder1, founder2), names_to = "Founder", values_to = "Fij") %>%
dplyr::group_by(Fi, Founder) %>%
dplyr::summarise(Fij = sum(Fij)) %>%
ggplot() +
geom_bar(aes(x = Fi, y = Fij, fill = Founder), stat = "identity", position = "fill") +
scale_x_continuous("Inbreeding coefficient (F)", limits = c(0.35, 0.625)) +
scale_y_continuous("Partial contribution to F (in %)", labels = scales::percent_format()) +
scale_fill_manual(values = c("darkgrey", "black")) +
panel.background = element_blank(),
legend.position = "bottom"
Alternatively, partial inbreeding can also be estimated via genedrop simulation (using option genedrop). This will however result in less precise estimation of \(F_{i(j)}\), and might only be
convenient to use in terms of performance for very large and complex pedigrees. In these cases, a value of genedrop = 100 might give results that are well correlated with exact estimates (\(r > 0.9\)
for the pedigree examples provided).
Ancestral inbreeding coefficient
The ancestral inbreeding coefficient (\(F_{a}\), Ballou 1997) measures the probability of IBD of an individual for an allele that has been in homozygosity in at least one ancestor.
This parameter provides information not only about inbreeding, but can also be used to detect purging, since individuals with inbreeding \(F\) and ancestral inbreeding \(F_{a}\) are expected to be
more fit than individuals with the same level of inbreeding but lower \(F_{a}\), given that the ancestors of the former have survived and reproduced despite their higher inbreeding (see Boakes & Wang
2005 and López-Cortegano et al. 2018 for analyses using this parameter).
Ancestral inbreeding can be estimated for an individual \(i\) with dam \(d\) and sire \(s\) as:
\[F_{a_{i}} = \frac{1}{2}[F_{a_{d}} + (1-F_{a_{d}})F_{d} + F_{a_{s}} + (1-F_{a_{s}})F_{s}]\] Alternatively, a gene-dropping simulation approach can be used, following Baumung et al. (2015), providing
unbiased estimates of \(F_{a}\). This is because, above expression assumes that \(F\) and \(F_{a}\) are uncorrelated, which is not true.
Both approaches can be used with the function ip_Fa. Note that the computation of \(F_{a}\) requires estimating \(F\) in advance. Use argument Fcol to declare a column with \(F\) values if it has
been computed and saved in advance (this will save time), or leave it blank to compute it on the go.
# F was pre-computed above
darwin %>%
purgeR::ip_Fa(Fcol = "Fi") %>%
dplyr::filter(names == "William Erasmus Darwin")
#> id dam sire names Fi Fa
#> 1 60 54 52 William Erasmus Darwin 0.06298828 0.001953125
# Compute F on the go (it won't be saved in the output)
# And enable genedropping
atlas %>%
purgeR::ip_Fa(genedrop = 1000, seed = 1234) %>%
dplyr::select(id, dam, sire, Fa) %>%
#> id dam sire Fa
#> 943 943 882 737 0.6475
#> 944 944 653 822 0.6000
#> 945 945 653 822 0.6000
#> 946 946 740 822 0.6050
#> 947 947 740 822 0.6075
#> 948 948 897 737 0.6340
\(F_{a}\) can also be estimated based on population estimates of \(N_{e}\) and generation numbers, using the expression from López-Cortegano et al. (2018):
\[F_{a(t)} = 1 - (1-\frac{1}{2N})^{\frac{1}{2}t(t-1)}\] This can be achieved with the function exp_Fa (e.g. exp_Fa (Ne = 50, t = 50)).
Purged inbreeding coefficient
The purged inbreeding coefficient (\(g\)) gives the probability of IBD for deleterious recessive alleles. The reduction of \(g\) when compared to standard inbreeding depends on the magnitude of a
purging coefficient (\(d\)) that measures the strength of the effective deleterious recessive component of the genome (García-Dorado 2012), so that \(d=0\) implies \(F=g\), and higher \(d\) (up to
0.5) means lower \(g\) in more inbred individuals. It can be calculated in pedigreed populations from the purged kinship coefficient (\(\gamma\)), in a similar way as standard inbreeding, following
the methods described in García-Dorado (2012) and García-Dorado et al. (2016):
\[\gamma_{i,i} = \frac{1}{2}(1+g_{i})(1-2dF_{i})\]
\[\gamma_{i,j} = \frac{1}{2}(\gamma_{i,j_{d}}+\gamma_{i,j_{s}})(1-dF_{j})\] Where \(j_{d}\) and \(j_{s}\) are \(j\)’s mother and father respectively, and \(i\) is older than \(j\).
The function ip_g computes the purged inbreeding coefficient, given a value of \(d\). The choice of a proper value of \(d\) can however be complex. A separate vignette titled “Inbreeding and Purging
Estimates” describes methods to help computing the inbreeding load as well as the purging coefficient.
atlas %>%
ip_F() %>%
ip_g(d = 0.48, Fcol = "Fi") %>%
dplyr::select(id, dam, sire, Fi, tidyselect::starts_with("g")) %>%
#> id dam sire Fi g0.48
#> 943 943 882 737 0.2350380 0.06066578
#> 944 944 653 822 0.2452226 0.08464775
#> 945 945 653 822 0.2452226 0.08464775
#> 946 946 740 822 0.2409467 0.07522799
#> 947 947 740 822 0.2409467 0.07522799
#> 948 948 897 737 0.2345642 0.06343757
\(g\) can also be estimated based on population estimates of \(N_{e}\) and generation numbers, given a value of \(d\), using the expression from García-Dorado (2012):
\[g_{t} = [(1-\frac{1}{2N})g_{t-1}+\frac{1}{2N}](1-2dF_{t-1})\]
This can be achieved with the function exp_g (e.g. exp_g (Ne = 50, t = 50, d = 0.2)).
This is the last of functions related to inbreeding coefficients. Plotting together expected values of \(F\), \(F_{a}\) and \(g\) (assuming \(N_{e} = 25\) and an intermediate value \(d=0.25\)), the
differences between the three coefficients become apparent.
data.frame(t = 0:50) %>%
dplyr::rowwise() %>%
dplyr::mutate(Fi = exp_F(Ne = 50, t),
Fa = exp_Fa(Ne = 50, t),
g = exp_g(Ne = 50, t, d = 0.25)) %>%
tidyr::pivot_longer(cols = c(Fi, Fa, g), names_to = "Type", values_to = "Inbreeding") %>%
ggplot(aes(x = t, y = Inbreeding, color = Type)) +
geom_line(size = 2) +
scale_x_continuous("Generations (t)") +
theme(legend.position = "bottom")
Opportunity of purging
Whereas previous purging methods focus on inbreeding measurements, opportunity of purging parameters calculate the potential reduction in individual inbreeding load, as a consequence of it having
inbred ancestors (Gulisija and Crow 2007). The (total) opportunity of purging for an individual \(i\) (\(O_{i}\)) can be computed as:
\[O_{i} = \sum_{j}\sum_{k} (1/2)^{n-1} F_{j}\] Where \(j\) is every inbred ancestor of \(i\), \(k\) is every path from \(i\) to \(j\), and \(n\) is the number of individuals in the path (including \
(i\) and \(j\)).
The expressed opportunity of purging depends on the probability of a given allele to be transmitted from an inbred ancestor \(j\) to \(i\), and thus on \(F_{i(j)}\). This is measured by the expressed
opportunity of purging (\(O_{e}\)) as:
\[O_{ei} = \sum_{j} 2F_{i(j)} F_{j}\] In complex pedigrees (involving more than one inbred ancestor per path), these measures need to be corrected to discount the probability of purging measured in a
close ancestor from that already calculated in a more distant ancestor. The function ip_op(complex=TRUE) does not perform Gulisija and Crow (2007) corrections, but instead applies an heuristic
approach, only accounting for close ancestors \(j\), and ignoring contributions from far ancestors \(k\) such that \(F_{j(k) > 0}\).
The function ip_op can be used as:
arrui %>%
ip_op(Fcol = "Fi") %>%
dplyr::filter(target == 1) %>%
tidyr::pivot_longer(cols = c(Oe, Oe_raw)) %>%
ggplot() +
geom_point(aes(x = Fi, y = (value), fill = name), pch = 21, size = 3, alpha = 0.5) +
scale_y_continuous(expression(paste("Expressed opportunity of purging (", O[e], ")", sep=""))) +
scale_x_continuous("Inbreeding coefficient (F)") +
#> Computing partial kinship matrix. This may take a while.
The plot shows the increase of the expressed opportunity of purging with the inbreeding coefficient for individuals in the reference population (last two cohorts). For individuals with the lowest
inbreeding values, the corrected (\(O_{e}\)) and uncorrected (\(O_{e \_raw}\)) have the same value, but as time progresses \(O_{e \_raw}\) becomes larger than \(O_{e}\). Both values are useful when
determining potential reduction in the individual inbreeding load (see more exhaustive examples in López-Cortegano 2022, and the ‘Inbreeding and Purging Estimates’ vignette).
Population parameters
The package purgeR is mainly focused on estimating inbreeding and purging parameters, but accessory functions are included to compute other population parameters that might be useful when
interpreting inbreeding and purging results. All functions for computing population parameters are prefixed with pop_.
Effective population size
The effective population size (\(N_{e}\)) can be computed from the individual increase in inbreeding (\(\Delta F\)) as defined by Gutiérrez et al. (2008, 2009):
\[N_{e} = \frac{1}{2\Delta F}\] Where the individual \(\Delta F\) can be computed as:
\[\Delta F_{i} = 1 - \sqrt[t_{i}-1]{1-F_{i}}\] Being \(F_{i}\) individual’s i coefficient of inbreeding, and \(t_{i}\) the generation number it belongs to.
The previous expression can be averaged to obtain \(\Delta F\) and used to estimate \(N_{e}\). Thus, all that is needed to compute \(N_{e}\) in a pedigree is the individual values of inbreeding and
generation time. The function pop_Ne will read a pedigree file and calculate \(N_{e}\) using accessory columns containing inbreeding and time information (named ‘F’ and ‘t’ here). Note that the
generation number is estimated here with pop_t as the number of equivalents to complete generations (see below).
atlas %>%
purgeR::ip_F() %>%
purgeR::pop_t() %>%
purgeR::pop_Ne(Fcol = "Fi", tcol = "t")
#> $Ne
#> [1] 8.184803
#> $se_Ne
#> [1] 0.2498695
However, we must warn caution when estimating \(N_{e}\) this way. Note that the previous estimate includes all individuals in the pedigree, but only the most recent individuals should be used, as
they already account for the inbreeding in their ancestors. In the following data set, the column target indicates the individuals that belonged to the reference population used to estimate \(N_{e}\)
(see details in López-Cortegano et al. 2021). Thus, \(N_{e}\) should be estimated in this case as:
atlas %>%
purgeR::ip_F() %>%
purgeR::pop_t() %>%
dplyr::filter(target == 1) %>%
purgeR::pop_Ne(Fcol = "Fi", tcol = "t")
#> $Ne
#> [1] 14.01041
#> $se_Ne
#> [1] 0.1707653
It is worth mentioning that this method to estimate \(N_{e}\) is of course equivalent to that using the classical formula \(F=1-(1-\frac{1}{2N_{e}})^t\).
Number of equivalents to complete generations
Generation times are easily computed for populations with discrete generations, but overlapping generations are the rule in most real world populations, and methods are required to estimate
generation times in such circumstances. The function pop_t computes the number of equivalents to complete generations (\(t_{eq}\)) following Boichard et al (1997). This is calculated for an
individual i as:
\[t_{eq} = \sum^{J}_{j=1} (\frac{1}{2})^{n}\] Where the sum is over all known ancestors, and \(n\) is the number of discrete generations that separate individual i from its ancestor j.
Of course, in populations with discrete generations, \(t_{eq} = t\), and in those with overlapping generations the estimates of \(t_{eq}\) strongly correlates with time. Consider as an example the
plot below showing the increase of \(t_{eq}\) with the year of birth (\(yob\)) of Gazella cuvieri:
atlas %>%
purgeR::pop_t() %>%
dplyr::mutate(t = plyr::round_any(t, 0.5)) %>%
ggplot() +
geom_boxplot(aes(x = yob, y = t, group = yob)) +
Not only is the correlation strong, but the total number of generations estimated (~10) matches the expectation given the total number of years of management (45) and the mean age for breeding
females (4.31 years, Moreno and Espeso 2008).
Number of founders and ancestors
Functions are also included to compute the total and effective number of founders and ancestors, as well as the number of founder genomes equivalents (\(N_{g}\)). These parameters can provide
information on early population bottlenecks due to unbalanced founder or ancestor contributions, as well as drift. Their estimation is based on probability of gene origin computations, following
Boichard et al (1997), but Caballero and Toro (2000) and Tahmoorespur and Sheikhloo (2011) are also recommended lectures in this regard. All these parameters are referred to a reference population
(RP) of interest that must be defined, e.g. it could be the latest cohort, or even the entire population.
The total number of founders (\(N_{f}\)) is calculated simply as the count of founders of the RP, while the effective number of founders (\(N_{fe}\)) is the number of equally contributing founders
that account for the observed genetic diversity in the RP. Founders are defined as individuals with not known parents (i.e. dam = 0 and sire = 0).
The total number of ancestors (\(N_{a}\)) is the count of all ancestors that contribute descendants to the RP, founders or not, while the effective number of ancestors (\(N_{ae}\)) is calculated as
the minimum number of ancestors, founders or not, required to account for the genetic diversity observed in the RP.
The number of founder genome equivalents (\(N_{g}\)) is defined in a similar way as (\(N_{fe}\)), but its estimated via Monte Carlo simulation of allele segregation, effectively accounting not only
for reductions in genetic diversity as consequence of bottlenecks in founders or ancestors contributions to the descent, but also to random sources of diversity loss, such as drift (Boichard et al
1997, Caballero and Toro 2000).
Thus, \(N_{ae}\) is always smaller than \(N_{f}\), and their ratio can inform on the diversity loss due to bottlenecks between the base population and the RP (Tahmoorespur and Sheikhloo 2011). On the
other hand, \(N_g\) is always the smallest parameter among these, since it accounts not only for diversity loss due to unbalanced founder or ancestor contributions, but also to genetic drift.
The function pop_Nancestors computes all these parameters, and returns them in a dataframe:
list("A. lervia" = arrui,
"G. cuvieri" = atlas,
"G. dorcas" = dorcas,
"N. dama" = dama) %>%
purrr::map_dfr(~ pop_Nancestors(., reference = "target", seed = 1234), .id = "Species")
#> Species Nr Nf Nfe Na Nae Ng se_Ng
#> 1 A. lervia 80 2 1.769424 63 1.769424 1.034102 0.2468827
#> 2 G. cuvieri 176 4 3.583086 249 3.583086 2.002087 0.4601456
#> 3 G. dorcas 283 20 13.386468 400 12.990604 5.817682 1.0154111
#> 4 N. dama 251 4 2.614750 349 2.614750 1.868039 0.4506204
Convenience functions are also available, named after the parameters they estimate. For example, the function pop_Ng will just estimate the number of founder genome equivalents, and return that value
as a numeric value. Similarly, pop_Nae will only estimate the effective number of ancestors, and so on. See more examples in ?pop_Nancestors.
Hardy-Weinberg deviation
In some cases it might be of interest to measure the degree of non-random mating in the population. This is given by deviation from Hardy-Weinberg equilibrium (\(\alpha\), Caballero and Toro 2000),
that can be calculated as:
\[\alpha = \frac{F-f}{1.0-f}\] Where \(F\) is the mean inbreeding coefficient of the population, and \(f\) the mean coancestry coefficient.
The function pop_hwd allows to estimate the previous coefficient, for the entire population, or preferably for a RP:
Note that in the example above \(\alpha\) is negative, as usually attributed to populations undergoing management. A value of zero would indicate random mating, and a positive one assortative mating
among relatives.
Fitness functions
Purging analyses may benefit from an interpretation in terms of fitness change. Fitness measurements are expected to be provided by the users, and could be for example ‘early survival’, or any other
trait known to be related to fitness in the studied species. A small set of functions is given however to help users to infer fitness measurements from the pedigree structure itself.
We warn, however, to make use of these with caution, as they might not always reflect true fitness. First, because measures of fitness based on contributions to the offspring (usually named as
‘breeding success’ or ‘productivity’) are limited to individuals present in the pedigree, and that information could be incomplete; Second, if the population is under active management, offspring
contributions may not represent actual biological fitness; Third, ‘reproductive values’ give expectations based on additive genetic relatedness and do not account for selective effects. Thus, they
may be unappropriated for downstream analysis considering purging effects; Finally, younger individuals in the pedigree might have lower fitness than older ones, because they haven’t had time to
generate offspring!
Fitness functions are prefixed with w_. A first measure of fitness given by the pedigree is individual breeding success, measured as the number of offspring present in the pedigree. The function
w_offspring can be used for this:
# Maximum overall breeding success
arrui %>%
purgeR::w_offspring(name_to = "P") %>%
.$P %>%
#> [1] 69
# Maximum female breeding success
arrui %>%
purgeR::w_offspring(name_to = "P", sire_offspring = FALSE) %>%
.$P %>%
#> [1] 21
Similarly, the number of grandoffspring can also be used as a proxy for fitness, with the function w_grandoffspring:
# Maximum overall grandoffspring productivity
arrui %>%
purgeR::w_grandoffspring(name_to = "GP") %>%
.$GP %>%
#> [1] 198
Finally, fitness can also be estimated as ‘reproductive values’, following the method developed by Hunter et al. (2019). Under this model, fitness is based on how well a gene originated in a set of
reference individuals is represented in their descendants. We do not go into the details of the algorithm used in this model, but it allows to correct genetic contributions by changes in population
size and migration (which is used by default). This method can be used with the function w_reproductive_value:
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How Do Dominoes Fall?
What caused the domino at the end of the row to move?
The student placed one hundred dominoes upright in a row. He knocked over the first domino and the rest all fell over; one after another.
Newton's first law; an object will stay at rest until a force is added onto it. Since the student knocked over the first domino that force the student added continues on until the row ends.
When a student places one hundred dominoes upright in a row and knocks over the first domino, it sets off a chain reaction where each domino falls onto the next one, causing a continuous movement
until the last domino falls. This phenomenon can be explained by Newton's first law of motion, which states that an object will stay at rest until a force is applied to it. In this case, the force
applied by knocking over the first domino initiates the motion of all the other dominoes in the row.
The reason why the domino at the end of the row moves is because of the transfer of energy from one domino to the next. As each domino falls, it releases potential energy that is converted into
kinetic energy, causing the next domino to fall. This process continues until all the dominos have fallen.
So, the next time you see a row of dominoes falling one after the other, remember that it's the result of Newton's first law in action, as well as the transfer of energy from domino to domino!
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How do I calculate the density of a super critical fluid?
• Thread starter Laurencet
• Start date
In summary, the conversation discusses the calculation of density for a supercritical fluid, specifically xenon. The density can be determined using the law of corresponding states and the
compressibility factor, which is a function of reduced temperature and pressure. The NIST website provides information on the fluid properties of xenon, and a generalized correlation can be used to
calculate the compressibility factor. This method gives a result of 200 kg/m3, which is close to the density of 2015 kg/m3 found on the NIST website. The conversation also mentions other methods for
calculating the compressibility factor, such as using "a" and "b" values, which may result in a different z value.
TL;DR Summary
The density of a supercritical fluid
How do I calculate the density of a supercritical fluid? If I have 100 Litres of Xenon at 23°C and 150bar, What will the Xenon in the tank weigh?
The phase diagram is here
Science Advisor
Homework Helper
Look on the NIST website for Xe, under Fluid properties. The answer for your case is 2015 kg/m^3.(There is no general equation like PV = nRT to calculate it.)
Staff Emeritus
Science Advisor
Homework Helper
2023 Award
The answer to your question can be determined using the law of corresponding states, based on the compressibility factor z. For xenon, the so-called as centric factor is zero, so the compressibility
factor is a function of the reduced temperature and pressure. For xenon, the critical temperature is 289.7 K and 58.4 bars, respectively. So, in this case, the reduced temperature is 296.2/289.7 =
1.02 and the reduced pressure is 150/58.4 = 2.57. From the generalized correlation of compressibility factor as a function of reduced temperature and pressure, this gives a compressibility factor of
z = 0.40. Therefore, the number of moles of xenon is $$n=\frac{PV}{zRT}=\frac{(15000000)(0.1)}{(0.40)(8.314)(296.2)}=1523\ moles = 200\ kg$$
So the estimated density is 2000 ##kg/m^3##.
Thankyou for your answers great to see you came up with the same answer.
mjc123 said:
Look on the NIST website for Xe, under Fluid properties. The answer for your case is 2015 kg/m^3.(There is no general equation like PV = nRT to calculate it.)
Do you have a link for where on the NIST website? I searched for XE on the NIST website and came up with 5109 documents... too many to read through.
Chestermiller said:
n=PVzRT=(15000000)(0.1)(0.40)(8.314)(296.2)=1523 moles=200 kgn=PVzRT=(15000000)(0.1)(0.40)(8.314)(296.2)=1523 moles=200 kg
n=\frac{PV}{zRT}=\frac{(15000000)(0.1)}{(0.40)(8.314)(296.2)}=1523\ moles = 200\ kg So the estimated density is 2000 kg/m3kg/m3kg/m^3.
I came up with 201 kg so very close, is there a good website or paper that explains this, that I can refer to.
Staff Emeritus
Science Advisor
Homework Helper
2023 Award
Google “compressibility factor”
Science Advisor
Homework Helper
Did you try searching for "xenon"? (I know I said "Xe" above, but I meant the element not the letters. I actually found it by googling "xenon density".)
I just worked through the above method example calculating "a" "b" and Vm to calculate Z
This gave me a z of 0.389 and a final density of 205.4kg
Further away than the method you used.
Is there more than one way to calculate z? did you do Reduced Temperature / Reduced Pressure?
FAQ: How do I calculate the density of a super critical fluid?
1. How do I determine the density of a super critical fluid?
The density of a super critical fluid can be calculated using the following formula: density = mass / volume. This means that you need to measure the mass of the fluid and its volume in order to
calculate its density.
2. What units should I use when calculating the density of a super critical fluid?
The units used for density are typically grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3). However, you can use any units as long as they are consistent for both mass and
3. Can I use the ideal gas law to calculate the density of a super critical fluid?
No, the ideal gas law is only applicable to gases at low pressures and high temperatures. Super critical fluids do not behave like ideal gases, so this equation cannot be used to calculate their
4. How does temperature and pressure affect the density of a super critical fluid?
In general, as temperature and pressure increase, the density of a super critical fluid decreases. This is because at higher temperatures and pressures, the molecules of the fluid are more spread out
and have more space between them, resulting in a lower density.
5. Is there a specific method for measuring the density of a super critical fluid?
Yes, there are several methods for measuring the density of a super critical fluid, including pycnometry, vibrating tube densitometry, and pressure-density correlation methods. Each method has its
own advantages and limitations, so it is important to choose the most appropriate method for your specific fluid and conditions.
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Evaluate ∫2x2+5x+7dx... | Filo
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Question Text Evaluate
Topic Integrals
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PID Autotuning · JuliaSimControl
The PID controller is a classical control method that is widely used due to its simplicity and ability to control many practically occurring systems. The parameters of a PID controller can be tuned
in multiple different ways:
• Trial and error, in simulation or on the real system
• Loop shaping
• Automatic tuning
This page details the automatic tuning features offered by JuliaSimControl. Automatic tuning offers the possibility of achieving an optimal response, subject to constraints on closed-loop robustness
with respect to uncertainty and noise amplification. While this page refers to the automatic tuning of PID controllers in isolation, we refer the reader to Automatic tuning of structured controllers
for tuning of more general structured controllers.
The PID autotuning in JuliaSimControl is based on disturbance step-response optimization, subject to robustness constraints on closed-loop sensitivity functions. We currently implement two methods,
one that optimizes a PID controller on the form
$$\[K(s) = (k_p + k_i/s + k_d s)\]$$
by solving
$$\[\operatorname{minimize}_K \int e(t) dt\]$$$$\[\text{subject to} \\ ||S(s)||_\infty \leq M_S \\ ||T(s)||_\infty \leq M_T \\ ||KS(s)||_\infty \leq M_{KS}\]$$
where $e(t) = r(t) - y(t)$ is the control error.
The second method performs joint optimization of a PID controller and a measurement filter on the form
$$\[K(s) = C(s) F(s) = (k_p + k_i/s + k_d s) \dfrac{1}{(sT)^2 + 2ζTs + 1}, ζ = 1/√2\]$$
by solving
$$\[\operatorname{minimize}_{C, F} \int |e(t)| dt\]$$$$\[\text{subject to} \\ ||S(s)||_\infty \leq M_S \\ ||T(s)||_\infty \leq M_T\\ ||KS(s)||_\infty \leq M_{KS}\]$$
The autotuning functions operate on SISOLTISystems from ControlSystems.jl. If you have a ModelingToolkit model, you may obtain a linear system model using linearization. The general workflow for
autotuning is
1. Define a plant model, $P$
2. Define desired constraints on the maximum magnitudes $M_S, M_T, M_{KS}$ of the sensitivity functions $S = \dfrac{1}{1+ PK}$, $T = \dfrac{PK}{1+ PK}$ and $KS = \dfrac{K}{1+ PK}$.
3. Choose problem and solver parameters and create an AutoTuningProblem.
4. Call solve.
5. Plot the result.
If you want to use the optimized controller in a ModelingToolkit simulation, see OptimizedPID.
Examples of this are shown below.
JuliaSimControl contains a Pluto-based graphical app (GUI) for PID-controller tuning using the two methods below, usage of this app is documented under PID autotuning GUI. This example demonstrates
the non-graphical interface.
The following example optimizes a PID controller with a low-pass filter using the method from
K. Soltesz, C. Grimholt, S. Skogestad. Simultaneous design of proportional–integral–derivative controller and measurement filter by optimisation. Control Theory and Applications. 11(3), pp.
341-348. IET. 2017.
using JuliaSimControl, Plots
# Process model (continuous time LTI SISO system).
T = 4 # Time constant
L = 1 # Delay
K = 1 # Gain
P = tf(K, [T, 1.0])*delay(L) # process dynamics
Ts = 0.1 # Discretization time
Tf = 25 # Simulation time
# Robustness constraints
Ms = 1.2 # Maximum allowed sensitivity function magnitude
Mt = Ms # Maximum allowed complementary sensitivity function magnitude
Mks = 10.0 # Maximum allowed magnitude of transfer function from process output to control signal, sometimes referred to as noise sensitivity.
w = 2π .* exp10.(LinRange(-2, 2, 200)) # frequency grid
prob = AutoTuningProblem(; P, Ms, Mt, Mks, w, Ts, Tf, metric = :IAE)
# p0 = Float64[1, 1, 0, 0.001] # Initial parameter guess can be optionally supplied, kp, ki, kd, T_filter
res = solve(prob)
The figure shows the Nyquist curve of the loop-transfer function $P(s)K(s)$ using the optimized controller, as well as circles corresponding to the chosen constraints. The top figures show Bode plots
of the closed-loop sensitivity functions together with the constrains, and the lower left figure shows the response to a unit load-disturbance step as well as a reference-step response. Note, the
response to a reference step is not part of the optimization criterion, and optimized suppression of load disturbances often leads to a suboptimal response to reference steps. If steps are expected
in the reference signal, reference shaping using a pre-filter is recommended (called a 2-DOF design, realized, for example, by the introduction of $W_r$ in the diagram of the following design example
The following example optimizes a PID controller using the method from
M. Hast, K. J. Astrom, B. Bernhardsson, S. Boyd. PID design by convex-concave optimization. European Control Conference. IEEE. Zurich, Switzerland. 2013.
This method optimizes integrated error (not integrated absolute error). This problem is relatively easy to solve and corresponds well to IAE if the system is well damped. If convergence of the above
method (IAE) appears difficult, this method can be used as initialization by choosing metric = :IEIAE.
using JuliaSimControl, Plots
T = 4 # Time constant
K = 1 # Gain
P = tf(K, [T, 1.0]) # process dynamics
## Robustness constraints
Ms = 1.2 # Maximum allowed sensitivity function magnitude
Mt = Ms # Maximum allowed complementary sensitivity function magnitude
Mks = 10.0 # Maximum allowed magnitude of transfer function from process output to control signal, sometimes referred to as noise sensitivity.
w = 2π .* exp10.(LinRange(-2, 2, 50)) # frequency vector
p0 = Float64[1, 1, 0.1] # Initial guess. Use only two parameters to tune a PI instead of PID controller
prob = AutoTuningProblem(; P, Ms, Mt, Mks, w, Ts=0.1, Tf=25.0, metric = :IE) # Set the metric here
res = solve(prob, p0)
The metric = :IE problem optimizes integrated error $\int e(t) dt$ (not integrated absolute error). This problem is relatively easy and fast to solve and corresponds well to IAE if the system is well
damped. If this metric is chosen, a PI or PID controller is tuned, determined by the number of parameters in the initial guess. The method requires a stabilizing controller as an initial guess. If
the plant is stable, the zero controller is okay. If the initial guess is not stabilizing, an attempt at automatically finding a feasible initial guess is made.
If the response is oscillatory, the metric = :IE metric is expected to perform poorly and metric = :IAE is recommended. If metric = :IAE is chosen, a PID controller with a low-pass filter is tuned by
minimizing $\int |e(t)| dt$. This problem is nonconvex and can be difficult to solve. This method can be initialized with the IE method by selecting metric = :IEIAE.
AutoTuningProblem{S, W}
Optimizes a controller on the form
K(s) = C(s) * F(s) = (kp + ki/s + kd*s) * 1/((s*T)^2+2*ζ*T*s+1), ζ = 1/√2
$$\[K(s) = C(s) F(s) = (k_p + k_i/s + k_d s) \dfrac{1}{(sT)^2 + 2ζTs + 1}, ζ = 1/√2\]$$
Can be plotted after optimization by using Plots; plot(prob, C) where C is obtained from the optimization.
See also OptimizedPID
Keyword arguments:
• P::LTISystem: Plant model
• w::AbstractVector: Frequency grid vector
• Ts::Float64: Discretization time (sample time, arbitrary time unit)
• Tf::Float64: Simulation duration in the same time unit as Ts.
• Ms::Float64: Maximum allowed sensitivity
• Mt::Float64: Maximum allowed complimentary sensitivity
• Mks::Float64: Maximum allowed noise sensitivity (controller times sensitivity)
• pmax::Vector{Float64}: An optional vector of the same length as the number of estimated parameters that contains upper bounds on parameters, the default is fill(Inf, 4) (no bounds).
• metric = :IAE: The metric to optimize. Choices are :IAE, :IE, :IEIAE.
• disc: The discretization method to use when optimizinf metric = :IAE. Choices are :zoh, :foh, :tustin (delay systems only support :zoh).
res = solve(prob::AutoTuningProblem, p0; kwargs...)
res = solve(prob::AutoTuningProblem; kwargs...)
Computes PID parameters, that minimize load step IAE, and filter for noise attenuation.
$$\[K(s) = C(s) F(s) = (k_p + k_i/s + k_d s) \dfrac{1}{(sT)^2 + 2 ζ Ts + 1}\]$$
• p0 Parameter vector guess: [kp; ki; kd; T]. If p0 is not provied, attempts will be made to find one automatically.
See AutoTuningProblem for arguments. res is of type AutoTuningResult and can be plotted plot(res). It contains the optimized parameters as well as an ODESystem representing an optimized controller.
Based on K. Soltesz, C. Grimholt, S. Skogestad. Simultaneous design of proportional–integral–derivative controller and measurement filter by optimisation. Control Theory and Applications. 11(3), pp.
341-348. IET. 2017.
Solver options for metric :IAE include
• maxeval = 500
• maxtime = 20
• xtol_rel = 1e-3
• alg An optimizer supported by Optimization.jl. The default is IPOPT.
• random_start = 0 A positive integer indicates a number of random starts to try in order to find a good soluiton. If random_start = 0, only the provided initial guess is used.
Solver options for metric :IE include
• maxiter = 15: Maximum number of convex-concave optimization iterations.
• tol = 1e-3: Tolerance for the convex-concave procedure.
• solver = Hypatia.Optimizer: Any MOI compatible solver.
• verbose = false:
Extended help
The autotuner optimizes the reponse to load disturbances appearing on the process input. This typically leads to good regulation performance, but may result in a controller that produces large
overshoots for step changes in the reference. If step changes in the reference are expected, we advice using one of the following strategies
• Prefilter the reference using aditional lowpass filters.
• If the system is a force/torque controlled servo system, generate a feasible reference trajectory with a continuous acceleration profile and make use of computed force/torque feedforward.
• Let the proportional and derivative terms of the PID controller act on the measurement only. This can be achieved by setting the wp and wd keyword arguments to
ModelingToolkitStandardLibrary.Blocks.LimPID (and similar to OptimizedPID).
OptimizedPID(popt; name, kwargs...)
OptimizedPID(res::AutoTuningResult; name, kwargs...)
Takes optimized parameters popt and returns the following system
────►│ F ├───────┐ ┌─────┐
reference └───┘ └►│ │ ctr_output
│ PID ├─────►
┌───┐ ┌►│ │
┌─►│ F ├───────┘ └─────┘
│ └───┘
└───── measurement
• popt: Obtained by solving an AutoTuningProblem
• kwargs: Are passed to ModelingToolkitStandardLibrary.Blocks.LimPID
A structure containing the results of performing PID autotuning.
This structure can be plotted plot(res; stepsize = 1), where stepsize controls the size of the reference step to show (turn off by setting to 0). See also OptimizedPID.
• prob::AutoTuningProblem
• p::Vector: The optimal parameter vector.
• K::ControlSystemsBase.StateSpace: The optimal controller.
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Kapp’s Regulation
Kapp had designed a diagram shown in Figure 1.42 to determine the regulation at any power factor. The description of the construction of the diagram is shown below.
Figure 1.42 Kapp’s Diagram
Load current (I[2]) is taken as a reference phasor. OA representing V[2] is drawn at angle θ[2] with I[2]. AB represents I[2]R[02] drawn parallel to I[2], whereas BC represents I[2]X[02] drawn
perpendicular to AB, i.e., I[2]. Here OC represents secondary emf ([0]V[2] = E[2]) at no-load. The circle 1 known as often circuit EMF circle is drawn with O as centre and OC as radius. The line OO^
′ is drawn parallel to AC representing I[2]Z[02]. With O^′ as centre and OA as radius, the circle 2 known as terminal voltage circle is drawn, which intersects with circle 1 at the points D and E.
The region above and below the reference line represents the lagging and leading power factors region, respectively. The point D is the point corresponding to zero regulation. The intercept FG gives
the maximum regulation, which is drawn through O and drawn parallel to AC. The regulation at any power factor angle θ is obtained by extending OA to meet the outer circle at H. The regulation at the
required power factor cosθ is represented by AH, which is the intercept between the two circles.
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Group RNW
What's the RNW of a group of people?
The Raw Net Worth of a group of people1 is just the sum of the RNWs of the individuals in the group.
To illustrate this, here’s a group of people: Alice, Bob and Charlotte.
Thanks for reading Economics in One Lesson in the 21st Century! Subscribe for free to receive new posts and support my work.
Alice owns a house and a car. She is owed £2,000 by Bank X, and owes a loaf of bread to Bob.
Bob owns a house, and 2 loaves of bread. He is owed £1,000 by Bank Y and a loaf of bread by Alice, but owes £90,000 of a mortgage to Bank X.
Charlotte owns a car, 4 loaves of bread and 2 apples. She owes an apple to Dom.
Between them, Alice, Bob and Charlotte:
• Own Alice’s house and Bob’s house, Alice’s car and Charlotte’s car, 6 loaves of bread and 2 apples.
• Are owed £3,000 and a loaf of bread, and
• Owe £90,000, a loaf of bread and an apple.
So here are the RNWs of Alice, Bob, Charlotte, and the group of all 3 of them.
Notice that the group’s RNW is the sum of the individuals’ RNWs. In particular, look at the number of loaves: -1 for Alice, 3 for Bob, and 4 for Charlotte, making 6 in total for the group.
The loaf of bread which Alice owes to Bob doesn’t affect the group’s RNW because the decrease from Alice owing the loaf is exactly offset by the increase from Bob being owed the same loaf. Debts owed
within a group have no effect on the group’s combined RNW.
The whole world’s RNW
The whole world’s RNW is, naturally, the sum of the RNWs of everyone in the world.
What’s interesting about this is that every debt owed by someone in the world is owed to someone in the world (and vice-versa)2.
This means that debts have no effect on the whole world’s combined RNW. They are all owed within the group. And so the whole world’s RNW is just the sum of everyone’s tangible assets.
Understand the importance of this. Creating new debts (e.g. creating money) and writing off existing debts (e.g. people paying down their debts in an economic downturn) have no effect on the world’s
RNW. At most, they transfer RNW from one person to another. The only way to change the world’s total RNW is to produce or consume tangible assets.
Remember, when I say people, I mean either real people or corporations. Corporations, like people, can own, be owed and owe things.
Unless there are inter-planetary debts I’m not aware of, in which case we can talk about the whole universe’s RNW.
This post will come in handy for when we look at the effects of insolvency on the rest of the world's RNW.
Expand full comment
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Kth largest element in array - Coding Interviews Searching and Sorting
Kth largest element in array
Given an integer array nums and an integer k, return the kth largest element in the array.
Note that it is the kth largest element in the sorted order, not the kth distinct element.
class Solution:
def findKthLargest(self, nums, k):
if not nums: return
pivot = random.choice(nums)
left = [x for x in nums if x > pivot]
mid = [x for x in nums if x == pivot]
right = [x for x in nums if x < pivot]
L, M = len(left), len(mid)
if k <= L:
return self.findKthLargest(left, k)
elif k > L + M:
return self.findKthLargest(right, k - L - M)
return mid[0]
Posted by Jamie Meyer 7 months ago
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Every adjacent pair of words differs by a single letter.
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sk == endWord
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sequence should be returned as a list of the words [beginWord, s1, s2, ..., sk].
Given an array of distinct integers candidates and a target integer target, return a list of all unique combinations of candidates where the chosen numbers sum to target. You may return the
combinations in any order.
The same number may be chosen from candidates an unlimited number of times. Two combinations are unique if the frequency of at least one of the chosen numbers is different.
There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course
bi first if you want to take course ai.
For example, the pair [0, 1], indicates that to take course 0 you have to first take course 1.
Return true if you can finish all courses. Otherwise, return false.
Given an integer array nums, find the subarray with the largest sum, and return its sum.
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How do you simplify (4y^3+12y^2-y-3)/(2y^3+y^2-18y-9)? | HIX Tutor
How do you simplify #(4y^3+12y^2-y-3)/(2y^3+y^2-18y-9)#?
Answer 1
We can factor our way through this:
The first two terms in the numerator and denominator seem to be comparable to the final two terms when viewed together:
We can now factor down the terms with #y^2#:
We can now cancel on the following terms:
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Answer 2
To simplify the expression (4y^3+12y^2-y-3)/(2y^3+y^2-18y-9), you can factor both the numerator and denominator and then cancel out any common factors. Factoring the numerator and denominator, we get
(y+3)(2y-1)(2y+1)/(y+3)(y-1)(2y+9). Canceling out the common factors (y+3) and (2y-1), the simplified expression is (2y+1)/(y-1)(2y+9).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Equation of a Plane Parallel to a Given Plane
Suppose a plane
? passes through the point
and is parallel to the plane
How do we find the equation of
Because the planes are parallel they have the same normal. The normal to a plane given in Cartesian form is equal to the coefficients of the equation of the plane, written as a vector, in this case
\[\mathbf{n}= \begin{pmatrix}2\\-3\\\end{pmatrix}\]
, so both have the same coefficients in the Cartesian form of the plane equation, and the plane
will have equation
can be found by substituting the point above into the equation.
The equation of the plane is
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neural network
Options for training deep learning neural network
options = trainingOptions(solverName) returns training options for the optimizer specified by solverName. To train a neural network, use the training options as an input argument to the trainnet
options = trainingOptions(solverName,Name=Value) returns training options with additional options specified by one or more name-value arguments.
Specify Training Options
Create a set of options for training a network using stochastic gradient descent with momentum. Reduce the learning rate by a factor of 0.2 every 5 epochs. Set the maximum number of epochs for
training to 20, and use a mini-batch with 64 observations at each iteration. Turn on the training progress plot.
options = trainingOptions("sgdm", ...
LearnRateSchedule="piecewise", ...
LearnRateDropFactor=0.2, ...
LearnRateDropPeriod=5, ...
MaxEpochs=20, ...
MiniBatchSize=64, ...
options =
TrainingOptionsSGDM with properties:
Momentum: 0.9000
InitialLearnRate: 0.0100
MaxEpochs: 20
LearnRateSchedule: 'piecewise'
LearnRateDropFactor: 0.2000
LearnRateDropPeriod: 5
MiniBatchSize: 64
Shuffle: 'once'
CheckpointFrequency: 1
CheckpointFrequencyUnit: 'epoch'
SequenceLength: 'longest'
PreprocessingEnvironment: 'serial'
L2Regularization: 1.0000e-04
GradientThresholdMethod: 'l2norm'
GradientThreshold: Inf
Verbose: 1
VerboseFrequency: 50
ValidationData: []
ValidationFrequency: 50
ValidationPatience: Inf
ObjectiveMetricName: 'loss'
CheckpointPath: ''
ExecutionEnvironment: 'auto'
OutputFcn: []
Metrics: []
Plots: 'training-progress'
SequencePaddingValue: 0
SequencePaddingDirection: 'right'
InputDataFormats: "auto"
TargetDataFormats: "auto"
ResetInputNormalization: 1
BatchNormalizationStatistics: 'auto'
OutputNetwork: 'auto'
Acceleration: "auto"
Monitor Deep Learning Training Progress
This example shows how to monitor the training progress of deep learning networks.
When you train networks for deep learning, plotting various metrics during training enables you to learn how the training is progressing. For example, you can determine if and how quickly the network
accuracy is improving, and whether the network is starting to overfit the training data.
This example shows how to monitor training progress for networks trained using the trainnet function. If you are training a network using a custom training loop, use a trainingProgressMonitor object
instead to plot metrics during training. For more information, see Monitor Custom Training Loop Progress.
When you set the Plots training option to "training-progress" in trainingOptions and start network training, the trainnet function creates a figure and displays training metrics at every iteration.
Each iteration is an estimation of the gradient and an update of the network parameters. If you specify validation data in trainingOptions, then the figure shows validation metrics each time trainnet
validates the network. The figure plots the loss and any metrics specified by the Metrics name-value option. By default, the software uses a linear scale for the plots. To specify a logarithmic scale
for the y-axis, select the log scale button in the axes toolbar.
During training, you can stop training and return the current state of the network by clicking the stop button in the top-right corner. After you click the stop button, it can take a while for
training to complete. Once training is complete, trainnet returns the trained network.
Specify the OutputNetwork training option as "best-validation" to get finalized values that correspond to the iteration with the best validation metric value, where the optimized metric is specified
by the ObjectiveMetricName training options. Specify the OutputNetwork training option as "last-iteration" to get finalized metrics that correspond to the last training iteration.
On the right of the pane, view information about the training time and settings. To learn more about training options, see Set Up Parameters and Train Convolutional Neural Network.
To save the training progress plot, click Export as Image in the training window. You can save the plot as a PNG, JPEG, TIFF, or PDF file. You can also save the individual plots using the axes
Plot Training Progress During Training
Train a network and plot the training progress during training.
Load the training and test data from the MAT files DigitsDataTrain.mat and DigitsDataTest.mat, respectively. The training and test data sets each contain 5000 images.
load DigitsDataTrain.mat
load DigitsDataTest.mat
Create a dlnetwork object.
Specify the layers of the classification branch and add them to the network.
layers = [
imageInputLayer([28 28 1])
net = addLayers(net,layers);
Specify options for network training. To validate the network at regular intervals during training, specify validation data. Record the metric values for the accuracy and F-score. To plot training
progress during training, set the Plots training option to "training-progress".
options = trainingOptions("sgdm", ...
MaxEpochs=8, ...
Metrics = ["accuracy","fscore"], ...
ValidationData={XTest,labelsTest}, ...
ValidationFrequency=30, ...
Verbose=false, ...
Train the network.
net = trainnet(XTrain,labelsTrain,net,"crossentropy",options);
Stop Training Early Using Metrics
Use metrics for early stopping and to return the best network.
Load the training data, which contains 5000 images of digits. Set aside 1000 of the images for network validation.
[XTrain,YTrain] = digitTrain4DArrayData;
idx = randperm(size(XTrain,4),1000);
XValidation = XTrain(:,:,:,idx);
XTrain(:,:,:,idx) = [];
YValidation = YTrain(idx);
YTrain(idx) = [];
Construct a network to classify the digit image data.
net = dlnetwork;
layers = [
imageInputLayer([28 28 1])
net = addLayers(net,layers);
Specify the training options:
• Use an SGDM solver for training.
• Monitor training performance by specifying validation data and validation frequency.
• Track the accuracy and recall during training. To return the network with the best recall value, specify "recall" as the objective metric and set the output network to "best-validation".
• Specify the validation patience as 5 so training stops if the recall has not decreased for five iterations.
• Visualize network training progress plot.
• Suppress the verbose output.
options = trainingOptions("sgdm", ...
ValidationData={XValidation,YValidation}, ...
ValidationFrequency=35, ...
ValidationPatience=5, ...
Metrics=["accuracy","recall"], ...
ObjectiveMetricName="recall", ...
OutputNetwork="best-validation", ...
Plots="training-progress", ...
Train the network.
net = trainnet(XTrain,YTrain,net,"crossentropy",options);
Input Arguments
solverName — Solver for training neural network
"sgdm" | "rmsprop" | "adam" | "lbfgs" (since R2023b) | "lm" (since R2024b)
Solver for training neural network, specified as one of these values:
• "sgdm" — Stochastic gradient descent with momentum (SGDM). SGDM is a stochastic solver. For additional training options, see Stochastic Solver Options. For more information, see Stochastic
Gradient Descent with Momentum.
• "rmsprop" — Root mean square propagation (RMSProp). RMSProp is a stochastic solver. For additional training options, see Stochastic Solver Options. For more information, see Root Mean Square
• "adam" — Adaptive moment estimation (Adam). Adam is a stochastic solver. For additional training options, see Stochastic Solver Options. For more information, see Adaptive Moment Estimation.
• "lbfgs" (since R2023b) — Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS). L-BFGS is a batch solver. Use the L-BFGS algorithm for small networks and data sets that you can process in a
single batch. For additional training options, see Batch Solver Options. For more information, see Limited-Memory BFGS.
• "lm" (since R2024b) — Levenberg–Marquardt (LM). LM is a batch solver. Use the LM algorithm for regression networks with small numbers of learnable parameters, where you can process the data set
in a single batch. If solverName is "lm", then the lossFcn argument of the trainnet function must be "mse" or "l2loss". For additional training options, see Batch Solver Options. For more
information, see Levenberg–Marquardt.
The trainBERTDocumentClassifier (Text Analytics Toolbox) function supports the "sgdm", "rmsprop", and "adam" solvers only.
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: Plots="training-progress",Metrics="accuracy",Verbose=false specifies to disable the verbose output and display the training progress in a plot that also includes the accuracy metric.
Plots — Plots to display during neural network training
"none" (default) | "training-progress"
Plots to display during neural network training, specified as one of these values:
• "none" — Do not display plots during training.
• "training-progress" — Plot training progress.
The contents of the plot depends on the solver that you use.
• When the solverName argument is "sgdm", "adam", or "rmsprop", the plot shows the mini-batch loss, validation loss, training mini-batch and validation metrics specified by the Metrics option, and
additional information about the training progress.
• When the solverName argument is "lbfgs" or "lm", the plot shows the training and validation loss, training and validation metrics specified by the Metrics option, and additional information about
the training progress.
To programmatically open and close the training progress plot after training, use the show and close functions with the second output of the trainnet function. You can use the show function to view
the training progress even if the Plots training option is specified as "none".
To switch the y-axis scale to logarithmic, use the axes toolbar.
For more information about the plot, see Monitor Deep Learning Training Progress.
Metrics — Metrics to monitor
[] (default) | character vector | string array | function handle | deep.DifferentiableFunction object (since R2024a) | cell array | metric object
Since R2023b
Metrics to monitor, specified as one of these values:
• Built-in metric or loss function name — Specify metrics as a string scalar, character vector, or a cell array or string array of one or more of these names:
□ Metrics:
☆ "accuracy" — Accuracy (also known as top-1 accuracy)
☆ "auc" — Area under ROC curve (AUC)
☆ "fscore" — F-score (also known as F[1]-score)
☆ "precision" — Precision
☆ "recall" — Recall
☆ "rmse" — Root mean squared error
☆ "mape" — Mean absolute percentage error (MAPE) (since R2024b)
□ Loss functions:
☆ "crossentropy" — Cross-entropy loss for classification tasks. (since R2024b)
☆ "indexcrossentropy" — Index cross-entropy loss for classification tasks. (since R2024b)
☆ "binary-crossentropy" — Binary cross-entropy loss for binary and multilabel classification tasks. (since R2024b)
☆ "mae" / "mean-absolute-error" / "l1loss" — Mean absolute error for regression tasks. (since R2024b)
☆ "mse" / "mean-squared-error" / "l2loss" — Mean squared error for regression tasks. (since R2024b)
☆ "huber" — Huber loss for regression tasks (since R2024b)
Note that setting the loss function as "crossentropy" and specifying "index-crossentropy" as a metric or setting the loss function as "index-crossentropy" and specifying "crossentropy" as a
metric is not supported.
• Built-in metric object — If you need more flexibility, you can use built-in metric objects. The software supports these built-in metric objects:
When you create a built-in metric object, you can specify additional options such as the averaging type and whether the task is single-label or multilabel.
• Custom metric function handle — If the metric you need is not a built-in metric, then you can specify custom metrics using a function handle. The function must have the syntax metric =
metricFunction(Y,T), where Y corresponds to the network predictions and T corresponds to the target responses. For networks with multiple outputs, the syntax must be metric = metricFunction
(Y1,…,YN,T1,…TM), where N is the number of outputs and M is the number of targets. For more information, see Define Custom Metric Function.
When you have data in mini-batches, the software computes the metric for each mini-batch and then returns the average of those values. For some metrics, this behavior can result in a different
metric value than if you compute the metric using the whole data set at once. In most cases, the values are similar. To use a custom metric that is not batch-averaged for the data, you must
create a custom metric object. For more information, see Define Custom Deep Learning Metric Object.
• deep.DifferentiableFunction object (since R2024a) — Function object with custom backward function. For categorical targets, the software automatically converts the categorical values to one-hot
encoded vectors and passes them to the metric function. For more information, see Define Custom Deep Learning Operations.
• Custom metric object — If you need greater customization, then you can define your own custom metric object. For an example that shows how to create a custom metric, see Define Custom Metric
Object. For general information about creating custom metrics, see Define Custom Deep Learning Metric Object.
If you specify a metric as a function handle, a deep.DifferentiableFunction object, or a custom metric object and train the neural network using the trainnet function, then the layout of the targets
that the software passes to the metric depends on the data type of the targets, and the loss function that you specify in the trainnet function and the other metrics that you specify:
• If the targets are numeric arrays, then the software passes the targets to the metric directly.
• If the loss function is "index-crossentropy" and the targets are categorical arrays, then the software automatically converts the targets to numeric class indices and passes them to the metric.
• For other loss functions, if the targets are categorical arrays, then the software automatically converts the targets to one-hot encoded vectors and then passes them to the metric.
This option supports the trainnet and trainBERTDocumentClassifier (Text Analytics Toolbox) functions only.
Example: Metrics=["accuracy","fscore"]
Example: Metrics={"accuracy",@myFunction,precisionObj}
ObjectiveMetricName — Name of objective metric
"loss" (default) | string scalar | character vector
Since R2024a
Name of objective metric to use for early stopping and returning the best network, specified as a string scalar or character vector.
The metric name must be "loss" or match the name of a metric specified by the Metrics argument. Metrics specified using function handles are not supported. To specify the ObjectiveMetricName value as
the name of a custom metric, the value of the Maximize property of the custom metric object must be nonempty. For more information, see Define Custom Deep Learning Metric Object.
For more information about specifying the objective metric for early stopping, see ValidationPatience. For more information about returning the best network using the objective metric, see
Data Types: char | string
Verbose — Flag to display training progress information
1 (true) (default) | 0 (false)
Flag to display training progress information in the command window, specified as 1 (true) or 0 (false).
The content of the verbose output depends on the type of solver.
For stochastic solvers (SGDM, Adam, and RMSProp), the table contains these variables:
Variable Description
Iteration Iteration number.
Epoch Epoch number.
TimeElapsed Time elapsed in hours, minutes, and seconds.
LearnRate Learning rate.
TrainingLoss Training loss.
ValidationLoss Validation loss. If you do not specify validation data, then the software does not display this information.
For batch solvers (L-BFGS and LM), the table contains these variables:
Variable Description
Iteration Iteration number.
TimeElapsed Time elapsed in hours, minutes, and seconds.
TrainingLoss Training loss.
ValidationLoss Validation loss. If you do not specify validation data, then the software does not display this information.
GradientNorm Norm of the gradients.
StepNorm Norm of the steps.
If you specify additional metrics in the training options, then they also appear in the verbose output. For example, if you set the Metrics training option to "accuracy", then the information
includes the TrainingAccuracy and ValidationAccuracy variables.
When training stops, the verbose output displays the reason for stopping.
To specify validation data, use the ValidationData training option.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
VerboseFrequency — Frequency of verbose printing
50 (default) | positive integer
Frequency of verbose printing, which is the number of iterations between printing to the Command Window, specified as a positive integer.
If you validate the neural network during training, then the software also prints to the command window every time validation occurs.
To enable this property, set the Verbose training option to 1 (true).
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
OutputFcn — Output functions
function handle | cell array of function handles
Output functions to call during training, specified as a function handle or cell array of function handles. The software calls the functions once before the start of training, after each iteration,
and once when training is complete.
The functions must have the syntax stopFlag = f(info), where info is a structure containing information about the training progress, and stopFlag is a scalar that indicates to stop training early. If
stopFlag is 1 (true), then the software stops training. Otherwise, the software continues training.
The trainnet function passes the output function the structure info.
For stochastic solvers (SGDM, Adam, and RMSProp), info contains these fields:
Field Description
Epoch Epoch number
Iteration Iteration number
TimeElapsed Time since start of training
LearnRate Iteration learn rate
TrainingLoss Iteration training loss
ValidationLoss Validation loss, if specified and evaluated at iteration.
State Iteration training state, specified as "start", "iteration", or "done".
For batch solvers (L-BFGS and LM), info contains these fields:
Field Description
Iteration Iteration number
TimeElapsed Time elapsed in hours, minutes, and seconds
TrainingLoss Training loss
ValidationLoss Validation loss. If you do not specify validation data, then the software does not display this information.
GradientNorm Norm of the gradients
StepNorm Norm of the steps
State Iteration training state, specified as "start", "iteration", or "done".
If you specify additional metrics in the training options, then they also appear in the training information. For example, if you set the Metrics training option to "accuracy", then the information
includes the TrainingAccuracy and ValidationAccuracy fields.
If a field is not calculated or relevant for a certain call to the output functions, then that field contains an empty array.
For an example showing how to use output functions, see Custom Stopping Criteria for Deep Learning Training.
Data Types: function_handle | cell
Data Formats
InputDataFormats — Description of input data dimensions
"auto" (default) | string array | cell array of character vectors | character vector
Since R2023b
Description of the input data dimensions, specified as a string array, character vector, or cell array of character vectors.
If InputDataFormats is "auto", then the software uses the formats expected by the network input. Otherwise, the software uses the specified formats for the corresponding network input.
A data format is a string of characters, where each character describes the type of the corresponding data dimension.
The characters are:
• "S" — Spatial
• "C" — Channel
• "B" — Batch
• "T" — Time
• "U" — Unspecified
For example, consider an array containing a batch of sequences where the first, second, and third dimensions correspond to channels, observations, and time steps, respectively. You can specify that
this array has the format "CBT" (channel, batch, time).
You can specify multiple dimensions labeled "S" or "U". You can use the labels "C", "B", and "T" once each, at most. The software ignores singleton trailing "U" dimensions after the second dimension.
For a neural networks with multiple inputs net, specify an array of input data formats, where InputDataFormats(i) corresponds to the input net.InputNames(i).
For more information, see Deep Learning Data Formats.
Data Types: char | string | cell
TargetDataFormats — Description of target data dimensions
"auto" (default) | string array | cell array of character vectors | character vector
Since R2023b
Description of the target data dimensions, specified as one of these values:
• "auto" — If the target data has the same number of dimensions as the input data, then the trainnet function uses the format specified by InputDataFormats. If the target data has a different
number of dimensions to the input data, then the trainnet function uses the format expected by the loss function.
• String array, character vector, or cell array of character vectors — The trainnet function uses the data formats you specify.
A data format is a string of characters, where each character describes the type of the corresponding data dimension.
The characters are:
• "S" — Spatial
• "C" — Channel
• "B" — Batch
• "T" — Time
• "U" — Unspecified
For example, consider an array containing a batch of sequences where the first, second, and third dimensions correspond to channels, observations, and time steps, respectively. You can specify that
this array has the format "CBT" (channel, batch, time).
You can specify multiple dimensions labeled "S" or "U". You can use the labels "C", "B", and "T" once each, at most. The software ignores singleton trailing "U" dimensions after the second dimension.
For more information, see Deep Learning Data Formats.
Data Types: char | string | cell
Stochastic Solver Options
MaxEpochs — Maximum number of epochs
30 (default) | positive integer
Maximum number of epochs (full passes of the data) to use for training, specified as a positive integer.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
MiniBatchSize — Size of mini-batch
128 (default) | positive integer
Size of the mini-batch to use for each training iteration, specified as a positive integer. A mini-batch is a subset of the training set that is used to evaluate the gradient of the loss function and
update the weights.
If the mini-batch size does not evenly divide the number of training samples, then the software discards the training data that does not fit into the final complete mini-batch of each epoch. If the
mini-batch size is smaller than the number of training samples, then the software does not discard any data.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
For best performance, if you are training a network using a datastore with a ReadSize property, such as an imageDatastore, then set the ReadSize property and MiniBatchSize training option to the same
value. If you are training a network using a datastore with a MiniBatchSize property, such as an augmentedImageDatastore, then set the MiniBatchSize property of the datastore and the MiniBatchSize
training option to the same value.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Shuffle — Option for data shuffling
"once" (default) | "never" | "every-epoch"
Option for data shuffling, specified as one of these values:
• "once" — Shuffle the training and validation data once before training.
• "never" — Do not shuffle the data.
• "every-epoch" — Shuffle the training data before each training epoch, and shuffle the validation data before each neural network validation. If the mini-batch size does not evenly divide the
number of training samples, then the software discards the training data that does not fit into the final complete mini-batch of each epoch. To avoid discarding the same data every epoch, set the
Shuffle training option to "every-epoch".
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
InitialLearnRate — Initial learning rate
positive scalar
Initial learning rate used for training, specified as a positive scalar.
If the learning rate is too low, then training can take a long time. If the learning rate is too high, then training might reach a suboptimal result or diverge.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
When solverName is "sgdm", the default value is 0.01. When solverName is "rmsprop" or "adam", the default value is 0.001.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
LearnRateSchedule — Learning rate schedule
"none" (default) | character vector | string array | built-in or custom learning rate schedule object | function handle | cell array
Learning rate schedule, specified as a character vector or string scalar of a built-in learning rate schedule name, a string array of names, a built-in or custom learning rate schedule object, a
function handle, or a cell array of names, metric objects, and function handles.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
Built-In Learning Rate Schedule Names
Specify learning rate schedules as a string scalar, character vector, or a string or cell array of one or more of these names:
Name Description Plot
"none" No learning rate schedule. This schedule keeps the learning rate constant.
"piecewise" Piecewise learning rate schedule. Every 10 epochs, this schedule drops the learn rate by a factor of 10.
"warmup" (since R2024b) Warm-up learning rate schedule. For 5 iterations, this schedule ramps up the learning rate to the base learning rate.
"polynomial" (since Polynomial learning rate schedule. Every epoch, this schedule drops the learning rate using a power law with a unitary exponent.
"exponential" (since Exponential learning rate schedule. Every epoch, this schedule decays the learning rate by a factor of 10.
"cosine" (since R2024b) Cosine learning rate schedule. Every epoch, this schedule drops the learn rate using a cosine formula.
"cyclical" (since Cyclical learning rate schedule. For periods of 10 epochs, this schedule increases the learning rate from the base learning rate for 5 epochs and then decreases the
R2024b) learning rate for 5 epochs.
Built-In Learning Rate Schedule Object (since R2024b)
If you need more flexibility than what the string options provide, you can use built-in learning rate schedule objects:
Custom Learning Rate Schedule (since R2024b)
For additional flexibility, you can define a custom learning rate schedule as a function handle or custom class that inherits from deep.LearnRateSchedule.
• Custom learning rate schedule function handle — If the learning rate schedule you need is not a built-in learning rate schedule, then you can specify custom learning rate schedules using a
function handle. To specify a custom schedule, use a function handle with the syntax learningRate = f(baseLearningRate,epoch), where baseLearningRate is the base learning rate, and epoch is the
epoch number.
• Custom learn rate schedule object — If you need more flexibility that what function handles provide, then you can define a custom learning rate schedule class that inherits from
Multiple Learning Rate Schedules (since R2024b)
You can combine multiple learning rate schedules by specifying multiple schedules as a string or cell array and then the software applies the schedules in order, starting with the first element. At
most one of the schedules can be infinite (schedules than continue indefinitely, such as "cyclical" and objects with the NumSteps property set to Inf) and the infinite schedule must be the last
element of the array.
Momentum — Contribution of previous step
0.9 (default) | scalar from 0 to 1
Contribution of the parameter update step of the previous iteration to the current iteration of stochastic gradient descent with momentum, specified as a scalar from 0 to 1.
A value of 0 means no contribution from the previous step, whereas a value of 1 means maximal contribution from the previous step. The default value works well for most tasks.
This option supports the SGDM solver only (when the solverName argument is "sgdm").
For more information, see Stochastic Gradient Descent with Momentum.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
GradientDecayFactor — Decay rate of gradient moving average
0.9 (default) | nonnegative scalar less than 1
Decay rate of gradient moving average for the Adam solver, specified as a nonnegative scalar less than 1. The gradient decay rate is denoted by β[1] in the Adaptive Moment Estimation section.
This option supports the Adam solver only (when the solverName argument is "adam").
For more information, see Adaptive Moment Estimation.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
SquaredGradientDecayFactor — Decay rate of squared gradient moving average
nonnegative scalar less than 1
Decay rate of squared gradient moving average for the Adam and RMSProp solvers, specified as a nonnegative scalar less than 1. The squared gradient decay rate is denoted by β[2] in [4].
Typical values of the decay rate are 0.9, 0.99, and 0.999, corresponding to averaging lengths of 10, 100, and 1000 parameter updates, respectively.
This option supports the Adam and RMSProp solvers only (when the solverName argument is "adam" or "rmsprop").
The default value is 0.999 for the Adam solver. The default value is 0.9 for the RMSProp solver.
For more information, see Adaptive Moment Estimation and Root Mean Square Propagation.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Epsilon — Denominator offset
1e-8 (default) | positive scalar
Denominator offset for Adam and RMSProp solvers, specified as a positive scalar.
The solver adds the offset to the denominator in the neural network parameter updates to avoid division by zero. The default value works well for most tasks.
This option supports the Adam and RMSProp solvers only (when the solverName argument is "adam" or "rmsprop").
For more information, see Adaptive Moment Estimation and Root Mean Square Propagation.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
LearnRateDropFactor — Factor for dropping the learning rate
0.1 (default) | scalar from 0 to 1
Factor for dropping the learning rate, specified as a scalar from 0 to 1. This option is valid only when the LearnRateSchedule training option is "piecewise".
LearnRateDropFactor is a multiplicative factor to apply to the learning rate every time a certain number of epochs passes. Specify the number of epochs using the LearnRateDropPeriod training option.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
To customize the piecewise learning rate schedule, use a piecewiseLearnRate object. A piecewiseLearnRate object is recommended over the LearnRateDropFactor and LearnRateDropPeriod training options
because it provides additional control over the drop frequency. (since R2024b)
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
LearnRateDropPeriod — Number of epochs for dropping the learning rate
10 (default) | positive integer
Number of epochs for dropping the learning rate, specified as a positive integer. This option is valid only when the LearnRateSchedule training option is "piecewise".
The software multiplies the global learning rate with the drop factor every time the specified number of epochs passes. Specify the drop factor using the LearnRateDropFactor training option.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
To customize the piecewise learning rate schedule, use a piecewiseLearnRate object. A piecewiseLearnRate object is recommended over the LearnRateDropFactor and LearnRateDropPeriod training options
because it provides additional control over the drop frequency. (since R2024b)
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Batch Solver Options
MaxIterations — Maximum number of iterations
1000 (default) | positive integer
Since R2023b
Maximum number of iterations to use for training, specified as a positive integer.
The L-BFGS solver is a full-batch solver, which means that it processes the entire training set in a single iteration.
This option supports batch solvers only (when the solverName argument is "lbfgs" or "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
GradientTolerance — Relative gradient tolerance
1e-5 (default) | positive scalar
Since R2023b
Relative gradient tolerance, specified as a positive scalar.
The software stops training when the relative gradient is less than or equal to GradientTolerance.
This option supports batch solvers only (when the solverName argument is "lbfgs" or "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
StepTolerance — Step size tolerance
1e-5 (default) | positive scalar
Since R2023b
Step size tolerance, specified as a positive scalar.
The software stops training when the step that the algorithm takes is less than or equal to StepTolerance.
This option supports batch solvers only (when the solverName argument is "lbfgs" or "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
LineSearchMethod — Method to find suitable learning rate
"weak-wolfe" (default) | "strong-wolfe" | "backtracking"
Since R2023b
Method to find suitable learning rate, specified as one of these values:
• "weak-wolfe" — Search for a learning rate that satisfies the weak Wolfe conditions. This method maintains a positive definite approximation of the inverse Hessian matrix.
• "strong-wolfe" — Search for a learning rate that satisfies the strong Wolfe conditions. This method maintains a positive definite approximation of the inverse Hessian matrix.
• "backtracking" — Search for a learning rate that satisfies sufficient decrease conditions. This method does not maintain a positive definite approximation of the inverse Hessian matrix.
This option supports the L-BFGS solver only (when the solverName argument is "lbfgs").
HistorySize — Number of state updates to store
10 (default) | positive integer
Since R2023b
Number of state updates to store, specified as a positive integer. Values between 3 and 20 suit most tasks.
The L-BFGS algorithm uses a history of gradient calculations to approximate the Hessian matrix recursively. For more information, see Limited-Memory BFGS.
This option supports the L-BFGS solver only (when the solverName argument is "lbfgs").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
InitialInverseHessianFactor — Initial value that characterizes approximate inverse Hessian matrix
1 (default) | positive scalar
Since R2023b
Initial value that characterizes the approximate inverse Hessian matrix, specified as a positive scalar.
To save memory, the L-BFGS algorithm does not store and invert the dense Hessian matrix B. Instead, the algorithm uses the approximation ${B}_{k-m}^{-1}\approx {\lambda }_{k}I$, where m is the
history size, the inverse Hessian factor ${\lambda }_{k}$ is a scalar, and I is the identity matrix. The algorithm then stores the scalar inverse Hessian factor only. The algorithm updates the
inverse Hessian factor at each step.
The initial inverse hessian factor is the value of ${\lambda }_{0}$.
For more information, see Limited-Memory BFGS.
This option supports the L-BFGS solver only (when the solverName argument is "lbfgs").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
MaxNumLineSearchIterations — Maximum number of line search iterations
20 (default) | positive integer
Since R2023b
Maximum number of line search iterations to determine the learning rate, specified as a positive integer.
This option supports the L-BFGS solver only (when the solverName argument is "lbfgs").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
InitialStepSize — Approximate maximum absolute value of the first optimization step
[] (default) | "auto" | real finite scalar
Since R2024b
Initial step size, specified as one of these values:
• [] — Do not use an initial step size to determine the initial Hessian approximation.
• "auto" — Determine the initial step size automatically. The software uses an initial step size of $‖{s}_{0}{‖}_{\infty }=\frac{1}{2}‖{W}_{0}{‖}_{\infty }+0.1$, where W[0] are the initial
learnable parameters of the network.
• Positive real scalar — Use the specified value as the initial step size $‖{s}_{0}{‖}_{\infty }$.
If InitialStepSize is "auto" or a positive real scalar, then the software approximates the initial inverse Hessian using ${\lambda }_{0}=\frac{‖{s}_{0}{‖}_{\infty }}{‖abla J\left({W}_{0}\right){‖}_{\
infty }}$, where λ[0] is the initial inverse Hessian factor and $abla J\left({W}_{0}\right)$ denotes the gradients of the loss with respect to the initial learnable parameters. For more information,
see Limited-Memory BFGS.
This option supports the L-BFGS solver only (when the solverName argument is "lbfgs").
InitialDampingFactor — Initial damping factor
0.001 (default) | positive scalar
Since R2024b
Initial damping factor, specified as a positive scalar.
This option supports the LM solver only (when the solverName argument is "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
MaxDampingFactor — Maximum damping factor
1e10 (default) | positive scalar
Since R2024b
Maximum damping factor, specified as a positive scalar.
This option supports the LM solver only (when the solverName argument is "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
DampingIncreaseFactor — Factor for increasing damping factor
10 (default) | positive scalar greater than 1
Since R2024b
Factor for increasing damping factor, specified as a positive scalar greater than 1.
This option supports the LM solver only (when the solverName argument is "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
DampingDecreaseFactor — Factor for decreasing damping factor
0.1 (default) | positive scalar less than 1
Since R2024b
Factor for decreasing damping factor, specified as a positive scalar less than 1.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
ValidationData — Data to use for validation during training
[] (default) | datastore | table | cell array | minibatchqueue object (since R2024a)
Data to use for validation during training, specified as [], a datastore, a table, a cell array, or a minibatchqueue object that contains the validation predictors and targets.
During training, the software uses the validation data to calculate the validation loss and metric values. To specify the validation frequency, use the ValidationFrequency training option. You can
also use the validation data to stop training automatically when the validation objective metric stops improving. By default, the objective metric is set to the loss. To turn on automatic validation
stopping, use the ValidationPatience training option.
If ValidationData is [], then the software does not validate the neural network during training.
If your neural network has layers that behave differently during prediction than during training (for example, dropout layers), then the validation loss can be lower than the training loss.
The software shuffles the validation data according to the Shuffle training option. If Shuffle is "every-epoch", then the software shuffles the validation data before each neural network validation.
The supported formats depend on the training function that you use.
trainnet Function
Specify the validation data as a datastore, minibatchqueue object, or the cell array {predictors,targets}, where predictors contains the validation predictors and targets contains the validation
targets. Specify the validation predictors and targets using any of the formats supported by the trainnet function.
For more information, see the input arguments of the trainnet function.
trainBERTDocumentClassifier Function (Text Analytics Toolbox)
Specify the validation data as one of these values:
• Cell array {documents,targets}, where documents contains the input documents, and targets contains the document labels.
• Table, where the first variable contains the input documents and the second variable contains the document labels.
For more information, see the input arguments of the trainBERTDocumentClassifier (Text Analytics Toolbox) function.
ValidationFrequency — Frequency of neural network validation
50 (default) | positive integer
Frequency of neural network validation in number of iterations, specified as a positive integer.
The ValidationFrequency value is the number of iterations between evaluations of validation metrics. To specify validation data, use the ValidationData training option.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
ValidationPatience — Patience of validation stopping
Inf (default) | positive integer
Patience of validation stopping of neural network training, specified as a positive integer or Inf.
ValidationPatience specifies the number of times that the objective metric on the validation set can be worse than or equal to the previous best value before neural network training stops. If
ValidationPatience is Inf, then the values of the validation metric do not cause training to stop early. The software aims to maximize or minimize the metric, as specified by the Maximize property of
the metric. When the objective metric is "loss", the software aims to minimize the loss value.
The returned neural network depends on the OutputNetwork training option. To return the neural network with the best validation metric value, set the OutputNetwork training option to
Before R2024a: The software computes the validation patience using the validation loss value.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
OutputNetwork — Neural network to return when training completes
"auto" (default) | "last-iteration" | "best-validation"
Neural network to return when training completes, specified as one of the following:
• "auto" – Use "best-validation" if ValidationData is specified. Otherwise, use "last-iteration".
• "best-validation" – Return the neural network corresponding to the training iteration with the best validation metric value, where the metric to optimize is specified by the ObjectiveMetricName
option. To use this option, you must specify the ValidationData training option.
• "last-iteration" – Return the neural network corresponding to the last training iteration.
Regularization and Normalization
L2Regularization — Factor for L[2] regularization
0.0001 (default) | nonnegative scalar
Factor for L[2] regularization (weight decay), specified as a nonnegative scalar. For more information, see L2 Regularization.
This option does not support the LM solver (when the solverName argument is "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
ResetInputNormalization — Option to reset input layer normalization
1 (true) (default) | 0 (false)
Option to reset input layer normalization, specified as one of the following:
• 1 (true) — Reset the input layer normalization statistics and recalculate them at training time.
• 0 (false) — Calculate normalization statistics at training time when they are empty.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
BatchNormalizationStatistics — Mode to evaluate statistics in batch normalization layers
"auto" (default) | "population" | "moving"
Mode to evaluate the statistics in batch normalization layers, specified as one of the following:
• "population" — Use the population statistics. After training, the software finalizes the statistics by passing through the training data once more and uses the resulting mean and variance.
• "moving" — Approximate the statistics during training using a running estimate given by update steps
$\begin{array}{l}{\mu }^{*}={\lambda }_{\mu }\stackrel{^}{\mu }+\left(1-{\lambda }_{\mu }\right)\mu \\ {\sigma }^{2}{}^{*}={\lambda }_{{\sigma }^{2}}\stackrel{^}{{\sigma }^{2}}\text{}\text{+}\
text{}\text{(1-}{\lambda }_{{\sigma }^{2}}\right)\text{}{\sigma }^{2}\end{array}$
where ${\mu }^{*}$ and ${\sigma }^{2}{}^{*}$ denote the updated mean and variance, respectively, ${\lambda }_{\mu }$ and ${\lambda }_{{\sigma }^{2}}$ denote the mean and variance decay values,
respectively, $\stackrel{^}{\mu }$ and $\stackrel{^}{{\sigma }^{2}}$ denote the mean and variance of the layer input, respectively, and $\mu$ and ${\sigma }^{2}$ denote the latest values of the
moving mean and variance values, respectively. After training, the software uses the most recent value of the moving mean and variance statistics. This option supports CPU and single GPU training
• "auto" — Use the "moving" option.
Gradient Clipping
GradientThreshold — Gradient threshold
Inf (default) | positive scalar
Gradient threshold, specified as Inf or a positive scalar. If the gradient exceeds the value of GradientThreshold, then the gradient is clipped according to the GradientThresholdMethod training
For more information, see Gradient Clipping.
This option does not support the LM solver (when the solverName argument is "lm").
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
GradientThresholdMethod — Gradient threshold method
"l2norm" (default) | "global-l2norm" | "absolute-value"
Gradient threshold method used to clip gradient values that exceed the gradient threshold, specified as one of the following:
• "l2norm" — If the L[2] norm of the gradient of a learnable parameter is larger than GradientThreshold, then scale the gradient so that the L[2] norm equals GradientThreshold.
• "global-l2norm" — If the global L[2] norm, L, is larger than GradientThreshold, then scale all gradients by a factor of GradientThreshold/L. The global L[2] norm considers all learnable
• "absolute-value" — If the absolute value of an individual partial derivative in the gradient of a learnable parameter is larger than GradientThreshold, then scale the partial derivative to have
magnitude equal to GradientThreshold and retain the sign of the partial derivative.
For more information, see Gradient Clipping.
This option does not support the LM solver (when the solverName argument is "lm").
SequenceLength — Option to pad or truncate sequences
"longest" (default) | "shortest"
Option to pad, truncate, or split input sequences, specified as one of these values:
• "longest" — Pad sequences in each mini-batch to have the same length as the longest sequence. This option does not discard any data, though padding can introduce noise to the neural network.
• "shortest" — Truncate sequences in each mini-batch to have the same length as the shortest sequence. This option ensures that no padding is added, at the cost of discarding data.
To learn more about the effect of padding and truncating sequences, see Sequence Padding and Truncation.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | char | string
SequencePaddingDirection — Direction of padding or truncation
"right" (default) | "left"
Direction of padding or truncation, specified as one of these options:
• "right" — Pad or truncate sequences on the right. The sequences start at the same time step and the software truncates or adds padding to the end of each sequence.
• "left" — Pad or truncate sequences on the left. The software truncates or adds padding to the start of each sequence so that the sequences end at the same time step.
Because recurrent layers process sequence data one time step at a time, when the recurrent layer OutputMode property is "last", any padding in the final time steps can negatively influence the layer
output. To pad or truncate sequence data on the left, set the SequencePaddingDirection argument to "left".
For sequence-to-sequence neural networks (when the OutputMode property is "sequence" for each recurrent layer), any padding in the first time steps can negatively influence the predictions for the
earlier time steps. To pad or truncate sequence data on the right, set the SequencePaddingDirection option to "right".
To learn more about the effects of padding and truncating sequences, see Sequence Padding and Truncation.
SequencePaddingValue — Value by which to pad input sequences
0 (default) | scalar
Value by which to pad the input sequences, specified as a scalar.
Do not pad sequences with NaN, because doing so can propagate errors through the neural network.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Hardware and Acceleration
ExecutionEnvironment — Hardware resource for training neural network
"auto" (default) | "cpu" | "gpu" | "multi-gpu" | "parallel-auto" | "parallel-cpu" | "parallel-gpu"
Hardware resource for training neural network, specified as one of these values:
• "auto" – Use a local GPU if one is available. Otherwise, use the local CPU.
• "cpu" – Use the local CPU.
• "gpu" – Use the local GPU.
• "multi-gpu" – Use multiple GPUs on one machine, using a local parallel pool based on your default cluster profile. If there is no current parallel pool, the software starts a parallel pool with
pool size equal to the number of available GPUs.
• "parallel-auto" – Use a local or remote parallel pool. If there is no current parallel pool, the software starts one using the default cluster profile. If the pool has access to GPUs, then only
workers with a unique GPU perform training computation and excess workers become idle. If the pool does not have GPUs, then training takes place on all available CPU workers instead. (since
Before R2024a: Use "parallel" instead.
• "parallel-cpu" – Use CPU resources in a local or remote parallel pool, ignoring any GPUs. If there is no current parallel pool, the software starts one using the default cluster profile. (since
• "parallel-gpu" – Use GPUs in a local or remote parallel pool. Excess workers become idle. If there is no current parallel pool, the software starts one using the default cluster profile. (since
The "gpu", "multi-gpu", "parallel-auto", "parallel-cpu", and "parallel-gpu" options require Parallel Computing Toolbox™. To use a GPU for deep learning, you must also have a supported GPU device. For
information on supported devices, see GPU Computing Requirements (Parallel Computing Toolbox). If you choose one of these options and Parallel Computing Toolbox or a suitable GPU is not available,
then the software returns an error.
For more information on when to use the different execution environments, see Scale Up Deep Learning in Parallel, on GPUs, and in the Cloud.
To see an improvement in performance when training in parallel, try scaling up the MiniBatchSize and InitialLearnRate training options by the number of GPUs.
The "multi-gpu", "parallel-auto", "parallel-cpu", and "parallel-gpu" options support stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
PreprocessingEnvironment — Environment for fetching and preprocessing data
"serial" (default) | "background" | "parallel"
Since R2024a
Environment for fetching and preprocessing data from a datastore during training, specified as one of these values:
• "serial" – Fetch and preprocess data in serial.
• "background" – Fetch and preprocess data using the background pool.
• "parallel" – Fetch and preprocess data using parallel workers. The software opens a parallel pool using the default profile, if a local pool is not currently open. Non-local parallel pools are
not supported. Using this option requires Parallel Computing Toolbox. This option is not supported when training in parallel (when the ExecutionEnvironment option is "parallel-auto",
"parallel-cpu", "parallel-gpu", or "multi-gpu").
To use the "background" or "parallel" options, the input datastore must be subsettable or partitionable. Custom datastores must implement the matlab.io.datastore.Subsettable class.
The "background" and "parallel" options are not supported when the Shuffle option is "never".
If you use the "background" and "parallel" options, then training is non-deterministic even if you use the deep.gpu.deterministicAlgorithms function.
Use the "background" option when your mini-batches require significant preprocessing. If your preprocessing is not supported on threads, or if you need to control the number of workers, use the
"parallel" option. For more information about the preprocessing environment, see Preprocess Data in the Background or in Parallel.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
Before R2024a: To preprocess data in parallel, set the DispatchInBackground training option to 1 (true).
Acceleration — Performance optimization
"auto" (default) | "none"
Since R2024a
Performance optimization, specified as one of these values:
• "auto" – Automatically apply a number of optimizations suitable for the input network and hardware resources.
• "none" – Disable all optimizations.
CheckpointPath — Path for saving checkpoint neural networks
"" (default) | string scalar | character vector
Path for saving the checkpoint neural networks, specified as a string scalar or character vector.
• If you do not specify a path (that is, you use the default ""), then the software does not save any checkpoint neural networks.
• If you specify a path, then the software saves checkpoint neural networks to this path and assigns a unique name to each neural network. You can then load any checkpoint neural network and resume
training from that neural network.
If the folder does not exist, then you must first create it before specifying the path for saving the checkpoint neural networks. If the path you specify does not exist, then the software throws
an error.
Data Types: char | string
CheckpointFrequency — Frequency of saving checkpoint neural networks
positive integer
Frequency of saving checkpoint neural networks, specified as a positive integer.
If solverName is "lbfgs" or CheckpointFrequencyUnit is "iteration", then the software saves checkpoint neural networks every CheckpointFrequency iterations. Otherwise, the software saves checkpoint
neural networks every CheckpointFrequency epochs.
When solverName is "sgdm", "adam", or "rmsprop", the default value is 1. When solverName is "lbfgs" or "lm", the default value is 30.
This option only has an effect when CheckpointPath is nonempty.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
CheckpointFrequencyUnit — Checkpoint frequency unit
"epoch" (default) | "iteration"
Checkpoint frequency unit, specified as "epoch" or "iteration".
If CheckpointFrequencyUnit is "epoch", then the software saves checkpoint neural networks every CheckpointFrequency epochs.
If CheckpointFrequencyUnit is "iteration", then the software saves checkpoint neural networks every CheckpointFrequency iterations.
This option only has an effect when CheckpointPath is nonempty.
This option supports stochastic solvers only (when the solverName argument is "sgdm", "adam", or "rmsprop").
Output Arguments
options — Training options
TrainingOptionsSGDM | TrainingOptionsRMSProp | TrainingOptionsADAM | TrainingOptionsLBFGS | TrainingOptionsLM
Training options, returned as a TrainingOptionsSGDM, TrainingOptionsRMSProp, TrainingOptionsADAM, TrainingOptionsLBFGS, TrainingOptionsLM object. To train a neural network, use the training options
as an input argument to the trainnet function.
• For most deep learning tasks, you can use a pretrained neural network and adapt it to your own data. For an example showing how to use transfer learning to retrain a convolutional neural network
to classify a new set of images, see Retrain Neural Network to Classify New Images. Alternatively, you can create and train neural networks from scratch using the trainnet and trainingOptions
If the trainingOptions function does not provide the training options that you need for your task, then you can create a custom training loop using automatic differentiation. To learn more, see
Train Network Using Custom Training Loop.
If the trainnet function does not provide the loss function that you need for your task, then you can specify a custom loss function to the trainnet as a function handle. For loss functions that
require more inputs than the predictions and targets (for example, loss functions that require access to the neural network or additional inputs), train the model using a custom training loop. To
learn more, see Train Network Using Custom Training Loop.
If Deep Learning Toolbox™ does not provide the layers you need for your task, then you can create a custom layer. To learn more, see Define Custom Deep Learning Layers. For models that cannot be
specified as networks of layers, you can define the model as a function. To learn more, see Train Network Using Model Function.
For more information about which training method to use for which task, see Train Deep Learning Model in MATLAB.
Initial Weights and Biases
For convolutional and fully connected layers, the initialization for the weights and biases are given by the WeightsInitializer and BiasInitializer properties of the layers, respectively. For
examples showing how to change the initialization for the weights and biases, see Specify Initial Weights and Biases in Convolutional Layer and Specify Initial Weights and Biases in Fully Connected
Stochastic Gradient Descent
The standard gradient descent algorithm updates the network parameters (weights and biases) to minimize the loss function by taking small steps at each iteration in the direction of the negative
gradient of the loss,
${\theta }_{\ell +1}={\theta }_{\ell }-\alpha abla E\left({\theta }_{\ell }\right),$
where $\ell$is the iteration number, $\alpha >0$ is the learning rate, $\theta$ is the parameter vector, and $E\left(\theta \right)$ is the loss function. In the standard gradient descent algorithm,
the gradient of the loss function, $abla E\left(\theta \right)$, is evaluated using the entire training set, and the standard gradient descent algorithm uses the entire data set at once.
By contrast, at each iteration the stochastic gradient descent algorithm evaluates the gradient and updates the parameters using a subset of the training data. A different subset, called a
mini-batch, is used at each iteration. The full pass of the training algorithm over the entire training set using mini-batches is one epoch. Stochastic gradient descent is stochastic because the
parameter updates computed using a mini-batch is a noisy estimate of the parameter update that would result from using the full data set.
Stochastic Gradient Descent with Momentum
The stochastic gradient descent algorithm can oscillate along the path of steepest descent towards the optimum. Adding a momentum term to the parameter update is one way to reduce this oscillation
[2]. The stochastic gradient descent with momentum (SGDM) update is
${\theta }_{\ell +1}={\theta }_{\ell }-\alpha abla E\left({\theta }_{\ell }\right)+\gamma \left({\theta }_{\ell }-{\theta }_{\ell -1}\right),$
where the learning rate α and the momentum value $\gamma$ determine the contribution of the previous gradient step to the current iteration.
Root Mean Square Propagation
Stochastic gradient descent with momentum uses a single learning rate for all the parameters. Other optimization algorithms seek to improve network training by using learning rates that differ by
parameter and can automatically adapt to the loss function being optimized. Root mean square propagation (RMSProp) is one such algorithm. It keeps a moving average of the element-wise squares of the
parameter gradients,
${v}_{\ell }={\beta }_{2}{v}_{\ell -1}+\left(1-{\beta }_{2}\right){\left[abla E\left({\theta }_{\ell }\right)\right]}^{2}$
β[2] is the squared gradient decay factor of the moving average. Common values of the decay rate are 0.9, 0.99, and 0.999. The corresponding averaging lengths of the squared gradients equal 1/(1-β
[2]), that is, 10, 100, and 1000 parameter updates, respectively. The RMSProp algorithm uses this moving average to normalize the updates of each parameter individually,
${\theta }_{\ell +1}={\theta }_{\ell }-\frac{\alpha abla E\left({\theta }_{\ell }\right)}{\sqrt{{v}_{\ell }}+ϵ}$
where the division is performed element-wise. Using RMSProp effectively decreases the learning rates of parameters with large gradients and increases the learning rates of parameters with small
gradients. ɛ is a small constant added to avoid division by zero.
Adaptive Moment Estimation
Adaptive moment estimation (Adam) [4] uses a parameter update that is similar to RMSProp, but with an added momentum term. It keeps an element-wise moving average of both the parameter gradients and
their squared values,
${m}_{\ell }={\beta }_{1}{m}_{\ell -1}+\left(1-{\beta }_{1}\right)abla E\left({\theta }_{\ell }\right)$
${v}_{\ell }={\beta }_{2}{v}_{\ell -1}+\left(1-{\beta }_{2}\right){\left[abla E\left({\theta }_{\ell }\right)\right]}^{2}$
The β[1] and β[2] decay rates are the gradient decay and squared gradient decay factors, respectively. Adam uses the moving averages to update the network parameters as
${\theta }_{\ell +1}={\theta }_{\ell }-\frac{\alpha {m}_{l}}{\sqrt{{v}_{l}}+ϵ}$
The value α is the learning rate. If gradients over many iterations are similar, then using a moving average of the gradient enables the parameter updates to pick up momentum in a certain direction.
If the gradients contain mostly noise, then the moving average of the gradient becomes smaller, and so the parameter updates become smaller too. The full Adam update also includes a mechanism to
correct a bias that appears in the beginning of training. For more information, see [4].
Limited-Memory BFGS
The L-BFGS algorithm [5] is a quasi-Newton method that approximates the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. Use the L-BFGS algorithm for small networks and data sets that you can
process in a single batch.
The algorithm updates learnable parameters W at iteration k+1 using the update step given by
${W}_{k+1}={W}_{k}-{\eta }_{k}{B}_{k}^{-1}abla J\left({W}_{k}\right),$
where W[k] denotes the weights at iteration k, ${\eta }_{k}$ is the learning rate at iteration k, B[k] is an approximation of the Hessian matrix at iteration k, and $abla J\left({W}_{k}\right)$
denotes the gradients of the loss with respect to the learnable parameters at iteration k.
The L-BFGS algorithm computes the matrix-vector product ${B}_{k}^{-1}abla J\left({W}_{k}\right)$ directly. The algorithm does not require computing the inverse of B[k].
To save memory, the L-BFGS algorithm does not store and invert the dense Hessian matrix B. Instead, the algorithm uses the approximation ${B}_{k-m}^{-1}\approx {\lambda }_{k}I$, where m is the
history size, the inverse Hessian factor ${\lambda }_{k}$ is a scalar, and I is the identity matrix. The algorithm then stores the scalar inverse Hessian factor only. The algorithm updates the
inverse Hessian factor at each step.
To compute the matrix-vector product ${B}_{k}^{-1}abla J\left({W}_{k}\right)$ directly, the L-BFGS algorithm uses this recursive algorithm:
1. Set $r={B}_{k-m}^{-1}abla J\left({W}_{k}\right)$, where m is the history size.
2. For $i=m,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}1$:
1. Let $\beta =\frac{1}{{s}_{k-i}^{\top }{y}_{k-i}}{y}_{k-i}^{\top }r$, where ${s}_{k-i}$ and ${y}_{k-i}$ are the step and gradient differences for iteration $k-i$, respectively.
2. Set $r=r+\text{}{s}_{k-i}\text{}\left({a}_{k-i}-\beta \right)$, where $a$ is derived from $s$, $y$, and the gradients of the loss with respect to the loss function. For more information,
see [5].
3. Return ${B}_{k}^{-1}abla J\left({W}_{k}\right)=r$.
The LM algorithm [6] interpolates between gradient descent and Gauss-Newton methods, and can be more robust for small neural networks. It approximates second order derivatives using a Jacobian outer
product. Use the LM algorithm for regression networks with small numbers of learnable parameters, where you can process the data set in a single batch.
The algorithm updates the learnable parameters W at iteration k+1 using the update step given by
${W}_{k+1}={W}_{k}+\Delta {W}_{k},$
where ΔW[k] the change of the weights at iteration k given by
$\Delta {W}_{k}=-{\left({H}_{k}\right)}^{-1}abla {E}_{k}.$
Here, H[k] is the approximated Hessian at iteration k and $abla {E}_{k}$ is the gradient of the loss at iteration k with respect to the learnable parameters. The algorithm approximates the Hessian
${H}_{k}={J}_{k}^{\top }{J}_{k}+{\mu }_{k}I,$
where J[k] is the Jacobian matrix at iteration k, μ[k] is the damping factor at iteration k, and I is the identity matrix.
The solver uses the damping factor to adjust the step size taken each iteration and adaptively updates it each iteration. It increases and decreases the damping factor when iterations increase and
decrease the loss, respectively. These adjustments make the optimizer take larger and smaller steps when the loss is increasing and decreasing, respectively.
When the loss increases or decreases, the solver adaptively increases or decreases the damping factor by multiplying it by DampingIncreaseFactor and DampingDecreaseFactor, respectively.
Gradient Clipping
If the gradients increase in magnitude exponentially, then the training is unstable and can diverge within a few iterations. This "gradient explosion" is indicated by a training loss that goes to NaN
or Inf. Gradient clipping helps prevent gradient explosion by stabilizing the training at higher learning rates and in the presence of outliers [3]. Gradient clipping enables networks to be trained
faster, and does not usually impact the accuracy of the learned task.
There are two types of gradient clipping.
• Norm-based gradient clipping rescales the gradient based on a threshold, and does not change the direction of the gradient. The "l2norm" and "global-l2norm" values of GradientThresholdMethod are
norm-based gradient clipping methods.
• Value-based gradient clipping clips any partial derivative greater than the threshold, which can result in the gradient arbitrarily changing direction. Value-based gradient clipping can have
unpredictable behavior, but sufficiently small changes do not cause the network to diverge. The "absolute-value" value of GradientThresholdMethod is a value-based gradient clipping method.
L[2] Regularization
Adding a regularization term for the weights to the loss function $E\left(\theta \right)$ is one way to reduce overfitting [1], [2]. The regularization term is also called weight decay. The loss
function with the regularization term takes the form
${E}_{R}\left(\theta \right)=E\left(\theta \right)+\lambda \Omega \left(w\right),$
where $w$ is the weight vector, $\lambda$ is the regularization factor (coefficient), and the regularization function $\Omega \left(w\right)$ is
$\Omega \left(w\right)=\frac{1}{2}{w}^{T}w.$
Note that the biases are not regularized [2]. You can specify the regularization factor $\lambda$ by using the L2Regularization training option. You can also specify different regularization factors
for different layers and parameters.
The loss function that the software uses for network training includes the regularization term. However, the loss value displayed in the command window and training progress plot during training is
the loss on the data only and does not include the regularization term.
[1] Bishop, C. M. Pattern Recognition and Machine Learning. Springer, New York, NY, 2006.
[2] Murphy, K. P. Machine Learning: A Probabilistic Perspective. The MIT Press, Cambridge, Massachusetts, 2012.
[3] Pascanu, R., T. Mikolov, and Y. Bengio. "On the difficulty of training recurrent neural networks". Proceedings of the 30th International Conference on Machine Learning. Vol. 28(3), 2013, pp.
[4] Kingma, Diederik, and Jimmy Ba. "Adam: A method for stochastic optimization." arXiv preprint arXiv:1412.6980 (2014).
[5] Liu, Dong C., and Jorge Nocedal. "On the limited memory BFGS method for large scale optimization." Mathematical programming 45, no. 1 (August 1989): 503-528. https://doi.org/10.1007/BF01589116.
[6] Marquardt, Donald W. “An Algorithm for Least-Squares Estimation of Nonlinear Parameters.” Journal of the Society for Industrial and Applied Mathematics 11, no. 2 (June 1963): 431–41. https://
Version History
Introduced in R2016a
R2024b: Train neural networks using more learning rate schedules
Train neural networks using these learning rate schedules by specifying them as the LearnRateSchedule argument of the trainingOptions function:
• "warmup" — Warm-up learning rate schedule
• "polynomial" — Polynomial learning rate schedule
• "exponential" — Exponential learning rate schedule
• "cosine" — Cosine learning rate schedule
• "cyclical" — Cyclical learning rate schedule
To customize these learning rate schedules, use these objects:
In previous versions, you could train using a piecewise learning rate schedule or no learning rate schedule.
To customize the existing piecewise learning rate schedule, use a piecewiseLearnRate object.
To specify a custom schedule, use a function handle with the syntax learnRate = f(initialLearnRate,epoch), or define your own custom learn rate schedule object by defining a class that inherits from
R2024b: Train using Levenberg–Marquardt solver
Train a neural network using the Levenberg–Marquardt (LM) solver. Use the LM algorithm for regression networks with small numbers of learnable parameters, where you can process the data set in a
single batch.
To use the LM solver with the trainnet function, create a TrainingOptionsLM object by specifying the solverName argument as "lm". You can customize the LM solver using these new training options:
R2024b: Monitor and plot more metrics during training
Use new and updated metric objects during training and testing.
You can also directly specify these new built-in metric and loss names:
• "mape"— Mean absolute percentage error (MAPE)
• "crossentropy" — Cross-entropy loss
• "index-crossentropy" — Index cross-entropy loss
• "binary-crossentropy" — Binary cross-entropy loss
• "mse" / "mean-squared-error" / "l2loss" — Mean squared error
• "mae" / "mean-absolute-error" / "l1loss" — Mean absolute error
• "huber" — Huber loss
R2024b: Specify initial step size for L-BFGS solver
Specify the initial step size for the L-BFGS solver using the InitialStepSize argument.
R2024a: Specify validation data using minibatchqueue object
Specify validation data as a minibatchqueue object using the ValidationData argument.
R2024a: Automatic performance optimization
Accelerate training with automatic performance optimization. When you train a network using the trainnet function, automatic performance optimization is enabled by default. You can disable
performance optimization by setting the Acceleration option to "none" using the trainingOptions function.
R2024a: Specify metrics as deep.DifferentiableFunction object
Specify the metrics as deep.DifferentiableFunction object.
R2024a: Setting SequenceLength to an integer is not recommended
Setting SequenceLength to an integer is not recommended, set SequenceLength to "longest" or "shortest" instead.
For trainNetwork workflows (not recommended), you can set SequenceLength to an integer. If SequenceLength is an integer, then for each mini-batch, the software pads the sequences to the length of the
longest sequence in the mini-batch, and then split the sequences into smaller sequences of the specified length. If splitting occurs, then the software creates extra mini-batches and updates the
network recurrent state between these mini-batches. If the specified sequence length does not evenly divide the sequence lengths of the data, then the mini-batches containing the ends those sequences
have length shorter than the specified sequence length.
R2024a: DispatchInBackground training option is not recommended
The DispatchInBackground training option is not recommended. Use the PreprocessingEnvironment option instead.
The PreprocessingEnvironment option provides the same functionality and also allows you to use the backgroundPool for preprocessing when you set PreprocessingEnvironment to "background".
This table shows how to update your code:
Not recommended Recommended
trainingOptions(solverName,DispatchInBackground=false) (default) trainingOptions(solverName,PreprocessingEnvironment="serial") (default)
trainingOptions(solverName,DispatchInBackground=true) trainingOptions(solverName,PreprocessingEnvironment="parallel")
There are no plans to remove the DispatchInBackground option.
R2024a: OutputNetwork default is "auto"
Starting in R2024a, the OutputNetwork training option default value is "auto". If you have specified validation data, then the software returns the network corresponding to the best validation metric
value. If you have not specified validation data, then the software returns the network corresponding to the last training iteration. If you have validation data and want to replicate the previous
default, then set OutputNetwork to "last-iteration".
This change applies when using the training options with trainnet only. If you are using the training options with the trainNetwork function, then there is no behavior change and by default the
software returns the network corresponding to the last training iteration.
R2024a: OutputNetwork value "best-validation-loss" is not recommended
Specifying OutputNetwork as "best-validation-loss" is not recommended. If you have code that set OutputNetwork to "best-validation-loss", then use "best-validation" instead. The software returns the
network corresponding to the best validation metric value as specified by the ObjectiveMetricName option. By default, the ObjectiveMetricName value is set to "loss". This behavior applies when using
the training options with the trainnet function only.
When using the training options with the trainNetwork function, if you specify OutputNetwork as "best-validation", then software always returns the network with the best validation loss value.
R2024a: ExecutionEnvironment value "parallel" is not recommended
Starting in R2024a, specifying the ExecutionEnvironment option as "parallel" is not recommended. Use "parallel-auto" instead.
"parallel-auto" has these advantages over "parallel":
• The name of the option more accurately describes the execution environment, as the software trains in parallel automatically using whatever hardware is available.
• The name of the option is consistent with the serial equivalent, "auto".
There are no plans to remove the "parallel" option. "parallel-auto" supports the trainnet function only. If you are using the training options with the trainNetwork function, then continue to use
R2024a: WorkerLoad training option is not recommended
Starting in R2024a, specifying the WorkerLoad training option is not recommended. Use spmd (Parallel Computing Toolbox) or the CUDA_VISIBLE_DEVICES environment variable instead.
There are no plans to remove support for WorkerLoad for training networks using the trainNetwork function. WorkerLoad is not supported for training networks using the trainnet function.
This table shows some typical usages of WorkerLoad and how to update your code to use spmd or the CUDA_VISIBLE_DEVICES environment variable instead.
Not Recommended Recommended
% Alternative 1
pool = parpool(3);
if spmdIndex == 3
gpuDevice(spmdIndex + 1);
options = trainingOptions(solver, ... end
ExecutionEnvironment="multi-gpu", ... end
WorkerLoad=[1 1 0 1]);
options = trainingOptions(solver, ...
% Alternative 2
% Set this environment variable immediately after your start MATLAB.
options = trainingOptions(solver, ...
pool = parpool(3);
if spmdIndex == 3
options = trainingOptions(solver, ... gpuDevice(spmdIndex + 1);
ExecutionEnvironment="parallel", ... else
WorkerLoad=[1 1 0 1]); gpuDevice(spmdIndex);
options = trainingOptions(solver, ...
If you were previously using the WorkerLoad option to reserve a worker to preprocess your data, consider also preprocessing you data in the background by specifying the PreprocessingEnvironment
option as "background".
R2023b: Train neural network using L-BFGS solver
Train a neural network using the L-BFGS solver by specifying solverName as "lbfgs". Use the L-BFGS algorithm for small networks and data sets that you can process in a single batch. To customize the
L-BFGS solver, use the Batch Solver Options properties.
This option supports the trainnet function only.
R2023b: Specify input and target data formats
Specify the input and target data formats using the InputDataFormats and TargetDataFormats options, respectively.
This option supports the trainnet function only.
R2023b: Train neural network in parallel using only CPU or only GPU resources
Train a neural network in parallel using specific hardware resources by specifying the ExecutionEnvironment as "parallel-cpu" or "parallel-gpu".
This option supports the trainnet function only.
R2023b: BatchNormalizationStatistics default is "auto"
Starting in R2023b, the BatchNormalizationStatistics training option default value is "auto".
This change does not affect the behavior of the function. If you have code that checks the BatchNormalizationStatistics property, then update your code to account for the "auto" option.
R2022b: trainNetwork pads mini-batches to length of longest sequence before splitting when you specify SequenceLength training option as an integer
Starting in R2022b, when you train a neural network with sequence data using the trainNetwork function and the SequenceLength option is an integer, the software pads sequences to the length of the
longest sequence in each mini-batch and then splits the sequences into mini-batches with the specified sequence length. If SequenceLength does not evenly divide the sequence length of the mini-batch,
then the last split mini-batch has a length shorter than SequenceLength. This behavior prevents the neural network training on time steps that contain only padding values.
In previous releases, the software pads mini-batches of sequences to have a length matching the nearest multiple of SequenceLength that is greater than or equal to the mini-batch length and then
splits the data. To reproduce this behavior, use a custom training loop and implement this behavior when you preprocess mini-batches of data.
R2018b: ValidationPatience training option default is Inf
Starting in R2018b, the default value of the ValidationPatience training option is Inf, which means that automatic stopping via validation is turned off. This behavior prevents the training from
stopping before sufficiently learning from the data.
In previous versions, the default value is 5. To reproduce this behavior, set the ValidationPatience option to 5.
R2018b: Different file name for checkpoint networks
Starting in R2018b, when saving checkpoint networks, the software assigns file names beginning with net_checkpoint_. In previous versions, the software assigns file names beginning with
If you have code that saves and loads checkpoint networks, then update your code to load files with the new name.
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This 2-dimensional shape has 5 sides; it's also a name for an important government building
What is the pentagon?
The absolute value of -5
What is 5?
1 ton = this many pounds
What is 2000?
A toilet paper roll would be a perfect example of this kind of 3-dimensional shape
What is a cylinder?
The square root of 225
What is 15?
1 meter = this many centimeters
What is 100?
When someone's story seems full of holes or when you are unsure of what actually happened, you may claim that the story is not doing "this"
What is adding up?
This is what Mr. Carpenter wore a lot of during the colder months (very comfy!)
What are flannels?
This shape forms many stop signs (and unfortunately is the result of many traffic tickets)
What is an octagon?
54 / 6
What is 9?
1 feet = this many inches
What is 12?
When sending an invite, you might tell someone to "be there or be 'this'"
What is square?
When we use a letter to represent an unknown number, we call it this
What is a variable?
When trying to solve a difficult problem, math or not, you may be told to approach it at a different "this"
What is angle?
This is what the E in PEMDAS stands for
What is exponent?
The pyramid can have many different kinds of bases at the bottom; for example, the pyramids in Egypt have this 2-dimensional shape as their base
What is square?
What is 64?
1 kilogram = this many pounds
What is 2.2?
When a comment on someone's social media post gets more engagement than the post itself, it is known as this
What is a ratio?
This math term for a collection of inputs and outputs (known as x and y) is also something people can't do without their morning coffee
What is function?
This 2-dimensional shape has 10 sides
What is a decagon?
The cubed root of 8
What is 2?
1 mile = this many feet
What is 5280?
A popular "scam" business model where many people contribute cash but only one person at the top actually profits is known as this (don't fall for them!)
What is a pyramid scheme?
This term represents the steepness of a line
What is a slope?
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Space Plasma Physics Revisited (2)
2)Magnetic mirrors and solar loops
The basic plasma physics for solar active regions usually starts by examining solar coronal loops to see where they conform to the typical magnetic mirror profile used in standard plasma physics. In
space physics, one uses the sine of the loss cone angle to obtain the mirror ratio relating the magnetic inductions at the loop ends:
sin (q [L] ) = ± Ö (B[min] / B [max] )
If one finds that there are particles within the “mirrors” for which the “pitch angle” (a) has:
sin (a ) > Ö (B[min] / B [max] )
then these will be reflected within the tube, On the other hand, those particles for which the “less than” condition applies will be lost, i.e. on transmission out of the mirror configuration.
Since the adiabatic invariant for particle motion is a constant of the motion:
m= ½ [mv ^2 /B]
we have:
[v [^]^2 /B[max]] = [v [||][ ]^2 /B[z]] = const. or
[v [^]^2 / v [||][ ]^2 ] = B[z ]/B[max]
Where we take B[z] = B[min]
A simple model for a mirror system is shown below with M1 and M2 the magnetic mirrors separated by a distance 2 L.
If we are careful to change L slowly, then one finds:
v [||][ ]L = const.
Now, assume M1 is stationary and M2 moves toward M1 at a velocity, v [m] , then the incident velocity relative to the wall is:
[(v [||][ ] + v [m]) - v [m] ]
D v [||] = - [- (v [||][ ] + v [m]) - v [m] ] + v [|] = 2 v [m]
Thus, with each reflection, the velocity changes by 2 v [m] and the number of reflections per second will be: v [||][ ]/ 2L
dv [||][ ] /dt = 2 v [m ] (v [||][ ]/ 2L) = v [||][ ]/L (-dL/dt)
= - v [||][ ]/L (dL/dt)
So that: d/dt (v [||][ ]L) = 0
For kinetic energy of particles we must have:
E = ½ m (v[⊥] ^2 + v [||][ ]^2 ) = const.
Note also that: v[⊥]/ Ω = v[⊥]/(qB/m) = m v[⊥]/qB
Let the guiding center lie on the z-axis (for simplicity) then the average force experienced will be written:
F [z] = + ½ q v[⊥ ]r [L] (¶B [z]/ ¶z) = + ½ q v[⊥ ]/ Ω [c ]¶B [z]/ ¶z
Since: v[⊥ ]= r [L ]Ω [c ]so: r [L] = v[⊥]/ Ω [c]
Where r [L ]is the guiding center Larmor radius.
With appropriate substitutions we get:
F [z ]= + ½ q v[⊥ ]r [L] (m v[⊥]/qB) (¶B [z]/ ¶z) =
+ ½ m v^2[⊥ ]/B (¶B [z]/ ¶z)
But by previous definition for the adiabatic invariant:
m [m ] = m(v⊥)^2/ 2B = const .
So: F [z] = m ¶B [z]/ ¶z
Where I have dropped the subscript ‘m’.
In relation to solar flare prognostication we are interested in the criterion for the hydrodynamic loss cone instability which requires that the particular condition for the ratio of untrapped to
trapped particles (Pearlstein et al, 1966)[1] :
n/ n[o] > 2 W [e ]/ p w[ e] = 0.1
where W [e] is the electron cyclotron frequency; and w[ e] is the electron plasma frequency:
w[ e] = [n[e] e^2/ m[e] ε[o]] ^½
Where n[e] is the electron number density, e is the electron charge, m[e] is the electron mass and ε[o ] is the permittivity of free space.
1)A proton moves in a uniform electric and magnetic field, with fields given by: E = 10 V/m (x^) and B = 0.0001 T (z^)where '^' denotes vector direction. (Take m [m ] = 8.5 x 10 ^-22 J/T)
a) Find the gyrofrequency and the gyro-radius
b) Find the proton's E X B drift speed
c)Find the gyration speed v and compare it to the drift speed
d)Find the gyro-period, gyration energy and magnetic moment of the proton.
2) Consider a plasma mirror machine of length 2L with a mirror ratio of 10 so that B(L) = B(-L) = 10 B(0). A group of N (N > 1) electrons with an isotropic velocity distribution is released at the
center of the machine. Ignoring collisions and the effect of space charge, how many electrons escape?
3) Consider a similar mirror configuration for a solar coronal loop for which B[min] = 0.2 B[max] . Find the loss cone angle for this loop and also determine whether particles will remain within it.
Find the velocity ratio :
v [^] / v [||][ ] if B[max] = 0.95 B [z]
4) The basic plasma frequency equation relating the ion to the electron plasma frequency can be written:
w[ ]^3[ ] = - ½[ ]( w[ i] )^ 2 w[ e]
Find the roots of this equation in terms of the ratio of masses, m[ e] /m[ i]. Show that the real part (or frequency seen in the ion rest frame) will be:
w[ r ] = (½) ^5/3 (m[ e] /m[ i]) ^1/3 w[ e]
Hint: You may make use of the fact that:
0 = 1 - w[ i] ^ 2 / w[ ]^2 -- ^ w[ e ]^2 /( w - w[ e ])^ 2
Where w[ i] is the ion plasma frequency and w[ e] is the electron plasma frequency
[1] L.D. Pearlstein, M.N. Rosenbluth, and D.B. Chang, 1966, Phys. Fluids., Vol. 9, p. 53.
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Solving Born-Oppenheimer using group-equivariant deep learning
By Sander Vandenhaute, Mieke De Schepper, Veronique Van Speybroeck
Quantum mechanics allows us to understand and predict the properties of complex molecules and materials. A key simplification that is made in order to be able to solve the quantum mechanical
equations for realistic systems of practical importance is the Born-Oppenheimer approximation. Effectively, it implies that the answer to questions such as “what is the total reaction rate of this
chemical process” to “what is the maximum pressure this material can sustain” and even “how much hydrogen gas can we store in this porous framework” may be found without having to solve the
nigh-unsolvable Schrödinger’s equation – the central differential equation in quantum mechanics. No, for many systems of practical importance, Born and Oppenheimer stated that the individual atoms
may actually be considered as simple, classical point-like particles which behave according to Newton’s equation of motion: F = ma.
Figure 1. Atomic structures of two prototypical metal-organic frameworks; the class of nanoporous materials that were studied in this work. Important macroscopic properties are e.g. the maximum
absorbed capacity of hydrogen gas or the threshold pressure which triggers the closing of the winerack structure on the right. The latter property is explicitly computed in this work and shows
excellent agreement with the experimental value. Image reproduced from [1]
For this approximation to make sense, the effective force F needs to capture most of the quantum tomfoolery of the system, and this is absolutely nontrivial. The more ‘complete’ a specific force
approximation is, the more expensive it is to compute and the more impractical it becomes when using it for real systems. To overcome this computational challenge, researchers began to develop
machine learning models which are able to predict these forces based on the positions of each of the atoms. Essentially, this allows us to bypass the computationally expensive quantum mechanical
calculations and reduce the financial and environmental cost of computational research on molecules and materials by orders of magnitude. At the moment, the state of the art in such force prediction
models — also referred to as machine learning (interatomic) potentials – is achieved by equivariant neural networks. These models are specifically designed to incorporate the intrinsic physical
symmetries that are present in an arrangement of atoms – rotational, translational, and even permutation invariance. By doing so, they achieve exceptional levels of accuracy and are now used in a
myriad of applications.
"Psiflow allows us to perform large amounts of quantum mechanical calculations on CPUs and model training on GPUs in a single workflow. [...] specific configurations for VSC systems are included
by default."
However, these machine learning potentials are only as accurate as the dataset of reference calculations they were trained on – and generating high-quality datasets has turned out to be nontrivial in
many cases. In particular, a reliable parameterization of these models requires a training dataset of QM reference calculations that covers all the relevant atomic configurations of interest. This is
often very challenging because the intrinsic time scale in most of the relevant processes in biology or chemistry is much larger than what is achievable using traditional simulation methods, and as a
result, it is very expensive and sometimes even impossible to generate sufficient training data in the first place.
Enter incremental learning; a new methodology that was developed at the Center for Molecular Modeling, in which the machine learning potential is trained on the fly while new atomic configurations
are continuously generated and added to the training set. A key component is the use of enhanced sampling to accelerate the simulation dynamics, which allows it to include more diverse samples in the
training set, leading to more accurate estimates of the interatomic force F and, ultimately, more accurate predictions of physical properties (Figure 2). The algorithm is implemented in psiflow, a
modular online learning library for interatomic potentials that is scalable towards hundreds or even thousands of compute nodes. It is configurable on a wide range of high-performance computing
platforms, and specific configurations for VSC systems are included by default. Psiflow allows us to perform large amounts of quantum mechanical calculations on CPUs and model training on GPUs in a
single workflow. For more information, check out our open-access article in npj Computational Materials.
Figure 2. Explicit simulation of the structural transition of the framework shown in Figure 1 (right). We were able to train our interaction potential with negligible amounts of compute resources
while predicting a threshold transition pressure of 18 – 20 MPa, which is in very good agreement with experimental data (and a drastic improvement with respect to previous computational estimates).
Image reproduced from [1].
[1] Sander Vandenhaute, Maarten Cools-Ceuppens, Simon DeKeyser, Toon Verstraelen, Veronique Van Speybroeck, npj Computational Materials, 9, 19 (2023).
Read the full publication in the Nature Portfolio here.
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UChicago Instructional Physics Laboratories
Table of Contents
Magnetic Fields I: $e/m$ of Electrons
In the late 19th century, J. J. Thompson began working with a device known as a cathode ray tube (CRT) to investigate the nature of the mysterious “cathode rays” that the device produced.
When scientists passed electricity through a glass tube filled with a low-density gas, they saw a beam of glowing light – the so-called cathode rays. Thompson discovered that he could deflect the
path of the beam by applying a magnetic field, and in doing so showed that the beam was a stream of negatively charged particles – what we today know are electrons. (The explanation for why the gas
glowed when the energetic electrons passed through it would not become clear until quantum mechanics explained the ideas of gas ionization and transitions of electrons between energy levels… but that
will have to wait for another experiment.)
Thompson's experiment is considered the “discovery” of the electron, but it could determine neither the magnitude of the charge nor the mass of the particle… only the ratio of the two: $e/m$. It was
not until Millikan's oil drop experiment – performed between 1900 and 1911 here at the University of Chicago – established the value of the fundamental charge $e$ that Thompson's ratio could be
solved for the mass of the electron on its own.
The force on a charge moving in a magnetic field is
$\mathbf{F} = q\mathbf{v}\times\mathbf{B}$, $(1)$
where $q$ is the charge, $\mathbf{v}$ is the velocity of the moving charge, and $\mathbf{B}$ is the magnetic field. The direction of the force is given by the right-hand rule and is perpendicular to
both the velocity and magnetic field.
The magnitude of the force is given by the scalar form of Eq. (1),
where $\phi$ is the angle between the direction of the magnetic field and the direction of motion of the moving charge.
Suppose a beam of electrons is directed into a magnetic field at right angles to the field as shown in Fig. 1.
Figure 1a: Electron trajectory geometry
In this special case, Eq. (2) becomes
where $e$ is the charge of the electron. The electron beam will follow a circular trajectory within the field with a centripetal force
$F=\dfrac{mv^2}{R}$, $(4)$
where $m$ is the mass of the electron and $R$ is the radius of the circle.
For a non-relativistic electron accelerated through a potential $V$, the kinetic energy is
$K = \frac{1}{2}mv^2 = eV$. $(5)$
Setting Eqs. (3) and (4) equal, solving for $v$, and substituting into Eq. (5), we have
$\dfrac{e}{m} = \dfrac{2V}{R^2B^2}$. $(6)$
Figure 1b: Electron trajectory geometry, highlighting the relevant similar triangles
Since $R$ is not measurable in this experiment, we wish to express it in terms of other variables which are measurable. Referring to Fig. 1b, by similar triangles (dashed green triangle and dotted
white triangle) we have
\begin{equation*} \dfrac{\overline{AB}}{R} = \dfrac{D}{\overline{OB}}. \end{equation*} However, $\overline{AB} = \overline{OB}/2$. Therefore, \begin{equation*} R = \dfrac{\left(\overline{OB}\right)^
2}{2D}. \end{equation*} For small angles $\theta$ (such as those in our cathode ray tube), $\overline{OB} \approx L$. Therefore, \begin{equation*} R = \dfrac{L^2}{2D} \end{equation*} Substituting
this expression for $R$ into Eq. (6) gives
$\dfrac{e}{m} = \dfrac{8V_aD^2}{L^4B^2}$. $(7)$
Since one can measure all of these quantities but $L$, it is now possible to arrive at a value of $e/m$. To make this more amenable to plotting, we may rearrange this as
$D^2 = \dfrac{e}{m} \dfrac{L^4B^2}{8V_a}$. $(8)$
In this way, the ratio $e/m$ is proportional to the slope of a line plotting either $D^2$ vs. $B^2$ or $D^2$ vs. $1/V_A$.
Experimental setup
Lab notebook template
One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to
share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.
Constructing the circuits
Construct the two separate parts of the apparatus as shown in Fig. 2. The portion of each schematic shows the wiring for the cathode ray tube which produces the beam of electrons and accelerates
them toward the tube’s screen. The right side shows the circuit which produces the magnetic field which will deflect the electron beam.
Physics 122
Physics 132
Figure 2b: $e/m$ wiring diagram using 3B Scientific power supply
Figure 2a: $e/m$ wiring diagram using Heathkit power supply model IP-17
When wiring your setup, you should start by connecting the yellow wires, and then connected to the $+$ terminal of your supply.
Click to enlarge either diagram
Wiring photos & connections
Photo of the red wire, brown wire, and yellow paired wires from the electron tube. Note that Photo of the twin leads from the electron tube. Each controls one of the deflection plates. While we
the red and brown wires do not stack, they will have to be plugged in last. won't be using the plates, these still need to be grounded to prevent unwanted behavior.
Physics 132 Physics 122
Photo of Coil connections around the cathode ray tubes. Note that the 132 course uses connectors on the back right of the setup, whereas the 122 course uses connectors beneath the front of the tube.
Magnetic field
Equation (7) giving $e/m$ for the electron from the measured quantities is derived using the following simplifying assumptions:
• The magnetic field $B$ is assumed to be perfectly constant over the well-defined path length $L$.
• The electrons are assumed to be moving at constant speed along $L$.
In our experiment, however, $B$ is not perfectly constant over the electron beam trajectory and $L$ is not well-defined. Also, the electrons are not moving at constant speed for the first $4\text{
cm}$ of travel.
Figure 3 shows the measured magnetic field profile along the tube length. The maximum magnetic field is normalized to $1.0$.
The relation between the current in the coils surrounding the tube $I$ and the magnetic field along the electron beam trajectory $B$ is
PHYS 132 $B = \left(8.3 \times 10^{-5} ~\dfrac{\mathrm{T}}{\mathrm{A}}\right) \times I$. $(9a)$
PHYS 122 $B = \left(7.8 \times 10^{-4} ~\dfrac{\mathrm{T}}{\mathrm{A}}\right) \times I$. $(9b)$
A note on $L$
Equation (8) is derived using the following simplifying assumptions:
• The magnetic field is assumed to be perfectly constant over the well-defined path length $L$.
• The electrons are assumed to be moving at constant speed along $L$.
In our experiment, however, $B$ is not perfectly constant over the electron beam trajectory and $L$ is not well-defined. Also, the electrons are not moving at constant speed for the first 4 cm of
Looking back at Fig. 3, we see the following:
• the electrons do not begin accelerating at position $x = 0$,
• the electrons are accelerating (but have not yet reached full velocity) in the region between about $x = 4$ cm and $x = 8$ cm, and
• the electrons experience a decreasing magnetic field over the final region from about $x = 20$ cm to $x = 23$ cm.
The range of possible values for $L$ is therefore between 15 cm and 19 cm. This will be an important point that we will return to during the analysis.
Taking data
From Eq. (8), we see that we have two independent variables we can control – $V_A$ and $B$ – and one dependent variable that we can measure – $D$. Therefore, you can either…
• …hold $V_A$ fixed while varying $B$
• …hold $B$ fixed while varying $V_A$.
The Colab notebook provides two methods for using a constant $B$ field. The first requires you to measure the difference in the spot's position both with and without the magnetic field for each
$V_A$. The second method instead lets this displacement be part of the fit equation.
Think about what advantages (or disadvantages) each method has and decide how you would like to take your data in order to test Eq. (8).
We provide a Google Colab notebook that you can use to plot and fit your data.
Physics 132: My voltage controls don't do anything!
You may have tripped the overcurrent protection feature of your power supply. See the video below for how to fix this.
Reminder: Using Logger Pro
Here's a reminder on how to make a plot and fit data in Logger Pro:
1. Make a new file in Logger Pro.
2. Enter your $x$-axis data in the first column.
3. Enter your $y$-axis data in the second column.
4. Double-click on a column header to rename it.
5. Use the “Scale” button to resize your screen to show your data.
6. Select the “Linear Fit” button to fit an equation to your line.
7. Use the top menu “Data → New Data Set” to make additional plots.
1. Click on a $y$-axis column and drag it to the plot area to display it
Submit your lab notebook
Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to
submit to Canvas, but make sure everyone's name is on the document!
When you're finished, don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!
Post-lab assignment
Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these
questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.
The conclusion is your interpretation and discussion of your data.
• What do your data tell you?
• How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making
qualitative comparisons.)
• Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
• Do your results lead to new questions?
• Can you think of other ways to extend or improve the experiment?
In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or
anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.
REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.
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Mass concentration in rescaled first order integral functionals
Published Paper
Inserted: 2 may 2021
Last Updated: 22 feb 2024
Journal: Journal de l'École Polytechnique
Volume: 11
Pages: pp. 431-472
Year: 2024
Doi: 10.5802/jep.257
Links: journal website
HAL repository
arXiv repository
We consider first order local minimization problems of the form $\min \int_{\mathbb{R}^N}f(u,\nabla u)$ under a mass constraint $\int_{\mathbb{R}^N}u=m$. We prove that the minimal energy function $H
(m)$ is always concave, and that relevant rescalings of the energy, depending on a small parameter $\varepsilon$, $\Gamma$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_i\
delta_{x_i}$ as $\sum_i H(m_i)$. We also consider Lagrangians depending on $\varepsilon$, as well as space-inhomogeneous Lagrangians and $H$-masses. Our result holds under mild assumptions on $f$,
and covers in particular $\alpha$-masses in any dimension $N\geq 2$ for exponents $\alpha$ above a critical threshold, and all concave $H$-masses in dimension $N=1$. Our result yields in particular
the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.
Keywords: Branched transport, Concentration-compactness, semicontinuity, integral functionals, \(\Gamma\)-convergence, convergence of measures, Cahn-Hilliard fluids
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Cliff Burgess
I was born in Manitoba and was raised in various places around Western Canada, Ontario and Europe. I received my B.Sc., with a joint honours in Physics and Applied Math from the University of
Waterloo, and continued for doctoral work in Theoretical Particle Physics at the University of Texas in Austin under the supervision of Steven Weinberg. After doing a postdoctoral stint at the
Institute for Advanced Study in Princeton, I became a professor at McGill University, ending up being appointed as a James McGill professor there in 2003. I am presently an Associate Member at PI
with a joint appointment as a Professor within McMaster University's department of Physics and Astronomy. I was elected a Fellow to the Royal Society of Canada in 2008, and awarded the CAP/CRM medal
for Theoretical Physics in 2010.
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Gr 6_RP_ProportionalReasoning_Problem_Critique_CarlylesPlants | Bridging Practices Among Connecticut Mathematics Educators
Gr 6_RP_ProportionalReasoning_Problem_Critique_CarlylesPlants
I used this task with a group of 5 students that I selected for my evaluation. When I started with this group in mid/late January they were chosen by the teacher and myself to receive intervention as
well. This year the students were learning about ratios and proportions. I tried to build their understanding of ratios by doing Number Talks before giving them this particular task. The students
were required to use a rubric I found on the Exemplars website to help guide them in solving and writing an argument for this problem. The students were instructed to use more than one model to solve
this task
Microsoft Word version: 6_RP_ProportionalReasoning_Problem_Critique_CarlylesPlants
PDF version: 6_RP_ProportionalReasoning_Problem_Critique_CarlylesPlants
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how many cubic units is a box that is 3 units high, 3 units wide, and 2 units deep?
How many cubic units is a box that is 3 units high, 3 units wide, and 2 units deep?
From www.answers.com :
To calculate volume, you simply have to multiply the length times the width times the depth. In this case, you'd calculate 3 x 3 x 2, which is 18.
From answers.yahoo.com :
You just multiply the dimensions.
2 x 3 x 3 = 18
So the box should be 18 cubic units. To find the volume of any rectangular prism, or cube, you just multiply the 3 sides by each other.
From bengislife.com :
How many cubic unit is a box that is 3 units high, 3 units wide, and 2 units deep?
This same how to calculate volume, you need only how many units
So, how to calculate unit?
You must multiply = unit high x unit wide x unit deep
3 unit high x 3 unit wide x 2 units deep = 18 units in box
How it hard for you? Please comment in under this article, thanks
what is
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Elementary Technical Mathematics Chapter 5 - Section 5.5 - Multiplication of Polynomials - Exercise - Page 225 7
By multiplying each term of the polynomial by the monomial, and then adding the products, we have $$x(3x^2-2x+5)=x(3x^2)+x(-2x)+x(5)\\ =3x^3-2x^2+5x. $$
You can help us out by revising, improving and updating this answer.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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Master the Art of Finding the Sum of a List – A Comprehensive Guide to Calculating the Sum of List Elements
The ability to calculate the sum of list elements is a fundamental skill in programming. Whether you’re working with numerical data, analyzing data sets, or solving mathematical problems, being able
to quickly and accurately find the sum of a list can greatly simplify your code and improve its efficiency. In this blog post, we will explore different methods of calculating the sum of a list, from
basic iterative approaches to advanced built-in functions. By the end of this post, you will have a clear understanding of how to find the sum of a list and be equipped with practical tips and
strategies for handling various scenarios.
Understanding Lists and List Elements
Before we dive into calculating the sum of a list, let’s first understand what lists are and how they are structured in Python. In Python, a list is an ordered collection of elements, where each
element is identified by its position or index. Lists can contain elements of different data types, such as numbers, text, or even other lists.
For example, consider the following list:
my_list = [1, 2, 3, 4, 5]
In this case, my_list contains five elements: 1, 2, 3, 4, and 5. The elements are indexed starting from 0, so we can access each element using its index.
Accurately identifying list elements is crucial for calculating the sum. It ensures that we include all the relevant elements and exclude any elements that are not part of the sum.
Basic Method: Iterative Summation
One of the simplest and most straightforward methods of finding the sum of a list is by using an iterative approach. The iterative summation method involves iterating through each element in the list
and adding it to a running total or sum variable.
Step-by-step walkthrough of iterative summation
Let’s break down the iterative summation method into simple steps:
1. Initializing the sum variable: Before we start iterating through the list, we need to initialize a sum variable to store the running total.
2. Looping through each element in the list: Using a loop (such as a for loop or a while loop), we iterate through each element in the list.
3. Adding each element to the sum variable: For each iteration, we add the current element to the sum variable.
4. Returning the final sum value: Once we have iterated through all the elements in the list, we return the final value of the sum variable.
Let’s take a look at an example to understand the iterative summation method better:
def calculate_sum(my_list): sum = 0 for num in my_list: sum += num return sum
result = calculate_sum([1, 2, 3, 4, 5]) print(result) # Output: 15
In this example, we define a function calculate_sum that takes a list my_list as an input. We initialize the sum variable to 0 and then iterate through each element in my_list using a for loop. For
each iteration, we add the current element to the sum variable. Finally, we return the value of the sum variable.
Pros and cons of using iterative summation method
The iterative summation method has several advantages:
1. Easy to understand and implement: The iterative summation method is relatively straightforward and easy to grasp, even for beginners.
2. Flexibility: This method allows for customization, as you can easily modify the algorithm to fit your specific requirements.
3. Control over the process: With the iterative method, you have complete control over the summation process, allowing you to add additional logic or conditions as needed.
However, there are also some limitations to consider:
1. Performance and efficiency: The iterative method may not be the most efficient solution for large lists or scenarios where performance is critical. For large lists, it can be time-consuming to
iterate through each element.
2. Increased complexity for complex operations: If you need to perform complex operations on each element before adding it to the sum, the iterative method may become more complex and less
Despite these limitations, the iterative summation method remains a valuable tool for finding the sum of a list, particularly for smaller lists or scenarios where performance is not a significant
Advanced Method: Built-in Functions
In addition to the iterative method, Python provides built-in functions that can be used to calculate the sum of a list. One such function is the sum() function, which is specifically designed for
finding the sum of numerical elements within a list.
Introduction to built-in sum() function
The sum() function takes an iterable (such as a list) as its argument and returns the sum of all the elements. It eliminates the need for manual iteration and addition, as it performs these
operations internally.
How to use sum() function for finding the sum of a list
Using the sum() function to calculate the sum of a list is simple. Let’s see an example:
my_list = [1, 2, 3, 4, 5] result = sum(my_list) print(result) # Output: 15
In this example, we directly pass the list my_list as an argument to the sum() function. The function internally iterates through each element of the list and adds them together to compute the sum.
Advantages and limitations of using built-in functions
The use of built-in functions like sum() offers several advantages:
1. Simplicity and readability: Built-in functions simplify the code and make it more readable by eliminating the need for manual iteration and addition.
2. Efficiency: The built-in functions are optimized for performance and are generally faster than manual iteration for large lists.
3. Reduced potential for errors: With built-in functions, there is less risk of introducing errors during manual iteration or summation.
However, it’s important to be aware of the limitations of built-in functions:
1. Less control and customization: Built-in functions may not offer the same level of control as manual iteration, limiting any additional or customized logic you may want to apply during the
summation process.
2. Not suitable for non-numeric elements: The sum() function is specifically designed for numerical elements and may throw an error if non-numeric elements are present in the list.
Despite these limitations, the use of built-in functions can significantly simplify the code and improve overall efficiency when calculating the sum of a list.
Handling Special Cases and Edge Cases
While the basic and advanced methods discussed above cover the majority of scenarios for finding the sum of a list, it’s important to consider special cases and edge cases that may require additional
Empty lists and handling zero elements
When dealing with an empty list or a list with zero elements, the sum of the list will naturally be zero. However, it’s important to handle these cases appropriately to avoid potential errors or
incorrect results.
When using the iterative summation method, an empty list would result in the initial value of the sum variable being returned as the final sum. To handle this case, you can add an initial check to
return 0 if the list is empty:
def calculate_sum(my_list): if not my_list: return 0 sum = 0 for num in my_list: sum += num return sum
Similarly, when using the sum() function, an empty list would return 0 by default. Therefore, no additional handling is required in this case.
Non-numeric elements in the list
Although both the iterative method and the sum() function are designed to handle numerical elements, they may throw an error when non-numeric elements are present in the list.
To handle cases where the list contains non-numeric elements, you can add additional logic to either skip these elements during iteration or filter them out before performing the summation.
def calculate_sum(my_list): sum = 0 for num in my_list: if isinstance(num, (int, float)): sum += num return sum
In this modified version of the iterative summation method, we use the isinstance() function to check if each element is of type int or float. If the element is numeric, we add it to the sum.
In the case of the sum() function, filtering out non-numeric elements can be done using a list comprehension or a generator expression:
my_list = [1, 2, 'a', 4, 5] result = sum(num for num in my_list if isinstance(num, (int, float))) print(result) # Output: 12
In this example, we use a generator expression to iterate through each element of the list and check if it is numeric. Only the numeric elements are considered in the summation.
Lists with large number of elements and computational efficiency
When dealing with lists that have a large number of elements, computational efficiency becomes a concern. Both the iterative method and the use of built-in functions can become time-consuming for
large lists.
To improve the efficiency in these scenarios, consider using optimized algorithms or techniques that take advantage of the specific characteristics of the problem. For example, if the list is sorted,
you can use binary search instead of linear search to find elements during iteration.
It’s also worth exploring parallel processing techniques or utilizing libraries that offer optimized functions for computing the sum of large lists.
Strategies and considerations for different scenarios
Handling special cases and edge cases requires careful consideration and a tailored approach based on specific requirements. Here are a few strategies and considerations to keep in mind:
1. Define clear rules and requirements: Define guidelines and expectations for handling special cases, such as empty lists or non-numeric elements. This will ensure consistency and accuracy in your
2. Document your code: Clearly document any assumptions, requirements, or special handling techniques you employ for different scenarios. This will make your code more maintainable and
understandable for others.
3. Refactor and optimize when necessary: Regularly review your code and seek opportunities to optimize algorithms or leverage libraries for improved efficiency. Keep an eye on the computational
complexity of your code to avoid unnecessary performance bottlenecks.
By following these strategies, you can develop robust and efficient solutions for handling special cases and edge cases when calculating the sum of a list.
Practical Applications and Use Cases
Now that we have explored different methods of calculating the sum of a list, let’s take a look at some practical applications and use cases where this skill can be useful.
Real-world scenarios where finding the sum of a list is useful
Calculating the sum of a list is a common operation in various real-world scenarios, including:
• Financial analysis and accounting: Summing income and expenses, budget tracking, and financial modeling.
• Data analysis and statistics: Aggregating data, calculating averages and totals, and generating summary reports.
• Scientific research: Analyzing experimental data, processing measurements, and calculating cumulative values.
• Mathematical modeling: Evaluating mathematical functions, performing integrations, and estimating areas under curves.
These are just a few examples, but the applications of summing a list are virtually endless. Mastering this skill will greatly enhance your ability to analyze and manipulate data effectively in
various domains.
Examples of problem-solving using sum of list elements
To further illustrate the practical applications, let’s consider a few problem-solving scenarios where finding the sum of a list plays a critical role:
• Problem: You are given a list of sales figures for a series of products. Find the total revenue generated by summing the sales figures.
• Problem: In a game, you need to calculate the total score of players by summing their individual scores stored in a list.
• Problem: A stock market analyst wants to calculate the percentage change in a stock’s price by summing the daily price differences stored in a list.
These examples highlight the importance of calculating the sum of a list in various problem-solving scenarios. By applying the methods and techniques discussed earlier, you can confidently tackle
these types of problems.
Advantages of mastering the art of finding the sum of a list
Mastering the skill of finding the sum of a list offers several advantages:
1. Efficiency and code optimization: By utilizing optimized methods and algorithms, you can significantly improve the efficiency of your code, especially for large lists or computational-intensive
2. Code readability and simplicity: Access to efficient methods like built-in functions allows for more concise and readable code, making it easier for you and others to understand, maintain, and
3. Increased problem-solving capabilities: The ability to calculate the sum of a list equips you with an essential element of problem-solving, enabling you to tackle a wide range of challenges
across various domains.
As you continue to develop your programming skills, mastering the art of finding the sum of a list will undoubtedly prove beneficial in your journey as a programmer.
In this blog post, we explored different methods of calculating the sum of list elements. Starting from the basic iterative summation method, we walked through step-by-step instructions and provided
examples to illustrate the process. We then delved into the advanced method of utilizing built-in functions like sum() to calculate the sum of a list more efficiently.
We also discussed strategies and considerations for handling special cases and edge cases, such as empty lists, non-numeric elements, and lists with a large number of elements. By adapting the
methods and techniques to suit specific requirements, you can develop robust and efficient solutions.
Finally, we explored the practical applications and use cases of finding the sum of a list, emphasizing its relevance in various domains and problem-solving scenarios. By mastering this skill, you
can increase your efficiency, improve code readability, and enhance your problem-solving capabilities.
So, start practicing and honing your skills in calculating the sum of a list, and unlock your potential as a proficient and resourceful programmer.
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Marie-Louise Von Franz
Marie-Louise Von Franz
IN REMEMBRANCE OF MARIE-LOUISE VON FRANZ
Charles R. Card
Department of Physics and Astronomy, Box 3055, University of Victoria, Victoria, BC,
Canada V8W 3P6
E-mail: card@UVic.CA
Vasile V. Morariu
Department of Biophysics, Institute of Isotopic and Molecular Technology, 3400 Cluj – Napoca,
P.O.Box 700, tel. 0040 64 184037; fax 0040 64 420042;
E – mail: vvm@L40.itim-cj.ro
On February 17 of this year, Marie-Louise von Franz, psychotherapist and classical scholar, completed her passage through life—an extraordinary life made all the more remarkable by her many years of
close association with the Swiss psychologist, C.G. Jung. von Franz was born in Munich in 1915, the daughter of an Austrian baron. When she was three years old, her family relocated to Switzerland.
She first met Jung when she was an 18-year-old student–shy, introverted, keenly intelligent, and struggling to understand her relationship with her parents and to determine which course of studies
best suited her. She chose to study classical languages, and by 1940 she had received a Ph.D. from the University of Zurich, with a specialty in medieval Latin. With this expertise, she provided Jung
with translations of the medieval alchemical texts which he needed to pursue his investigation of the psychological basis of alchemy, in exchange for psychoanalytic sessions and training. By 1949 she
had become an analyst in her own right, and she soon emerged as one of Jung’s closest collaborators in the development of analytical psychology. Within the Jungian community she is best known for her
studies of the psychological significance of fairytales and of alchemy, but her deepest and most difficult work is concerned with the archetypal aspects of the natural numbers.
About two years before his death, Jung handed to von Franz a small slip of paper upon which he had begun to gather notes about the mathematical properties of the first four integers, saying, “I am
too old to be able to write this now, so I hand it over to you.” At the time she did not know whether he intended for her to pursue his study of number archetypes, or if he simply wanted her to hand
it over to someone whom she might meet that would be suitable for such a project. After Jung’s death she preferred to assume the latter because she felt incapable of doing it herself. However, as a
long interval passed without the appearance of anyone to take up the task, she was ‘bitten’ by her conscience and subsequently entered into a long and intensive period of research and writing that
culminated in the publication in 1970 of her treatise on number archetypes, Zahl und Zeit (Number and Time, 1974).
As indicated by the subtitle of this work —“Reflections Leading toward a Unification of Depth Psychology and Physics”—von Franz’s intention with Number and Time was to continue to explore the ideas
that had grown out of the collaboration of Jung with the Nobel laureate quantum physicist, Wolfgang Pauli. von Franz was, despite her reticence, well positioned to take on this work, for she had
worked closely with both men during the decade of their collaboration—helping each by translating passages from alchemical texts and working with Pauli to explore the symbolism of his dreams and
visions that were related to his intellectual quest.
In the course of this collaboration, which in its most active phase spanned the years from 1946-54, Jung and Pauli arrived at a set of propositions about the nature of reality that mark a fundamental
departure from the tenets of the worldview of modern science that has prevailed since Descartes. Jung and Pauli came to hold that the realm of mind, psyche, and the realm of matter, physis, are
complementary aspects of the same transcendental reality, the unus mundus. They asserted that archetypes act as the fundamental dynamical patterns whose various representations characterize all
processes, whether mental or physical. In the realm of psyche, archetypes organize images and ideas; in the realm of physis, they organize the structure and transformations of matter and energy and
account for acausal orderedness, as well. Furthermore, archetypes acting simultaneously in both the realms of psyche and physis were held to account for instances of synchronistic phenomena.
Jung and Pauli’s collaboration came to an end in what were to be the later years of their lives, and neither man was able to pursue these propositions further. Pauli, however, expressed an interest
in exploring the archetypal ideas that form the basis of mathematics, particularly the idea in arithmetic of an infinite series of integers and the idea in geometry of the continuum ( Pauli, 1994),
and Jung was drawn to the archetypal nature of natural numbers.
Starting from Jung’s initial hints, von Franz investigated number archetypes as dynamical ordering factors active both in psyche and in matter. In Number and Time, she examined aspects of number and
numeration drawn from a wide variety of cultures both ancient and modern, primitive and technologically advanced. She discussed in particular detail the qualitative aspects of the structure of the
number archetypes that give rise to the first four integers. As well, she investigated the dynamical aspects of the number archetypes and their relationship to physical and psychic energy, and she
discussed historical and mathematical models of the unus mundus and the role of number archetypes in synchronistic phenomena.
From her investigation of number archetypes, von Franz concluded that the primarily collective, quantitative aspects of number that preoccupy Western number theory are complemented by individual,
qualitative aspects. To illustrate these aspects of number, she cited examples of the treatment of numbers in ancient Chinese number systems, and concluded that the Chinese did not use numbers as
quantitative sets but as emblems or symbols: “Numbers thus serve chiefly to make visible the circumstantial individual aspects of the cosmic unity or whole.”(p. 41) Chinese numbers also contained an
essential relation with time: “In China, numbers signify organizations which vary in time, or transient ‘ensembles’ of inner and outer factors within the world-totality.”(p. 41-2)
Common to both Western and ancient Chinese approaches to number, however, is the use of number in establishing regularity and order. Jung had stated that ‘[number] may well be the most primitive
element of order in the human mind…thus we define number psychologically as an archetype of order which has become conscious.” (p. 45) As with all archetypes, the number archetypes have an inherent
dynamical quality–that is, they represent abstract patterns of rhythmical behavior. von Franz held that:
The archetypes primarily represent dynamical units of psychic energy. In preconscious processes they assimilate representational material originating in the phenomenal world to specific images and
models, so that they become introspectively perceptible as ‘psychic’ happenings.(p. 155)
In Number and Time, the quaternio of archetypes that underlie the first four integers are discussed in particular detail. Summarizing their archetypal behavior, von Franz explained that,
Numbers then become typical psychological patterns of motion about which we can make the following statements: One comprises wholeness, two divides, repeats and engenders symmetries, three centers
the symmetries and initiates linear succession, four acts as a stabilizer by turning back to the one as well as bringing forth observables by creating boundaries, and so on.(p. 74) von Franz
postulated that representations of this quaternio provide the dynamical patterns which underlie all processes of perception and symbol formation in the psyche and account for the structure and
transformation of matter and energy in the physical world.
Natural numbers appear to represent the typical universally recurring, common motion patterns of both psychic and physical energy. Because these motion patterns (numbers) are identical for both forms
of energy, the human mind can, on the whole, grasp the phenomena of the outer world. This means that the motion patterns engender “thought and structure models” in man’s psyche, which can be applied
to physical phenomena and achieve relative congruence.
The existence of such numerical nature constants in the outer world, on the one hand, and in the preconscious psyche, on the other (e.g., in the quaternary structures of the “psychic center”, the
triadic structure of dynamic processes, the dualistic structure of threshold phenomena, and so forth) is probably what finally makes all conscious knowledge of nature possible.(p. 166-7)
The dynamical behavior of the number archetypes, in particular the quaternio, is thus held to characterize all physical processes and all mental acts of perception and symbolic representation. Thus,
the number archetypes are thought to be universal aspects of symbol formation. Consequently, as von Franz has pointed out, the number archetypes should provide the means to construct what Pauli had
called a language which is ‘neutral’ with respect to psycho-physical distinction. Such a language, as yet undeveloped, would offer an archetypally-invariant basis upon which representations of all
physical and mental processes could be established.
The cluster of propositions that grew out of the collaboration of Jung and Pauli constituted a hypothesis about the role of archetypes in the structuring of reality. Through her research into number
archetypes, von Franz has significantly clarified and extended their archetypal hypothesis such that it may be restated as follows:
All mental and physical phenomena are complementary aspects of the same unitary, transcendental reality.
At the basis of all physical and mental phenomena there exist certain fundamental dynamical forms or patterns of behavior called number archetypes.
Any specific process, physical or mental, is a particular representation of certain of these archetypes. In particular, the number archetypes provide the basis for all possible symbolic expression.
Therefore, it is possible that a neutral language constructed from abstract symbolic representations of the number archetypes may provide highly unified, although not unique, descriptions of all
mental or physical phenomena.
von Franz was not satisfied with Number and Time; she called it, “a rather unreadable book” and regretted that it had failed to communicate and provoke discussion of its central tenets. With it, she
had tried to take Jung’s initial hints somewhat further and to show that a “real, absolute isomorphism is present,” between representations of the number archetypes as they appear in the psyche and
in the physical world. She said, “I was able to take this up to the number four. Then it became too complicated, and at that point I also hit my head on the ceiling,” (von Franz, 1992, p. 37) just as
Jung, too, had hit his head on the ceiling prior to turning the project over to her.
With Number and Time, von Franz unquestionably leads the reader through forbidding terrain–assuming extensive knowledge of Jungian psychology, as well as knowledge of major developments and issues in
twentieth century physics and mathematics. She has attempted to create a discourse between the areas of depth psychology and physics, with the intent of working toward their ultimate unification—a
task which can only be seen as Herculean.
At present there are two factors which have helped to improve the accessibility of Number and Time. The first is the publication in 1992 of von Franz’s Psyche and Matter, a collection of twelve of
her essays and lectures which clarify and amplify much of the content of Number and Time, and as such it comprises a suitable companion volume to it.
The second factor is the emergence in the past decade of a growing interest in the collaboration of Jung and Pauli, including the publication of their correspondence and much valuable secondary
source material.(for references see Card, 1991, 1992, 1998; Robertson, 1995)
From the vantage point offered by these works, it is now possible to reassess the importance of von Franz’s work on number archetypes: by developing and refining the central ideas of the Jung-Pauli
collaboration, she has pointed, through her examination of the number archetypes, to the way by which Pauli’s psycho-physically neutral language might be obtained. If this should ever be achieved, it
could become the means for the development of a post-Cartesian archetypal science in which a unified inquiry into the nature of mind and matter could take place. If this is so, then von Franz’s most
obscure work would easily become her most important. It is hoped that the essay which follows will help to advance von Franz’s work and be seen as a fitting tribute to her.
Further reading
CARD, C.R., (1991), The Archetypal View of Jung and Pauli, Psychological Perspectives, #24 and #25 ( Los Angeles: C.G. Jung Institute)
CARD, C.R., (1992), The Archetypal Hypothesis of Wolfgang Pauli and C.G. Jung: Origins, Development and Implications, in K.V.Laurikainen and C.Mononen, eds. Symposia on the Foundations of Modern
Physics, Singapore: World Scientific Publishing Co., p.382.
CARD, C.R., (1998), The Emergence of Archetypes in Present-Day Science and its Significance for a Contemporary Philosophy of Nature, Mind in Time, (Hampton Press, forthcoming). The German version of
this paper was published in T. Artz, et al., Philosophia Naturalis, Wuerzburg, Koenigshausen & Neumann, 1996, and the English version is published in the e-journal, Dynamical Psychology (http://
von FRANZ, M.L. (1974), Number and Time, Northwestern University Press, Evanston
von FRANZ, M.-L. (1992), Psyche and Matter, Shambhala, Boston.
JUNG, C.G. and W. Pauli, (1955), The Interpretation of Nature and the Psyche, Pantheon, N.Y.
PAULI, W.,(1994), “Ideas of the Unconscious from the Standpoint of Natural Science and Epistemology”, Wolfgang Pauli: Writings on Physics and Philosophy, C.P. Enz and K. Von Meyenn, ed.,
Springer-Verlag, 1994, Berlin, p.149.
ROBERTSON, R. ( 1995) Jungian Archetypes: Jung, Goedel, and the History of Archetypes, Nicolas-Hays, York Beach, Maine.
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Electric Fields
• Electric field strength is the force per unit charge
• Units NC⁻¹ or Vm⁻¹
• It is a vector quantity
• The direction of the field is the direction a positive charge would move in
• Density of field lines indicates field strength (closer together = stronger field)
The diagram above shows a uniform electric field between two parallel plates. The straight field lines are equal distances apart. Equipotential lines are also equally spaced and perpendicular to the
field lines.
Potential difference is the energy transferred per unit charge so, a volt is a joule per coulomb (JC⁻¹).
To move a charge q from X to Y, the force on the charge due to the electric field F = E x Q (where E is the electric field strength) must be equal in magnitude to the work done in moving the charge.
Work done = F x d.
Work done = EQ x d
The work done is also equal to the energy gained by the charge moving through a potential difference, work done = QV.
EQd = QV
Ed = V
E = V / d
This shows that electric field strength can also be measured in Vm ⁻¹.
Applications of Uniform Electric Fields
The diagram below shows a charged particle (e.g an electron) suspended in a vertical electric field. The charged plates have a distance d between them and a potential difference of V. Here, the
gravitational field strength would be equal in magnitude but opposite in direction to the force on the charge due to the electric field.
F[E] = F[g]
Ee = m[e]/g (F = EQ, in this case Q = charge on electron = e)
g/E = e/m[e] (e/m[e] is the specific charge of the electron)
By substituting in E = V/d, we get e/m[e] = gd/V
Here we have an electron being accelerated in a uniform electric field. The electron has an initial velocity u and mass m[e]. The force on the electron can be found using F = EQ and E = V/d. It will
travel in a straight line as the as the acceleration due to the electric field is constant.
Find an equation for the final velocity v:
As the work done on the electron by the electric field is equal to the change in kinetic energy of the electron, we can set them equal to each other.
Work done on electron = Fd = QV
1/2 mv² = eV/d
Rearrange to find v:
v = √(2eV/mₑd)
As E = V/d
v = √(2Ee/mₑ)
A charged sphere is suspended in a uniform electric field:
F[E] = force due to the electric field
For a charged sphere, charge may be considered to be at the centre.
We can resolve vertically and horizontally to form an equation:
R(H): Tsinθ = F[E] = qV/d
R(V): Tcosθ = F[g] = mg
Tsinθ/Tcosθ = F[E]/F[g]
tanθ = qV/mgd
Projectile Motion in Uniform Electric Fields
The diagram below shows the plan view of an electron entering a uniform electric field along the centre line with initial velocity u.
In the x-direction, the electron experiences no force so, Vₓ is constant = u. Using the suvat equation: s = ut+ 1/2 at², in the x-direction, x = ut.
In the y-direction, F = QV/d = eV/d (e is the charge on the electron).
As F = ma (Newton’s Second Law), a = eV/mₑd
(mₑ is the mass of the electron).
Since the force is constant and directed towards the positive plate, the electron will undergo uniform acceleration. Due to the constant acceleration in the vertical direction, and no forces acting
in the horizontal direction, the electron will follow a parabolic path as it travels between the plates.
s = y
u = 0
v =
a = eV/mₑd
t = t
s = ut+ 1/2 at²
y = eVt²/2mₑd
Subbing in x = ut:
y = eVx²/2mₑdu²
Once the electron clears the plates and leaves the electric field, it no longer experiences any force due to the field. Since no forces are acting on the electron, it will continue moving in a
straight line at a constant velocity (Newton’s First Law of Motion).
The velocity will have a horizontal and vertical component. The horizontal component will be the same as when it entered the field and the vertical component will have increased due to the
acceleration in the field.
What is the minimum velocity e⁻ must enter with to clear the plates?
When the electron has travelled a length L in the x-direction, it has travelled d/2 up (in the y-direction).
We can substitute x = L and y = d/2 into our equation:
This is the minimum velocity with which the electron would need to enter the field to clear the plates.
Coulomb’s Law is very similar to Newton’s Law of Gravitation which may make it easier to remember.
The force between 2 point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It is a force of attraction between
charges of opposite sign and a force of repulsion if the charges are of the same sign.
Air can be treated as a vacuum when calculating force between charges.
F ∝ Q₁Q₂
F ∝ 1/r²
ε₀ is the permittivity of free space (how well a medium can allow an electric field to develop) and has the value 8.85 x 10⁻¹² Fm⁻¹
1/4πε₀ is a constant (think of it is as the gravitational constant G)
Coulomb's Law
The force between 2 point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
Comparing the Magnitude of the Gravitational and Electromagnetic Force
To determine the ratio of the electromagnetic force to the gravitational force between two protons at any separation, we can divide one of the equations by the other (we want our answer to be in the
form F[E] : F[g]).
e is the charge on a proton = 1.60 x 10⁻¹⁹
mₚ is the mass of a proton = 1.67 x 10⁻²⁷
r² cancels out and we can start rearranging and substituting in:
(9.58 x 10⁷)² x 1/(4π x 8.85 x 10⁻¹² x 6.67 x 10⁻¹¹) : 1
1.24 x 10³⁶ : 1
The electromagnetic force is roughly 10³⁶ times greater than the gravitational force. F[E] >> F[g]
When a hydrogen atom is in its ground state, the radius of orbit of the electron is 5.30 x 10⁻¹¹ m. Calculate the linear speed of the electron and the number of orbits it completes in one second.
Centripetal force = Force between two charges
mₑv²/r = e²/4πε₀r²
So, v = √(e²/4πε₀rmₑ)
By square rooting e²/4, we can take them out of the square root sign (this may make calculations easier):
v = e/2 x √(1/πε₀rmₑ)
Substitute in the values:
v = (1.6 x 10⁻¹⁹)/2 x √(1/(π x 8.85 x 10⁻¹² x 5.30 x 10⁻¹¹ x 9.11 x 10⁻³¹))
v = 2.18 x 10⁶ ms⁻¹ = speed of inner orbital electron
The number of orbits it completes in one second = frequency
v = 2πrf
f = v/2πr
f = 2.18 x 10⁶/(2π x 5.30 x 10⁻¹¹)
f = 6.56 x 10¹⁵ Hz
What is the de Broglie wavelength?
λ = h/mv
λ = 6.63 x 10⁻³⁴ / (9.11 x 10⁻³¹ x 2.18 x 10⁶)
λ = 3.33 x 10⁻¹⁰ m
How does this compare with the circumference of orbit?
Circumference = 2πr = 2π x 5.30 x 10⁻¹¹
Circumference = 3.33 x 10⁻¹⁰ m
The diagram below shows a square with sides of length a. There are four charged points on each corner of the square. You can imagine the point P is a particle with charge +q.
Find the magnitude and direction of the electric field at the centre of the square (P).
We can use Coulomb’s Law for each force (F₁, F₂, F₃, F₄) and then resolve to find the resultant force.
To find r (the distance between the centres of the points), we can use Pythagoras’s theorem:
Now we have a value for r² which we can use in our equation.
F₁ = 2Qq/4πε₀a² = Qq/2πε₀a²
We are subbing in r² = a²/2, dividing by r² flips the fraction.
F₂ = Qq/πε₀a²
This is double F₁.
F₃ = Qq/2πε₀a²
F₄ = -Qq/πε₀a²
As F₄ is negative, resolving in one direction results in F₁ – (-F₄) = F₁ + F₄.
F₁+ F₄ = Qq/2πε₀a² + Qq/πε₀a² = 3Qq/2πε₀a²
Now we need to subtract F₃ from F₂.
F₂ – F₃ = Qq/πε₀a² – Qq/2πε₀a² = Qq/2πε₀a²
To find the resultant force we can use Pythagoras again, it will be easier to let Y = Qq/2πε₀a². Y is a random variable.
√((3Qq/2πε₀a²)² + (Qq/2πε₀a²)²)
This is the same as:
√((3Y)² + (Y)²) = √(9Y² + Y²) = √(10Y²)
We can sub Y = Qq/2πε₀a² back into the resultant force:
Qq/2πε₀a² x √(10)
Finally, to find the magnitude of the electric field strength we can use the equation: F = EQ
Eₙₑₜ = Fₙₑₜ / q
Eₙₑₜ = Q/2πε₀a² x √(10)
To find the direction we would use trig.
What is the electric field strength at each point? (A, B, C)
Imagine a point charge (+q) at different positions:
Point A:
Qq/4πε₀R² (to the left)
2Qq/4πε₀(3R)² = Qq/18πε₀R² (to the right)
F[A] = 1/4 – 1/18 = 7/36
F[A] = 7Qq/36πε₀R²
As F = EQ,
E[A] = F[A]/q
E[A] = 7Q/36πε₀R² (to the left)
Point B:
Now imagine there is a point charge (+q) at point B.
Qq/4πε₀R² (to the right)
2Qq/4πε₀R² = Qq/2πε₀R² (to the right)
F[B] = 1/4 + 1/2 = 3/4
F[B] = 3Qq/4πε₀R²
E[B] = 3Q/4πε₀R² (to the right)
Point C:
Now imagine there is a point charge (+q) at point C.
2Qq/4πε₀R² = Qq/2πε₀R² (to the left)
Qq/4πε₀(3R)² = Qq/36πε₀R² (to the right)
F[C] = 1/2 – 1/36 = 17/36
F[C] = 17Qq/36πε₀R²
E[C] = 17Q/36πε₀R² (to the left)
In moving a charge q from an equipotential V to another at V + ΔV, the work done is qΔV and is independent of the path taken.
Absolute electric potential: the work done per unit positive charge in moving it to a point from infinity.
Graphical representations of Electric Potential
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George Burger
WILL OF JACOB BÜRGER
General account for the five children of what is due to them as interest of property.
First to my four guardians children named:
From the auction sale of Jacob's property:
First: According to an article passed in WOERTH contracts the house in GOERSDORF was sold to Christoph FRANCK on the 11th day of January, 1766 for the amount of 166 Gulden.
Second: According to another contract in WOERTH notary records dated the 30th September, 1765, another property in the village of GOERSDORF was sold for the amount of 199 Gulden, 5 Schillings.
Third: The father's clothes were sold by auction on the 23rd May, 1765 for 11 Gulden, 8 schillings and 1½ d.
Hay in the fields: none
Also: concerning the property that Jacob had loaned to Andreas SCHENCK until his death, and sold on auction, under condition that the loaner would receive compensation for the crops in the
Credit Debts:
According to the father's inventory is first concerned:
A credit made to Lienhard STOCKER in WEILER according to a letter of credit written on the 7th January, 1764 for an amount of 30 Gulden.
On this amount my 4 guardians children claim for one third in the name of their deceased mother as this credit had been made in her lifetime and only included in the father's inventory, so 10
Gulden, and each one receives 2 Gulden 5 Schillings, and the five children each 20 Gulden, but 3 Gulden has already been paid on this sum, so remains 17 Gulden.
On my side, I have tried to obtain the interest on this credit in front of the Justice Court in WISSEMBOURG and obtained decision on the 11th July, 1765 there but this was taxed with the amount
of 1 Gulden, 1 Schilling and 6 d.
The same is still due by Andreas SCHENCK for the loan of house and land : 15 Gulden for year 1764.
Also due by JOHANNES, shepherd in PFAFFENBRONN for potatoes he sold: 1Gulden, 6 Schillings, 5 d.
Total amount is: 414 Gulden, 9 Schillings, 6½ d.
Follow now the debts that the five children have on this inventory:
First due to the guardians, Andreas MÖSSNER, Hans Michael KEIFER in name of Gottfried GREINER for the glaziers works in MATTSTALL, a sum of 5 Gulden, 3 Schillings, 2½ d according to receipt dated
the 11th January, 1766.
Also due to Hans Michael KOBLER the shoe mender for work, the sum of 2 Gulden, 7 Schillings, according to receipt dated 13th June, 1765.
Also due to Philipp SCHAFFNER for conveyance work, the sum of 2 Gulden, 7 Schillings according to receipt dated 18th July, 1765.
Also due to MEYER the Jew in GUNSTETT according to Notary record and decision rendered on the 28th August, 1764 on a sum with interests of 113 Gulden, 1 Schilling, 6 d.
Also due to Georg THOMANN senior, according to foundation made by the deceased to the churches of GOERSDORF, PREUSCHDORF and LAMPERTSLOCH as from receipt made on the 12th December, 1765 in the
father's inventory which equals 60 Gulden.
Also due to the widow in this inventory according to two receipts on the 3rd October, 1765 and 15th June, 1765, 24 Gulden.
Same to Eva Dorothea, daughter, an amount of 22 Gulden, 5 Schillings according to a receipt on the 12th August, 1763.
Same to daughter Anna Maria according to receipt on the 13th December, 1764, 9 Gulden.
Same to vicar KÜGELE for the burial as from receipt on the 14th July, 1765, 2 Gulden, 7 Schillings, 6 d.
Same to the schoolteacher as from hand written note, 5 Schillings, 6 d.
Same the cost of redaction of the father's inventory, 8 Gulden, 1 Schilling, 6 d
Same to Hans Georg BRUCKER for war's taxes to be paid on the 14th January, 1766, (no amount given)
Same to Gottfried LORENTZ for crop delivered and omitted in the previous inventory, as from receipt dated 15th September, 1765, 8 Gulden, 9 Schillings.
Same to Georg Nicolaus MÜLLER, the Mayor, for the yearly town tax on the father's property as from receipt dated 7th February, 1765. (no amount given)
Same to the vicar as asked by the children that he would read Holy masses for the father, 4 Gulden.
Same due to Stephan, the Mayor in GOERSDORF, on request of the deceased father, to be admitted to leave GOERSDORF for PFAFFENBRONN, and then back again as protected citizen in GOERSDORF, but upon
this request, he happened to die, dated 4th August 1765 and for an amount of 9 Gulden.
Same due on occasion of trips to WISSEMBOURG to obtain a decision concerning the debt of Leonhard STOCKER in WEILER of 30 Gulden and time spent to try to obtain there the interest of this amount
has cost of 1 Gulden, 6 Schillings, 6 d.
Same for publication of the auction of the father's property, spent: 6 Gulden.
Same, my four guardians children are endowed to receive on the father's inventory on their mother's property:
Maria Eva Dorothea. 25 Gulden.
Maria Anna. 23 Gulden, 4 Schillings.
Johann Nicolaus. 22 Gulden, 2 Schillings.
Catharina Magdalena. 23 Gulden, 2 Schillings.
Cost of the account rendered here, to be deducted off the general account is: 6 Gulden, 3 Schillings, 7½d.
Note: What is considered as debts will be deducted here after from the detailed account of what each of the 4 children should receive after this.
Total of this page is 99 Gulden, 3 Schillings, 7½ d.
Item: each of the children must receive from their deceased brother one share of the mother's property, on what the father has sold, 25 Gulden, 5 Schillings and each of them receives on this a
sum of one quarter, so, 6 Gulden, 3 Schillings, 9 d. (Using this calculation, 12 d = 1 Schilling, 10 Schilling = 1 Gulden)
Same is due to the Justice Accountant for 2½ days work on this account, a total of 2 Gulden, 4 Schillings.
Same is due to Johannes GÜLG, the guardian of the son in the second marriage (Mathias), 33 Gulden and 9 Schillings.
All 5 children also owe to the Bailiff's authority a sum of 4 Gulden and to the same for one years accounting of 2 Gulden, 6 d.
Same to the Bailiff for assisting this account, 7 Schillings, 6d.
Same for the Bailiff for 3 trips to WISSEMBOURG in order to obtain payment due by Lienhard STOCKER, 2 Gulden, 2 Schillings, 6 d.
Same due to the guardian from time of the father's death, 5 Schillings.
The rest will be named afterwards.
General payment of the 5 children is: 397 Gulden, 6 d. Remains to be paid to the guardian, Andreas MÖSSNER for his four guardians children and to Johannes GÜLG for the 5th son, Mathias, totalling
17 Gulden, 9 Schillings, 6½ d.
On this is due a fifth part to Johannes GÜLG, so: 3 Gulden, 5 Schillings, 10 d.
To be added, the share of the sale of the father's clothes: 8 Schillings, 6d. on a total of 1 Gulden, 7 Schillings. Total is: 4 Gulden, 4 Schillings, 4 d.
This will be considered by Johannes GÜLG when he renders his accounts as a received sum from Andreas MÖSSNER. And to Andreas MÖSSNER as guardian of the other 4 children: four fifths or 3
Schillings and 7 and 3/5 d.
Now follows what is due to MARIA EVA DOROTHEA, eldest daughter.
First as before named what she owes to her guardian: 1/5th or 3 Gulden, 5 schillings, 10 d.
Interest on sales:
According to auction made 2nd January, 1760 on the mother's property, it had been agreed that the property in GOERSDORF would be loaned to Philipp SCHAFFNER for a yearly interest of 3 Gulden,
with condition that the auction buyer would pay the interests yearly on St. Martins Day (11th November), so: For year 1760: 3 Gulden and interest on 5 years, 7 Schillings 6d.
For year 1761: 3 Gulden and interest on 4 years, 6 Schillings.
For year 1762: 3 Gulden and interest on 3 years , 4 Schillings 6 d.
For year 1763: 3 Gulden and interest on 2 years, 3 Schillings.
For year 1764: 3 Gulden and interest on 1 year, 1 Schilling 6 d.
For year 1765: 3 Gulden.
This amount at the time of the mother's death had been left to the deceased father on condition that he would pay the taxes on this amount of 20 Gulden, 2 Schillings, 6 d.
Crops in the fields: These were also left to the father on the death of the mother with no conditions.
Same is due to Maria Eva Dorothea on her share of the debt of Lienhard STOCKER which amounts to 30 Gulden and the interests on this of 10 Gulden which is 1/4, so 2 Gulden, 5 Schillings and
interests for a year of 1 Schilling, 3 d.
She is to receive on her mother's general inventory: 24 Gulden, which makes a total of 26 Gulden, 6 Schillings, 3 d.
She is to receive one quarter of the inheritance of her deceased brother Georg Jacob's share of her mother's inventory which equals 6 Gulden, 3 Schillings, 9 d.
These last amounts are included into the general account by Andreas MÖSSNER.
This makes a total for Maria Eva Dorothea to receive: 63 Gulden, 3 Schillings, 1 and 9/10ths d.
She purchased a pair of stockings which cost 7 Schillings, 6 d and made receipt for this on the 6th August, 1765, plus interest for 11 months of 6½ d.
She also received at the time that she was named as Godmother a sum of 2 Gulden, 1 Schilling and has to pay the interest on this for 1 year of 1 Schilling, ¾ d.
She also received for herself and Maria Anna some grocers ware as receipt made on the 10th January, 1766 for 14 Gulden, 5 Schillings, 6 d.
And on tailor's work in 1765 for receipt of 7 Schillings 6 d.
This makes a total of 18 Gulden, 3 Schillings, 1¼ d already spent by her and part for her sister of 7 Gulden, 6 Schillings, 6 d with interests added to this leaves a total of 10 Gulden, 6
Schillings, 7¼ d.
Paid to the Tax receiver BALLIS at this time to assist the inventory, as by receipt on the 31st March, 1756, 1 Schilling, 6 d.
And to myself, Andreas MÖSSNER, guardian at time of oath taken, 4 Schillings.
And interest on both articles is: 10 d.
And for 14 years of guardians account at 5 Schillings per year, 7 Gulden. plus 5 Schillings.
Gives a total of 8 Gulden, 1 Schilling, 4 d.
So: one quarter equals 2 Gulden, 4 d.
The total of her debts being: 12 Gulden, 9 Schillings, 1/10 d.;
Her share left on the general account: 50 Gulden, 4 Schillings, 2/10 d.
Now follows what is due to MARIA ANNA, second daughter.
First as before named what she is owed from her guardian: 1/5th or 3 Gulden, 5 schillings, 10 d.
According to auction of 1761 she has sold on her mothers share in GOERSDORF and with interest to Andreas BÜRI:
For the year 1761: 2 Gulden.
For the year 1762: 2 Gulden, 5 Schillings and the interest for 3 years, 3 Schillings, 9 d.
For the year 1763: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1764: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1765: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1766 and 1767 a new account will be made later.
Total of page is: 12 Gulden, 7 Schillings, 6 d.
As did her sister, Maria Anna has to receive on her share of the debt from Lienhard STOCKER, from WEILER, 2 Gulden, 5 Schillings and interests for a year of 1 Schilling, 3 d.
On her mother's property her share of 25 Gulden plus 5 Schillings as all children by the first marriage.
She is to receive one quarter of the inheritance of her deceased brother Georg Jacob's share of her mother's inventory which equals 6 Gulden, 3 Schillings, 9 d.
Received these articles by the guardian, Andreas MÖSSNER in his general account.
Maria Anna's share is: 50 Gulden, 8 Schillings, 5 d.
She purchased a pair of stockings which cost 7 Schillings, 6 d and made receipt for this on the 6th August, 1765, plus interest for 11 months of 6½ d.
She also received for herself with Maria Eva Dorothea some grocers ware as receipt made on the 10th January, 1766 for 14 Gulden, 5 Schillings, 6 d. her share being 7 Gulden, 6 Schillings, 6 d.
And on tailor's work in 1765 for receipt of 7 Schillings 6 d.
For Taxes and accounting, 2 Gulden, 10½ d.
And so she has to compensate the others for this and is left a sum of 39 Gulden, 9 Schillings, 9 d.
Now follows what is due to JOHANN NICOLAUS, eldest son.
First as before named what he is owed from his guardian: 1/5th or 3 Gulden, 5 schillings, 10 d.
According to auction of 1761 he has sold on his mothers share in GOERSDORF and with interest to Andreas BÜRI:
For the year 1761: 2 Gulden.
For the year 1762: 2 Gulden, 5 Schillings and the interest for 3 years, 3 Schillings, 9 d.
For the year 1763: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1764: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1765: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1766 and 1767 a new account will be made later.
Total of page is: 12 Gulden, 7 Schillings, 6 d.
As did his sister, Johann Nicolaus has to receive on his share of the debt from Lienhard STOCKER, from WEILER, 2 Gulden, 5 Schillings and interests for a year of 1 Schilling, 3 d.
On his mother's property his share of 25 Gulden plus 5 Schillings as all children by the first marriage.
He is to receive one quarter of the inheritance of his deceased brother Georg Jacob's share of his mother's inventory which equals 6 Gulden, 3 Schillings, 9 d.
Received these articles by the guardian, Andreas MÖSSNER in his general account.
Johann Nicolaus’s share is: 50 Gulden, 8 Schillings, 5 d.
Now follows what is due to CATHARINA MAGDALENA, third daughter.
First as before named what she is owed from her guardian: 1/5th or 3 Gulden, 5 schillings, 10 d.
According to auction of 1761 she has sold on her mothers share in GOERSDORF and with interest to Andreas BÜRI:
For the year 1761: 2 Gulden.
For the year 1762: 2 Gulden, 5 Schillings and the interest for 3 years, 3 Schillings, 9 d.
For the year 1763: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1764: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1765: 2 Gulden, 5 Schillings and interest of 1 Schilling, 3 d
For the year 1766 and 1767 a new account will be made later.
Total of page is: 12 Gulden, 7 Schillings, 6 d.
As did her sisters, Catharina Magdalena has to receive on her share of the debt from Lienhard STOCKER, from WEILER, 2 Gulden, 5 Schillings and interests for a year of 1 Schilling, 3 d.
On her mother's property her share of 25 Gulden plus 5 Schillings as all children by the first marriage.
She is to receive one quarter of the inheritance of her deceased brother Georg Jacob's share of her mother's inventory which equals 6 Gulden, 3 Schillings, 9 d.
Received these articles by the guardian, Andreas MÖSSNER in his general account.
Catharina Magdalena's share is: 50 Gulden, 8 Schillings, 5 d.
Andreas MÖSSNER. Johannes GÜLG. Guardians.
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Teaching | Kimbk
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[Summer 2024] Machine Learning for Social Scientists
This course provides an overview of machine learning methods and their applications to social science research. In this course, we will learn the basic framework of machine learning, popular machine
learning methods -- supervised, semi-supervised, and unsupervised learning -- and their social science applications. The topics covered in this course include the bias-variance trade-off, the curse
of dimensionality, decision trees, random forests, naive Bayes, boosting models, support vector machines, PCA, k-nearest neighbors, hierarchical clustering, kernel-based clustering techniques, and
topic models for text data.
[Summer 2024] Text Analysis for Social Scientists
In this course, we learn statistical/computational theories and tools for text analysis. The course has three parts. First, we will cover statistical/computational preliminaries such as Dirichlet
distribution, multinomial distribution, and cosine similarity. The second part of the course is dedicated to learning core concepts and frameworks of text analysis. In this part, we learn how to
represent text data for quantitative analysis. Specifically, students will be introduced to concepts such as tokens, document-feature matrix, bag-of-words, word embed- ding, tf-idf, and etc. Finally,
we cover statistical/computational models for text data. These include LDA-based methods, word2vec, and a soft introduction of large language models such as ChatGPT.
[Spring 2024] R Programming for Social Scientists
This course provides introductory and intermediate-level practice on R programming for Social Science applications. The topics include data cleaning, data wrangling, data visualization, tools for
data analysis, loops, basic web scraping, and etc.
[Fall 2024] Statistical Foundations for Data Science
This course covers essential mathematical and statistical theories for data science. The topics include integral, probability theory, vector space, eigenvalue, gradient, and numerical optimization.
Together with R Programming for Social Scientists, this course serves as one of the prerequisites for other data science courses such as Machine Learning for Social Scientists and Text Analysis for
Social Scientists.
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t-Test, Chi-Square, ANOVA, Regression, Correlation...
3-Way ANOVA calculator
Load ANOVA data set
If you want to calculate a 3-way analysis of variance online, simply select three nominal variables and one metric variable. The results of a three-way ANOVA will then be calculated.
You will then see the results of the calculated 3-way ANOVA. First the hypotheses are displayed and then the table with the calculated main effects and the interaction effects.
3-Way ANOVA
A 3-way ANOVA, also known as a three-factor ANOVA, is a statistical test used to determine the effects of three independent categorical variables (or factors) on a continuous dependent variable. It
also assesses the interactions between these three factors. In other words, a 3-way ANOVA is an extension of the one-way and two-way ANOVA calculator for situations where there are three independent
Possible Outcomes:
1. Main effects for each of the three factors:
□ Effect of factor A on the dependent variable, ignoring factors B and C.
□ Effect of factor B on the dependent variable, ignoring factors A and C.
□ Effect of factor C on the dependent variable, ignoring factors A and B.
2. Two-way interaction effects between pairs of factors:
□ Interaction effect between factors A and B, ignoring factor C.
□ Interaction effect between factors A and C, ignoring factor B.
□ Interaction effect between factors B and C, ignoring factor A.
3. Three-way interaction effect among all three factors:
□ Interaction effect between factors A, B, and C. This tests whether the two-way interactions are consistent across levels of the third factor.
The idea behind these tests is to assess not only how each factor independently affects the dependent variable but also how combinations of factors do. For instance, a two-way interaction indicates
that the effect of one factor depends on the level of another factor.
The results of a 3-way ANOVA can be complex and require careful interpretation. If significant interactions are detected, it is common to conduct post-hoc tests or plot interaction graphs to better
understand the nature of these interactions.
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sagag ball and rod mills
WEBRod mills are less common than ball mills for grinding minerals. The rods used in the mill, usually a highcarbon steel, can vary in both the length and the diameter. However, the smaller the
rods, the larger is the total surface area and hence, the greater the grinding efficiency. Principle of SAG Mill operation SAG mill. SAG is an acronym for ...
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WEBJan 30, 2022 · In terms of breakage mechanisms, while abrasion and impact prevail in ball mills, rod mills are used for their type of preferential grinding over larger particles . In general,
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WEBA treatise on the internal mechanics of ball, tube and rod mills | Semantic Scholar. DOI: / (59)903011. Corpus ID: .
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WEBRod mills are very similar to ball mills but instead of using grinding balls they use long steel rods to do the media grinding. Some miners believe that rod mills are more efficient than ball
mills and that the desired product result is obtained at a lower cost per ton. But it depends on the material you are mining for that will ultimately ...
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WEBCHANGSHA TIANCHUANG POWDER TECHNOLOGY CO., LTD. was established in 2006 with headquarter in Changsha City, Hunan Province, China. It is an entity enterprise integrating production, sales and
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WEBJan 16, 2018 · NEW Eaton Electronic Controls for Grinding Mill Pneumatic VC Clutches (Mining and Cement Processing Plants) Oct 17, 2017 Kaizen Systems News / Eaton Airflex Distributor
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WEBFeb 21, 2015 · Cytec Handbook. Why have a large circulation load:. If a product all finer than a certain critical size is required, the capacity of the ball mill is increased considerably by
using it in closed circuit with a classifier and this increase is made still greater by increasing the circulating load in between the ball mill and the classifier (Fig. 70).
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WEBBall and Rod Mills. JeanPaul Duroudier, in Size Reduction of Divided Solids, 2016. Operation principle. The ball mill is a cylindrical drum (or cylindrical conical) turning around its
horizontal axis. It is partially filled with grinding bodies: cast iron or steel balls, or even flint (silica) or porcelain bearings. Spaces between balls ...
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WEBTega Mill Linings provide optimal grinding solutions in major mineral processing plants all over the world. Tega rubber lining system is the preferred lining system for secondary ball mills;
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WEBThe ball/rod mill is designed to accept a ball/rod charge to 30% of the total volume. The barrel is charged with balls/rods and sample and allowed to grind for the desired time. The milling
chamber of the LMBM150 is surrounded by a safety cabinet, with a safety locking system. Additional information. Function:
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WEBJan 5, 2016 · For 60 mm (″) and smaller top size balls for cast metal liners use double wave liners with the number of lifters to the circle approximately D in meters (for D in feet, divide D
by ). Wave height above the liners from to 2 times the liner thickness. Rubber liners of the integral molded design follow the cast metal design.
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WEBGrinding Mills. Westpro's heavy duty grinding mills are designed for durability and excellent grinding performance in mining appliions. 6ft Diameter x10ft Ball and Rod Mills at the Westpro
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WEBBuy used Ballmills from King Industries. We can help guide you to the best solution for your equipment needs. Contact; Request Quote; Save your ... Rod or Ball Mill PARTS UNIT. Inventory ID:
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WEBFeb 13, 2017 · Ball Mills. In all ore dressing and milling Operations, including flotation, cyanidation, gravity concentration, and amalgamation, the Working Principle is to crush and grind,
often with rod mill or ball mill, the ore in order to liberate the minerals. In the chemical and process industries, grinding is an important step in preparing raw ...
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WEBBall mill and rod mill solutions are available here at APT, providing an effective means for grinding and blending materials in preparation for mineral liberation, from rock to dust. APT
specifies and designs the entire circuit around the mill, to ensure that the target product size is achieved. Available ranging in size ering for ...
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WEBA Treatise on the Internal Mechanics of Ball, Tube, and Rod Mills. Horace Edgar Rose, Ralph Major Edward Sullivan. ... input power required power to drive pulp radius rate of grinding read
from Fig reduced relationship required to drive Rittinger's rod mill Rose and Evans shown in Fig smooth mill specific surface speed of rotation surface ...
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WEBChina Rod Mills wholesale Select 2024 high quality Rod Mills products in best price from certified Chinese Grinding Mill manufacturers, Mining Machine suppliers, wholesalers and factory on ...
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WEBTypically R = 8. Rod Mill Charge: Typically 45% of internal volume; 35% – 65% range. Bed porosity typically 40%. Height of bed measured in the same way as ball mills. Bulk density of rods =
tons/m3. In wet grinding, the solids concentration 1s typically 60% – 75% by mass. A rod in situ and a cutaway of a rod mill interior.
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WEBPromas Engineers is a wellknown name in ball mills manufacturers in India. We are manufacturer, supplier and exporter of ball mills according to ISI guidelines and user's requirements since
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WEBBALL Mills, SAG Mills, AG Mills ROD Mills. We refurbish, design and manufacture an extensive range of highquality mills and mill components to exacting standards, offering you a turnkey
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WEBAug 6, 2015 · The Ball mills are much like the Rod mills except that they use balls as their grinding media. The size of these balls may range from three quarters of an inch to over five
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WEBSep 1, 2018 · The article presents the results of laboratoryscale research on the determination of the impact of ball mill parameters and the feed directed to grinding on its effectiveness and
comparing it with the efficiency of grinding in a rod mill. The research was carried out for grinding copper ore processed in O/ZWR KGHM PM
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WEBJan 25, 2012 · At feed rate of g/s, rotational speed of the drum 72 rpm, and 40% stirring media filling, the dispersion coefficient was cm 2 /s and cm 2 /s in the case of using balls and rods
as stirring media, respectively. This means that the dispersion in the ball mill is about 8 times that in the rod mill.
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WEBApr 9, 2015 · Fig 1 Cross sections of a rod mill and a bar mill. Ball mills. Ball mill (Fig 1) is same kind of mill as rod mill, except that it is filled with balls instead of rods. Because of
balls have greater ratio of surface area than rods they are more suitable for fine grinding. Balls are also lighter, so the kinetic energy of a single dropping ball is ...
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WEBSelect™ Rod mills {{decodeHtmlEntities()}} Select™ mills resource center ... Ball feeder. Metso's Select™ ball feeder utilizes a simple and robust design for continuous and automatic grinding
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WEBJun 19, 2015 · We can calculate the steel charge volume of a ball or rod mill and express it as the % of the volume within the liners that is filled with grinding media. While the mill is
stopped, the charge volume can be gotten by measuring the diameter inside the liners and the distance from the top of the charge to the top of the mill.
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WEBJul 4, 2017 · In both ball and rod mills, the material holdup is an increasing function of mill speed in the commercial operating range of mill speeds. Effect of material feed rate on RTD:
Figure 11 shows the Peclet number and the dispersion coefficient of the material flowing through the rod mill as a function of material feed rate. The Peclet number ...
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WEBYou can not select more than 25 topics Topics must start with a letter or number, can include dashes ('') and can be up to 35 characters long.
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WEBSummary. The Rod Mill is the same as the Ball Mill in the construction and available to both of wet and dry processing. Unlike the Ball Mill, however, the Rod Mill uses rods instead of balls
as the grinding medium. As a result, its uses are different from those with the Ball Mill. In other words, the Rod Mill gives impact on and grinds coarse ...
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WEBOct 20, 2016 · Grinding Balls. Steel balls ranging from ¾ to 5 in. in diameter are used. Rods range from 1½ to 4 in. in diameter and should be 3 to 4 in. shorter than the inside mill length.
Tube mills are usually fed balls smaller than 2 in., whereas 4 or 5in. balls are more commonly used for ballmill grinding.
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WEBBall/Rod mill Literature . The Ball/Rod mills are meant for producing fine particle size reduction through attrition and compressive forces at the grain size level. They are the most effective
laboratory mills for batchwise, rapid grinding of mediumhard to very hard samples down to finest particle sizes.
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WEBThe rod mill feeds a wet ball mill at a feed size of mm (1000 μm) and produces a product with 80% passing a 150 μm screen. The rod mill is in an open grinding circuit. Determine: 1. the shaft
power of the rod mill, 2. the size of the industrial mill. Data: Laboratory Standard Bond Test:
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WEBCIC has the ability to offer solutions for all kinds of mills, such as cement mill, mine mill, ball mill, rod mill liners for rod mill, AG mill, SAG mill, etc. With several decades of
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riddles about
Every time you lose something, you always find it in the very last place you would look. Why is this?
It is the last place you would look because once you find it there is no need to keep looking.
Two planes take off at the same exact moment. They are flying across the Atlantic. One leaves New York and is flying to Paris at 500 miles per hour. The other leaves Paris and is flying to New York
at only 450 miles per hour. Which one will be closer to Paris when they meet?
They will both the same distance from Paris when they meet!
In the land of Brainopia, there are three races of people: Mikkos, who tell the truth all the time, Kikkos, who always tell lies, and Zikkos, who tell alternate false and true statements, in which
the order is not known (i.e. true, false, true or false, true, false). When interviewing three Brainopians, a foreigner received the following statements: Person 1: I am a Mikko. Person 2: I am a
Kikko. Person 3: a. They are both lying. b. I am a Zikko. Can you help the very confused foreigner determine who is who, assuming each person represents a different race?
Person 1 is a Miko. Person 2 is a Ziko. Person 3 is a Kikko.
As I was going to St. Ives I met a man with seven wives The seven wives had seven sacks The seven sacks had seven cats The seven cats had seven kits Kits, cats, sacks and wives How many were going to
St. Ives?
One person is going to St. Ives (the narrator). Because the narrator "met" all of the others mentioned in the poem, this implies that they walked past each other in opposite directions, and thus none
of the wives, sacks, cats, or kits was actually headed to St. Ives. If you (like many) think this answer is a bit silly, you can assume that all the people, sacks, and animals mentioned were heading
for St. Ives. In this case, we would have 1 narrator + 1 man + 7 wives + 49 sacks + 343 cats + 2401 kits = 2802 total going to St. Ives. However, this isn't the traditional answer.
You are somewhere on Earth. You walk due south 1 mile, then due east 1 mile, then due north 1 mile. When you finish this 3-mile walk, you are back exactly where you started. It turns out there are an
infinite number of different points on earth where you might be. Can you describe them all? It's important to note that this set of points should contain both an infinite number of different
latitudes, and an infinite number of different longitudes (though the same latitudes and longitudes can be repeated multiple times); if it doesn't, you haven't thought of all the points.
One of the points is the North Pole. If you go south one mile, and then east one mile, you're still exactly one mile south of the North Pole, so you'll be back where you started when you go north one
mile. To think of the next set of points, imagine the latitude slighty north of the South Pole, where the length of the longitudinal line around the Earth is exactly one mile (put another way,
imagine the latitude slightly north of the South Pole where if you were to walk due east one mile, you would end up exactly where you started). Any point exactly one mile north of this latitude is
another one of the points you could be at, because you would walk south one mile, then walk east a mile around and end up where you started the eastward walk, and then walk back north one mile to
your starting point. So this adds an infinite number of other points we could be at. However, we have not yet met the requirement that our set of points has an infinite number of different latitudes.
To meet this requirement and see the rest of the points you might be at, we just generalize the previous set of points. Imagine the latitude slightly north of the South Pole that is 1/2 mile in
distance. Also imagine the latitudes in this area that are 1/3 miles in distance, 1/4 miles in distance, 1/5 miles, 1/6 miles, and so on. If you are at any of these latitudes and you walk exactly one
mile east, you will end up exactly where you started. Thus, any point that is one mile north of ANY of these latitudes is another one of the points you might have started at, since you'll walk one
mile south, then one mile east and end up where you started your eastward walk, and finally, one mile north back to where you started.
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Check For Harshad Number In Python - PythonForBeginners.com
Numbers have many specialties. Based on the specialties, they are given unique names. One such special number is the Harshad number or Niven number. In this article, We will discuss a program to
check if a given number is a Harshad number or not.
What is a Harshad Number?
A number is said to be a Harshad number or Niven number if it is divisible by the sum of its digits. In other words, If we are given a number that is divisible by the sum of its own digits, the
number will be called a Harshad Number.
For example, let us consider the number 4320. The sum of its digits is 9 that can be obtained by adding the digits 4, 3, 2, and 0. We can see that 4320 is divisible by 9. Hence, 4320 is a Harshad
number. On the other hand, the sum of digits in 4321 is 10. Here, 4321 is not completely divisible by 10. Hence, it is not a Harshad number.
Program To Check For A Harshad Number in Python
To check whether a number is a Harshad number or not, we will first calculate the sum of digits of the given number. After that, we will divide the number by the sum of digits to see if it is
completely divisible or not. If the number is divisible by the sum of the digits, we will say that it is a Harshad Number. Otherwise not.
To write the complete program, we will first write a function to calculate the sum of digits of the given number. For this, we will keep dividing the number by 10 until the number becomes 0. Each
time we divide the number by 10, we get the rightmost digit as the remainder. We can use this remainder to find the sum of digits by adding all the remainders till the number becomes 0.
The following function for calculating the sum of digits of a number in python accepts a number and returns the sum of digits of the number.
def calculate_sum_of_digits(N):
sumOfDigits = 0
while N > 0:
digit = N % 10
sumOfDigits = sumOfDigits + digit
N = N // 10
return sumOfDigits
input_number = 4320
output = calculate_sum_of_digits(input_number)
print("Sum of digits of {} is {}.".format(input_number, output))
Sum of digits of 4320 is 9.
After finding the sum of digits, we will divide the number by the sum of digits. If the reminder for the division is zero, we will say that a number is a Harshad number or Niven number. Otherwise, we
will print that the number is not a Harshad Number.
def calculate_sum_of_digits(N):
sumOfDigits = 0
while N > 0:
digit = N % 10
sumOfDigits = sumOfDigits + digit
N = N // 10
return sumOfDigits
def check_for_harshad_number(N):
sumOfDigits = calculate_sum_of_digits(N)
if N % sumOfDigits == 0:
return True
return False
input_number = 4320
output = check_for_harshad_number(input_number)
print("{} is a Harshad Number:{}".format(input_number, output))
input_number = 4321
output = check_for_harshad_number(input_number)
print("{} is a Harshad Number:{}".format(input_number, output))
4320 is a Harshad Number:True
4321 is a Harshad Number:False
In this article, we have discussed what a Harshad number or Niven number is. We have also implemented a program in python to check if a given number is a Harshad number or not. To learn more about
numbers in python, you can read this article on decimal numbers in python. You might also like this article on complex numbers in python.
Recommended Python Training
Course: Python 3 For Beginners
Over 15 hours of video content with guided instruction for beginners. Learn how to create real world applications and master the basics.
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ACT Math Help
Students in need of ACT Math help will benefit greatly from our interactive syllabus. We break down all of the key elements so you can get adequate ACT Math help. With the imperative study concepts
and relevant practice questions right at your fingertips, you’ll have plenty of ACT Math help in no time. Get help today with our extensive collection of essential ACT Math information.
With colleges becoming increasingly more selective, admissions tests like the ACT have become more high-stakes. Tested in multiple sections, the math portion of the ACT seeks to understand how high
school students perform in various algebra, geometry, and pre-calculus skills. With proper study, the ACT math section becomes more straightforward. Additionally, because most students have initial
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Students taking the ACT need to understand a fair number of topics, including:
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crucial to understand, as they form the basis of later required knowledge.
Algebra: Evaluating algebraic expressions through substitution, expressing factorial relationships, and solving quadratic equations.
Coordinate Geometry: Recognizing and defining lines, planes, segments, polynomials, circles, and other curves; inequalities; determining slope; and finding the midpoint of parallel and perpendicular
lines and segments.
Plane Geometry: Parallel and perpendicular lines form the basis for plane geometry. Students are asked to understand the relationship between the various angles defined by transecting parallel lines.
Additionally, students learn how to graph triangles, rectangles, and other geometric figures on the coordinate plane system and use processes like translation, rotation, and reflection to move these
figures. Finally, students learn the basics of two column and paragraph proofs. Proving congruence and similarity between triangles is the primary endpoint of plane geometry, as aspects of the
coordinate plane system, parallel lines, and transecting lines call all be incorporated into a single problem.
Trigonometry: Understanding of right angles and trigonometric functions like sine, cosine, and tangent are tested in the trigonometry section. Students are often asked to find the value of an unknown
variable that requires background knowledge of the trigonometry functions.
Succeeding in the ACT math section often requires visualization of the problem at hand. For students beginning to prepare for the ACT, these are often complex skill sets that take time to build.
Beginning with hands-on models like wooden blocks of various shapes or using colored construction paper to denote parts of a shape or line can provide a visual upon which to build the necessary
information. You may also benefit from ACT Math tutoring or the free digital ACT prep book offered by Varsity Tutors.
However, the primary means of succeeding on the ACT math section is protected time to study and do practice problems. Students should plan to review well in advance and review the content before
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Ming Sun – Silicon achitect, design lead, researcher.
Gvg derivation of Boost converter
Ming Sun / November 29, 2022
13 min read • ––– views
Step 1 - construct small-signal equations
Fig. 1Boost power stage block diagram^[1]
Voltage-second balance equation
Fig. 1 shows a synchronous Boost power stage, where it contains a low side switch S[1] and a high side switch S[2].
For the inductor, we can write the voltage-second balance as^[1]:
`L{dI}/{dt} = D*V_g + D^'*(V_g-V) = V_g - D^'*V`
Where, I is the inductor current, V[g] is the Boost converter's input voltage, and V is Boost converter's output voltage V[out]. Next, let us perturb and linearize Eq. 1 by introducting the small
signal perturbation as:
`L{d(I+hat(i))}/{dt} = V_g + hat(v)_g - D^'*(V+hat(v))`
Here we are trying to derive the transfer function of G[vg]. As a result, we can assume D is constant. Removing the DC terms from Eq. 2, we have:
`L{dhat(i)}/{dt} = hat(v)_g - D^'*hat(v)`
Eq. 3 can be written in s domain as:
`sL*hat(i) = hat(v)_g - D^'*hat(v)`
charge balance equation
For the capacitor, we can write the charge balance as^[1]:
`C{dV}/{dt} = D^'*I-V/R`
Next, let us perturb and linearize Eq. 5 by introducting the small signal perturbation as:
`C{d(V+hat(v))}/{dt} = D^'*(I+hat(i))-(V+hat(v))/R`
Removing the DC terms from Eq. 6, we have:
`C{dhat(v)}/{dt} = D^'*hat(i) - hat(v)/R`
Eq. 7 can be written in s domain as:
`sC*hat(v) = D^'*hat(i) - hat(v)/R`
Step 2 - solve the G[vg] in Matlab
The Matlab script used to derive the G[vg] transfer function is as shown below:
clc; clear; close all;
syms s
syms v i vg
syms R L C V Dp
syms Gvg Gig
eqn1 = s*L*i == vg - Dp*v;
eqn2 = s*C*v == Dp*i - v/R;
eqn3 = Gvg == v/vg;
eqn4 = Gig == i/vg;
results = solve(eqn1, eqn2, eqn3, eqn4, [v i Gvg Gig]);
Gvg = simplify(results.Gvg)
Gig = simplify(results.Gig)
Fig. 2 shows the G[vg] derived result from Matlab.
Fig. 2Gvg derived result from Matlab^
From Fig. 2, we have:
`G_{vg} = 1/D^' * 1/{1+s*L/(RD^('2)) + (LC)/D^('2)*s^2}`
Simplis for verification of Gvg transfer function
To simulate G[vg] transfer function in Simplis, the open loop Boost converter model is as shown in Fig. 3.
Fig. 3Open-loop Boost converter model for Gvg simulation^
To set the property of the Laplace Transfer Function block, Eq. 9 can be re-written as:
`G_{vg} = 1/D^' * 1/{1+s*L/(RD^('2)) + (LC)/D^('2)*s^2} = D^'/(LC) * 1/{D^('2)/(LC)+s/(RC) + s^2}`
We can plug in the inductor, capacitor, resistor, and D' values into Eq. 10. We have:
`G_{vg} = D^'/(LC) * 1/{D^('2)/(LC)+s/(RC) + s^2} = 800G* 1/(640G + 160k*s + s^2)`
Based on Eq. 11, the property of the 2nd-order Laplace Transfer Function is as shown in Fig. 4.
Fig. 42nd-order Laplace Transfer Function block property^
The Simplis simulation results are as shown in Fig. 5. From Fig. 5, we can see that the mathematical Laplace transfer function matches with the AC simulation results of G[vg] very well.
Fig. 5Simulation results comparison between mathematical derivation and AC analysis^
Gig verification
From Fig. 2, the G[ig] transfer function can be written as:
`G_{ig} = 1/(RD^('2))*(1+sRC)/(1+s*L/(RD^('2)) + s^2*(LC)/(D^('2)))`
To verify the G[ig] transfer function, the Simplis test bench can be modified as shown in Fig. 6.
Fig. 6Gig test bench in Simplis for Boost converter^
To set the property of the Laplace Transfer Function block, Eq. 12 can be re-written as:
`G_{ig} = 1/(RD^('2))*(1+sRC)/(1+s*L/(RD^('2)) + s^2*(LC)/(D^('2))) = 1/(RLC) * (1+sRC)/((D^('2))/(LC)+s/(RC) + s^2)`
We can plug in the inductor, capacitor, resistor and Vg values into Eq. 13. We have:
`G_{ig} = 1/(RLC) * (1+sRC)/((D^('2))/(LC)+s/(RC) + s^2 = 160G*(1+s*1µ)/(640G+s*160k+s^2)`
Based on Eq. 14, the property of the 2nd-order Laplace Transfer Function is as shown in Fig. 7.
Fig. 72nd-order Laplace Transfer Function block property for Gig simulation^
The Simplis simulation results are as shown in Fig. 9. From Fig. 9, we can see that the mathematical Laplace transfer function matches with the AC simulation results of G[ig].
Fig. 8Gig comparison between mathematical derivation and AC analysis^
References and downloads
[1] Fundamentals of power electronics - Chapter 2
[2] Open-loop Boost converter model for Gvg simulation in Simplis - pdf
[3] Open-loop Boost converter model for Gvg simulation in Simplis - download
[4] Open-loop Boost converter model for Gig simulation in Simplis - pdf
[5] Open-loop Boost converter model for Gig simulation in Simplis - download
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Calculate Compound Interest in Python
To calculate compound interest in Python, you can use the formula to calculate compound interest and create a function.
def compound_interest(p,r,n,t):
a = p*(1+r/100/n)**(n*t)
return a - p
If you have continuous compounding and want to calculate the compound interest, then you can use the continuous compounding equation.
import math
def compound_interest(p,r,t):
a = p * math.exp(r/100*t)
return a - p
Compound interest is the eighth wonder of the world, according to Albert Einstein.
The ability for us to calculate compound interest is valuable and with Python, you can easily create a function which will calculate compound interest and the total amount gained after a certain
period of compounding.
The equation for periodic compounding is as follows.
amount = principal * (1 + rate / number of periods) ^ (number of periods * time)
If you want to get compound interest from this equation, you can subtract the starting principal from amount and get the total interest accrued.
In Python, this is easy to implement because it is just multiplying and dividing.
Below is an example showing you how to perform periodic compounding and calculate the compound interest in Python.
def compound_interest(p,r,n,t):
a = p*(1+r/100/n)**(n*t)
return a - p
print(compound_interest(1000,5,1,10)) #annually
print(compound_interest(1000,5,2,10)) #biannually
print(compound_interest(1000,5,4,10)) #quarterly
print(compound_interest(1000,5,12,10)) #monthly
Calculating Compound Interest in Python for Continuous Compounding
Another case of compounding is when you have continuous compounding. The equation for continuous compounding is as follows.
amount = principal * e ^ (rate * time)
In Python, this is easy to implement with the help of the math module.
Below is an example showing you how to perform continuous compounding and calculate the compound interest in Python.
import math
def compound_interest(p,r,t):
a = p * math.exp(r/100*t)
return a - p
Hopefully this article has been useful for you to learn how to calculate compound interest in Python.
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To study resistors in series circuit
In this experiment, we investigate the behavior of resistors in a series circuit. By connecting resistors end-to-end in a single path, we observe the total resistance and voltage drops across
individual resistors.
A series circuit consists of components connected in a single loop, where the current passes through each component in succession. When resistors are connected in series, their resistances add up to
produce the total resistance of the circuit.
Experiment Details:
The experiment setup includes a DC power supply, resistors of known values, an ammeter to measure current, and a voltmeter to measure voltage.
1. Connect the resistors in series by connecting one end of each resistor to the next.
2. Connect the series circuit to the DC power supply.
3. Measure the total voltage (V) across the series circuit using a voltmeter.
4. Measure the total current (I) passing through the circuit using an ammeter.
5. Calculate the total resistance (R) using Ohm's Law: R = V / I.
6. Measure the voltage drop across each resistor using the voltmeter.
7. Repeat the experiment with different combinations of resistors.
Observations and Calculations:
Let's assume we have three resistors connected in series with resistances R1, R2, and R3.
• Total voltage across the circuit (V):
• Total current passing through the circuit (I):
• Total resistance of the circuit (R):
• Voltage drop across each resistor (V1, V2, V3):
Through this experiment, we understand the concept of resistors in series circuits and how their resistances add up. The total resistance in a series circuit is the sum of individual resistances.
This experiment also reinforces the application of Ohm's Law in calculating voltage, current, and resistance in a series circuit.
• Ensure proper connections to avoid short circuits.
• Handle resistors and equipment carefully to prevent damage.
• Double-check connections and readings to ensure accuracy.
• Use appropriate safety measures when working with electricity.
Short Questions with Answers:
1. What is a series circuit?
A series circuit is a circuit where components are connected in a single path, so the same current flows through each component.
2. How do resistors add up in a series circuit?
In a series circuit, resistors add up to produce the total resistance, which is the sum of all individual resistances.
3. What is Ohm's Law?
Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points.
4. What happens to the total resistance in a series circuit when more resistors are added?
The total resistance increases as more resistors are added in series.
5. Why is it important to handle resistors and equipment carefully during the experiment?
Handling equipment carefully prevents damage and ensures accurate results.
6. How can you calculate the total resistance in a series circuit?
The total resistance can be calculated by summing up the individual resistances of all the resistors connected in series.
7. What is the purpose of measuring voltage drops across each resistor?
Measuring voltage drops helps in understanding how voltage is distributed across different components in the circuit.
8. What is the unit of resistance?
The unit of resistance is the ohm (Ω).
9. Why is it necessary to repeat the experiment with different combinations of resistors?
Repeating the experiment with different combinations helps in verifying the consistency of results and understanding the behavior of resistors in series under varying conditions.
10. What safety measures should be taken when working with electricity?
Use insulated tools, wear protective gear, and avoid working with electricity in wet conditions.
11. Explain the concept of voltage drop.
Voltage drop is the decrease in voltage across a component in a circuit due to the resistance of that component.
12. What happens to the current in a series circuit if one resistor fails?
If one resistor fails in a series circuit, the current flow in the entire circuit may be affected, depending on the nature of the failure.
13. What is the significance of measuring current in a series circuit?
Measuring current helps in determining the flow of electric charge through the circuit and verifying the principles of Kirchhoff's laws.
14. How does the total resistance of a series circuit compare to the individual resistances?
The total resistance of a series circuit is greater than any individual resistance in the circuit.
15. What is the relationship between voltage, current, and resistance in a series circuit?
According to Ohm's Law, voltage (V) is equal to the product of current (I) and resistance (R) in a series circuit (V = IR).
16. Explain the role of an ammeter and a voltmeter in this experiment.
An ammeter measures the current flowing through the circuit, while a voltmeter measures the voltage across components in the circuit.
17. Why is it important to ensure proper connections in the circuit?
Proper connections prevent short circuits and ensure accurate readings.
18. What factors affect the total resistance of a series circuit?
The total resistance is affected by the individual resistances of the components and their arrangement in the circuit.
19. How can you verify Ohm's Law experimentally in a series circuit?
By measuring voltage, current, and resistance in a series circuit and confirming that they follow the relationship defined by Ohm's Law.
20. What is the purpose of repeating the experiment with different combinations of resistors?
Repeating the experiment helps in understanding how the total resistance varies with different configurations of resistors in series.
Multiple Choice Questions:
1. What happens to the total resistance in a series circuit when more resistors are added?
□ a) Increases
□ b) Decreases
□ c) Remains the same
□ d) None of the above
Answer: a) Increases
2. Which instrument is used to measure current in a circuit?
□ a) Voltmeter
□ b) Ammeter
□ c) Ohmmeter
□ d) Rheostat
Answer: b) Ammeter
3. What is the unit of resistance?
□ a) Ampere
□ b) Watt
□ c) Ohm
□ d) Volt
Answer: c) Ohm
4. Which law governs the relationship between voltage, current, and resistance in a circuit?
□ a) Newton's Law
□ b) Boyle's Law
□ c) Ohm's Law
□ d) Faraday's Law
Answer: c) Ohm's Law
5. What is the purpose of measuring voltage drops across resistors in a series circuit?
□ a) To calculate power
□ b) To verify Kirchhoff's laws
□ c) To understand the distribution of voltage
□ d) To measure current
Answer: c) To understand the distribution of voltage
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2D Periodic Triangulation
Nico Kruithof
The periodic 2D-triangulation class of CGAL is designed to represent the triangulation of a set of points in the two-dimensional flat torus. The triangulation forms a partition of the space it is
computed in. It is a simplicial complex, i.e. it contains all incident \( j\)-simplices ( \( j<k\)) of any \( k\)-simplex and two \( k\)-simplices either do not intersect or share a common \( j\)
-face, \( j<k\). The occurring simplices of dimension up to two are called vertex, edge and face, respectively.
The Flat Torus
The 2D Periodic Triangulation package computes triangulations in the space \( \mathbb T_c^2\), which is defined as follows: Let \( c\in\mathbb R\setminus\{0\}\) and \( G\) be the group \( (c\cdot\
mathbb Z^2, +)\), where \( c\cdot\mathbb Z\) denotes the set containing all integer multiples of \( c\). The flat torus is the quotient space: \( \mathbb T_c^2:=\mathbb R^2/G\). The parameter \( c\)
defines the period.
The elements of \( \mathbb T_c^2\) are the equivalence classes of sets of points in \( \mathbb R^2\). We call these points representatives of an element of \( \mathbb T_c^2\). The implementation does
not work directly on elements of \( \mathbb T_c^2\) but on some representatives in \( \mathbb R^2\). So there need to be distinguished representatives to work on. Given \( \alpha\) and \( \beta\),
the square \( [\alpha,\alpha+c)\times[\beta,\beta+c)\) contains exactly one representative of each element in \( \mathbb T_c^2\). We call it original domain. From now on, when we talk about points,
we generally mean representatives of elements of \( \mathbb T_c^2\) that lie inside the original domain. Note that any input point is required to be an element of the half-open square representing
the original domain as defined above.
There are simplices containing points inside the original domain but also points outside it. The points outside the original domain are periodic copies of points inside the original domain. So, to
specify a simplex we need points together with some additional information that determines the respective periodic copy of each point. The set of representatives of an element of \( \mathbb T_c^2\)
is a square point grid. We address each representative by a two-dimensional integer vector \( (o_x,o_y)\), called offset. It represents the number of periods a representative in the original domain
must be translated in \( x\)- and \( y\)-direction. The vector \( (0,0)\) corresponds to the representative in the original domain. To specify a \( k\)-simplex we need \( k+1\) point-offset pairs
(cf. Fig. Figure 43.1).
A triangulation is a collection of vertices and faces that are linked together through incidence and adjacency relations. Each face gives access to its three incident vertices, their corresponding
offsets, and to its three adjacent faces. Each vertex gives access to one of its incident faces.
The three vertices of a face are indexed with 0, 1 and 2 in positive orientation. The orientation of a simplex in \( \mathbb T_c^2\) is defined as the orientation of the corresponding simplex in \( \
mathbb R^2\) given by representatives determined by the respective offsets (see Figure 43.2).
As in the underlying combinatorial triangulation (see Chapter Chapter_2D_Triangulation_Data_Structure), the neighbors of a faces are indexed with 0, 1 and 2 in such a way that the neighbor indexed by
\( i\) is opposite to the vertex with the same index. Edges ( \( 1\)-faces) are not explicitly represented: an edge is given by a face and an index (the edge i of a face f is the edge of f that is
opposite to the vertex with index i). See Figure 43.2.
Some point sets do not admit a triangulation in \( \mathbb T_c^2\). In this case we use 9 periodic copies of the point set arranged in a square of edge length \( 3c\). Any point set constructed in
this way has a triangulation in \( \mathbb R^2/G'\) with \( G'=(3c\cdot\mathbb Z)^2\) [1]. So we compute the triangulation in this space, which is a 9-sheeted covering space of \( \mathbb T_c^2\)
(see Figure 43.3).
The machinery that manages the copies is largely hidden from the user. However there are some effects that cannot be ignored. For example if the point set does not permit a triangulation in \( \
mathbb T_c^2\) then the combinatorial iterators (Face_iterator, Edge_iterator and Vertex_iterator) return all simplices that are internally stored, which correspond to 9 periodic copies of each
geometric primitive (Triangle, Segment, and Point). This is necessary to ensure consistency in the adjacency relations. In case it is desired to have only one periodic copy of each primitive, we
provide geometric iterators. They return geometric primitives of the triangulation without relations between them. Another effect is that when the algorithm switches from 9-sheeted covering to
1-sheeted covering the Vertex_handles and Face_handles referencing deleted items becomes invalid.
In the data structure each vertex stores the input point it corresponds to. If we are computing in 9-sheeted covering space, each vertex stores the representative inside the original domain it
corresponds to. So, the 9 vertices corresponding to the same element of \( \mathbb T_c^2\) all store the same representative in \( \mathbb R^2\), and not different periodic copies.
A periodic triangulation is said to be locally valid iff
(a)-(b) Its underlying combinatorial graph, the triangulation data structure, is locally valid (see Chapter Chapter_2D_Triangulation_Data_Structure)
(c) Any face has its vertices ordered according to positive orientation. See Figure 43.2.
Delaunay Triangulation
The class Periodic_2_Delaunay_triangulation_2 implements Delaunay triangulations of point sets in \( \mathbb T_c^2\).
Delaunay triangulations have the empty circle property, that is, the circumscribing circle of each face does not contain any other vertex of the triangulation in its interior. These triangulations
are uniquely defined except in degenerate cases where four points are co-circular. Note however that the CGAL implementation computes a unique triangulation even in these cases [3].
This implementation is fully dynamic: it supports both insertions of points and vertex removal.
Triangulation Hierarchy
The class Periodic_2_triangulation_hierarchy_2 is the adaptation of the hierarchical structure described in Chapter Chapter_2D_Triangulations, Section The Triangulation Hierarchy to the periodic
The class Periodic_2_triangulation_hierarchy_2<Tr> inherits from the triangulation type passed as template parameter Tr. The insert, move, and remove member functions are overwritten to update the
data structure at each operation. The locate queries are also overwritten to take advantage of the data structure for a fast processing.
Software Design
We have chosen the prefix `‘Periodic_2’' to emphasize that the triangulation is periodic in all two directions of space. There are also `‘cylindrical’' periodicities where the triangulation is
periodic only in one direction of the space.
The two main classes Periodic_2_Delaunay_triangulation_2 and Periodic_2_triangulation_2 provide high-level geometric functionality and are responsible for the geometric validity.
Periodic_2_Delaunay_triangulation_2 contains all the functionality that is special to Delaunay triangulations, such as point insertion and vertex removal, the side-of-circle test, finding the
conflicting region of a given point, dual functions etc. Periodic_2_triangulation_2 contains all the functionality that is common to triangulations in general, such as location of a point in the
triangulation [4], access functions, geometric queries like the orientation test etc.
They are built as layers on top of a triangulation data structure, which stores their combinatorial structure. This separation between the geometry and the combinatorics is reflected in the software
design by the fact that the triangulation classes take two template parameters:
The Geometric Traits Parameter
The first template parameter of the periodic triangulation class Periodic_2_triangulation_2<Traits, Tds> is the geometric traits class, described by the concept Periodic_2TriangulationTraits_2.
Similar, the first template parameter of the Delaunay triangulation class Periodic_2_Delaunay_triangulation_2<Traits,Tds> is the geometric traits class, described by the concept
Periodic_2DelaunayTriangulationTraits_2. These concepts are different from the TriangulationTraits_2 and DelaunayTriangulationTraits_2 (see chapter The Geometric Traits) in that they also implement
all objects, predicates and constructions with using offsets.
The class Periodic_2_Delaunay_triangulation_traits_2<Traits,Periodic_2Offset_2> provides the required functionality. It expects two template parameters: a model of the concept
DelaunayTriangulationTraits_2 and a model of the concept Periodic_2Offset_2. Since the concept TriangulationTraits_2 refines the concept DelaunayTriangulationTraits_2, the class
Periodic_2_Delaunay_triangulation_traits_2<Traits,Periodic_2Offset_2> is also a model for the concept TriangulationTraits_2.
The kernels Cartesian, Homogeneous, Simple_cartesian, Simple_homogeneous and Filtered_kernel can all be used as models for Traits. Periodic_2_triangulation_traits_2 provides exact predicates and
exact constructions if Traits does. It provides exact predicates but not exact constructions if Filtered_kernel<CK> with CK an inexact kernel is used as its first template parameter. Using
Exact_predicates_inexact_constructions_kernel as Traits provides fast and exact predicates and not exact constructions, using Exact_predicates_exact_constructions_kernel provides fast and exact
predicates and exact constructions. The latter is recommended if the dual constructions and constructions of points, segments, triangles, and tetrahedra are used.
The second parameter Periodic_2Offset_2 defaults to Periodic_2_offset_2.
The Triangulation Data Structure Parameter
The second template parameter of the main classes Periodic_2_triangulation_2 and Periodic_2_Delaunay_triangulation_2 is a triangulation data structure class. This class must be a model of the concept
TriangulationDataStructure_2, which describes requirements for the class to be a container for the faces and vertices maintaining incidence and adjacency relations (see Chapter
Chapter_2D_Triangulation_Data_Structure). In addition, the concepts TriangulationDataStructure_2::Vertex and TriangulationDataStructure_2::Face are extended to support periodicity: the vertex and
face must be models of Periodic_2TriangulationVertexBase_2 and Periodic_2TriangulationFaceBase_2. A model of such concept is CGAL::Triangulation_data_structure_2. It is parameterized by a vertex base
class and a face base class, which gives the possibility to customize the vertices and faces used by the triangulation data structure, and hence by the geometric triangulation using it. Basic models
of the vertex and face concepts are provided: CGAL::Periodic_2_triangulation_vertex_base_2 and CGAL::Periodic_2_triangulation_face_base_2.
A default value for the triangulation data structure parameter is provided in all the triangulation classes, so it does not need to be specified by the user unless he wants to use a different
triangulation data structure or a different vertex or face base class.
Flexibility of the Design
Periodic_2_triangulation_2 uses the TriangulationDataStructure_2 in essentially the same way as Triangulation_2. That is why the flexibility described in Software Design is applicable in exactly the
same way. Also the classes Triangulation_vertex_base_with_info_2 and Triangulation_face_base_with_info_2 can be reused directly, see also Example Adding a color.
Basic example
This example shows the incremental construction of a periodic 2D Delaunay triangulation, the location of a point and how to perform elementary operations on indices in a face. It uses the default
parameter of the Periodic_2_Delaunay_triangulation_2 class for the triangulation data structure.
File Periodic_2_triangulation_2/p2t2_simple_example.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
#include <fstream>
#include <cassert>
#include <list>
#include <vector>
typedef PDT::Face_handle Face_handle;
typedef PDT::Vertex_handle Vertex_handle;
typedef PDT::Locate_type Locate_type;
typedef PDT::Point Point;
typedef PDT::Iso_rectangle Iso_rectangle;
int main()
Iso_rectangle domain(-1, -1, 2, 2); // The cube for the periodic domain
// construction from a list of points :
std::list<Point> L;
L.push_front(Point(0, 0));
L.push_front(Point(1, 0));
L.push_front(Point(0, 1));
PDT T(L.begin(), L.end(), domain); // Put the domain with the constructor
size_t n = T.number_of_vertices();
// insertion from a vector :
std::vector<Point> V(3);
V[0] = Point(0, 0);
V[1] = Point(1, 1);
V[2] = Point(-1, -1);
n = n + T.insert(V.begin(), V.end());
assert( n == 5 ); // 6 points have been inserted, one is a duplicate
assert( T.is_valid() ); // checking validity of T
Locate_type lt;
int li;
Point p(0, 0);
Face_handle fh = T.locate(p, lt, li);
// p is the vertex of c of index li :
assert( lt == PDT::VERTEX );
assert( fh->vertex(li)->point() == p );
Vertex_handle v = fh->vertex( (li + 1) % 3 );
// v is another vertex of c
Face_handle nb = fh->neighbor(li);
// nb = neighbor of fh opposite to the vertex associated with p
// nb must have vertex v :
int nli;
assert( nb->has_vertex( v, nli ) );
// nli is the index of v in nc
std::ofstream oFileT("output.tri", std::ios::out);
// writing file output;
oFileT << T;
PDT T1;
std::ifstream iFileT("output.tri", std::ios::in);
// reading file output;
iFileT >> T1;
assert( T1.is_valid() );
assert( T1.number_of_vertices() == T.number_of_vertices() );
assert( T1.number_of_faces() == T.number_of_faces() );
return 0;
The class Periodic_2_Delaunay_triangulation_2 represents a Delaunay triangulation in two-dimensional ...
Definition: Periodic_2_Delaunay_triangulation_2.h:52
The class Periodic_2_Delaunay_triangulation_traits_2is designed as a default traits class for the cla...
Definition: Periodic_2_Delaunay_triangulation_traits_2.h:29
Changing the vertex base
The following two examples show how the user can plug his own vertex base in a triangulation. Changing the face base is similar.
Adding a color
If the user does not need to add a type in a vertex that depends on the TriangulationDataStructure_2 (e.g. a Vertex_handle or Face_handle), he can use the Triangulation_vertex_base_with_info_2 class
to add his own information easily in the vertices. The example below shows how to add a CGAL::IO::Color this way.
File Periodic_2_triangulation_2/p2t2_colored_vertices.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
#include <CGAL/Triangulation_vertex_base_with_info_2.h>
#include <CGAL/IO/Color.h>
typedef PDT::Point Point;
int main()
PDT T;
T.insert(Point(.1, 0));
T.insert(Point(0, .1));
T.insert(Point(.2, .2));
T.insert(Point(.9, 0));
// Set the color of vertices with degree 6 to red.
PDT::Vertex_iterator vit;
for (vit = T.vertices_begin(); vit != T.vertices_end(); ++vit)
if (T.degree(vit) == 6)
vit->info() = CGAL::IO::red();
return 0;
Vertex_handle insert(const Point &p, Face_handle start=Face_handle())
Inserts point p in the triangulation and returns the corresponding vertex.
The class Periodic_2_triangulation_face_base_2 is a model of the concept Periodic_2TriangulationFaceB...
Definition: Periodic_2_triangulation_face_base_2.h:40
The class Periodic_2_triangulation_vertex_base_2 is a model of the concept Periodic_2TriangulationVer...
Definition: Periodic_2_triangulation_vertex_base_2.h:31
Adding handles
If the user needs to add a type in a vertex that depends on the TriangulationDataStructure_2 (e.g. a Vertex_handle or Face_handle), he has to derive his own vertex base class, as the following
example shows.
File Periodic_2_triangulation_2/p2t2_adding_handles.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
#include <CGAL/Triangulation_vertex_base_2.h>
template < class GT, class Vb >
class My_vertex_base
: public Vb
typedef typename Vb::Vertex_handle Vertex_handle;
typedef typename Vb::Face_handle Face_handle;
typedef typename Vb::Point Point;
template < typename Tds2 >
struct Rebind_TDS
typedef typename Vb::template Rebind_TDS<Tds2>::Other Vb2;
typedef My_vertex_base<GT, Vb2> Other;
My_vertex_base() {}
My_vertex_base(const Point& p)
: Vb(p) {}
My_vertex_base(const Point& p, Face_handle c)
: Vb(p, c) {}
Vertex_handle vh;
Face_handle fh;
typedef My_vertex_base<GT, VbDS> Vb;
typedef PDT::Vertex_handle Vertex_handle;
typedef PDT::Point Point;
int main()
PDT T;
Vertex_handle v0 = T.
(Point(0, 0));
Vertex_handle v1 = T.insert(Point(.1, 0));
Vertex_handle v2 = T.insert(Point(0, .1));
Vertex_handle v3 = T.insert(Point(0, 0.2));
Vertex_handle v4 = T.insert(Point(.2, .2));
Vertex_handle v5 = T.insert(Point(.9, 0));
// Now we can link the vertices as we like.
v0->vh = v1;
v1->vh = v2;
v2->vh = v3;
v3->vh = v4;
v4->vh = v5;
v5->vh = v0;
return 0;
9-sheeted covering
The user can check at any time whether a triangulation would be a simplicial complex in \( \mathbb T_c^2\) and force a conversion if so. However this should be done very carefully in order to be sure
that the internal structure always remains a simplicial complex and thus a triangulation.
In this example we construct a triangulation that can be converted to the 1-sheeted covering. However, we can insert new points such that the point set does not have a Delaunay triangulation in the
1-sheeted covering anymore, so the triangulation is not extensible.
File Periodic_2_triangulation_2/p2t2_covering.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
#include <iostream>
#include <vector>
typedef PDT::Point Point;
typedef PDT::Covering_sheets Covering_sheets;
int main()
PDT T;
// Input point grid (27 points)
for (double x = 0. ; x < .9 ; x += 0.4)
for (double y = 0. ; y < .9 ; y += 0.4)
Covering_sheets cs = T.number_of_sheets();
std::cout << "Current covering: " << cs[0] << ' ' << cs[1] << std::endl;
if ( T.is_triangulation_in_1_sheet() ) // = true
bool is_extensible = T.is_extensible_triangulation_in_1_sheet_h1()
|| T.is_extensible_triangulation_in_1_sheet_h2(); // = false
cs = T.number_of_sheets();
std::cout << "Current covering: " << cs[0] << ' ' << cs[1] << std::endl;
if ( is_extensible ) // = false
std::cout << "It is safe to change the triangulation here." << std::endl;
std::cout << "It is NOT safe to change the triangulation here!" << std::endl;
cs = T.number_of_sheets();
std::cout << "Current covering: " << cs[0] << ' ' << cs[1] << std::endl;
std::cout << "It is (again) safe to modify the triangulation." << std::endl;
return 0;
Large point set
For large point sets there are two optimizations available. Firstly, there is spatial sorting that sorts the input points according to a Hilbert curve, see chapter Spatial Sorting. The second one
inserts 12 appropriately chosen dummy points to avoid the use of a 9-sheeted covering in the beginning. The 12 dummy points are deleted in the end. If the point set turns out to not have a Delaunay
triangulation in 1-sheeted covering, the triangulation is converted to 9-sheeted covering during the removal of the 12 dummy points. This might take even longer than computing the triangulation
without using this optimization. In general, uniformly distributed random point sets of more than 1000 points have a Delaunay triangulation in 1-sheeted covering.
It is recommended to run this example only when compiled in release mode because of the relatively large number of points.
File Periodic_2_triangulation_2/p2t2_large_point_set.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
#include <CGAL/Random.h>
#include <CGAL/point_generators_2.h>
#include <CGAL/Timer.h>
#include <iostream>
#include <vector>
typedef PDT::Point Point;
int main()
CGAL::Timer t;
CGAL::Random random(7);
CGAL::Random_points_in_square_2<Point, Creator> in_square(.5, random);
int n = 10000;
std::vector<Point> pts;
PDT PT1, PT2, PT3;
// Generating n random points
for (int i = 0 ; i < n ; i++)
Point p = *in_square;
pts.push_back(Point(p.x() + .5, p.y() + .5));
// Standard insertion
for (int i = 0 ; i < n ; i++)
std::cout << " Time: " << t.time() << " sec. (Standard insertion)" << std::endl;
// Iterator range insertion using spatial sorting but no dummy points
PT2.insert(pts.begin(), pts.end()); // third parameter defaults to false
std::cout << " Time: " << t.time() << " sec. (with spatial sorting)" << std::endl;
// Iterator range insertion using spatial sorting and dummy point heuristic
PT3.insert(pts.begin(), pts.end(), true);
std::cout << " Time: " << t.time() << " sec. (Dummy point heuristic)" << std::endl;
return 0;
Geometric access
There might be applications that need the geometric primitives of a triangulation as an input but do not require a simplicial complex. For these cases we provide the geometric iterators that return
only the geometric primitives fulfilling some properties. In the following example we use the Periodic_triangle_iterator with the option UNIQUE_COVER_DOMAIN. This means that only those triangles are
returned that have a non-empty intersection with the original domain of the 1-sheeted covering space, see Figure P2Triangulation2figgeom_iterators. The Periodic_triangle is actually a two-dimensional
array of point-offset pairs. We check for all three entries of the periodic triangle whether the offset is (0,0,0) using the method is_null. If so, we convert the periodic triangle to a
PK::Triangle_2, which requires exact constructions.
File Periodic_2_triangulation_2/p2t2_geometric_access.cpp
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
typedef PK::Point_2 Point;
typedef PK::Triangle_2 Triangle;
typedef P2DT2::Periodic_triangle Periodic_triangle;
typedef P2DT2::Periodic_triangle_iterator Periodic_triangle_iterator;
typedef P2DT2::Iterator_type Iterator_type;
int main()
P2DT2 T;
T.insert(Point(0, 0.5));
T.insert(Point(0.5, 0));
Periodic_triangle pt;
Triangle t_bd;
// Extracting the triangles that have a non-empty intersection with
// the original domain of the 1-sheeted covering space
for (Periodic_triangle_iterator ptit = T.periodic_triangles_begin(P2DT2::UNIQUE_COVER_DOMAIN);
ptit != T.periodic_triangles_end(P2DT2::UNIQUE_COVER_DOMAIN); ++ptit)
pt = *ptit;
if (! (pt[0].second.is_null() && pt[1].second.is_null() && pt[2].second.is_null()) )
// Convert the current Periodic_triangle to a Triangle if it is
// not strictly contained inside the original domain.
// Note that this requires EXACT constructions to be exact!
t_bd = T.triangle(pt);
The performance of the 2D periodic Delaunay triangulation is compared to the Euclidean 2D Delaunay triangulation. The points are inserted in the Euclidean 2D Delaunay triangulation using spatial
sorting. In the Periodic triangulation the points are first inserted in random order until the triangulation is valid in the 1 sheeted covering space. The remaining points are then inserted using
spatial sorting. For the large point set, first dummy points are inserted to create a valid triangulation in the 1 sheeted covering space. Then all points are inserted using spatial sorting. As a
final step, the dummy points are removed again.
The plot shows the running time in seconds for different numbers of batch inserted points. The points are uniformly randomly distributed in the unit rectangle. The tests were done on an Intel i7 @
Draw a 2D Periodic Triangulation
A 2D periodic triangulation can be visualized by calling the CGAL::draw<P2T2>() function as shown in the following example. This function opens a new window showing the given Periodic Triangulation.
Elements of the periodic triangulation can be viewed in four different modes:
File Periodic_2_triangulation_2/draw_periodic_2_triangulation_2.cpp
#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/draw_periodic_2_triangulation_2.h>
#include <fstream>
typedef PDT::Point Point;
int main(int argc, char* argv[])
// Declare periodic triangulation 2D
PDT T;
// Read points and insert in T
Point p;
std::ifstream ifs((argc > 1) ? argv[1] : "data/data1.dt.cin");
if (ifs)
while (ifs >> p)
{ T.insert(p); }
if( T.is_triangulation_in_1_sheet())
{ T.convert_to_9_sheeted_covering(); }
// Draw the periodic triangulation
return EXIT_SUCCESS;
void draw(const T2 &at2, const GSOptions &gso)
• STORED Display all geometric primitives as they are stored in Triangulation_data_structure_2;
• UNIQUE Display only one representative of each geometric primitive even if the triangulation is computed in multiply sheeted covering space;
• STORED_COVER_DOMAIN Same as STORED but also display all primitives whose intersection with the original domain of the current covering space is non-empty;
• UNIQUE_COVER_DOMAIN Same as UNIQUE but also display all primitives whose intersection with the original domain of the current covering space is non-empty.
The domain can also be visualized by a key press. To see how to visualize the Periodic Triangulation in various modes, press key H when the viewer window is active and go to Keyboard tab. See Figure
43.4 and Figure 43.5.
Design and Implementation History
The periodic 2D-triangulation is based on the 2D triangulation package developed by Mariette Yvinec and inspired by the periodic 3D-triangulation package developed by Manuel Caroli and Monique
Teillaud. The periodic 3D-triangulation package is described in Manuel's PhD thesis [2] Triangulating Point Sets in Orbit Spaces and [1].
In 2009, Nico Kruithof started implementation of the Periodic_2_triangulation_2 package.
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Pre-Calculus Course Content, 12. Complex Numbers, Polar Form of Complex Numbers, Euler's Notation, DeMoivre's Theorem, Applications
Complex Numbers and their Polar Form
In this module, complex numbers are represented in terms of polar coordinates. This representation helps in evaluating complex number arithmetic, particularly their powers and roots as an application
of DeMoivre's Theorem.
Learning Objectives:
• Complex number - real and imaginary part
• Complex Plane
• Modulus and Argument of a complex number, and their properties
• Polar form of complex number
• Euler's Formula for a complex number
• Products, Powers and Quotients of complex numbers, DeMoivre's Theorem
• nth roots of complex numbers
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Bio-Design Fabrication and Soft Robots
9.1.12.B.4 Visual Arts
9.1.12.B.4.1 Paint
CCSS.Math.Content.K.MD.A.1 Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
CCSS.Math.Content.K.MD.A.2 Directly compare two objects with a measurable attribute in common, to see which object has “more of”/”less of” the attribute, and describe the difference.
CCSS.Math.Content.1.MD.B.3 Tell and write time in hours and half-hours using analog and digital clocks.
CCSS.Math.Content.1.MD.C.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many
more or less are in one category than in another.
CCSS.Math.Content.2.MD.A.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
CCSS.Math.Content.2.MD.A.2 Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit
CCSS.Math.Content.2.MD.A.3 Estimate lengths using units of inches, feet, centimeters, and meters.
CCSS.Math.Content.2.MD.A.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
CCSS.Math.Content.3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and
exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
CCSS.Math.Content.4.MD.B.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using
information presented in line plots.
CCSS.Math.Content.4.MD.C.5a An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points
where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
CCSS.Math.Content.4.MD.C.5b An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
CCSS.Math.Content.4.MD.C.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
CCSS.Math.Content.4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.
Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
CCSS.Math.Content.K.G.A.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of,
behind, and next to.
CCSS.Math.Content.K.G.A.2 Correctly name shapes regardless of their orientations or overall size.
CCSS.Math.Content.K.G.A.3 Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
CCSS.Math.Content.K.G.B.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g.,
number of sides and vertices/”corners”) and other attributes (e.g., having sides of equal length).
CCSS.Math.Content.K.G.B.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
CCSS.Math.Content.K.G.B.6 Compose simple shapes to form larger shapes.
CCSS.Math.Content.2.G.A.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons,
hexagons, and cubes.
CCSS.Math.Content.2.G.A.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
CCSS.Math.Content.2.G.A.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole
as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
CCSS.Math.Content.5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each
line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction
of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and
x-coordinate, y-axis and y-coordinate).
CCSS.Math.Content.5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of
the situation.
CCSS.Math.Content.6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply
these techniques in the context of solving real-world and mathematical problems.
CCSS.Math.Content.6.G.A.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the
volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in
the context of solving real-world and mathematical problems.
CCSS.Math.Content.6.G.A.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the
same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
CCSS.Math.Content.6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the
context of solving real-world and mathematical problems.
CCSS.Math.Content.7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a
different scale.
CCSS.Math.Content.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or
sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
CCSS.Math.Content.7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
CCSS.Math.Content.7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and
area of a circle.
CCSS.Math.Content.7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
CCSS.Math.Content.7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons,
cubes, and right prisms.
CCSS.Math.Content.HSG-SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
CCSS.Math.Content.HSG-SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
CCSS.Math.Content.HSG-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the
meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
CCSS.Math.Content.HSG-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
CCSS.Math.Content.HSG-SRT.B.4 Prove theorems about triangles.
CCSS.Math.Content.HSG-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
CCSS.Math.Content.HSG-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute
CCSS.Math.Content.HSG-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
CCSS.Math.Content.HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
CCSS.Math.Content.HSG-SRT.D.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
CCSS.Math.Content.HSG-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
CCSS.Math.Content.HSG-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant
CCSS.Math.Content.HSG-C.A.1 Prove that all circles are similar.
CCSS.Mat.Content.HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords.
CCSS.Math.Content.HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
CCSS.Math.Content.HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.
K-PS2-1 Plan and conduct an investigation to compare the effects of different strengths or different directions of pushes and pulls on the motion of an object.
K-PS2-2 Analyze data to determine if a design solution works as intended to change the speed or direction of an object with a push or a pull.
3-PS2-1 Plan and conduct an investigation to provide evidence of the effects of balanced and unbalanced forces on the motion of an object.
3-PS2-2 Make observations and/or measurements of an object’s motion to provide evidence that a pattern can be used to predict future motion.
Barth R. (2004). Learning By Heart. San Francsico: Jossey Bass
Browne H. (1974). How I found Freedom In An Unfree World: A Handbook For Personal Liberty. Great Falls,Mont: Liamworks.
Crawford M. (2009). Shop Class as Soulcraft: An Inquiry Into the Value of Work. New York,New York: Penguin Press.
Fox J. (2008). Your Children’s Strengths: Discover Them, Develop Them, Use Them. . New York,New York: Viking Adult.
Gardner H. (1993). Multiple Intelligences: New Horizons in Theory and Practice:Basic Books
Jerold W. Apps (1996). Teaching from the Heart: Krieger Publishing Company.
Kozol J. (1991). Savage Inequalities.New York: Crown Publishers, Inc.
Maslow A (1962). Toward a Psychology of Being: Martino Books.
McDermott J. (1973). The Philosophy of Dewey/The Lived Experience.Capricorn Books.
Palmer P. (2007). The Courage to Teach: Exploring the Inner Landscape of a Teacher’s Life: Jossey-Bass.
Perricone J (2005). Zen and the Art of Public School Teaching. Publish America.
Pirsig R. (1974). Zen and the Art of Motorcycle Maintenance: An Inquiry Into Values.Bantam Books.
Sterling P. (1972). The Real Teachers New York: Random House.
Suzuki S. (1970). Zen Mind, Beginner Mind.Weatherhill.
White-Clark R. (2005). Training Teachers to Succeed in a Multicultural Classroom The Education Digest 70 no8 23-6 New York: National Association of Elementary School Principals.
Fostr M.Lewis J., Onafowora L. (2005). Grooming Great Urban Teachers 62 no6 28-32 Durham, North Carolina: Educational Leadership.
Simpson, Judith W. (July, 1995). Choices for urban art education New York: Arts Education Policy Review.
Bisgard D. (2005) Diversity [15 Years From Now] Do we have 2020 vision?
H.W. Wilson Company:http://www.nais.org
Charles G. (2005) The Challenge of Diversity and Choice 101
Educ Horiz 83 no2 ,H.W. Wilson Company: http://www.pilambda.org
Raymond A. and S. (1993,2005) I Am A Promise: The Children of the Stanton Elementary School DVD: New York: New Video Group, Inc.: http://www.videoverite.tv/pages/timelinemain.html#promise
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Assessing Interventions
9 Assessing Interventions
Since its outbreak in late 2019, COVID-19 affected millions of people worldwide. One of the ways to track the spread of the disease is through the collection and analysis of data on daily cases,
which has become a crucial component of new digital footprint trends. Contact tracing apps and tools were developed and implemented in many countries to help track the spread of COVID-19.
Data on daily cases can help us evaluate the effectiveness of policies implemented during the pandemic, such as lockdowns and other restrictions. By analyzing these data, policymakers and public
health experts have gained insights into the impact of these policies on the spread of the disease Brodeur et al. (2021) and (Zhou and Kan 2021). For instance, if daily cases decrease significantly
during a lockdown, it could indicate that the policy was successful in controlling the spread of the disease.
Data on daily cases can also help policymakers to make informed decisions about when to implement or lift certain restrictions. For example, if daily cases start to rise again after a lifting of
restrictions, it may indicate that it is too early to do so, and that more caution is necessary. In summary, data on daily cases was an essential tool for evaluating the effectiveness of policies
implemented during the COVID-19 pandemic and making informed decisions about future policies. As we will see in this chapter, evaluating COVI9-19 policies based onn cases is, unfortunately not so
To assess government interventions during the pandemic, we will be using difference-in-differences (diff-n-diff), a widely-adopted quasi-experimental design, used to evaluate the effect of an
intervention or exposure on an outcome of interest. This strategy involves comparing the change in the outcome before and after the intervention or exposure in the treated group, relative to the
change in the outcome over the same time period in a control group. By doing so, diff-n-diff can help isolate the effect of the intervention or exposure from other factors that may be driving the
This chapter is based on :
9.2 Data
First let’s import the Greater London COVID-19 data:
# import csv
covid_cases_london <- read.csv("data/longitudinal-2/covid_cases_london.csv", header = TRUE)
# check out the variables
[1] "areaType" "areaName"
[3] "areaCode" "date"
[5] "newCasesBySpecimenDate" "cumCasesBySpecimenDate"
[7] "newFirstEpisodesBySpecimenDate" "cumFirstEpisodesBySpecimenDate"
[9] "newReinfectionsBySpecimenDate" "cumReinfectionsBySpecimenDate"
Then we do the same for the Lazio area data, which is the Region of the capital of Italy, Rome. We are choosing this region because it did not see sharp peaks in COVID-19 cases during the winter of
# import csv
covid_cases_lazio <- read.csv("data/longitudinal-2/covid_cases_lazio.csv", header = TRUE)
# check out the variables
[1] "data" "stato"
[3] "codice_regione" "denominazione_regione"
[5] "codice_provincia" "denominazione_provincia"
[7] "sigla_provincia" "lat"
[9] "long" "totale_casi"
First we need to clean up the data somewhat and rename some variables in both dataframes to have 4 variables:
• date: year-month-day
• geo: geographical region
• cases: number of COVID-19 cases that day
• area : Lazio (Rome) or London
# Rename the variables in the Lombardia data frame
covid_cases_lazio_ren <- covid_cases_lazio %>%
rename(date = data , geo = denominazione_provincia, totalcases = totale_casi)
# Group the data by region and calculate total cases for each day (not cumulative cases)
covid_cases_lazio_daily <- covid_cases_lazio_ren %>%
group_by(geo) %>%
mutate(cases = totalcases - lag(totalcases, default = 0)) %>%
select(date, geo, cases) %>%
mutate(area = "Rome (Lazio)") %>%
filter(cases >= 0)
#df_milan <- covid_cases_lombardia_new %>%
# filter(geo == "Milano", new_cases >= 0)
# Rename the variables in the first data frame
covid_cases_london_ren <- covid_cases_london %>%
rename(date = date , geo = areaName, cases = newCasesBySpecimenDate) %>%
select(date, geo, cases) %>%
mutate(area = "London")
# Correct date format
covid_cases_london_ren$date <- as.Date(covid_cases_london_ren$date)
covid_cases_lazio_daily$date <- as.Date(covid_cases_lazio_daily$date)
# Append the renamed data frame to the second data frame
covid_combined <- rbind(covid_cases_london_ren, covid_cases_lazio_daily)
# Add a variable of log of cases
covid_combined <- covid_combined %>%
mutate(log_cases = log(cases))
9.3 Data Exploration
Similarly to the previous chapter. Let’s start by eyeballing the data.
# Visualizing Cases in London
covid_cases_1 <- ggplot(data = covid_cases_london_ren, aes(x = date, y = cases, color=geo)) +
geom_line() +
scale_color_viridis(discrete = TRUE, option="magma") +
theme_minimal() +
legend.position = "bottom")
x = "",
y = "",
title = "Evolution in Covid-19 cases",
color = "Region"
) +
scale_x_date(date_breaks = "6 months")
# Visualizing Cases in Lazio (Rome)
covid_cases_2 <- ggplot(data = covid_cases_lazio_daily, aes(x = date, y = cases, color=geo)) +
geom_line() +
scale_color_viridis(discrete = TRUE, option="magma") +
theme_minimal() +
legend.position = "bottom")
x = "",
y = "",
title = "Evolution in Covid-19 cases",
color = "Region"
) +
scale_x_date(date_breaks = "6 months")
To identify whether there is a timeperiod where a lockdown was implemented in one location but not the other, and how cases evolved, we can plot aggregates of both locations in one plot.
# Aggregate the data by region for each day
covid_combined_agg <- aggregate(cases ~ area + date, data = covid_combined, FUN = sum)
# Visualizing aggregated
covid_cases_3 <- ggplot(data = covid_combined_agg, aes(x = date, y = cases, color=area)) +
geom_line() +
scale_color_manual(values=c("darkblue", "darkred")) + # set individual colors for the areas
theme_minimal() +
legend.position = "bottom"
) +
x = "",
y = "",
title = "Evolution in Covid-19 cases",
color = "Region"
) +
scale_x_date(date_breaks = "6 months")
From an initial look at the data, the 2020/2021 winter period seems interesting as there is a high increase in London cases but not as much as a peak in Lazio cases. In fact, after a quick review of
COVID-19 lockdowns, we found that:
• On the 5th of November 2020, the UK Prime Minister announced a second national lockdown, coming into force in England
• On 4 November 2020, Italian Prime Minister Conte announced a new lockdown as well, however this lockdown divided the country into three zones depending on the severity of the pandemic,
corresponding to red, orange and yellow zones. The Lazio region, was a yellow zone for the duration of this second lockdown. In yellow zones, the only restrictions included compulsory closing for
restaurant and bar activities at 6 PM, and online education for high schools only.
# Usual chart
covid_cases_4 <- ggplot(data = covid_combined_agg, aes(x = date, y = cases, color=area)) +
geom_line() +
scale_color_manual(values=c("darkblue", "darkred")) + # set individual colors for the areas
theme_minimal() +
legend.position = "bottom"
) +
x = "",
y = "",
title = "Evolution in Covid-19 cases",
color = "Region"
) +
scale_x_date(limit=c(as.Date("2020-08-01"), as.Date("2021-01-15"))) +
geom_vline(xintercept=as.numeric(as.Date("2020-11-05")), linetype="dashed") +
annotate("text", x=as.Date("2020-11-06"), y=25000, label="Lockdown",
color="black", fontface="bold", angle=0, hjust=0, vjust=0)
We could make some assumptions and set this up as a quasi experiment. In social science, researchers are often using natural or quasi experimental setting as randomized experiments can rarely be
conducted. This involves splitting the population at hand into a treatment and control group.
9.4 Difference in Difference
Plotting Means
For a diff-in-diff analysis using COVID data, possible shocks that would make this type of quasi-experiment possible could be the following:
• National lockdown: The first national lockdown in the UK was announced on March 23, 2020. This sudden shock to the economy and society could be used as a treatment group for the diff-in-diff
analysis, with the pre-lockdown period as the control group.
• Regional lockdowns: The UK also implemented regional lockdowns throughout the pandemic, with different regions experiencing restrictions at different times. These regional lockdowns could be used
as treatment groups, with regions that did not experience lockdowns as the control group.
• School closures: In response to the pandemic, schools in the UK were closed from March 20, 2020, until June 1, 2020, and then again from January 5, 2021, until March 8, 2021. The impact of school
closures on education outcomes could be studied using a diff-in-diff approach, with the period before school closures as the control group.
• Travel restrictions: The UK implemented various travel restrictions throughout the pandemic, including quarantine requirements for travelers from certain countries. The impact of these travel
restrictions on the tourism industry or the spread of the virus could be studied using a diff-in-diff approach.
• Vaccine rollout: The UK began its COVID-19 vaccination program in December 2020. The impact of the vaccine rollout on various health and economic outcomes could be studied using a diff-in-diff
approach, with the period before the rollout as the control group.
These are just a few examples of shocks that could be used for a diff-in-diff analysis using COVID data. The choice of shock will depend on the research question and the data available.
The DiD approach includes a before-after comparison for a treatment and control group. In our example:
• A cross-sectional comparison (= compare a sample that was treated (London) to an non-treated control group (Rome))
• A before-after comparison (= compare treatment group with itself, before and after the treatment (5th of November))
The main assumption is that without the change in the natural environment the outcome variable would have remained constant!
First, we create a dummy variable to indicate the time when the treatment started. In our case this will be the 5th of November 2020. We will also limit the time-span of our data.
# keep data from 2020-09-01 to 2021-01-01
covid_combined_filtered <- covid_combined %>%
filter(date >= "2020-09-01" & date <= "2021-01-01")
# create a dummy variable to indicate the time when the treatment started (5 Nov 2020)
covid_combined_filtered <- covid_combined_filtered %>%
mutate(after_5nov = ifelse(date >= "2020-11-05", 1, 0)) #changed to 05 Nov
# Create a frequency table of area and treatment
freq_table <- table(covid_combined_filtered$area, covid_combined_filtered$after_5nov)
# Print the frequency table
London 3150 540
Rome (Lazio) 1436 240
We then want to plot averages to see differences between treatment/control groups and before/after. But we can also calculate the mean and 95% confidence interval. We can also use group_by() and
summarize() to figure out group means before sending the data to ggplot.
plot_data <- covid_combined_filtered %>%
# Make these categories instead of 0/1 numbers so they look nicer in the plot
mutate(after_5nov = factor(after_5nov, labels = c("Before 5 November 2020", "After 5 November 2020"))) %>%
group_by(area, after_5nov) %>%
summarize(mean_cases = mean(cases),
se_cases = sd(cases) / sqrt(n()),
upper = mean_cases + (1.96 * se_cases),
lower = mean_cases + (-1.96 * se_cases))
ggplot(plot_data, aes(x = area, y = mean_cases)) +
geom_pointrange(aes(ymin = lower, ymax = upper),
color = "darkred", size = 1) +
Here, we can start to see a diff-in-diff plot, where there is little to no difference in means with our control city (Rome-Lazio) and a substancial jump in means in our treatment city (London). It
looks there were many more cases of COVID-19 after the 5th of march in London, indicating the lockdown did not have an effect, at least in this time-frame. Why could that be?
We can also plot a more standard diff-in-diff format:
ggplot(plot_data, aes(x = after_5nov, y = mean_cases, color = area)) +
geom_pointrange(aes(ymin = lower, ymax = upper), size = 1) +
geom_line(aes(group = area)) +
scale_color_manual(values = c("darkblue", "darkred"))
This second plot shows us it is probable that our diff-n-diff set up will not work. A clean classic diff-n-diff would look more like the following. Please note the following plot is theoretical.
# import csv
covid_perfect_example <- read_csv("data/longitudinal-2/example_covid.csv")
# label pre/post labels
covid_perfect_example <- covid_perfect_example %>%
mutate(after_5nov = factor(after_5nov, labels = c("Before 5 November 2020", "After 5 November 2020")))
# plot
ggplot(covid_perfect_example, aes(x = after_5nov, y = mean_cases, color = area)) +
geom_pointrange(aes(ymin = lower, ymax = upper), size = 1) +
geom_line(aes(group = area)) +
scale_color_manual(values = c("darkblue", "darkred"))
Difference in Difference by hand
We can find the exact difference by filling out the 2x2 before/after treatment/control table:
Before After Difference
Treatment A B B - A
Control C D D - C
Difference C - A D - B (D − C) − (B − A)
A combination of group_by() and summarize() makes this really easy. We can pull each of these numbers out of the table with some filter()s and pull():
before_treatment <- covid_perfect_example %>%
filter(after_5nov == "Before 5 November 2020", area == "London") %>%
before_control <- covid_perfect_example %>%
filter(after_5nov == "Before 5 November 2020", area == "Lazio") %>%
after_treatment <- covid_perfect_example %>%
filter(after_5nov == "After 5 November 2020", area == "London") %>%
after_control <- covid_perfect_example %>%
filter(after_5nov == "After 5 November 2020", area == "Lazio") %>%
diff_treatment_before_after <- after_treatment - before_treatment
The diff-in-diff estimate is 131.99, which means that the lockdown here caused an increase in cases in the time-window we are analysing. Not it’s intended effect!
We can visualise this really well with a bit of extra code:
ggplot(covid_perfect_example, aes(x = after_5nov, y = mean_cases, color = area)) +
geom_point() +
#geom_pointrange(aes(ymin = lower, ymax = upper), size = 1) +
#geom_line(aes(group = area)) +
geom_line(aes(group = as.factor(area))) +
scale_color_manual(values = c("darkblue", "darkred")) +
# If you use these lines you'll get some extra annotation lines and
# labels. The annotate() function lets you put stuff on a ggplot that's not
# part of a dataset. Normally with geom_line, geom_point, etc., you have to
# plot data that is in columns. With annotate() you can specify your own x and
# y values.
annotate(geom = "segment", x = "Before 5 November 2020", xend = "After 5 November 2020",
y = before_treatment, yend = after_treatment - diff_diff,
linetype = "dashed", color = "grey50") +
annotate(geom = "segment", x = "After 5 November 2020", xend = "After 5 November 2020",
y = after_treatment, yend = after_treatment - diff_diff,
linetype = "dotted", color = "blue") +
annotate(geom = "label", x = "After 5 November 2020", y = after_treatment - (diff_diff / 2),
label = "Program effect", size = 3)
It is important for all diff-in-diff analyses to give careful attention to possible violations of the common trends assumption, especially considering the COVID-19 situation where many of these
violations are likely to occur. Furthermore, due to the unique dynamics of COVID-19 such as lags between exposure and recorded infections, nonlinearities from person-to-person transmission, and the
possibility of policies having differential effects over time, it further complicates the potential risks to the diff-in-diff research design.
(Goodman-Bacon and Marcus 2020) comment on the following problems which can be consulted in their paper:
• Packaged Policies
• Reverse Causality
• Voluntary Precautions
• Difference Data collection
• Anticipation Spillovers
• Variation in Policy Timing
(Goodman-Bacon and Marcus 2020) also give great recommendations on how to address these problems, but this is far beyond the objective of this chapter.
Difference-in-Difference with regression
Calculating all the pieces by hand like that is tedious, so we can use regression to do it instead! Remember that we need to include indicator variables for treatment/control and for before/after, as
well as the interaction of the two.
This is the equation:
\(\Delta Y_{gt} = \beta_0 + \beta_1 London_{g} + \beta_2 Post5Nov_{t} + \beta_3 London_{g} \times Post5Nov_{t} + \beta_4 Rome_{g} + \epsilon_{gt}\)
The output will show the diff-in-diff coefficient estimate, standard error, t-value, and p-value, which can be used to determine whether there was a significant effect of the second lockdown 4 on
Covid cases in November 2020.
model_small <- lm(cases ~ area + after_5nov + area * after_5nov,
data = covid_combined_filtered)
# Tidy the model output
diffndiff1 <- tidy(model_small)
# Add significance stars using stars.pval from gtools
diffndiff1$stars <- stars.pval(diffndiff1$p.value)
# View the results
# A tibble: 4 × 6
term estimate std.error statistic p.value stars
<chr> <dbl> <dbl> <dbl> <dbl> <chr>
1 (Intercept) 55.7 4.47 12.5 3.62e- 35 ***
2 areaRome (Lazio) 126. 7.99 15.8 8.81e- 55 ***
3 after_5nov 301. 11.7 25.7 7.63e-138 ***
4 areaRome (Lazio):after_5nov -269. 21.0 -12.8 5.37e- 37 ***
# Create a model summary table for the model
summary_table <- modelsummary(list("Simple" = model_small), estimate =c("{estimate}{stars}"))
# View the results
(Intercept) 55.694***
areaRome (Lazio) 125.910***
after_5nov 300.656***
areaRome (Lazio) × after_5nov −269.302***
Num.Obs. 5366
R2 0.130
R2 Adj. 0.130
AIC 74524.8
BIC 74557.7
Log.Lik. −37257.375
F 267.482
RMSE 250.71
9.5 Questions
For the assignment, you will continue to use Google Mobility data for the UK for 2021. For details on the timeline you can have a look here. You will need to do a bit of digging on when lockdowns or
other COVID-19 related shock happened in 2021 to set up a diff-in-diff strategy. Have a look at Brodeur et al. (2021) to get some inspiration. They used Google Trends data to test whether COVID-19
and the associated lockdowns implemented in Europe and America led to changes in well-being.
Start by loading both the csv
1. Visualize the data with ggplot and identify what section of the data could be used to evaluate a COVID-19 intervention. Examples of these interventions could be a regional lockdown, school
closures, travel restrictions or vaccine rollouts. Generate a clean ggplot which indicates which intervention you are going to examine.
2. Explore differences in means through a frequency table and a graph of these averages. Chose whichever suits your purposes best.
3. Define and estimage a diff-in-diff regression. What do the results suggest? Was the intervention you chose effective? Discuss the reasons why it was or was not.
4. Discuss how the unique dynamics of COVID-19 and the possibility of policies having differential effects over time complicate the interpretation of your results.
Analyse and discuss what insights you obtain into people’s changes in behaviors during the pandemic in responde to an intervention.
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A Phenomenological Extension of the Newtonian Gravity
^1We use the tensor calculus including the sum convention and restrict ourselves to Cartesian coordinates for simplicity; $|i$ means the partial derivative $\frac{\partial }{\partial {x}_{i}}$ .
1. Introduction
In a previous paper [1] we have proposed in view of the dark matter problem a modification of the Newtonian gravity theory in such a way that the assumption of the existence of dark matter is not
necessary. In detail we have given a theoretical explanation of MOND [2] following the structure of the electrodynamics assuming also in the case of Newtonian gravity a difference between intensive
and extensive field quantities. The intensive field quantity described by ${F}_{i}$ obeys the homogeneous field equation^1
${F}_{i|j}-{F}_{j|i}=0⇒{F}_{i}=-{\varphi }_{|i},$(1)
where $\varphi$ is the gravitational potential; simultaneously ${F}_{i}$ determines the gravitational force on a massive point particle (mass m) at the position ${x}_{i}\left(t\right)$
according to ( $\cdot =\frac{\text{d}}{\text{d}t}$ )
using the weak equivalence principle. Thus ${F}_{i}$ is the field strength. In consequence of (1) and (2) energy conservation is guaranteed for ${\varphi }_{|t}\equiv 0$ . On the other hand there
exists the extensive field quantity ${G}_{i}$ determined by the mass density $\rho$ of the matter distribution according to the inhomogeneous field equation
(G Newtonian gravitational constant). Accordingly ${G}_{i}$ is the field excitation, because it is determined by the excitation Equation (3). Between both quantities the “material” equation is
${F}_{i}=\gamma {G}_{i},$(4)
with the “material” quantity $\gamma$ , which may depend on the field strength itself in consequence e.g. of induced vacuum polarisations. In our previous paper we have assumed
$\gamma =\frac{|{F}_{0}|}{|F|}+1,$(5)
where $|{F}_{0}|$ is a critical field strength, under which the value of $\gamma$ increases drastically, so that the field strength $|F|$ increases compared with the Newtonian case ( $\gamma =1$ ),
and the assumption of dark matter will be superfluous.
In the present paper we show that the ansatz (5) can be enlarged in a very simple way in such a direction that the field strength ${F}_{i}$ can also not be larger than a critical field strength $|{F}
_{1}|$ also in consequence of vacuum polarisations so that singularities (e. g. black holes) may be avoided. Doing this we enlarge relation (5) to
$\gamma =\frac{|{F}_{0}|}{|F|}+\sqrt{1-{\left(\frac{|F|}{|{F}_{1}|}\right)}^{2}}\text{ }\text{ }\text{with}\text{ }\text{\hspace{0.17em}}|{F}_{1}|\gg |{F}_{0}|.$(6)
This procedure corresponds exactly to the idea of Born and Infeld in the electrodynamics [3] avoiding there electromagnetic field strengths larger than a critical value. Similar impressive influences
of vacuum polarisations and fluctuations on the classical physics are e.g. the Casimir effect [4] and the Scharnhorst-Barton effect [5] .
^1Usually Lenz’s rule is connected with time-variable electromagnetic fields. But it has a deeper meaning for all physical systems. If induced quantities, induced by time-variations or variable field
strengths in space, would reinforce the original cause, the system would be unstable. Thus Lenz’s rule is rather a general stability law.
According to our proposal (6) we assume that the gravitational field strength will be weakened by increasing field-strength and enforced by decreasing field-strength in consequence e.g. of vacuum
polarisations induced by the gravitational field-strength itself following Lenz’s rule. Herewith Lenz’s fundamental rule is implemented in the whole range of field strengths.^2 The Newtonian gravity
is valid in the very large intermediate range $|{F}_{0}|\ll |F|\ll |{F}_{1}|$ where $\gamma \simeq 1$ is valid and where also the Newtonian gravitational constant G is determined.^3 The value of $|
{F}_{0}|$ amounts to 2 × 10^−8 cm/sec^2 in view of the flat rotation curves of the spiral galaxies (see discussions in [1] and footnote 5), whereas the value of $|{F}_{1}|$ may be of the order of 10^
30 cm/sec^2, where the vacuum will be unstable e.g. with respect to spontaneous particle-antiparticle generations.^4 Of course an exact theoretical derivation of the ansatz (6) does not exist until
now because a quantum theory of gravity is missing. In a quantum theory of gravity vacuum polarisation effects would be included, so that a relation like (6), comparable with the running coupling
constant in the non-Abelian QCD, would be expectable in the classical limit. In the electromagnetic case Heisenberg and Euler could show that the Born-Infeld ansatz follows from the quantum
electrodynamics [6] . In the same sense we consider also our proposal as a phenomenological extension of gravity taking into account expectable quantum effects on the classical level.
2. The Integration Procedure
For solving the equation of motion (2) for a point-like test particle the knowledge of the field strength $F$ is necessary. For this we find from the field Equation (1) and Equation (3) with the use
of (4) for the potential $\varphi$ the differential equation:
${\varphi }_{|i|i}-\frac{{\gamma }_{|i}}{\gamma }{\varphi }_{|i}=4\text{π}G\rho \gamma .$(7)
Herein $\gamma$ is given by (see (6))
$\gamma =\frac{|{F}_{0}|}{\sqrt{{\varphi }_{|j}{\varphi }_{|j}}}+\sqrt{1-\frac{{\varphi }_{|j}{\varphi }_{|j}}{{|{F}_{1}|}^{2}}}.$(8)
^3General Relativity would be also valid only in this intermediate range.
^4By this acceleration an electron reaches approximately a distance of a Compton wave length during a Compton time necessary for spontaneous electron-positron pair creation.
Now we restrict ourselves for simplicity to the centrally symmetric case of a mass-sphere of mass M with constant density $\rho$ and radius R and consider at first the area outside the sphere, i.e.
$r>R$ . Than the field Equation (7) reads
( $"=\frac{\partial }{\partial r}$ ):
${\varphi }^{″}+\frac{2}{r}{\varphi }^{\prime }=\frac{{\gamma }^{\prime }}{\gamma }{\varphi }^{\prime }$(9)
with the first integral:
${\varphi }^{\prime }{r}^{2}=A\gamma ,\text{ }A=\text{const}.$(10)
(A integration constant). With
$\gamma =\frac{|{F}_{0}|}{{\varphi }^{\prime }}+\sqrt{1-\frac{{{\varphi }^{\prime }}^{2}}{{|{F}_{1}|}^{2}}}$(11)
according to (8) we obtain from (10) the following quadratic equation for ${\varphi }^{"2}$ :
${\left({{\varphi }^{\prime }}^{2}{r}^{2}-|{F}_{0}|A\right)}^{2}={A}^{2}{{\varphi }^{\prime }}^{2}\left(1-\frac{{{\varphi }^{\prime }}^{2}}{{|{F}_{1}|}^{2}}\right)$(12)
with the solution:
${{\varphi }^{\prime }}^{2}=\frac{1}{2}\frac{{A}^{2}+2|{F}_{0}|A{r}^{2}}{{r}^{4}+{A}^{2}/{|{F}_{1}|}^{2}}+\sqrt{\frac{1}{4}{\left(\frac{{A}^{2}+2|{F}_{0}|A{r}^{2}}{{r}^{4}+{A}^{2}/{|{F}_{1}|}^{2}}\
For $r\to \infty$ we obtain
${\varphi }^{\prime }=\sqrt{A|{F}_{0}|}\frac{1}{r},$(14)
so that the flat rotation curves of the spiral galaxies are guaranteed with the constant orbital velocity $v\left(r\to \infty \right)={\left(A|{F}_{0}|\right)}^{1/4}$ .^5 This is in accordance with
our previous paper and confirms also the Tully-Fisher law [7] . The exact value of the constant A can be only determined by the connection of the solution (13) with the inner one for $r<R$ , see
Equation (28).
Thus we consider now the case $r<R$ . With the abbreviation
we obtain from (7) the differential equation
${\varphi }_{|i|i}-\frac{{\gamma }_{|i}}{\gamma }{\varphi }_{|i}-C\gamma =0,$(16)
which goes over in the centrally symmetric case into:
${\left(\frac{{\varphi }^{\prime }}{\gamma }\right)}^{\prime }+\frac{2}{r}\frac{{\varphi }^{\prime }}{\gamma }=C.$(17)
We solve this inhomogeneous differential equation by the method of the variation of the constants. The solution of the homogeneous equation belonging to (17) is already known and given by (10) as
${\varphi }^{\prime }=\frac{\stackrel{˜}{A}\gamma }{{r}^{2}},$(18)
^5Herewith the value $|{F}_{0}|$ can be estimated, c.f. (28) and [1] .
where $\stackrel{˜}{A}$ is now the integration constant, which is to vary for solving (17). This gives the differential equation for $\stackrel{˜}{A}$
${\stackrel{˜}{A}}^{\prime }=C{r}^{2}$(19)
with the solution
(B new integration constant), so that Equation (18) goes over into:
${\varphi }^{\prime }=\left(\frac{1}{3}C{r}^{3}+B\right)\frac{\gamma }{{r}^{2}},$(21)
where $\gamma$ is finally given by (11). Herewith we obtain once more a quadratic equation for ${{\varphi }^{\prime }}^{2}$ with the solution:
$\begin{array}{c}{{\varphi }^{\prime }}^{2}=\frac{1}{2}\frac{\left(\frac{1}{3}C{r}^{3}+B\right)\left(2|{F}_{0}|{r}^{2}+\frac{1}{3}C{r}^{3}+B\right)}{{r}^{4}+{\left(\frac{1}{3}C{r}^{3}+B\right)}^{2}/
{|{F}_{1}|}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left[\frac{1}{4}\frac{{\left(\frac{1}{3}C{r}^{3}+B\right)}^{2}{\left(2|{F}_{0}|{r}^{2}+\frac{1}{3}C{r}^{3}+B\right)}^{2}}{{\left({r}^
Setting $C=0$ we obtain the solution (13), where B plays the role of A. Accordingly B has the meaning of
where ${M}_{0}$ is the mass value of an additional central point mass (see Equation (28)). For $r\to 0$ one gets for the case $Be 0$
${{\varphi }^{\prime }}^{2}\left(r\to 0\right)=\frac{1}{2}{|{F}_{1}|}^{2}\left[1+\sqrt{1-4{|{F}_{0}|}^{2}/{|{F}_{1}|}^{2}}\right].$(24)
Obviously this is the solution for an additional point mass at $r=0$ , where the field-strength possesses no singularity but reaches its maximum possible value $|{F}_{1}|$ for $|{F}_{0}|/|{F}_{1}|\ll
1$ independently of the mass value ${M}_{0}$ . Therefore also the existence of black holes should be avoidable.
Avoiding in the following this central point mass we have to set $B=0$ . Herewith it follows from (22):
$\begin{array}{c}{{\varphi }^{\prime }}^{2}=\frac{1}{2}\frac{\frac{1}{3}C{r}^{3}\left(2|{F}_{0}|{r}^{2}+\frac{1}{3}C{r}^{3}\right)}{{r}^{4}+{\left(\frac{1}{3}C{r}^{3}\right)}^{2}/{|{F}_{1}|}^{2}}\\ \
Now we obtain for $r\to 0$ after inserting of C according to (15) ( $M=\frac{\text{4π}}{3}\rho {R}^{3}$ )
${\varphi }^{\prime }\left(r\to 0\right)={\left(\frac{MG}{{R}^{3}}|{F}_{0}|\right)}^{1/2}\sqrt{r},$(26)
which means that the orbital velocity v has the value
$v\left(r\to 0\right)={\left(MG|{F}_{0}|\right)}^{1/4}{\left(\frac{r}{R}\right)}^{3/4}.$(27)
This is greater than the Newtonian one by the factor ${\left(\frac{|{F}_{0}|{R}^{3}}{MGr}\right)}^{1/4}$ . Therefore
the high star velocities near the center of the galaxy should be discussed newly avoiding the assumption of a black hole in the galaxy center.
Finally we have to determine the meaning of the integration constant A in the solution (13). It follows from the condition, that the solution (25) for $r<R$ goes steadily over into the solution (13)
for $r>R$ at $r=R$ . On this way one finds immediately
Now we can give the complete solution for the massive sphere of radius R and mass M. It is valid for $r\le R$ (see (25)):
$\begin{array}{c}{{\varphi }^{\prime }}^{2}=\frac{1}{2}\frac{MG\frac{r}{{R}^{3}}\left(MG\frac{r}{{R}^{3}}+2|{F}_{0}|\right)}{1+{\left(MG\frac{r}{{R}^{3}}/|{F}_{1}|\right)}^{2}}\\ \text{\hspace
and for $r\ge R$ (see (13)):
$\begin{array}{c}{{\varphi }^{\prime }}^{2}=\frac{1}{2}\frac{\frac{MG}{{r}^{2}}\left(\frac{MG}{{r}^{2}}+2|{F}_{0}|\right)}{1+{\left(\frac{MG}{{r}^{2}}/|{F}_{1}|\right)}^{2}}\\ \text{\hspace{0.17em}}\
The square root of (29) and (30) is given by (32) and (33). In the case of $R\to 0$ the solution (29) results in the finite value (compare (24))
${{\varphi }^{\prime }}^{2}={|{F}_{1}|}^{2}\text{ }\text{ }\text{for}\text{ }\text{\hspace{0.17em}}|{F}_{0}|/|{F}_{1}|\ll 1,$(31)
so that black holes may be avoided, although the field strength is very high. But for confirming this consequence finally a general relativistic investigation of the situation may be necessary.
3. Alternative Solution of the Centrally Symmetric Case
Considering the Equation (4) and Equation (6) one can solve these with respect to ${F}_{i}$ in the centrally symmetric case, i.e. with respect to ${\varphi }^{\prime }$ . After that one can insert
for $G$ the Newtonian expression $|{F}_{N}|$ according to (3). In the centrally symmetric case the condition (1) is fulfilled as one can test easily. In this way one finds:
$\begin{array}{l}{\varphi }^{\prime }=\sqrt{\frac{1}{2}}\left[{\left\{\frac{1}{2}\frac{|{F}_{N}|\left(|{F}_{N}|+2|{F}_{0}|\right)}{1+{\left(|{F}_{N}|/|{F}_{1}|\right)}^{2}}+\frac{|{F}_{N}||{F}_{0}|}
{\sqrt{1+{\left(|{F}_{N}|/|{F}_{1}|\right)}^{2}}}\right\}}^{1/2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}
For the mass-sphere with constant mass density $\rho$ and radius R one has to insert
$\text{ }\text{for}\text{ }\text{\hspace{0.17em}}r<R:\text{ }|{F}_{N}|=\frac{MG}{{R}^{3}}r\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\
text{for}\text{ }\text{\hspace{0.17em}}r>R:\text{ }|{F}_{N}|=\frac{MG}{{r}^{2}}.$(33)
Squaring of (32) results into (29) and (30).
We mention however explicitly that this procedure is in general only applicable in the centrally symmetric case because of the condition (1) and in the case, where the mass density distribution $\
rho$ is not determined by the gravitational field strength ${F}_{i}$ itself. For the latter case, where $\rho$ is determined e.g. by the Euler equation
$\rho \left({v}_{i|t}+{v}_{i|k}{v}_{k}\right)=\rho {F}_{i}-{p}_{|i}$(34)
( $p\left(\rho \right)$ pressure of the substratum) as e.g. in stars and galaxies or polytropic gas spheres the procedure of chapt. 2 is to be used.
4. Conclusion
Considering induced vacuum polarisations we could show in the framework of Newtonian gravity, that the assumption of dark matter and the existence of gravitational field singularities can be avoided.
However our proposal discussed in this paper should be translated finally into a general relativistic form. This is not possible in an immediate way, because in the general theory of relativity the
distinction between extensive and intensive field quantities is impossible. Therefore the modification of the theory has to start from the very beginning by a modification of the Lagrangian in the
form of a $f\left(R\right)$ -theory [8] as it is done also in the Born-Infeld electrodynamics. This will be our further aim.
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Barbara Barabasz
Jan 25, 2022
Abstract:Convolutional neural networks (CNNs) have dramatically improved the accuracy of tasks such as object recognition, image segmentation and interactive speech systems. CNNs require large
amounts of computing resources because ofcomputationally intensive convolution layers. Fast convolution algorithms such as Winograd convolution can greatly reduce the computational cost of these
layers at a cost of poor numeric properties, such that greater savings in computation exponentially increase floating point errors. A defining feature of each Winograd convolution algorithm is a set
of real-value points where polynomials are sampled. The choice of points impacts the numeric accuracy of the algorithm, but the optimal set of points for small convolutions remains unknown. Existing
work considers only small integers and simple fractions as candidate points. In this work, we propose a novel approach to point selection using points of the form {-1/c , -c, c, 1/c } using the full
range of real-valued numbers for c. We show that groups of this form cause cancellations in the Winograd transform matrices that reduce numeric error. We find empirically that the error for different
values of c forms a rough curve across the range of real-value numbers helping to localize the values of c that reduce error and that lower errors can be achieved with non-obvious real-valued
evaluation points instead of integers or simple fractions. We study a range of sizes for small convolutions and achieve reduction in error ranging from 2% to around 59% for both 1D and 2D
convolution. Furthermore, we identify patterns in cases when we select a subset of our proposed points which will always lead to a lower error. Finally we implement a complete Winograd convolution
layer and use it to run deep convolution neural networks on real datasets and show that our proposed points reduce error, ranging from 22% to 63%.
* 19 pages, 3 figures, 9 tables and 32 equations
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DEtools[rifsimp] Algorithm Output
• The output of the rifsimp command is a table. A number of possible table formats (or entries) may or may not be present as they depend on specific options. The format changes from a table
containing the simplified system to a nested table when casesplit is activated.
Solved This entry gives all equations that have been solved in terms of their leading indeterminate.
Pivots This entry gives all inequations (equations of the form expression <> 0) that hold for this case. These may be part of the input system, or decisions made by the program during the
calculation (see rifsimp[cases] for information on pivots introduced by the algorithm).
Case This entry describes what assumptions were made to arrive at this simplified system (see rifsimp[cases].)
Constraint This table contains all equations that are nonlinear in their leading indeterminate. The equations in this list form a Groebner basis and can be viewed as purely algebraic because any
differential consequences that result from these equations are already taken care of through spawning (see rifsimp[nonlinear]). If the initial option has been specified, then these
equations are isolated for the highest power of their leading derivatives, otherwise they are in the form .
DiffConstraint This table entry contains all equations that are nonlinear in their leading indeterminate, but either are not in Groebner basis form or have differential consequences that are not
accounted for (see spoly and spawn in rifsimp[nonlinear]). Whenever equations appear in this entry, the system is in incomplete form and must be examined with care.
UnSolve This table entry contains all equations that rifsimp did not attempt to solve (see unsolved in rifsimp[adv_options]). Whenever equations appear in this entry, the system is in
incomplete form and must be examined with care.
UnClass This table entry contains all equations that have not yet been examined (i.e. Unclassified). This entry is only present when looking at partial calculations using rifread, or when a
computation is halted by mindim.
status If this entry is present, then the output system is missing due to either a restriction or an error. The message in this entry indicates what the restriction or error is.
dimension This entry is only present when the mindim option is used (see rifsimp[cases]), or for maxdimsystems. For the case where a single constraint is in effect (such as through use of the
option ), the right-hand side is a single number (the dimension for the case). For multiple constraints, it is a list of dimension counts, one for each constraint in the mindim
• Here are examples of status messages:
"system is inconsistent" No solution exists for this system.
"object too large" Expression swell has exceeded Maple's ability to calculate.
"time expired" Input time limit has been exceeded (see ctl, stl and itl in rifsimp[options]).
"free count fell below mindim" Free parameters have fallen below the minimum (see mindim in rifsimp[adv_options]).
Of the above, only the "object too large" message actually indicates an error.
• To summarize, if the input system is fully linear in all indeterminates (including unknown constants), then only the Solved entry will be present. If the system is (and remains) linear in its
leading indeterminates throughout the calculation, but has indeterminate expressions in the leading coefficients, then Solved, Pivots, and Case will be present. If equations that are nonlinear in
their leading indeterminates result during a calculation, then Constraint will also be present. If the status entry is present, then not all information for the case will be given.
If mindim is used, then the dimension entry will be set to the dimension of the linear part of the system for the case when status is not set, and an upper bound for the dimension of the case if
the count fell below the minimum dimension requirement.
For multiple cases (using the casesplit option), numbered entries appear in the output table, each of which is itself a table of the form described above.
For example, if a calculation resulted in 10 cases, the output table would have entries '1=...',..., '10=...', where each of these entries is itself a table that contains Solved, Pivots, Case, or
other entries from the single case description.
In addition to the numbered tables is the entry casecount, which gives a count of the number of cases explored. Cases that rejected for reasons other than inconsistency will have the Case entry
assigned, in addition to the status entry. Inconsistent cases, for multiple case solutions, are removed automatically.
• So what is the difference between Pivots and Case?
The Pivots entry contains the inequations for the given case in simplified form with respect to the output system. The Case entry is a list with elements of the form [assumption,leading derivative]
or [assumption,leading derivative,"false split"]. It describes the actual decision made for the case split in unsimplified form (i.e. as it was encountered in the algorithm). The assumption will be
of the form or , where depends on the dependent variables, derivatives and/or constants of the problem. The leading derivative is the indeterminate the algorithm was isolating that required the
assumption. If the third "false split" entry is present, then it was later discovered that one branch of the split is entirely inconsistent, so the actual splitting was a false splitting, as the
displayed assumption is always true with respect to the rest of the system. For example, if the algorithm were to split on an equation of the form , the Case entries that correspond to this case
split are , and . If it was found later in the algorithm that leads to a contradiction, then the Case entry would be given by [a<>0,diff(f(y),y),"false split"].
Note that when faclimit or factoring are used (of which factoring is turned on by default), it is possible to introduce a splitting that does not isolate a specific derivative. When this occurs,
the case entry will be of the form or .
Occasionally both the Case and Pivots entries contain the same information, but it should be understood that they represent different things.
As discussed above, some options have the effect of preventing rifsimp from fully simplifying the system. Whenever DiffConstraint or UnSolve entries are present in the output, some parts of the
algorithm have been disabled by options, and the resulting cases must be manually examined for consistency and completeness.
As a first example, we take the overdetermined system of two equations in one dependent variable f(x), and two constants a and b.
Call rifsimp for a single case only (the default).
We see that under the given assumptions for the form of a and b (from Pivots), the only solution is given as f(x)=0 (from Solved). Now, run the system in multiple case mode using casesplit.
We see that we have four cases:
All cases except 2 have f(x)=0.
Looking at case 2 in detail, we see that under the constraint a = b^2 (from Solved) and b <> 0 from Pivots, the solution to the system will be given by the remaining ODE in f(x) (in Solved). Note
here that the constraint on the constants a and b, together with the assumption b <> 0, imply that a <> 0, so this constraint is not present in the Pivots entry due to simplification. It is still
present in the Case entry because Case describes the decisions made in the algorithm, not their simplified result. Also, case 4 has no Pivots entry. This is because no assumptions of the form were
used for this case.
One could look at the caseplot with the command:
As a final demonstration involving this system, suppose that we are only interested in nontrivial cases where f(x) is not identically zero. We can simply include this assumption in the input system,
and rifsimp will take it into account.
We see that the answer is returned in a single case with two false split Case entries. This means the computation discovered that the and cases lead to contradictions, so the entries in the Case
list are labelled as false splits, and the alternatives for the binary case splittings (cases with or ) are not present.
For the next example, we have a simple inconsistent system:
So there is no solution u(x) to the above system of equations.
The next example demonstrates the UnSolve list, while also warning about leaving indeterminates in unsolved form.
So we run rifsimp, but only solve for f(x), leaving g(x) in unsolved form. Unfortunately, the resulting system is inconsistent, but this is not recognized because equations containing only g(x) are
left unsolved. As discussed earlier in the page, these equations come out in the UnSolve list.
When equations are present in the UnSolve list, they must be manually examined.
Here is a nonlinear example.
By default rifsimp spawns the nonlinear equation to obtain a leading linear equation, and performs any required simplifications. The end result gives the following output:
We have only one consistent case. Attempting to perform this calculation with the spawn=false option gives the following:
So it is clear that by disabling spawning, the system is not in fully simplified form (as indicated by the presence of the DiffConstraint entry), and we do not obtain full information about the
See Also
caseplot, rifsimp, rifsimp[adv_options], rifsimp[cases], rifsimp[nonlinear], rifsimp[options]
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Publications from 2022
Benchmarking quantum annealing dynamics: The spin-vector Langevin model
The classical spin-vector Monte Carlo (SVMC) model is a reference benchmark for the performance of a quantum annealer. Yet, as a Monte Carlo method, SVMC is unsuited for an accurate description of
the annealing dynamics in real-time. We introduce the spin-vector Langevin (SVL) model as an alternative benchmark in which the time evolution is described by Langevin dynamics. The SVL model is
shown to provide a more stringent test than the SVMC model for the identification of quantum signatures in the performance of quantum annealing devices, as we illustrate by describing the
Kibble-Zurek scaling associated with the dynamics of symmetry breaking in the transverse field Ising model, recently probed using D-Wave machines. Specifically, we show that D-Wave data are
reproduced by the SVL model.
Bound states in the continuum in a fluxonium qutrit
Heavy fluxonium at zero external flux has a long-lived state when coupled capacitively to any other system. We analyze it by projecting all the fluxonium-relevant operators into the qutrit subspace,
as this long-lived configuration corresponds to the second excited fluxonium level. This state becomes a bound state in the continuum (BIC) when the coupling occurs to an extended system supporting a
continuum of modes. In the case without noise, we find BIC lifetimes T_1 that can be much larger than seconds when the fluxonium is coupled to a superconducting waveguide, while typical device
frequencies are on the order of gigahertz. We have performed a detailed study of the different sources of decoherence in a realistic experiment, finding that upwards transitions caused by a finite
temperature in the waveguide and decay induced by 1/f flux noise are the most dangerous ones. Even in their presence, BIC decay times could reach the range of T_1∼10^{−1} ms, while preparation times
are of the order of 10^2 ns
Bridging the gap between topological non-Hermitian physics and open quantum systems
We relate observables in open quantum systems with the topology of non-Hermitian models using the Keldysh path-integral method. This allows to extract an effective Hamiltonian from the Green’s
function which contains all the relevant topological information and produces ω-dependent topological invariants, linked to the response functions at a given frequency. Then, we show how to detect a
transition between different topological phases by measuring the response to local perturbations. Our formalism is exemplified in a one-dimensional Hatano-Nelson model, highlighting the difference
between the bosonic and the fermionic case.
Bridging the gap between topological non-Hermitian physics and open quantum systems
Cold Atoms meet lattice gauge theories
The central idea of this review is to consider quantum field theory models relevant for particle physics and replace the fermionic matter in these models by a bosonic one. This is mostly motivated by
the fact that bosons are more ‘accessible’ and easier to manipulate for experimentalists, but this ‘substitution’ also leads to new physics and novel phenomena. It allows us to gain new information
about among other things confinement and the dynamics of the deconfinement transition. We will thus consider bosons in dynamical lattices corresponding to the bosonic Schwinger or Z2 Bose–Hubbard
models. Another central idea of this review concerns atomic simulators of paradigmatic models of particle physics theory such as the Creutz–Hubbard ladder, or Gross–Neveu–Wilson and Wilson–Hubbard
models. This article is not a general review of the rapidly growing field—it reviews activities related to quantum simulations for lattice field theories performed by the Quantum Optics Theory group
at ICFO and their collaborators from 19 institutions all over the world. Finally, we will briefly describe our efforts to design experimentally friendly simulators of these and other models relevant
for particle physics.
Complete physical characterization of QND measurements via tomography
We introduce a self-consistent tomography for arbitrary quantum nondemolition (QND) detectors. Based on this, we build a complete physical characterization of the detector, including the measurement
processes and a quantification of the fidelity, ideality, and backaction of the measurement. This framework is a diagnostic tool for the dynamics of QND detectors, allowing us to identify errors, and
to improve their calibration and design. We illustrate this on a realistic Jaynes-Cummings simulation of a superconducting qubit readout. We characterize nondispersive errors, quantify the backaction
introduced by the readout cavity, and calibrate the optimal measurement point.
Connecting steady-states of driven-dissipative photonic lattices with spontaneous collective emission phenomena
Critical quantum metrology with fully-connected models: from Heisenberg to Kibble–Zurek scaling
Decimation technique for open quantum systems: A case study with driven-dissipative bosonic chains
The unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (nonunitary) dynamics, which can be radically different from closed-system scenarios. Such open quantum
system dynamics is generally described by Lindblad master equations, whose dynamical and steady-state properties are challenging to obtain, especially in the many-particle regime. Here, we introduce
a method to deal with these systems based on the calculation of a (dissipative) lattice Green’s function with a real-space decimation technique. Compared to other methods, such a technique enables us
to obtain compact analytical expressions for the dynamics and steady-state properties, such as asymptotic decays or correlation lengths. We illustrate the power of this method with several examples
of driven-dissipative bosonic chains of increasing complexity, including the Hatano-Nelson model. The latter is especially illustrative because its surface and bulk dissipative behavior are linked
due to its nontrivial topology, which manifests in directional amplification.
Decimation technique for open quantum systems: A case study with driven-dissipative bosonic chains
Devil’s staircase of topological Peierls insulators and Peierls supersolids
We consider a mixture of ultracold bosonic atoms on a one-dimensional lattice described by the XXZ-Bose-Hubbard model, where the tunneling of one species depends on the spin state of a second deeply
trapped species. We show how the inclusion of antiferromagnetic interactions among the spin degrees of freedom generates a Devil’s staircase of symmetry-protected topological phases for a wide
parameter regime via a bosonic analog of the Peierls mechanism in electron-phonon systems. These topological Peierls insulators are examples of symmetry-breaking topological phases, where long-range
order due to spontaneous symmetry breaking coexists with topological properties such as fractionalized edge states. Moreover, we identify a region of supersolid phases that do not require long-range
interactions. They appear instead due to a Peierls incommensurability mechanism, where competing orders modify the underlying crystalline structure of Peierls insulators, becoming superfluid. Our
work show the possibilities that ultracold atomic systems offer to investigate strongly-correlated topological phenomena beyond those found in natural materials.
Devil’s staircase of topological Peierls insulators and Peierls supersolids
We consider a mixture of ultracold bosonic atoms on a one-dimensional lattice described by the XXZ-Bose-Hubbard model, where the tunneling of one species depends on the spin state of a second deeply
trapped species. We show how the inclusion of antiferromagnetic interactions among the spin degrees of freedom generates a Devil’s staircase of symmetry-protected topological phases for a wide
parameter regime via a bosonic analog of the Peierls mechanism in electron-phonon systems. These topological Peierls insulators are examples of symmetry-breaking topological phases, where long-range
order due to spontaneous symmetry breaking coexists with topological properties such as fractionalized edge states. Moreover, we identify a region of supersolid phases that do not require long-range
interactions. They appear instead due to a Peierls incommensurability mechanism, where competing orders modify the underlying crystalline structure of Peierls insulators, becoming superfluid. Our
work show the possibilities that ultracold atomic systems offer to investigate strongly-correlated topological phenomena beyond those found in natural materials.
Dispersive readout of molecular spin qudits
We study the physics of a magnetic molecule described by a “giant” spin with multiple (d>2) spin states interacting with the quantized cavity field produced by a superconducting resonator. By means
of the input-output formalism, we derive an expression for the output modes in the dispersive regime of operation. It includes the effect of magnetic anisotropy, which makes different spin
transitions addressable. We find that the measurement of the cavity transmission allows us to uniquely determine the spin state of the qudits. We discuss, from an effective Hamiltonian perspective,
the conditions under which the qudit readout is a nondemolition measurement and consider possible experimental protocols to perform it. Finally, we illustrate our results with simulations performed
for realistic models of existing magnetic molecules.
Dynamical photon-photon interaction mediated by a quantum emitter
Single photons role in the development of quantum science and technology. They can carry quantum information over extended distances to act as the backbone of a future quantum internet and can be
manipulated in advanced photonic circuits, enabling scalable photonic quantum computing. However, more sophisticated devices and protocols need access to multi-photon states with particular forms of
entanglement. Efficient light–matter interfaces offer a route to reliably generating these entangled resource states. Here we utilize the efficient and coherent coupling of a single quantum emitter
to a nanophotonic waveguide to realize a quantum nonlinear interaction between single-photon wavepackets. We demonstrate the control of a photon using a second photon mediated by the quantum emitter.
The dynamical response of the two-photon interaction is experimentally unravelled and reveals quantum correlations controlled by the pulse duration. Further development of this platform work, which
constitutes a new research frontier in quantum optics, will enable the tailoring of complex photonic quantum resource states.
Experimental validation of the Kibble-Zurek mechanism on a digital quantum computer
The Kibble-Zurek mechanism (KZM) captures the essential physics of nonequilibrium quantum phase transitions with symmetry breaking. KZM predicts a universal scaling power law for the defect density
which is fully determined by the system’s critical exponents at equilibrium and the quenching rate. We experimentally tested the KZM for the simplest quantum case, a single qubit under the
Landau-Zener evolution, on an open access IBM quantum computer (IBM-Q). We find that for this simple one-qubit model, experimental data validates the central KZM assumption of the adiabatic-impulse
approximation for a well isolated qubit. Furthermore, we report on extensive IBM-Q experiments on individual qubits embedded in different circuit environments and topologies, separately elucidating
the role of crosstalk between qubits and the increasing decoherence effects associated with the quantum circuit depth on the KZM predictions. Our results strongly suggest that increasing circuit
depth acts as a decoherence source, producing a rapid deviation of experimental data from theoretical unitary predictions.
Fermionic Gaussian states: an introduction to numerical approaches
This document is meant to be a practical introduction to the analytical and numerical manipulation of Fermionic Gaussian systems. Starting from the basics, we move to relevant modern results and
techniques, presenting numerical examples and studying relevant Hamiltonians, such as the transverse field Ising Hamiltonian, in detail. We finish introducing novel algorithms connecting Fermionic
Guassian states with matrix product states techniques. All the numerical examples make use of the free Julia package F_utilities.
Harnessing nonadiabatic excitations promoted by a quantum critical point: Quantum battery and spin squeezing
Improving quantum state transfer: Correcting non-Markovian and distortion effects
Quantum state transfer is a key operation for quantum information processing. The original pitch-and-catch protocols rely on flying qubits or single photons with engineered wavepacket shapes to
achieve a deterministic, fast and high-fidelity transfer. Yet, these protocols overlook two important factors, namely, the distortion of the wavepacket during the propagation and non-Markovian
effects during the emission and reabsorption processes due to time-dependent controls. Here we address both difficulties in a general quantum-optical model and propose a correction strategy to
improve quantum state transfer protocols. Including non-Markovian effects in our theoretical description, we show how to derive control pulses that imprint phases on the wavepacket that compensate
the distortion caused by propagation. Our theoretical results are supported by detailed numerical simulations showing that a suitable correction strategy can improve state transfer fidelities up to
three orders of magnitude.
Locality of spontaneous symmetry breaking and universal spacing distribution of topological defects formed across a phase transition
The crossing of a continuous phase transition results in the formation of topological defects with a density predicted by the Kibble-Zurek mechanism (KZM). We characterize the spatial distribution of
pointlike topological defects in the resulting nonequilibrium state and model it using a Poisson point process in arbitrary spatial dimensions with KZM density. Numerical simulations in a
one-dimensional ϕ4 theory unveil short-distance defect-defect corrections stemming from the kink excluded volume, while in two spatial dimensions, our model accurately describes the vortex spacing
distribution in a strongly coupled superconductor indicating the suppression of defect-defect spatial correlations.
Manipulating generalized Dirac cones in subwavelength dipolar arrays
Multiqudit interactions in molecular spins
We study photon-mediated interactions between molecular spin qudits in the dispersive regime of operation. We derive from a microscopic model the effective interaction between molecular spins,
including their crystal field anisotropy (i.e., the presence of non-linear spin terms) and their multi-level structure. Finally, we calculate the long time dynamics for a pair of interacting
molecular spins using the method of multiple scales analysis. This allows to find the set of 2-qudit gates that can be realized for a specific choice of molecular spins and to determine the time
required for their implementation. Our results are relevant for the implementation of logical gates in general systems of qudits with unequally spaced levels or to determine an adequate computational
subspace to encode and process the information.
Optimal simulation of quantum dynamics
Tensor networks are mathematical structures that efficiently compress the data required to describe quantum systems. An algorithm for the optimal simulation of quantum dynamics based on tensor
networks has now been implemented on a trapped-ion processor.
Phase Diagram of 1+1D Abelian-Higgs Model and Its Critical Point
We determine the phase diagram of the Abelian-Higgs model in one spatial dimension and time (1+ 1 D) on a lattice. We identify a line of first order phase transitions separating the Higgs region from
the confined one. This line terminates in a quantum critical point above which the two regions are connected by a smooth crossover. We analyze the critical point and find compelling evidence for its
description as the product of two noninteracting systems: a massless free fermion and a massless free boson. However, we find also some surprising results that cannot be explained by our simple
picture, suggesting this newly discovered critical point is an unusual one.
Phase Diagram of Abelian-Higgs Model and Its Critical Point
We determine the phase diagram of the Abelian-Higgs model in one spatial dimension and time (1+1D) on a lattice. We identify a line of first order phase transitions separating the Higgs region from
the confined one. This line terminates in a quantum critical point above which the two regions are connected by a smooth crossover. We analyze the critical point and find compelling evidences for its
description as the product of two non-interacting systems, a massless free fermion and a massless free boson. However, we find also some surprizing results that cannot be explained by our simple
picture, suggesting this newly discovered critical point to be an unusual one.
Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations
In this work we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum
algorithm to solve static, Schrödinger-type, Hamiltonian partial differential equations (PDEs), using space-efficient variational circuits, including the symmetries of the problem, and global and
gradient-based optimizers. We use this algorithm to benchmark the performance of the representation techniques by means of the computation of the ground state in three PDEs, i.e., the one-dimensional
quantum harmonic oscillator and the transmon and flux qubits, studying how they would perform in ideal and near-term quantum computers. With the Fourier methods developed here, we obtain low
infidelities of order 10^{−4}–10^{−5} using only three to four qubits, demonstrating the high compression of information in a quantum computer. Practical fidelities are limited by the noise and the
errors of the evaluation of the cost function in real computers, but they can also be improved through error mitigation techniques.
Quenches to the critical point of the three-state Potts model: Matrix product state simulations and conformal field theory
Conformal field theories (CFTs) have been used extensively to understand the physics of critical lattice models at equilibrium. However, the applicability of CFT calculations to the behavior of the
lattice systems in the out-of-equilibrium setting is not entirely understood. In this work, we compare the CFT results of the evolution of the entanglement spectrum after a quantum quench with
numerical calculations of the entanglement spectrum of the three-state Potts model using matrix product state simulations. Our results lead us to conjecture that CFT does not describe the
entanglement spectrum of the three-state Potts model at long times, contrary to what happens in the Ising model. We thus numerically simulate the out-of-equilibrium behavior of the Potts model
according to the CFT protocol, i.e., by taking a particular product state and “cooling” it, then quenching to the critical point and finding that, in this case, the entanglement spectrum is indeed
described by the CFT at long times.
Quenches to the critical point of the three-state Potts model: Matrix product state simulations and conformal field theory
Conformal Field Theories (CFTs) have been used extensively to understand the physics of critical lattice models at equilibrium. However, the applicability of CFT calculations to the behavior of the
lattice systems in the out-of-equilibrium setting is not entirely understood. In this work, we compare the CFT results of the evolution of the entanglement spectrum after a quantum quench with
numerical calculations of the entanglement spectrum of the three-state Potts model using matrix product state simulations. Our results lead us to conjecture that CFT does not describe the
entanglement spectrum of the three-state Potts model at long times, contrary to what happens in the Ising model. We thus numerically simulate the out-of-equilibrium behaviour of the Potts model
according to the CFT protocol – i.e. by taking a particular product state and “cooling” it, then quenching to the critical point and find that, in this case, the entanglement spectrum is indeed
described by the CFT at long times.
Reconfigurable photon localization by coherent drive and dissipation in photonic lattices
Role of boundary conditions in the full counting statistics of topological defects after crossing a continuous phase transition
In a scenario of spontaneous symmetry breaking in finite time, topological defects are generated at a density that scales with the driving time according to the Kibble-Zurek mechanism (KZM).
Signatures of universality beyond the KZM have recently been unveiled: The number distribution of topological defects has been shown to follow a binomial distribution, in which all cumulants inherit
the universal power-law scaling with the quench rate, with cumulant rations being constant. In this work, we analyze the role of boundary conditions in the statistics of topological defects. In
particular, we consider a lattice system with nearest-neighbor interactions subject to soft antiperiodic, open, and periodic boundary conditions implemented by an energy penalty term. We show that
for fast and moderate quenches, the cumulants of the kink number distribution present a universal scaling with the quench rate that is independent of the boundary conditions except for an additive
term, which becomes prominent in the limit of slow quenches, leading to the breaking of power-law behavior. We test our theoretical predictions with a one-dimensional scalar theory on a lattice.
Spin Many-Body Phases in Standard-and Topological-Waveguide QED Simulators
Topology detection in cavity QED
We explore the physics of topological lattice models immersed in c-QED architectures for arbitrary coupling strength with the photon field. We propose the use of the cavity transmission as a
topological marker and study its behavior. For this, we develop an approach combining the input–output formalism with a Mean-Field plus fluctuations description of the setup. We illustrate our
results with the specific case of a fermionic Su–Schrieffer–Heeger (SSH) chain coupled to a single-mode cavity. Our findings confirm that the cavity can indeed act as a quantum sensor for topological
phases, where the initial state preparation plays a crucial role. Additionally, we discuss the persistence of topological features when the coupling strength increases, in terms of an effective
Hamiltonian, and calculate the entanglement entropy. Our approach can be applied to other fermionic systems, opening a route to the characterization of their topological properties in terms of
experimental observables.
Tunable Directional Emission and Collective Dissipation with Quantum Metasurfaces
Tunable photon-mediated interactions between spin-1 systems
Tuning Long-Range Fermion-Mediated Interactions in Cold-Atom Quantum Simulators
Ultrastrong capcitive coupling of flux qubits
A flux qubit can interact strongly when it is capacitively coupled to other circuit elements. This interaction can be separated into two parts, one acting on the qubit subspaces and one in which
excited states mediate the interaction. The first term dominates the interaction between the flux qubit and an LC-resonator, leading to ultrastrong couplings of the form σ_y(a+a†), which complement
the inductive iσ_x(a†−a) coupling. However, when coupling two flux qubits capacitively, all terms need to be taken into account, leading to complex nonstoquastic ultrastrong interaction of types
σ_yσ_y, σ_zσ_z, and σ_xσ_x. Our theory explains all these interactions, describing them in terms of general circuit properties—coupling capacitances, qubit gaps, inductive, Josephson and capacitive
energies—that apply to a wide variety of circuits and flux qubit designs.
Unconventional mechanism of virtual-state population through dissipation
Universal deterministic quantum operations in microwave quantum links
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Lectures: 20 h - Preceptorship: 10 h
Why do some materials conduct electricity while other do not? Why do metals shine and dielectric materials are translucent or transparent? A material hosting more electrons, is it always a better
conductor? Why do materials composed of the same atoms may have different electric or magnetic properties? Semi-conductors: what is hidden behind this term? How do the electronic devices we use every
day work? To address these simple questions, the quantum origin of matter needs to be considered.
1. Introduction
□ Solid State Physics as a science that addresses properties and phenomena in the condensed matter at all relevant scales. Link to applications.
□ Example 1 : electronic data processors. Moor’s « law » of miniaturisation, FET transistors.
□ Example 2 : electronic memory devices. HDD, SSD etc.
□ History of the solid state physics : A short overview.
2. Drude model of the electron conduction in metals (classic approach)
□ Electric conduction phenomenon: knowledges at the beginning of XXth century, Drude’s hypotheses.
□ Drude formula of conductivity. Orders of magnitude of relevant parameters.
□ Temperature variation of the electric conductance.
□ Specific heat.
□ Applications of Drude model.
□ High-frequency response of Drude’s electron gas (20 min.): AC-conductivity; local equations, propagation.
3. Hall effect
□ Description of the phenomenon. Equation of motion of an electron.
□ The Hall constant.
□ Applications
4. Phonons (crystal lattice vibrations), Brillouin zones
□ Modelling the crystal potential (Lennard-Jones).
□ Harmonic approximation.
□ Harmonic vibrations of a 1D atomic chain (one atom per unit cell).
□ Harmonic vibrations of a 1D atomic chain (two atoms per unit cell).
□ Brillouin zones : Bravais lattice, Vigner-Seitz cell, construction of Brillouin zones of a solid.
5. Quantum model of a non-interaction electron gas (Sommerfeld)
□ Limitations and problems with classic Drude model.
□ Schrödinger equation. Physical meanning.
□ Born von Karman cyclic boundary conditions. Momentum (wave vector) and energy quantization.
□ K-space filling. Fermi energy, Fermi sphere.
□ Total energy of the system. Density of electronc states (DOS) vs system dimensions.
□ TD properties of the Sommerfeld’s electron gas. Specific heat. Strengths and weaknesses of the model.
6. Quantum near-free electrons model
□ Introduction. Historical context.
□ A single electron in a periodic potential. Central equation.
□ Gap opening (forbidden bands) at the limits of the Brillouin zone. Relation between the gap width and the crystal potential V(r).
□ Reduced-zone representation: translation of E(k) branches inside the reduced (1st) Brillouin zone.
□ Band occupation. Metals, insulators (semiconductors).
7. Tight-binding model. Electronic band dispersion
□ Introduction. General ideas.
□ Construction of the wave function.
□ Energy eigenvalues.
□ Dispersion. Group velocity, effective mass.
□ Consequence of the existence of the electronic bands on the electronic properties of materials.
8. Specific heat of a crystal
□ Classical limit : Dulong and Petit law (1812)
□ Quantum limit. Phonons.
□ Specific heat of a crystal lattice. Einstein model. Debye model.
9. Occupation of electronic bands: insulators, semiconductors, metals
□ Intrinsic semi-conductors. Fermi level. Effective mass action law. Applications.
□ Doped semi-conductors. Microscopic model of a single dopant atom in a solid.
□ Examples of applications.
10. Introduction to superconductivity
□ A bit of history. Discovery of the zero-resistance state.
□ Perfect diamagnetism.
□ Consequences of the Meissner-Ochsenfeld effect (1933). TD considerations.
□ Phase diagram of a superconductor. Vortex.
□ Examples of applications
11. Conclusions : recent trends and challenges in the Condensed Matter physics
□ Novel quantum materials and nano-structured materials/ Example : low-dimensional semiconducting heterostructures, graphene, topological insulators, surface and interface phenomena).
Applications (example: photovoltaics).
□ Strongly correlated electron systems (example: cuprates HTSC).
□ Mott metal-insulator phase transition materials.
• Vibrations of crystal lattice (phonons 2D).
• Nearly free electrons in a square 2D potential.
• Electronic properties of graphene.
• Doped semiconductors (p-n junctions).
• Upon tutor's choice: Field-Effect transistor; magnetism; Quantum Hall effect; Quantum corral.
Requirements :
preparatory classes (or L2) + basics of quantum mechanics.
Evaluation mechanism :
written exam (2 hours).
Last Modification :
Wednesday 6 September 2017
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A Statistical View of Deep Learning (IV): Recurrent Nets and Dynamical Systems
Recurrent neural networks (RNNs) are now established as one of the key tools in the machine learning toolbox for handling large-scale sequence data. The ability to specify highly powerful models,
advances in stochastic gradient descent, the availability of large volumes of data, and large-scale computing infrastructure, now allows us to apply RNNs in the most creative ways. From handwriting
generation, image captioning, language translation and voice recognition, RNNs now routinely find themselves as part of large-scale consumer products.
On a first encounter, there is a mystery surrounding these models. We refer to them under many different names: as recurrent networks in deep learning, as state space models in probabilistic
modelling, as dynamical systems in signal processing, and as autonomous and non-autonomous systems in mathematics. Since they attempt to solve the same problem, these descriptions are inherently
bound together and many lessons can be exchanged between them: in particular, lessons on large-scale training and deployment for big data problems from deep learning, and even more powerful
sequential models such as changepoint, factorial or switching state-space models. This post is an initial exploration of these connections.
Equivalent models: recurrent networks and state-space models.
Recurrent Neural Networks
Recurrent networks [cite key="bengioDLbook"] take a functional viewpoint to sequence modelling. They describe sequence data using a function built using recursive components that use feedback
from hidden units at time points in the past to inform computations of the sequence at the present. What we obtain is a neural network where activations of one of the hidden layers feeds back into
the network along with the input (see figures). Such a recursive description is unbounded and to practically use such a model, we unfold the network in time and explicitly represent a fixed number of
recurrent connections. This transforms the model into a feedforward network for which our familiar techniques can be applied.
If we consider an observed sequence x, we can describe a loss function for RNNs unfolded for T steps as:
The model and corresponding loss function is that of a feedforward network, with d(.) an appropriate distance function for the data being predicted, such as the squared loss. The difference from
standard feedforward networks is that the parameters θ of the recursive function f are the same for all time points, i.e. they are shared across the model. We can perform parameter estimation by
averaging over a mini-batch of sequences and using stochastic gradient descent with application of the backpropagation algorithm. For recurrent networks, this combination of unfolding in time and
backpropagation is referred to as backpropagation through time (BPTT) [cite key=werbos1990backpropagation].
Since we have simplified our task by always considering the learning algorithm as the application of SGD and backprop, we are free to focus our energy on creative specifications of the recursive
function. The simplest and common recurrent networks use feedback from one past hidden layer— earlier examples include the Elman or Jordan networks. But the true workhorse of current recurrent deep
learning is the Long Short-Term Memory (LSTM) network [cite key=gersLSTM]. The transition function in an LSTM produces two hidden vectors: a hidden layer h, and a memory cell c, and applies the
function f composed of soft-gating using sigmoid functions σ(.) and a number of weights and biases (e.g., A, B, a, b):
Probabilistic dynamical systems
We can also view the recurrent network construction above using a probabilistic framework (relying on reasoning used in part I of this series). Instead of viewing the recurrent network as a recursive
function followed by unfolding for T time steps, we can directly model a sequence of length T with latent (or hidden) dynamics and specify a probabilistic graphical model. Both the latent states h
and the observed data x are assumed to be probabilistic. The transition probability is the same for all time, so this is equivalent to assuming the parameters of the transition function are shared.
We could refer to these models as stochastic recurrent networks; the established convention is to refer to them as dynamical systems or state-space models.
In probabilistic modelling, the core quantity of interest is the probability of the observed sequence x, computed as follows:
Using maximum likelihood estimation, we can obtain a loss function based on the log of this marginal likelihood. Since for recurrent networks the transition dynamics is assumed to be deterministic,
we can easily recover the RNN loss function:
which recovers the original loss function with the distance function given by the log of the chosen likelihood function. It is no surprise that the RNN loss corresponds to maximum likelihood
estimation with deterministic dynamics.
As machine learners we never really trust our data, so in some cases we will wish to consider noisy observations and stochastic transitions. We may also wish to explore estimation beyond maximum
likelihood. A great deal of power is obtained by considering stochastic transitions that transform recurrent networks into probabilistic generative temporal models [cite key=barber2011bayesian][cite
key=sarkka2013bayesian] — models that account for missing data, allow for denoising and built-in regularisation, and that model the sequence density. We gain new avenues for creativity in our
transitions: we can now consider states that jump and random times between different operational modes, that might reset to a base state, or that interact with multiple sequences simultaneously.
But when the hidden states h are random, we are faced with the problem of inference. For certain assumptions such as discrete or Gaussian transitions, algorithms for hidden Markov models and Kalman
filters, respectively, demonstrate ways in which this can be done. More recent approaches use variational inference or particle MCMC [cite key=barber2011bayesian]. In general, efficient inference for
large-scale state-space models remains an active research area.
Prediction, Filtering and Smoothing
Dynamical systems are often described to make three different types of inference problems explicit: prediction, filtering and smoothing [cite key="sarkka2013bayesian"].
• Prediction (inferring the future) is the first use of most machine learning models. Having seen training data we are asked to forecast the behaviour of the sequence at some point k time-steps in
the future. Here, we compute the predictive distribution of the hidden state, since knowing this allows us to predict or generate what would be observed:
• Filtering (inferring the present) is the task of computing the marginal distribution of the hidden state given only the past states and observations.
• Smoothing (inferring the past) is the task of computing the marginal distribution of the hidden state given knowledge of the past and future observations.
These operations neatly separate the different types of computations that must be performed to correctly reason about the sequence with random hidden states. For RNNs, due to their deterministic
nature, computing predictive distributions and filtering are realised by the feedforward operations in the unfolded network. Smoothing is an operation that does not have a counterpart, but
architectures such as bi-directional recurrent nets attempt to fill this role.
Recurrent networks and state space models attempt to solve the same problem: how to best reason from sequential data. As we continue research in this area, it is the intersection of deterministic
and probabilistic approaches that will allow us to further exploit the power of these temporal models. Recurrent networks have been shown to be powerful, scalable, and applicable to an incredibly
diverse set of problems. They also have much to teach in terms of initialisation, stability issues, gradient management and the implementation of large-scale temporal models. Probabilistic approaches
have much to offer in terms of better regularisation, different types of sequences we can model, and the wide range of probabilistic queries we can make with models of sequence data. There is much
more that can be said, but these initial connections make clear the way forward.
[bibsource file=http://www.shakirm.com/blog-bib/rnnDynSys.bib]
4 thoughts on “A Statistical View of Deep Learning (IV): Recurrent Nets and Dynamical Systems”
1. Hey,
Thanks a lot for the great post, really enjoyed reading it. It opens up a new horizon for looking at RNNs which is super interesting. I was wondering if you could provide references that you've
mentioned in through the post.
2. Great post!
I really appreciate your multidisciplinary view on Deep Learning. I found particularly insightful when you show the similarities and common underlying mechanisms of seemingly diverse approaches
which have got different names, but in fact are dual views of the same thing.
You're breaking the jargon barrier!
3. Do you actually know of an application bidirectional RNNs to filtering *and* smoothing, in the probabilistic sense?
The problem is that the typical generative approach to decompose p(x_1:T) into a product of p(x_t|x_1:t-1) over all t already makes it impossible to incorporate information from the future. I
could imagine a an alternating cascade of forward/backward passes in time to achieve that, but I do not know of an RNN that simultaneously allows efficient and exact filtering and smoothing with
the same architecture and parameters.
We used a bidirectional RNN in our recent Nips VI Workshop paper, but that was only as an inference model in SGVB, not for the actual graphical model.
4. Really enjoyed reading this post. Thanks!
I totally share your view that recurrent networks can be a huge source of inspiration to make probabilistic models of time series more scalable and useful. It will be interesting to see if we can
come up with approximate inference methods that don't penalise us too much for carrying the extra burden of the probabilistic machinery. In particular, I am quite worried that to learn a model we
often have to do probabilistic inference (filtering/smoothing) on large state spaces with nonlinear dynamics. Not only this step can be computationally expensive, it can also be hard to know a
priori which approximate inference method will be more efficient and robust for the problem at hand.
Shameless plug: in a recent paper with Yutian Chen and Carl Rasmussen we learn nonlinear state-space models where we have put a Gaussian process prior over the state transition function. Since we
are using variational inference with sparse GPs (à la Titsias) the model becomes "parametric" and not dissimilar to a recurrent neural network. I'm currently exploring if initialisation
techniques borrowed from RNNs can be useful. But I think that the Achilles heel of our approach is the need to perform smoothing as part of the learning algorithm.
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1. π € Individual
To submit your assignment, please do the following:
• Submit the individual math-related tasks as a typeset PDF to GradeScope.
• Submit the teamβ s coding-related tasks to your team-submissions repository (instructions below).
Deadline: the VNAV staff will clone your repository on October 2nd at 1 PM ET.
π € Individual
π ¨ Deliverable 1 - Single-segment trajectory optimization (20 pts)
Consider the following minimum velocity ($r=1$) single-segment trajectory optimization problem:
\[\begin{eqnarray} \min_{P(t)} \quad \int_0^1 (P^{(1)}(t))^2 dt, \label{eq:minvel} \\\ s.t. \quad P(0) = 0, \label{eq:initpos} \\\ \quad P(1) = 1, \label{eq:finalpos} \end{eqnarray}\]
with $P(t) \in \mathbb{R}[t]$, i.e., $P(t)$ is a polynomial function in $t$ with real coefficients:
$$P(t) = p_N t^N + p_{N-1} t^{N-1} + \dots + p_1 t + p_0.$$
Note that because of constraint (\ref{eq:initpos}), we have $P(0)=p_0=0$, and we can parametrize $P(t)$ without a scalar part $p_0$.
1. Suppose we restrict $P(t) = p_1 t$ to be a polynomial of degree 1, what is the optimal solution of problem (\ref{eq:minvel})? What is the value of the cost function at the optimal solution?
2. Suppose now we allow $P(t)$ to have degree 2, i.e., $P(t) = p_2t^2 + p_1 t$.
(a) Write $\int_0^1 (P^{(1)}(t))^2 dt$, the cost function of problem (\ref{eq:minvel}), as $\boldsymbol{p}^T \boldsymbol{Q} \boldsymbol{p}$, where $\boldsymbol{p} = [p_1,p_2]^T$ and $\boldsymbol{Q} \
in \mathcal{S}^2$ is a symmetric $2\times 2$ matrix.
(b) Write $P(1) = 1$, constraint (\ref{eq:finalpos}), as $\boldsymbol{A}\boldsymbol{p} = \boldsymbol{b}$, where $\boldsymbol{A} \in \mathbb{R}^{1 \times 2}$ and $\boldsymbol{b} \in \mathbb{R}$.
(c) Solve the Quadratic Program (QP): $$\min_{\boldsymbol{p}} \boldsymbol{p}^T \boldsymbol{Q} \boldsymbol{p} \quad s.t. \quad \boldsymbol{A} \boldsymbol{p} = \boldsymbol{b}. \label{eq:QPtrajOpt}$$
You can solve it by hand, or you can solve it using numerical QP solvers (e.g., you can easily use the quadprog function in Matlab). What is the optimal solution you get for $P(t)$, and what is the
value of the cost function at the optimal solution? Are you able to get a lower cost by allowing $P(t)$ to have degree 2?
3. Now suppose we allow $P(t) = p_3t^3 + p_2 t^2 + p_1 t$:
(a) Let $\boldsymbol{p} = [p_1,p_2,p_3]^T$, write down $\boldsymbol{Q} \in \mathcal{S}^3, \boldsymbol{A} \in \mathbb{R}^{1\times 3}, \boldsymbol{b} \in \mathbb{R}$ for the QP (\ref{eq:QPtrajOpt}).
(b) Solve the QP, what optimal solution do you get? Do this example agree with the result we learned from Euler-Lagrange equation in class?
4. Now suppose we are interested in adding one more constraint to problem (\ref{eq:minvel}):
\begin{eqnarray} \min_{P(t)} \quad \int_0^1 (P^{(1)}(t))^2 dt, \label{eq:minveladd} \
s.t. \quad P(0) = 0, \
\quad P(1) = 1, \
\quad P^{(1)}(1) = -2. \end{eqnarray} Using the QP method above, find the optimal solution and optimal cost of problem (\ref{eq:minveladd}) in the case of:
(a) $P(t) = p_2t^2 + p_1 t$, and
(b) $P(t) = p_3t^3 + p_2 t^2 + p_1t$.
π ¨ Deliverable 2 - Multi-segment trajectory optimization (15 pts)
1. Assume our goal is to compute the minimum snap trajectory ($r= 4$) over $k$ segments. How many and which type of constraints (at the intermediate points and at the start and end of the trajectory)
do we need in order to solve this problem? Specify the number of waypoint constraints, free derivative constraints and fixed derivative constraints.
• Hint: According to Euler-Lagrange method, what is the degree of the polynomial of each segment?
• Hint: How many unknown parameters do we need to solve?
• Hint: How many constraints does each waypoint/free derivative/fixed derivative constraint provide?
• Hint: See figure for $k=3$ as described in the lecture notes.
2. Can you extend the previous question to the case in which the cost functional minimizes the $r$-th derivative and we have $k$ segments?
π ₯ Team
π ¨ Deliverable 3 - Drone Racing (65 pts)
For this lab we will be racing our simulated our simulated quadcopters in a drone racing course we prepared in our Unity simulator!
Graded GitHub files checklist
• Implement all the missing parts in the code:
□ Part 0 β planner_pkg/src/simple_traj_planner.cpp
□ Part 1.1 β trajectory_generation_pkg/src/trajectory_generation_node.cpp
□ Part 1.2 β trajectory_generation_pkg/src/trajectory_generation_node.cpp
□ Part 1.3 β trajectory_generation_pkg/src/trajectory_generation_node.cpp
• Create a file called links.txt with two publically links (i.e., hosted on either Google Drive or Dropbox) to the following deliverables:
□ A video (i.e., ideally a screen capture) showing your quadrotor completing the race course.
□ A ROS2 bag recording of your complete and fastest run. In ROS2 this a bag recording is a folder with a metadata.yaml and .db3 file(s) β i.e., it is no longer a single .bag files as in ROS1.
Instructions on how to record are below.
Setting up your codebase
Moving forward, we will be using the following file structure:
β β β starter-code β cloned from https://github.com/MIT-SPARK/VNAV-labs
β Β Β β β β ... β starter code for previous labs
β Β Β β β β lab4 β start code for this lab
β β β team-submissions β cloned from https://github.mit.edu/VNAV2024-submissions
β Β Β β β β ... β simulator files for previous labs
β Β Β β β β lab4 β this lab's graded code files need to be commited/pushed here
β β β personal-submissions β cloned from https://github.mit.edu/VNAV2024-submissions
β Β Β β β β ... β submission code from previous labs (no personal code for lab4)
β β β tesse β create this folder for downloading/unzipping simulator code
β Β Β β β β lab3 β download/unzip from https://vnav.mit.edu/material/lab3.zip
β Β Β β β β lab4 β download/unzip from https://vnav.mit.edu/material/lab4.zip
β β β ws β `colcon` workspaces for each assignment
β β β ... β ros2 workspaces for previous labs
β β β lab4 β ros2 workspace for lab4
β β β build β generated by `colcon build`
β β β install β generated by `colcon build`
β β β log β generated by `colcon build`
β β β src
β β β ... β packages placed/linked here by you
Letβ s get the new assignment setup!
First, make sure to install these two dependencies:
# dependency libraries
sudo apt install libnlopt-dev libgoogle-glog-dev
Next, letβ s get the new starter code:
# make sure you have the new starter code
cd ~/vnav/starter-code/
git pull
In ~/vnav/starter-code/lab4 we now have the planner_pkg and trajectory_generation_pkg, the two packages we will be modifying for this lab.
To prevent merge conflicts, only one team member should do this next code block:
# copy lab4 starter code to submissions
mkdir -p ~/vnav/team-submissions/lab4
cp -r ~/vnav/starter-code/lab4/* ~/vnav/team-submissions/lab4
# commit starter code to submissions repo
cd ~/vnav/team-submissions/lab4
git add .
git commit -m "added lab4 starter files"
git push
Now, the other team member(s) should pull the updated submission code with the following code block:
# pull starter code
cd ~/vnav/team-submissions
git pull
# confirm ~/vnav/team-submissions/lab4 exists and has the starter code in it
Everyone will now setup their colcon package for this assignment and link their solution code:
# create folder
mkdir -p ~/vnav/ws/lab4/src
cd ~/vnav/ws/lab4/src
# link lab4 solution packages
ln -s ../../../team-submissions/lab4/* .
# link needed packages from lab3
ln -s ../../../team-submissions/lab3/controller_pkg .
# link needed unmodified packages from lab3
ln -s ../../../starter-code/lab3/tesse_msgs .
ln -s ../../../starter-code/lab3/tesse_ros_bridge .
# clone needed packages
git clone https://github.com/fishberg/mav_trajectory_generation.git
git clone https://github.com/fishberg/mav_comm.git
# build the colcon package
cd ..
colcon build --symlink-install
Now update your ~/.bashrc:
# make sure these are in your ~/.bashrc
source /opt/ros/humble/setup.bash
source ~/vnav/ws/lab4/install/setup.bash
# remove ANY other VNAV workspaces you have been previously sourcing
and then source your updated ~/.bashrc with:
Note we are using a different mav_comm repo than from the previous assignment. We needed to correct an incompatibility with the previous repo. If you followed the above instructions closely, it
should all work.
To sanity check, your ~/vnav/ws/lab4/src folder should look like this:
β β β controller_pkg -> linked from lab3 team-submissions
β β β mav_comm -> cloned from https://github.com/fishberg/mav_comm
β β β mav_trajectory_generation -> cloned from https://github.com/MIT-SPARK/mav_trajectory_generation`
β β β planner_pkg -> linked from lab4 team-submissions
β β β tesse_msgs -> linked from lab3 starter-code
β β β tesse_ros_bridge -> linked from lab3 starter-code
β β β trajectory_generation_pkg -> linked from lab4 team-submissions
Next we will unpack the new simulator code. Please download the zip archive here.
# create folder for simulator
mkdir -p ~/vnav/tesse/lab4
# unzip contents of lab4.zip into ~/vnav/tesse/lab4
# make simulator executable
chmod +x ~/vnav/tesse/lab4/lab4.x86_64
Try launch the simulator. It should look like this:
Since we linked your controller_pkg from your lab3 team-submissions, your controller should be running with the gains you found in lab3. Feel free to further tune the gains in ~/vnav/ws/lab4/src/
Controller gains
It's possible that you may need to adjust your controller gains from lab3 to achieve good performance in lab4.
Before you start coding, letβ s take a look at the big picture. This is the system weβ re trying to construct:
Coding β Part 0
As a warm up exercise, letβ s just fly and hover at the first gate! Follow the instructions for Part 0 inside planner_pkg/src/simple_traj_planner.cpp.
To run your code, use the following commands:
# terminal 1
# run the simulator
cd ~/vnav/tesse/lab4
# terminal 2
# launches the bridge between tesse and ros2
# NOTE: make sure you always start the bridge AFTER the simulator
ros2 launch tesse_ros_bridge tesse_quadrotor_bridge.launch.yaml
# terminal 3
# launches the controller_node, traj_vertices_publisher, simple_traj_planner
ros2 launch planner_pkg static_point_test.launch.py
• Hint: you can press r to respawn your quadcopter
• Hint: review the handout from lab 3 if you have trouble running the simulator.
Waypoint publishing
We wrote the waypoint publishing node for you in planner_pkg/src/traj_vertices_publisher.cpp. Although you don't need to modify this code, it is important to understand what it is doing. This node
reads the position and orientation of the gates in the racing course and publishs them as waypoints for trajectory optimization. The topic to note here is /desired_traj_vertices, which should contain
a geometry_msgs::PoseArray type storing the position and orientation of the gates on the racing course.
Coding β Parts 1.1, 1.2, and 1.3
Next, letβ s implement the rest of the stack. Follow the instructions for Parts 1.1, 1.2, and 1.3 in trajectory_generation_pkg/src/trajectory_generation_node.cpp to get your quadcopter ready for
drone racing!
This node will subscribe to the waypoints published and use them for trajectory optimization using the mav_trajectory_generation library, and then based on the trajectory, publish the desired state
at time t to your controller.
• Hint: use vertex.addConstraint(POSITION, position) where position is of type Eigen::Vector3d to enforce a waypoint position.
• Hint: use vertex.addConstraint(ORIENTATION, yaw) where yaw is a double to enforce a waypoint yaw.
• Hint: remember angle wraps around 2$\pi$. Be careful!
• Hint: for the ending waypoint for position use end_vertex.makeStartOrEnd as seen with the starting waypoint instead of vertex.addConstraint as you would do for the other waypoints.
To run your code, use the following commands:
# terminal 1
# run the simulator
cd ~/vnav/tesse/lab4
# terminal 2
# launches the bridge between tesse and ros2
# NOTE: make sure you always start the bridge AFTER the simulator
ros2 launch tesse_ros_bridge tesse_quadrotor_bridge.launch.yaml
# terminal 3
# launches the controller_node and trajectory_generation_node (i.e., trajectory follower)
ros2 launch trajectory_generation traj_following.launch.py
# terminal 4
# launches traj_vertices_publisher node (i.e., waypoint publisher)
ros2 launch planner_pkg traj_gen.launch.py
If everything is working well, your drone should be gracefully going through each gate like this
Note: If the quadcopter flips over after launching the trajectory follower, press r in the Unity window to respwan, the quadcopter should just go back to the starting position.
Recording a Video and ROS2 Bag
Once you have your drone racing, you will need to submit a video and a ROS2 bag recording of your best trial.
To take a video, ideally use a screen recorder of your choice. If this isnβ t working with your setup, you make take a high quality video of your screen with a phone.
To collect a ROS2 bag recording, use the following command (i.e., it only records the /current_state and /desired_state topics):
ros2 bag record /current_state /desired_state
Note that a ROS2 bag recording is a folder with a metadata.yaml and .db3 file(s) β i.e., it is no longer a single .bag files as in ROS1.
Upload the both the video and ROS2 bag folder to Google Drive or Dropbox, generate two publicly viewable links, and place them in a textfile called links.txt within your team-submissions/lab4 folder.
Submitting your code
Navigate to your submisison directory:
cd ~/vnav/team-submissions/lab4
Sanity check the correct files/folders are in the lab4 submission directory:
$ ls
links.txt planner_pkg trajectory_generation_pkg
Add these files to your submission repo, commit, and push:
git add links.txt
git add planner_pkg
git add trajectory_generation
git commit -m "YOUR MESSAGE HERE"
git push
Confirm the submitted files appear correctly in your teamβ s submission repo.
[Optional] Faster, faster (Extra credit: 15 pts)
How can you make the drone go faster? We will award extra credit to the fastest 3 teams! Current record is 10.3s, achieved by Nikhil Singhal, Fernando Herrera and Chris Chang in 2020.
[Optional] Provide feedback (Extra credit: 3 pts)
This assignment involved a lot of code rewriting, so it is possible something is confusing or unclear. We want your feedback so we can improve this assignment (and the class overall) for future
offerings. If you provide actionable feedback to the Google Form link, you can get up to 3 pts of extra credit. Feedback can be positive, negative, or something in between β the important thing is
your feedback is actionable (i.e., β I didβ t like Xβ is vague, hard to address, and thus not actionable).
The Google Form is pinned in the course Piazza (we do not want to post the link publically, which is why it is not provided here).
Note that we cannot give extra points for anonymous feedback while maintaining your anonymity. That being said, you are still encouraged to provide anonymous feedback if youβ d like!
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Hopf ring
From Encyclopedia of Mathematics
A (graded) ring object in the category of (graded) co-commutative co-algebras (cf. Co-algebra). Such an object consists, first, of a sequence Abelian group objects in the category. These are better
known as commutative Hopf algebras with conjugation. Since they belong to the category, they have a coproduct:
As with any ring, there must be a distributive law relating the multiplication and the addition. Chasing diagrams in the category one sees that it is:
Hopf rings arise naturally in the study of the cohomology theory,
There are a number of Hopf rings which have been computed. Examples are [a9] (the basic reference for Hopf rings); [a14] and [a8]; [a13], § 8; [a10]; [a5]; [a11]; and the breakthrough description of
[a12], and its sequel for [a1] followed by corresponding results for odd primes in [a7]. Other references are [a2], [a3], [a4], and [a6].
Hopf rings have a very rich algebraic structure, useful in two distinct ways: descriptive and computational. All of the above examples have their Hopf rings described with just a few generators and
relations. The computations are generally carried out using Hopf ring techniques as well.
[a1] P.J. Eccles, P.R. Turner, W.S. Wilson, "On the Hopf ring for the sphere" Math. Z. , 224 (2) (1997) pp. 229–233
[a2] M.J. Hopkins, J.R. Hunton, "The structure of spaces representing a Landweber exact cohomology theory" Topology , 34 (1) (1995) pp. 29–36
[a3] J.R. Hunton, N. Ray, "A rational approach to Hopf rings" J. Pure Appl. Algebra , 101 (3) (1995) pp. 313–333
[a4] T. Kashiwabara, "Hopf rings and unstable operations" J. Pure Appl. Algebra , 194 (1994) pp. 183–193
[a5] R. Kramer, "The periodic Hopf ring of connective Morava Ph.D. Thesis, Johns Hopkins Univ. (1990)
[a6] T. Kashiwabara, N.P. Strickland, P.R. Turner, "Morava Algebraic Topology: New Trends in Localization and Periodicity , Progress in Mathematics , 139 , Birkhäuser (1996) pp. 209–222
[a7] Y. Li, "On the Hopf ring for the sphere" Ph.D. Thesis, Johns Hopkins Univ. (1996)
[a8] D.C. Ravenel, W.S. Wilson, "The Hopf ring for Canadian J. Math. , 48 (5) (1996) pp. 1044–1063
[a9] D.C. Ravenel, W.S. Wilson, "The Hopf ring for complex cobordism" J. Pure Appl. Algebra , 9 (1977) pp. 241–280
[a10] D.C. Ravenel, W.S. Wilson, "The Morava Amer. J. Math. , 102 (1980) pp. 691–748
[a11] N. Strickland, "Bott periodicity and Hopf rings" Ph.D. Thesis, Univ. Manchester (1992)
[a12] P.R. Turner, "Dickson coinvariants and the homology of Math. Z. , 224 (2) (1997) pp. 209–228
[a13] W.S. Wilson, "Brown–Peterson homology: an introduction and sampler" , CBMS , 48 , Amer. Math. Soc. (1982)
[a14] W.S. Wilson, "The Hopf ring for Morava Publ. RIMS Kyoto Univ. , 20 (1984) pp. 1025–1036
How to Cite This Entry:
Hopf ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_ring&oldid=14158
This article was adapted from an original article by W.S. Wilson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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The goal of SimMultiCorrData is to generate continuous (normal or non-normal), binary, ordinal, and count (Poisson or Negative Binomial) variables with a specified correlation matrix. It can also
produce a single continuous variable. This package can be used to simulate data sets that mimic real-world situations (i.e. clinical data sets, plasmodes, as in Vaughan et al., 2009). All variables
are generated from standard normal variables with an imposed intermediate correlation matrix. Continuous variables are simulated by specifying mean, variance, skewness, standardized kurtosis, and
fifth and sixth standardized cumulants using either Fleishman’s Third-Order or Headrick’s Fifth-Order Polynomial Transformation. Binary and ordinal variables are simulated using a modification of
GenOrd::ordsample. Count variables are simulated using the inverse cdf method. There are two simulation pathways which differ primarily according to the calculation of the intermediate correlation
matrix Sigma. In Correlation Method 1, the intercorrelations involving count variables are determined using a simulation based, logarithmic correlation correction (adapting Yahav and Shmueli’s 2012
method). In Correlation Method 2, the count variables are treated as ordinal (adapting Barbiero and Ferrari’s 2015 modification of GenOrd). There is an optional error loop that corrects the final
correlation matrix to be within a user-specified precision value. The package also includes functions to calculate standardized cumulants for theoretical distributions or from real data sets, check
if a target correlation matrix is within the possible correlation bounds (given the distributions of the simulated variables), summarize results (numerically or graphically), to verify valid power
method pdfs, and to calculate lower standardized kurtosis bounds.
There are several vignettes which accompany this package that may help the user understand the simulation and analysis methods.
1. Benefits of SimMultiCorrData and Comparison to Other Packages describes some of the ways SimMultiCorrData improves upon other simulation packages.
2. Variable Types describes the different types of variables that can be simulated in SimMultiCorrData.
3. Function by Topic describes each function, separated by topic.
4. Comparison of Correlation Method 1 and Correlation Method 2 describes the two simulation pathways that can be followed.
5. Overview of Error Loop details the algorithm involved in the optional error loop that improves the accuracy of the simulated variables’ correlation matrix.
6. Overall Workflow for Data Simulation gives a step-by-step guideline to follow with an example containing continuous (normal and non-normal), binary, ordinal, Poisson, and Negative Binomial
variables. It also demonstrates the use of the standardized cumulant calculation function, correlation check functions, the lower kurtosis boundary function, and the plotting functions.
7. Comparison of Simulation Distribution to Theoretical Distribution or Empirical Data gives a step-by-step guideline for comparing a simulated univariate continuous distribution to the target
distribution with an example.
8. Using the Sixth Cumulant Correction to Find Valid Power Method Pdfs demonstrates how to use the sixth cumulant correction to generate a valid power method pdf and the effects this has on the
resulting distribution.
Installation instructions
SimMultiCorrData can be installed using the following code:
## from GitHub
## from CRAN
This is a basic example which shows you how to solve a common problem: Compare a simulated exponential(mean = 2) variable to the theoretical exponential(mean = 2) density.
Step 1: Obtain the standardized cumulants
In R, the exponential parameter is rate <- 1/mean.
stcums <- calc_theory(Dist = "Exponential", params = 0.5)
#> mean sd skew kurtosis fifth sixth
#> 2 2 2 6 24 120
Step 2: Simulate the variable
Note that calc_theory returns the standard deviation, not the variance. The simulation functions require variance as the input.
H_exp <- nonnormvar1("Polynomial", means = stcums[1], vars = stcums[2]^2,
skews = stcums[3], skurts = stcums[4],
fifths = stcums[5], sixths = stcums[6], Six = NULL,
cstart = NULL, n = 10000, seed = 1234)
#> Constants: Distribution 1
#> Constants calculation time: 0 minutes
#> Total Simulation time: 0.001 minutes
#> [1] "constants" "continuous_variable" "summary_continuous"
#> [4] "summary_targetcont" "sixth_correction" "valid.pdf"
#> [7] "Constants_Time" "Simulation_Time"
# Look at constants
#> c0 c1 c2 c3 c4 c5
#> 1 -0.3077396 0.8005605 0.318764 0.03350012 -0.00367481 0.0001587077
# Look at summary
round(H_exp$summary_continuous[, c("Distribution", "mean", "sd", "skew",
"skurtosis", "fifth", "sixth")], 5)
#> Distribution mean sd skew skurtosis fifth sixth
#> X1 1 1.99987 2.0024 2.03382 6.18067 23.74145 100.3358
Step 3: Determine if the constants generate a valid power method pdf
Step 4: Select a critical value
Let alpha = 0.05.
Step 5: Solve for \(\\Large z'\)
Since the exponential(2) distribution has a mean and standard deviation equal to 2, solve \(\\Large 2 \* p(z') + 2 - y\_star = 0\) for \(\\Large z'\). Here, \(\\Large p(z') = c0 + c1 \* z' + c2 \* z'
^2 + c3 \* z'^3 + c4 \* z'^4 + c5 \* z'^5\).
f_exp <- function(z, c, y) {
return(2 * (c[1] + c[2] * z + c[3] * z^2 + c[4] * z^3 + c[5] * z^4 +
c[6] * z^5) + 2 - y)
z_prime <- uniroot(f_exp, interval = c(-1e06, 1e06),
c = as.numeric(H_exp$constants), y = y_star)$root
#> [1] 1.644926
Step 6: Calculate \(\\Large \\Phi(z')\)
This is approximately equal to the alpha value of 0.05, indicating the method provides a good approximation to the actual distribution.
Step 7: Plot graphs
plot_sim_pdf_theory(sim_y = as.numeric(H_exp$continuous_variable[, 1]),
overlay = TRUE, Dist = "Exponential", params = 0.5)
We can also plot the empirical cdf and show the cumulative probability up to y_star.
Calculate descriptive statistics.
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Research areas
A learning algorithm for fuzzy sets processing data labeled with their membership degrees has been proposed in [
Malchiodi and Pedrycz, 2013
Malchiodi, 2019a
] . Such algorithm has been applied to axiom mining within semantic Web [
Malchiodi and Tettamanzi, 2018
] and to negative examples selection in bioinformatics [
Frasca and Malchiodi, 2017
Frasca and Malchiodi, 2016
] . This approach has been extended in [
Cermenati et al., 2020
] to the simultaneous induction of several fuzzy sets, and in [
Malchiodi and Zanaboni, 2019
] to shadowed sets.
in collaboration with Prof. Zanaboni (Università degli Studi di Milano), Prof. Pedrycz (University of Alberta)
Knowledge induced via machine learning techniques is often encoded and stored in a distributed fashion withen models learnt from data. Thus it might be difficult to give a qualitative interpretation
of the obtained results. Moreover, this typically turns out in bandwidth and storage capacity issues when resources are limited. A possible solution to these problems consists in reducing the amount
of space necessary in order to store the above mentioned models after they have been trained. Some compression techniques for neural networks obtained via deep learning is currently under
investigation within the research project
Multicriteria Data Structures and Algorithms: from compressed to learned indexes, and beyond
, funded by the Italian Ministry of Education and Research under the PRIN initiative [
Marinò et al., 2021
] . Their implementation is described in [
Marinò et al., 2021
] .
in collaboration with Prof. Frasca (Università degli Studi di Milano)
Searching potential axioms within a set of formulas is a particularly demanding problem from a computational viewpoint. The solution of inducing such axioms starting from formulas labeled via a
precomputed fitness measure, obtained through processing of a knowledge base from the semabtic Web field, has been studied using learning algorithms for fuzzy sets [
Malchiodi and Tettamanzi, 2018
] and kernel-based regression techniques [
Malchiodi et al., 2018
] . The dependency of the problem on the used learning algorithm and on the dimensionality reduction technique employed in order to encode axioms as numerical vectors has been investigated in [
Malchiodi et al., 2020
] .
in collaboration with Prof. Da Costa Pereira, Prof. Tettamanzi (Université de la Côte d'Azur)
The application of supervised machine learning methods in bioinformatics requires the selection among non-positively labeled data of those representing reliable negative examples, that is excluding
entities on which no experiments have been conducted. In [
Frasca and Malchiodi, 2017
Frasca and Malchiodi, 2016
] such negative selection problem has been tackled using a ranking based on membership functions to fuzzy sets, while [
Frasca et al., 2017
Boldi et al., 2018
] propose an encoding for the available data promoting the negative selection process in the problem of protein functions prediction. Finally, a similar procedure has been proposed in [
Frasca et al., 2019
] for the problem of gene prioritization.
in collaboration with Prof. Frasca (Università degli Studi di Milano)
Casiraghi et al., 2020
] and [
Esposito et al., 2021
] describe the application of machine learning techniques to the problem of predicting the severity of COVID-19 in patients entering EDs.
in collaboration with Prof. Valentini (Università degli Studi di Milano) Prof. Casiraghi (Università degli Studi di Milano) Prof. Frasca (Università degli Studi di Milano)
Some machine learning and statistical data analysis techniques have been adapted in order to deal with problems in the veterinary and forensic fields. In particular, [
Galizzi et al., 2021
] and [
Bagardi et al., 2021
] describe the application of statistical methods in order to classify the incidence of cardiovascular factors in the death of dogs undergoing specific therapy, while [
Casali et al., 2021
] discusses a pilot study on the application of classification algorithms to predict the type of vehicle involved in a pedestrian hit.
in collaboration with Prof. Zanaboni (Università degli Studi di Milano)
Machine learning models have as starting point a labeled sample whose elements are processed homogeneously (that is, each element has the same importance). In [
Malchiodi, 2008
] the general model of data quality-based learning was proposed. In this model it is possible to associate each of the available data items a numerical quantification of its importance with reference
to the remaining data. This model was applied to the problem of classification through Support Vector Machines, both in its linear [
Apolloni and Malchiodi, 2006
] and kernel-based version [
Apolloni et al., 2007
] . A first analysis of the performance for these applications has been undertaken both theoretically [
Apolloni et al., 2007
] and experimentally [
Malchiodi, 2009
] . Some preliminary applications in the bioinformatics field is described in [
Malchiodi et al., 2010
] . A similar approach has also been applied to the regression problem in [
Apolloni et al., 2010
Malchiodi et al., 2009
Apolloni et al., 2005
] and to unbalanced learning in [
Malchiodi, 2013b
] .
Several types of learning algorithms have been designed, implemented and analyzed. In particular, [
Malchiodi and Legnani, 2014
] proposes an improvement of the support vector-based classification algorithms dealing both with partially labeled data and with uncertain labels, while [
Malchiodi and Pedrycz, 2013
] introduces a learning algorithm for membership functions of fuzzy sets. The latter approach has been extended in [
Malchiodi and Zanaboni, 2019
] to shadowed sets.
Concerning tertiary-level teaching, two publications have been produced: a manual for a software for automatic computations and a exercise textbook on operating systems [
Malchiodi, 2007
Malchiodi, 2015
] . Within a wider audience, [
Monga et al., 2017
] is centered around Alan Turing, and [
Malchiodi, 2019a
] describes possible future evolutions of fuzzy-based technologies.
in collaboration with laboratorio ALaDDIn (Università degli Studi di Milano)
The algomotorial approach has been introduced in [
Bellettini et al., 2014
] with the aim of teaching computing as the science studying the automatic elaboration of information, in contrast with the trend of tying computing to the working knowledge of specific technological
tools [
Lonati et al., 2015
Bellettini et al., 2014
] . The proposed approach has been evaluated in the realm of teaching habilitation [
Bellettini et al., 2015
] , with special focus to a constructivist perspective [
Bellettini et al., 2018
Bellettini et al., 2018
] . Furthermore, the relation within teaching and computational thinking competitions was studied in [
Lonati et al., 2017
] , evaluating the impact of the presentation of questions on the latter efficacy [
Lonati et al., 2017
] .
in collaboration with laboratorio ALaDDIn (Università degli Studi di Milano)
Starting from an analysis of computing education in Italian schools [
Bellettini et al., 2014
] and a criticism to the common identification of computer programming with the use of a language in order to encode an algorithm [
Lonati et al., 2015
] , the field of computer programming teaching has been studied from the viewpoint of its introduction via projects and specific tools [
Bulgheroni and Malchiodi, 2009
Paterson et al., 2015
] , of an interdisciplinary approach with musical subjects [
Ludovico et al., 2017
Baraté et al., 2017
Baratè et al., 2017
] , also considering advanced aspects of the discipline [
Lonati et al., 2016
Lonati et al., 2017
] . Finally, [
Monga et al., 2018
Lodi et al., 2019
] analyses a constructionist approach to computer programming and [
Condorelli and Malchiodi, 2022
] describes the joint design of a Master course on Big Data Architectures done with an Industrial partner.
in collaboration with laboratorio ALaDDIn (Università degli Studi di Milano)
Within the organization of non-competitive challenges on computational thinking at the national level [
Lissoni et al., 2012
Lissoni et al., 2013
Lissoni et al., 2014
Lissoni et al., 2015
] and the evaluation of their results [
Bellettini et al., 2015
Lonati et al., 2017
] , an analysis of the possibility to exploit this tools as a resource for learning in primary and secondary schools has been carried out [
Lonati et al., 2017
Calcagni et al., 2017
Morpurgo et al., 2018
] .
in collaboration with laboratorio ALaDDIn (Università degli Studi di Milano)
The algomotorial approach introduced in [
Bellettini et al., 2014
Bellettini et al., 2014
] has been applied to the introduction of core concepts of computing, such as information representation [
Bellettini et al., 2012
Bellettini et al., 2013
Baraté et al., 2017
] , basics of computer programming [
Baratè et al., 2017
] , as well as recursive and greedy strategies [
Lonati et al., 2016
Lonati et al., 2017
Lonati et al., 2017
] .
in collaboration with laboratorio ALaDDIn (Università degli Studi di Milano)
The granular computing model, giving information a granular meaning and allowing its analysis and its processing at different abstraction levels, is described in [
Apolloni et al., 2008
] , where its links with machine learning models are analysed. The effects of a fusion of these two models have been studied within the general field of regression, proposing new algorithms based on
Support Vector Machines [
Apolloni et al., 2008
Apolloni et al., 2006
] or on local search techniques [
Apolloni et al., 2005
] .
Bootstrap techniques are based on data resampling models with the aim of approximating the distribution of a population. A specialization of this kind of techniques, intially proposed in [
Apolloni et al., 2006
] and subsequently refined in [
Apolloni et al., 2009
Apolloni et al., 2007
] , gives as output confidence regions for regression curves, avoiding usual assumptions on the distribution of measurement drifts. The use of this technique to solve linear and nonlinear regression
problems is shown in [
Apolloni et al., 2008
] , while [
Apolloni et al., 2007
] describes some applications to the medical field.
The task of integrating under a unique theoretical model istances of inference problems from statistics (point and interval estimation of distribution parameters) and computer science (estimation of
approximation error in machine learning) is tackled in [
Apolloni et al., 2006
Apolloni et al., 2005
Apolloni et al., 2002
Apolloni et al., 2002
Apolloni and Malchiodi, 2001
Malchiodi, 2000
] , building on previously obtained results on sample complexity [
Apolloni and Malchiodi, 2001
] and describing the Algorithmic Inference model. This model was used with the aim of estimating the risk in classification problems based on Support Vector Machines [
Apolloni et al., 2007
Apolloni et al., 2005
Apolloni and Malchiodi, 2002
Apolloni and Malchiodi, 2001
] , learning confidence regions for regression lines avoiding the typical assumption requiring a Gaussian drift distribution [
Apolloni et al., 2005
Apolloni et al., 2002
] , and learning confidence regions for the risk function of re-occurrence distribution times in particular cancer pathologies [
Apolloni et al., 2007
Apolloni et al., 2005
Apolloni et al., 2002
] .
Systems for scientific computation can be used to run simulations and to analyze mathematical problems from an interactive and incremental point of view; To this effect, such systems offer
interesting cues in order to design educational activities [
Bulgheroni and Malchiodi, 2009
Malchiodi, 2008a
] . A commercial version of this kind of systems, thoroughly described in [
Malchiodi, 2007
] , has been extended so as to solve purely computational aspects associated to information encoding [
Malchiodi, 2006c
] , remote procedure invocation [
Malchiodi, 2006b
Malchiodi, 2006
] , production of scientific documentation [
Malchiodi, 2011
] , and solutions to optimization [
Malchiodi, 2006a
] and machine learning problems based on Support Vectors [
Malchiodi et al., 2009
Malchiodi et al., 2009
] , as well as to perform software validation techniques [
Malchiodi, 2013a
] . The related code has been used in order to build up the simulations in [
Apolloni et al., 2007
Apolloni and Malchiodi, 2006
] . Moreover, [
Malchiodi, 2010a
] describes a library handling machine learning problems within an open source system for scientific computation.
Hybrid learning systems are typically organized coupling sub-symbolic modules (typically based on the neural networks paradigm) with symbolic ones (described in terms of logic circuits). Such a
system, having as inputs a set of features describing the available data and extracting their boolean independent components, is described in [
Apolloni et al., 2005
Apolloni et al., 2004
] . These components, interpreted as truth values, are used in order to infer logical formulas describing in a symbolic ways the relations among original input data [
Apolloni et al., 2006
Apolloni et al., 2003
Apolloni et al., 2002
Apolloni et al., 2000
] . This system is applied in [
Apolloni et al., 2004
] to the problem of emotion recognition on the basis of voice signals, while [
Apolloni et al., 2004
Apolloni et al., 2004
Apolloni et al., 2003
Apolloni et al., 2003
Apolloni et al., 2003
] describes an applications to the monitoring of awareness in car driving in function of biosignals, within the research project IST-2000-26091 ORESTEIA (mOdular hybRid artEfactS wiTh adaptivE
functIonAlity, funded between 2001 and 2003 by the EC within the fifth framework programme, under the IST-FET initiative). Moreover, [
Apolloni and Malchiodi, 2006
Apolloni et al., 2005
] study two hybrid systems obtained through the integration of a fuzzy system for the measurement of quality in available data respectively with a linear Support Vector classifier and with a linear
regression model.
Whithin computational learning theory, the structural risk minimization principle investigates on the problem of balancing the complexity of a model with its accuracy in describing experimental data.
This principle has been applied to classifiers based on logic expressions built in terms of disjuctive and conjunctive boolean normal forms. A simplification algorithm for such forms was developed in
Apolloni et al., 2006
Apolloni et al., 2005
Apolloni et al., 2003
Apolloni et al., 2002
Apolloni et al., 2002
] , focusing on the stochastic optimization of parameters in fuzzy sets describing the above mentioned forms.
Within this subject the activities have been focused on the problem of modeling conflicting situations through an approach alternative to that of classical game theory. In particular, these conflicts
were modeled in terms of approximating the solution to an NP-hard problem [
Apolloni et al., 2006
Apolloni et al., 2003
Apolloni et al., 2002
Apolloni et al., 2002
] , applying the Algorithmic Inference model in order to assign limited computational resources to two players, subsequently extending this technique to team games [
Apolloni et al., 2006
] . This model is applied in [
Apolloni et al., 2007
Apolloni et al., 2005
] to the biologic field, while [
Apolloni et al., 2010
] uses this approach with the aim of correctly dimensioning the running time for learning algorithms based on local error minimization.
The research project ORESTEIA (mOdular hybRid artEfactS wiTh adaptivE functIonAlity, funded between 2001 and 2003 by the EC within the fifth framework programme, under the IST-FET initiative) was
grounded on the design, implementation and analysis of intelligent systems for pervasive and ubiquitous computing. These fields are characterized by highly specialized computers devoted to execute
specific tasks. These special computers can be produced so as to significantly reduce their size and cost, consequently being able to immerse them inside an environment. Focusing specifically on the
awareness detection problem [
Kasderidis et al., 2003
] , a prototype for the detection of driving awareness on the basis of biosignals [
Apolloni et al., 2004
Apolloni et al., 2004
Apolloni et al., 2003
Apolloni et al., 2003
Apolloni et al., 2003
] have been developed.
Within the progress of reserach project PHYSTA (Principled Hybrid Systems: Theory and Applications, funded between 1998 and 2000 by the EC within the fourth framework programme, within the TMR
initiative), the Algorithmic Inference model described in [
Apolloni et al., 2006
Malchiodi, 2000
] was applied to the problem of automatic classification of emotions on the basis of vocal signals [
Apolloni et al., 2004
Apolloni et al., 2002
] . The obtained results were presented at an international school on computational learning within the same research project.
The availability of hardware circuits able to directly process information with the aim of synthesizing them through estimators allow a remarkable shortening in running times. Their use imply a set
of constraints basically linked to the architecture of the circuits themselves. The inference-among-gossips, developed in [
Malchiodi, 1996
] , has been applied within this scope with the aim of obtaining a family of estimators for bernoulli populations directly implementable on pRAM boards [
Apolloni et al., 1997
] . The same model has been applied in [
Apolloni et al., 2013
] to the study of information exchange in social networks.
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The intuition behind Word2Vec
Have you ever wondered how YouTube knows which videos to recommend, how Google Translate is able to translate whole texts into a decent version of the target language or how your Smartphone keyboard
knows which words and text snippets to suggest while you type your texts?
There’s a very high likelihood that so-called Embeddings were used behind the scenes. Embeddings are one of the central ideas behind modern Natural Language Processing models.
In the following writeup we’ll discover the main building blocks and basic intuition behind Embeddings. We’ll learn how and why they work and how Word2Vec, a method to turn words into vectors, can be
used to show that:
[king - man + woman = queen ]
All the code we’ll write here can be found in my “Lab” repository on GitHub. Feel free to code along while reading through this tutorial.
Basic Setup
Before jumping right into the code we need to make sure that all Python packages we’ll be using are installed on our machine.
We install Seaborn, a visualization tool which helps us to plot nice-looking charts and diagrams. We don’t really work with Seaborn directly but rather use its styles in conjunction with Matplotlib
to make our plots look a little bit more “modern”.
Next up we need to import the modules we’ll use throughout this tutorial (the last few lines configure Matplotlib to use Seaborn styles).
import json
from pathlib import Path
import pandas as pd
import seaborn as sns
import numpy as np
from IPython.display import HTML, display
# prettier Matplotlib plots
import matplotlib.pyplot as plt
import matplotlib.style as style
Since we’re dealing with different datasets we should create a separate directory to store them in.
!mkdir -p data
data_dir = Path('data')
Comparing Countries
Let’s start with our first data analysis task. Our goal is to compare and contrast different countries based on their surface area and population. The main idea being that we want to analyze which
countries are quite similar and which are rather different based on those two metrics.
The dataset we’ll use is part of the country-json project by @samayo. Make sure to take some time to browse through the different JSON files to get an idea about the structure of the data.
In our example we’re only interested in the country-by-surface-area.json and country-by-population.json files. Let’s go ahead and download the files to our data directory.
After that we can define 2 variables which will point to the files on our file system.
SURFACE_AREA_FILE_NAME = 'country-by-surface-area.json'
POPULATION_FILE_NAME = 'country-by-population.json'
!wget -nc https://raw.githubusercontent.com/samayo/country-json/master/src/country-by-surface-area.json -O data/country-by-surface-area.json
!wget -nc https://raw.githubusercontent.com/samayo/country-json/master/src/country-by-population.json -O data/country-by-population.json
surface_area_file_path = str(data_dir / SURFACE_AREA_FILE_NAME)
population_file_path = str(data_dir / POPULATION_FILE_NAME)
During our data analysis we’ll utilize Pandas, a great Python library which makes it dead simple to inspect and manipulate data.
Since our data is in JSON format we can use Pandas read_json function to load the data into a so-called DataFrame (think of it as an Excel spreadsheet on steroids).
The dropna function makes sure that we remove all entries which are undefined and therefore useless for further inspection.
df_surface_area = pd.read_json(surface_area_file_path)
df_population = pd.read_json(population_file_path)
You might’ve noticed that dealing with 2 separate files will get quite hairy if we want to compare countries based on their 2 metrics.
Since both files contain the same countries with the same names and only differ in terms of their area and population data we can use merge to create a new DataFrame containing all countries with
their respective area and population numbers.
Another tweak we perform here is to set the index to the country name. This way we can easily query for country data based on the country names rather than having to deal with non-expressive integer
df = pd.merge(df_surface_area, df_population, on='country')
df.set_index('country', inplace=True)
As you can see we have a total of 227 countries in our DataFrame. 227 are way too many countries for our need. Especially since we’re about to plot the data in the next step.
Let’s reduce our result set by performing some range-queries with the area and population data.
df = df[
(df['area'] > 100000) & (df['area'] < 600000) &
(df['population'] > 35000000) & (df['population'] < 100000000)
Great! 12 countries are way easier to analyze once plotted.
Speaking of which, let’s do a 2D scatterplot of our 12 countries. We decide to plot the area on the X axis and the population on the Y axis.
fig, ax = plt.subplots()
df.plot(x='area', y='population', figsize=(10, 10), kind='scatter', ax=ax)
for k, v in df.iterrows():
ax.annotate(k, v)
Looking at the plotted data we can immediately see some relationships. It appears that Vietnam has a high population compared to its area. Kenya on the other hand has a large surface area but a
smaller population compared to its size.
Plotting the data like this helps us to reason about it in a visual way. In addition to that we can also easily validate the integrity of our data.
While we as humans can immediately tell the relationships in our country data just by looking at our plot it’s necessary to translate our visual reasoning into raw numbers so our computer can
understand them too.
Looking at the plot again it seems like the distance between the data points of the countries is a good measure to determine how “similar” or “different” the countries are.
There are several algorithms to calculate the distance between two (or more) coordinates. The Euclidean distance is a very common formula to do just that. Here’s the Math notation:
[d(x, y) = d(y, x) = \sqrt{\sum_{i=1}^N (x_i - y_i)^2} ]
While the formula might look intimidating at first it’s rather simple to turn it into code.
def euclidean_distance(x, y):
x1, x2 = x
y1, y2 = y
result = np.sqrt((x1 - x2) **2 + (y1 - y2)** 2)
# we'll cast the result into an int which makes it easier to compare
return int(round(result, 0))
According to our plot it seems like Thailand and Uganda are 2 countries which are very different. Computing the Euclidean distance between both validates our hunch.
# Uganda <--> Thailand
uganda = df.loc['Uganda']
thailand = df.loc['Thailand']
x = (uganda['area'], thailand['area'])
y = (uganda['population'], thailand['population'])
euclidean_distance(x, y)
If we compare this result to the Euclidean distance between Iraq and Morocco we can see that those countries seem to be more “similar”.
# Iraq <--> Morocco
iraq = df.loc['Iraq']
morocco = df.loc['Morocco']
x = (iraq['area'], morocco['area'])
y = (iraq['population'], morocco['population'])
euclidean_distance(x, y)
While this exercise was quite simple and intuitive if one is fluent in geography it also introduced us to the basic concepts of Embeddings. With Embeddings we map data (e.g. words or raw numbers)
into multi-dimensional spaces and use Math to manipulate and calculate relationships between that data.
This might sound rather abstract and I agree that the relationship between our Country data analysis and Embeddings is still a little bit fuzzy.
Trust me, the upcoming example will definitely result in an “Aha Moment” and suddenly what we’ve learned so far will click!
Color Math
Now that we’ve seen some of the underlying principles of Embeddings let’s take another look at a slightly more complicated example. This time we’ll work with different colors and their representation
as a combination of Red, Green and Blue values (also known as RGB).
Before we jump right into our analysis we’ll define a helper function which lets us render the color according to its RGB representation.
The following code defines a function which takes the integer values of Red, Green and Blue (values in the range of 0 - 255) and renders a HTML document with the given color as its background.
def render_color(r, g, b):
<div style="background-color: rgba(%d, %d, %d, 1); height: 100px;"></div>
''' % (r, g, b)),
The color black is represented as 0 Red, 0 Green and 0 Blue. Let’s validate that our render_color function works as expected.
Great. It works!
Next up it’s time to download the dataset we’ll be using for our color analysis. We’ve decided to use the 256 Colors dataset by @jonasjacek. It lists the 256 colors used by xterm, a widely used
terminal emulator. Make sure to take a couple of minutes to familiarize yourself with the data and its structure.
Downloading the dataset follows the same instruction we’ve used in the beginning of this tutorial where we downloaded the Country data.
COLORS_256_FILE_NAME = 'colors-256.json'
!wget -nc https://jonasjacek.github.io/colors/data.json -O data/colors-256.json
colors_256_file_path = str(data_dir / COLORS_256_FILE_NAME)
Now that we have access to the data in our programming environment it’s time to inspect the structure and think about ways to further process it.
color_data = json.loads(open(colors_256_file_path, 'r').read())
[{'colorId': 0,
'hexString': '#000000',
'rgb': {'r': 0, 'g': 0, 'b': 0},
'hsl': {'h': 0, 's': 0, 'l': 0},
'name': 'Black'},
{'colorId': 1,
'hexString': '#800000',
'rgb': {'r': 128, 'g': 0, 'b': 0},
'hsl': {'h': 0, 's': 100, 'l': 25},
'name': 'Maroon'},
{'colorId': 2,
'hexString': '#008000',
'rgb': {'r': 0, 'g': 128, 'b': 0},
'hsl': {'h': 120, 's': 100, 'l': 25},
'name': 'Green'},
{'colorId': 3,
'hexString': '#808000',
'rgb': {'r': 128, 'g': 128, 'b': 0},
'hsl': {'h': 60, 's': 100, 'l': 25},
'name': 'Olive'},
{'colorId': 4,
'hexString': '#000080',
'rgb': {'r': 0, 'g': 0, 'b': 128},
'hsl': {'h': 240, 's': 100, 'l': 25},
'name': 'Navy'}]
As we can see there are 3 different color representations available in this dataset. There’s a Hexadecimal, a HSL (Hue, Saturation, Lightness) and a RGB (Red, Green, Blue) representation. Furthermore
we have access to the name of the color via the name attribute.
In our analysis we’re only interested in the name and the RGB value of every color. Given that we can create a simple dict which key is the lowercased color name and its value is a tuple containing
the Red, Green and Blue values respectively.
colors = dict()
for color in color_data:
name = color['name'].lower()
r = color['rgb']['r']
g = color['rgb']['g']
b = color['rgb']['b']
rgb = tuple([r, g, b])
colors[name] = rgb
To validate that our data structure works the way we described above we can print out some sample colors with their RGB values.
print('Black: %s' % (colors['black'],))
print('White: %s' % (colors['white'],))
print('Red: %s' % (colors['red'],))
print('Lime: %s' % (colors['lime'],))
print('Blue: %s' % (colors['blue'],))
Black: (0, 0, 0)
White: (255, 255, 255)
Red: (255, 0, 0)
Lime: (0, 255, 0)
Blue: (0, 0, 255)
While our dict is a good starting point it’s often easier and sometimes faster to do computations on the data if it’s stored in a Pandas DataFrame. The from_dict function helps us to turn a simple
Python dictionary into a DataFrame.
df = pd.DataFrame.from_dict(colors, orient='index', columns=['r', 'g', 'b'])
Seeing the data formatted in this way we can think of its representation as a mapping of the Red, Green and Blue values into a 3-Dimensional space where for example Red is the X axis, Green is the Y
axis and Blue is the Z axis.
You might recall that we’ve used Euclidean distance in our Country example above to determine how “similar” countries are. The main idea was that similar countries have less distance between their
data points compared to dissimilar countries whose data points are farther apart.
Another very useful formula to calculate the similarity of data points is the so-called Cosine similarity. The Cosine similarity measures the angle between two vectors in a multi-dimensional space.
The smaller the angle, the more similar the underlying data.
Translating this to our color example we can think of every color being represented as a vector with 3 values (Red, Green and Blue) which (as stated above) can be mapped to the X, Y and Z axis in a
3D coordinate system. Using the Cosine similarity we can take one of such vectors and calculate the distance between it and the rest of the vectors to determine how similar or dissimilar they are.
And that’s exactly what we’ll be doing here.
The Math notation for the Cosine similarity looks like this:
[similarity = \cos(\Theta) = \frac{A \cdot B}{\left\lVert A\right\rVert \left\lVert B\right\rVert} ]
We’re taking the dot-product between the two vectors A and B and divide it by the product of their magnitudes.
The following code-snippet implements such formula. Again, it might look intimidating and rather complicated but if you take some time to read through it you’ll see that it’s not that hard to
In fact our implementation here does more than just calculating the Cosine similarity. In addition to that we copy our DataFrame containing the colors and add another column to it which will include
the distance as a value between 0 and 1. Once done we sort our copied DataFrame by such distance in descending order. We do this to see the computed values when querying for similar colors later on.
def similar(df, coord, n=10):
# turning our RGB values (3D coordinates) into a numpy array
v1 = np.array(coord, dtype=np.float64)
df_copy = df.copy()
# looping through our DataFrame to calculate the distance for every color
for i in df_copy.index:
item = df_copy.loc[i]
v2 = np.array([item.r, item.g, item.b], dtype=np.float64)
# cosine similarty calculation starts here
theta_sum = np.dot(v1, v2)
theta_den = np.linalg.norm(v1) * np.linalg.norm(v2)
# check if we're trying to divide by 0
if theta_den == 0:
theta = None
theta = theta_sum / theta_den
# adding the `distance` column with the result of our computation
df_copy.at[i, 'distance'] = theta
# sorting the resulting DataFrame by distance
df_copy.sort_values(by='distance', axis=0, ascending=False, inplace=True)
return df_copy.head(n)
To validate that our similar function works we can use it to find similar colors to red.
similar(df, colors['red'])
We can also pass in colors as a list of RGB values.
similar(df, [100, 20, 120])
Since it’s hard to imagine what color 100, 20 and 120 represent it’s worthwhile to use our render_color function to see it.
render_color(100, 20, 120)
Looking at the list of most similar colors from above it appears that darkvioletis quite similar to 100, 20, 120. Let’s see how this color looks like.
darkviolet = df.loc['darkviolet']
render_color(darkviolet.r, darkviolet.g, darkviolet.b)
And we can validate that darkviolet in fact looks quite similar to 100, 20, 120!
But it doesn’t end here. Our 3 color values are numbers in the range of 0 - 255. Given that, it should be possible to do some basic Math computations such as addition or subtraction on them.
Since we only have access to 256 different colors it’s highly unlikely that our resulting color values for Red, Green and Blue will exactly match one of our 256 colors. That’s where our similar
function comes in handy! The similarfunction should make it possible to calculate a new color and find its most similar representation in our 256 color dataset.
We can look at a Color Wheel to see that subtracintg a red color from purpleone should result in a Blueish color. Let’s do the Math and check whether that’s true.
blueish = df.loc['purple'] - df.loc['red']
similar(df, blueish)
And sure enough the most similar colors in our dataset are Blueish ones. We can validate that by rendering darkblue, one of the best matches.
darkblue = df.loc['darkblue']
render_color(darkblue.r, darkblue.g, darkblue.b)
Here’s a simple one. If we have Black and add some White to the mix we should get something Greyish, correct?
greyish = df.loc['black'] + df.loc['white']
similar(df, greyish)
And sure enough we do. Rendering grey93 shows a light grey color.
grey93 = df.loc['grey93']
render_color(grey93.r, grey93.g, grey93.b)
Let’s end our color exploration with a more complex formula. So far we’ve only done some very simple Math like subtracting and adding colors. But there’s more we can do. We can also express our
search for a color as a “solve for x” problem.
Mixing Yellow and Red will result in Orange. We can translate this behavior to other colors as well. Here we ask “Yellow is to Red as X is to Blue” and express it in Math notation to get the result
for X.
# yellow is to red as X is to blue
yellow_to_red = df.loc['yellow'] - df.loc['red']
X = yellow_to_red + df.loc['blue']
similar(df, X)
Our calculation shows us that lightseargreen is to Blue as Yellow is to Red. Intuitively that makes sense if you think about it.
lightseagreen = df.loc['lightseagreen']
render_color(lightseagreen.r, lightseagreen.g, lightseagreen.b)
In the beginnig of this tutorial I promised that once done we should understand the intuition behind Word2Vec, a key component for modern Natural Language Processing models.
The Word2Vec model does to words what we did with our colors represented as RGB values. It maps words into a multi-dimensional space (our colors were mapped into a 3D space). Once such words are
mapped into that space we can perform Math calculations on their vectors the same way we e.g. calculated the similarity between our color vectors.
Having a mapping of words into such a vector space makes it possible to do calculations resulting in:
[king - man + woman = queen ]
In this tutorial we took a deep dive into the main building blocks and intuitions behind Embeddings, a powerful concept which is heavily utilized in modern Natural Language Processing models.
The main idea is to map data into a multi-dimensional space so that Math calculations from the realm of Linear Algebra can be performed on it.
We started our journey with a simple example in which we mapped the surface area and population of different countries into a 2D vector space. We then used the Euclidean distance to verify that
certain countries are similar while others are dissimilar based on their metrics.
Another, more advanced example mapped colors and their RGB representation into a 3D vector space. We then used Cosine similarity and some basic Math to add and subtract colors.
With this knowledge we’re now able to understand how more advanced models such as Word2Vec or Doc2Vec make it possible to do calculations on words and texts.
The Lab
You can find more code examples, experiments and tutorials in my GitHub Lab repository.
Additional Resources
Eager to learn more? Here’s a list with all the resources I’ve used to write this post.
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NCERT Book for Class 12 Mathematics Download PDF
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syllabus issued by CBSE. Students of Class 12 are recommended to download and read latest NCERT books and also remember to refer to NCERT Solutions for Class 12 Mathematics
NCERT Mathematics Book Class 12 PDF Free Download
Students in Class 12 should strictly follow NCERT book for Class 12 Mathematics issued as per the syllabus designed by CBSE. These books have bee designed by the best Mathematics teachers and if you
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NCERT Book for Class 12 Maths Mathematics Part I
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NCERT Book for Class 12 Maths PDF Download
Chapter 1: Relations and Functions
Chapter 2: Inverse Trigonometric Functions
Chapter 5: Continuity and Differentiability
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Chapter 6: Application of Derivatives
Chapter 8: Application of Integrals
Chapter 9: Differential Equations
Chapter 11: Three Dimensional Geometry
Chapter 12: Linear Programming
NCERT Books For Class 12 Maths Part 1 In Hindi Medium
अध्याय 2: प्रतिलोम त्रिकोणमितीय फलन
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NCERT Books For Class 12 Maths Part 2 In Hindi Medium
अध्याय 8: समाकलनों के अनुप्रयोग
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Maths Ganit Purak Pathya Samagri
NCERT Books for Class 12 Mathematics are published by the National Council of Educational Research and Training (NCERT) for latest 2021 academic session for Class 12. These books issued are by NCERT
for Mathematics Class 12. They are recommended by all schools and is being implemented in almost all states in India as questions on exams for Class 12 Mathematics normally comes from Books by NCERT
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Advanced Continuous Optimization
M2 course at the "Institut Polytechnique de Paris"
Advanced Continuous Optimization
This page describes the 48 hour course given at "Institut Polytechnique de Paris" in the M2 Optimization, during the academic year 2020-2021, entitled Advanced Continuous Optimization. It contains
- the teacher internet pages,
- a short presentation of its contents,
- the detailed program of part I and part II.
The timetable of the other courses of the M2 is given here.
Jean Charles Gilbert (Inria Paris),
The first part of the module (30 hours) starts with the presentation of the optimality conditions of an optimization problem described in a rather general manner, so that these can be useful for
dealing with a large variety of problems: the constraints are expressed by $c(x)\in G$, where $c:\mathbb{E}\to\mathbb{F}$ is a nonlinear map between two Euclidean spaces $\mathbb{E}$ and $\mathbb{F}
$, and $G$ is a closed convex part of $\mathbb{F}$. Next, the course describes and analyzes various advanced algorithms to solve functional inclusions (of the kind $F(x)+N(x)\ni0$, where $F$ is a
function and $N$ is a multifunction) and optimization problems (nonsmooth methods, linearization methods, proximal and augmented Lagrangian methods, interior point methods) and shows how they can be
used to solve a few classical optimization problems (linear optimization, convex quadratic optimization, semidefinite optimization (SDO), nonlinear optimization). Along the way, various tools from
convex and nonsmooth analysis will be presented. Everything is conceptualized in finite dimension. The goal of the lectures is therefore to consolidate basic knowledge in optimization, on both
theoretical and algorithmic aspects.
The second part of the module (20 hours) focuses on the implementation of some of the previously seen algorithms in Matlab and allows the designer to understand its behavior and to evaluate its
efficiency on a concrete problem. A choice will have to be made between the following three projects:
• the implementation of an SQP solver to solve the hanging chain problem (viewed as a nonlinear optimization problem);
• the implementation of a self-dual conic optimization (SDCO) solver in real numbers to solve various academic/concrete problems, such as the global minimization of a univariate polynomial or a few
small size OPF problems (more precisely the rank relaxation of a QCQP version of this problem);
• the implementation of a semismooth-Newton-like algorithm to minimize a convex quadratic function on a convex polyhedron with application to the case when the polyhedron is the Birkhoff polytope.
To follow the second part, the student must have followed the first part, because the algorithms implemented in the second part are presented and anaylzed in the first part.
Detailed program
First part
• The course is composed of 7 lectures of 4h15 each, which makes it approximately 30h long. It is always given on Monday from 2:00 pm to 6:30 pm.
• Due to the corona virus pandemic, this year, the course is given online. You must contact Anne Richard (anne.richard@ensta-paris.fr) for having the authorization to enter ENSTA (this is
automatically ensured for those who are registered to the M2).
• See below for the schedule.
• J.F. Bonnans, J.Ch. Gilbert, C. Lemaréchal, C. Sagastizábal (2006). Numerical Optimization - Theoretical and Practical Aspects (second edition). Universitext, Springer Verlag, Berlin. [authors]
• J.F. Bonnans, A. Shapiro (2000). Perturbation Analysis of Optimization Problems. Springer Verlag, New York.
• A.L. Dontchev, R.T. Rockafellar (2009). Implicit Functions and Solution Mappings - A View from Variational Analysis. Springer Monographs in Mathematics. Springer.
• J.Ch. Gilbert (2015). Fragments d'Optimisation Différentiable - Théorie et Algorithmes. Syllabus de cours à l'ENSTA-ParisTech. [internet].
• J.-B. Hiriart-Urruty, C. Lemaréchal (1996). Convex Analysis and Minimization Algorithms (second edition). Grundlehren der mathematischen Wissenschaften, 305-306. Springer-Verlag.
• A.F. Izmailov, M.V. Solodov (2014). Newton-Type Methods for Optimization and Variational Problems. Springer Series in Operations Research and Financial Engineering, Springer.
• Y.E. Nesterov, A.S. Nemirovskii (1994). Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, 13. SIAM, Philadelphia, PA, USA.
• J. Renegar (2001). A Mathematical View of Interior-Point Methods in Convex Optimization. MPS-SIAM Series on Optimization 3, SIAM.
• R.T. Rockafellar (1970). Convex Analysis. Princeton Mathematics Ser., 28. Princeton University Press, Princeton, New Jersey.
• R.T. Rockafellar, R. Wets (1998). Variational Analysis. Grundlehren der mathematischen Wissenschaften, 317. Springer.
• C. Roos, T. Terlaky, J.-Ph. Vial (2006). Interior Point Methods for Linear Optimization (second edition). Springer.
• S.J. Wright (1997). Primal-Dual Interior-Point Methods. SIAM Publication, Philadelphia.
Actual program on a daily basis (the access to the Lecture materials requires the username "Student" and the given password)
Date Themes of the lectures Lecture Actual details of the contents of the lectures
Monday Presentation and recalls Manual Presentation of the course
November Scripts 1 Background
16 Video 1
2020 • Convex analysis: relative interior, absorbing point, dual cone and Farkas lemma, tangent and normal cones
• Nonlinear analysis: multifunction
• Optimization: tangent cone and Peano-Kantorovich necessary optimality condition of the first order for $(P_X)$, sufficient
optimality condition of the first order for a convex problem $(P_X)$, KKT conditions for $(P_{EI})$, sufficient conditions for
contraint qualification
Optimality conditions
• First order optimality conditions for $(P_G)$: definition of the problem and its convexity, tangent and linearizing cones to the
feasible set, constraint qualification, NC1, SC1 for a convex problem
Monday First order optimality conditions Manual Background
November of the general optimization problem $ Scripts 2
23 (P_G)$ Video 2 • Convex analysis: asymptotic cone
Optimality conditions
• First order optimality conditions for $(P_G)$: SC1 for a convex problem, Robinson's condition and metric regularity, open
multifunction theorem, metric regularity diffusion theorem, Robinson's constraint qualification, Gauvin's boundedness
Monday Second order optimality conditions Manual Background
November for the equality and inequality Scripts 3
30 constrained optimization problem $(P_ Video 3 • Optimization: second order optimality conditions for $(P_E)$, linear optimization duality
2020 {EI})$
Optimality conditions
• Second order optimality conditions for $(P_{EI})$: difficulties, solution strategy, NC2, SC2
Monday Josephy-Newton algorithm Manual Background
December for solving functional inclusions Scripts 4
7 Video 4 • Convex analysis: convex polyhedron
2020 • Algorithmics: speeds of convergence, Newton's method, quasi-Newton algorithms, Dennis and Moré superlinear criterion
Linearization methods
• Overview of linearization methods
• Josephy-Newton algorithm for functional inclusions: functional inclusion problems, semistable solutions of functional inclusions,
speed of convergence of the JN algorithm, semistable solutions of polyhedral variational inequalities
Monday The SQP algorithm Manual Background
December for solving problem $(P_{EI})$ Scripts 5
14 Video 5 • Short overview of algorithms for solving constrained optimization problems: primal, dual, and primal-dual methods
Linearization methods
• Josephy-Newton algorithm for functional inclusions: local convergence
• The SQP algorithm: the algorithm viewed as a Josephy-Newton method, semistability of a solution, local convergence, exact
penalization, globalization by linesearch
Monday The semismooth Newton algorithm Manual Linearization methods
January for solving nonsmooth equations Scripts 6
4 Video 6 is • The semismooth Newton algorithm for nonsmooth equations: Motivation, Clarke differential, semismoothness, local convergence
2020 lost
Monday Semidefinite optimization Manual Background
January Scripts 7
11 Video 7 • Convex analysis: asymptotic function, separation of convex sets
2020 Lecture Notes
(2020-02-17) Semidefinite optimization
Exercises • problem definition and examples, existence of solution, optimality conditions, central path, sketch of an interior point algorithm
Monday Examination
18 • Written examination of 3 hours (2pm-5pm), at Ensta (room 1315). Please, wear a mask and bring your own exam sheets.
2021 • The questions are similar to those given in the homework exercises, although they should be decomposed into more subquestions.
• The documents distributed during the course can be perused. You can put the lecture notes on your computer and look at them during the exam.
• The student will join to his/her script the solution to 2 exercises proposed during the lectures (presumably the hardest he/she was able to solve).
Second part
It is recommended to register to this course, by contacting "Jean-Charles.Gilbert@inria.fr". Indeed, some technical issues depend on the participants (like having an account at Ensta and establishing
access to Matlab through a VPN connection to Ensta, as explained on the page https://cascad.ensta.fr/BYOD).
The course is made up of 5 sessions of 3h45 each (given on Thursday, 2pm-6pm), which makes it 18h long, approximately.
The goal of the course is to guide the student in the implementation of one of some well known optimization/nonsmooth-system algorithms and in its use to solve a concrete problem. The student will
choose one project among the following ones.
• Project SQP+HC. Implementation of an SQP algorithm (a solver of nonlinear optimization) and its use to solve the hanging chain problem. This project is recommended to those who have never
implemented an optimization solver, since it is instructive on many aspects. The project is well organized and guided.
• Project SDCO+OPF. Implementation of an interior point algorithm to solve a self-dual conic optimization problem in real numbers (which concatenates several linear semidefinite optimization
problems and one linear optimization problem) and its use to solve the rank relaxation of a small-size OPF problem (expressed as a QCQP problem) and global minimization of univariate polynomials
on a finite union of intervals. The project is well organized and guided.
• Project SN+Proj. This last project is more exploratory with a large part dedicated to the understanding of all the aspects of a version of a semismooth Newton-like method for making the
projection on a convex polyhedron. This is then applied abstractly and numerically to make the projection on the Birkhoff polytope and, more generally, to minimize a convex quadratic function
(defined on the space of matrices) subject to the Birkhoff constraint. For the while (it may change during the course of the project), the project is completely left to the responsibility of the
student, which makes this project closer to a research subject. The role of the teacher is therefore to discuss with the student during the realization of the project.
Students having already realized (part of) the SQP+HC project must choose another one.
Presentation of the projects (2021-01-19, 19:06). It is recommended to make your choice before the first session (and send it to "Jean-Charles.Gilbert@inria.fr"), although more information on the
projects could be obtained at the beginning of the first session.
The material/concrete aspects of the course are still not determined, due to the pandemic (20-1-2021). Clearly, the course will be given online. Since it aims at realizing an optimization solver, the
student must have the possibility to implement it on his/her machine, preferably in Matlab (or Octave?). If an access to Matlab is not furnished through Ensta (this has not been determined yet),
there might be possible to use other free languages (preferably interpreters), like Python, Julia,..., but the teacher may not have the skill to guide the student with all these languages. The access
to Matlab should be possible through a VPN channel to Ensta.
Actual program on a daily basis
Date Project SQP+HC Project SDCO+OPF Project SN+Proj
Thursday Problem modeling and simulator design Directions of displacement Projection on the Birkhoff polytope
January 28 Oracle verification Notes
2021 (2pm-6pm) Notes
Thursday The local SQP algorithm Moving along the central path
February 4 Notes (7-2-2021) Notes
2021 (2pm-6pm)
Short report and code to submit before Sunday 7th of February, 10 pm Short report and code to submit before Sunday 7th of February, 10 pm
Thursday Globalization by linesearch Finding an appropriate starting point
February 11
2021 (2pm-6pm) Short report and code to submit before Sunday 14th of February, 10 pm Short report and code to submit before Sunday 14th of February, 10 pm
Thursday Quasi-Newton version Starting from an infeasible point
February 18 (Last session statement)
2021 (2pm-6pm)
Thursday A few applications
February 25
2021 (2pm-6pm)
Thursday Oral examination Oral examination Oral examination
March 4 (discussion in front of the screen) (discussion in front of the screen) (discussion in front of the screen)
2021 Examination
Timetable m2optim_2020_2021_a.m
Documents useful for the SQP+HC project:
• A script for drawing the floor: floor.m.
• A function doing a modified Cholesky factorization of a symmetric matrix: cholmod.m.
Documents useful for the SDCO+OPF project:
• Two Sedumi matrix-vector converters: mat.m, vec.m.
• A function translating in the SDCO format the problem of finding the global minimum of a univariate polynomial on an interval: polmin2sdco.m.
• A function translating in the SDCO format some small size OPF problems (m2o_testcase_5.m) and three data sets (matpower_casewb2_qcqp.txt, matpower_casewb5_qcqp.txt, matpower_case9_qcqp.txt).
Documents useful for the SN+Proj project:
• F.H. Clarke (1990). Optimization and Nonsmooth Analysis (second edition). Classics in Applied Mathematics 5. SIAM, Philadelphia, PA, USA. [DOI]
• Y. Cui, D.F. Sun, K.-C. Toh (2016). On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions Technical report. [arXiv]
• J.Y. Han, D.F. Sun (1997). Newton and quasi-Newton methods for normal maps with polyhedral sets. Journal of Optimization Theory and Applications, 94, 659-676. [DOI]
• J.Y. Han, D.F. Sun (1997). Newton and quasi-Newton methods for normal maps with polyhedral sets. Journal of Optimization Theory and Applications, 94, 659-676. [DOI]
• J.-B. Hiriart-Urruty, J.-J. Strodiot, V.H. Nguyen (1984). Generalized Hessian matrix and second-order optimality conditions for problems with $C^{1,1}$ data. Applied Mathematics and Optimization,
11, 43-56. [DOI]
• X. Li, D. Sun, K.-C. Toh (2016). QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming. Technical report. [arXiv]
• X. Li, D. Sun, K.-C. Toh (2020). On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope. Mathematical Programming, 179, 419-446. [DOI]
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Colette Calmelet – Department of Mathematics and Statistics
Colette Calmelet
Education: Ph.D. Vanderbilt University
Research Areas: Mathematical Biology, Differential Equations, Mathematical Modeling
Research Interests:
• Biomathematics - Mathematical modeling and simulations of biological problems
• Cell movements in developmental biology, zebra fish gastrulation, angiogenesis of the mouse retina
• Cancer stem cells hypothesis
• Dynamics in peritoneal dialysis
• Bioinformatics - Mathematical methods in sequence analysis
• Fluid dynamics - Study of multi-phase flows, micropolar fluids, porous media, blood flow through capillaries
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What is: Significance Level
What is Significance Level?
The significance level, often denoted as alpha (α), is a fundamental concept in statistics that determines the threshold for rejecting the null hypothesis in hypothesis testing. It represents the
probability of making a Type I error, which occurs when a true null hypothesis is incorrectly rejected. In practical terms, the significance level sets the standard for how much evidence is required
to conclude that an observed effect or relationship is statistically significant. Commonly used significance levels include 0.05, 0.01, and 0.10, with 0.05 being the most widely accepted in many
fields of research.
Understanding the Role of Significance Level in Hypothesis Testing
In hypothesis testing, researchers formulate two competing hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically posits that there is no effect or no
difference, while the alternative hypothesis suggests that there is an effect or a difference. The significance level is crucial because it defines the cutoff point for determining whether the
observed data is sufficiently inconsistent with the null hypothesis to warrant its rejection. If the p-value, which measures the strength of evidence against the null hypothesis, is less than or
equal to the significance level, the null hypothesis is rejected in favor of the alternative hypothesis.
Common Significance Levels and Their Implications
The choice of significance level can significantly impact the results of a study. A significance level of 0.05 implies that there is a 5% risk of concluding that a difference exists when there is
none. This level is often used in social sciences and biomedical research. A more stringent significance level, such as 0.01, reduces the likelihood of Type I errors but increases the risk of Type II
errors, where a false null hypothesis is not rejected. Conversely, a significance level of 0.10 may be used in exploratory research where the consequences of Type I errors are less severe, allowing
for a more lenient approach to hypothesis testing.
Significance Level and P-Values
The relationship between significance level and p-values is central to hypothesis testing. The p-value quantifies the probability of obtaining test results at least as extreme as the observed
results, given that the null hypothesis is true. When researchers calculate a p-value, they compare it to the predetermined significance level. If the p-value is less than or equal to the
significance level, the results are considered statistically significant, indicating strong evidence against the null hypothesis. This comparison is essential for making informed decisions based on
statistical analyses.
Choosing the Appropriate Significance Level
Selecting an appropriate significance level is a critical decision that depends on the context of the research and the potential consequences of errors. In fields where false positives can lead to
severe consequences, such as medical trials, a lower significance level (e.g., 0.01) is often preferred. Conversely, in exploratory studies or preliminary research, a higher significance level (e.g.,
0.10) may be acceptable to encourage the discovery of new hypotheses. Researchers must carefully consider the trade-offs between Type I and Type II errors when determining the significance level for
their studies.
Significance Level in Multiple Testing Scenarios
In situations where multiple hypotheses are tested simultaneously, the significance level must be adjusted to account for the increased risk of Type I errors. This phenomenon, known as the multiple
testing problem, can lead to misleading conclusions if not properly addressed. Techniques such as the Bonferroni correction or the Benjamini-Hochberg procedure can be employed to adjust the
significance level, ensuring that the overall error rate remains controlled. These adjustments are crucial in fields such as genomics and psychology, where large datasets often lead to multiple
Limitations of Significance Level
While the significance level is a widely used tool in statistical analysis, it is not without its limitations. One major criticism is that it can lead to a binary mindset, where results are
categorized as either significant or not significant, potentially oversimplifying the complexity of the data. Additionally, the reliance on arbitrary thresholds can result in different conclusions
depending on the chosen significance level. Researchers are encouraged to report effect sizes and confidence intervals alongside p-values to provide a more comprehensive understanding of their
Significance Level and Confidence Intervals
The significance level is closely related to confidence intervals, which provide a range of values within which the true population parameter is likely to fall. A confidence interval is constructed
based on the significance level; for example, a 95% confidence interval corresponds to a significance level of 0.05. This relationship underscores the importance of understanding both concepts in the
context of statistical inference. Reporting confidence intervals alongside significance levels allows researchers to convey the precision of their estimates and the uncertainty inherent in their
Conclusion on the Importance of Significance Level in Data Science
In the realm of data science, the significance level plays a pivotal role in guiding decision-making processes based on statistical evidence. As data-driven approaches become increasingly prevalent
across various industries, understanding the implications of significance levels is essential for interpreting results accurately. By grasping the nuances of significance levels, researchers and
practitioners can enhance the rigor of their analyses and contribute to more reliable and valid conclusions in their respective fields.
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Warwick Population (2024) - Total Population
Quick answer: Warwick is a local authority in England. The population of Warwick is 148,700. The population per square mile of Warwick is 1,362. The average salary in Warwick is £35,209 per year. The
average property price in Warwick is £398,117.
Books about Warwick: Numerous fiction and non-fiction books have been produced about the rich history of Warwick. Here is a link to some of the books about Warwick.
Warwick Demographics
The population of Warwick is 148,700. The population per square mile of Warwick is 1,362. The population has increased by 3,800 in the past year. The population has increased by 33,400 since 1981.
There are more females in Warwick. They account for 50.3% of the population.
Warwick Employment
78.6% of the people in Warwick are in employment. There are more males in employment. The definition of employment is being employed or self-employed and aged 16 to 64.
The unemployment rate in Warwick is 3.6%. The definition of unemployment is being economically active but unemployed. The average unemployment rate nationally is 3.8%.
Warwick Student Population
The student population of Warwick is 7,200. Students account for 44.2% of the economically inactive population aged 16 to 64.
91.7% of the people in Warwick have an NVQ1 qualification. 83.3% have an NVQ2 qualification. 65.6% have an NVQ3 qualification. 49.4% have an NVQ4 qualification.
3.6% of the people in Warwick have no qualifications. The national average for people with no qualifications is 6.6%.
Warwick Economy
There are 7,165 businesses in Warwick. 88.6% are micro businesses. 9.1% are small businesses. 1.7% are medium businesses. 0.6% are large businesses.
The biggest industry employers in Warwick are wholesale and retail trade; repair of motor vehicles and motorcycles businesses.
Warwick Salary
The average salary in Warwick is £35,209 per year. Males earn 27.6% more.
Salaries in Warwick are 5.47% more than the national average of £33,384.
Warwick Property
The average property price in Warwick is £398,117. The national average property price is £294,910.
Average property prices by type in Warwick: £661,896 for detached houses. £390,808 for semi-detached houses. £322,250 for terrace houses. £217,515 for flats.
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Origin and purpose of this book
«Music is arithmetical work of the mind,
to which it remains concealed that it thinks in numbers».
Gottfried Wilhelm von Leibniz (17th century).
Since the classical antiquity music and mathematics have been described as a wonderful pair. Yet until now the lucky relation between the two disciplines has been associated neither with primary
school mathematics nor with the topics of primary school musical instruction. Topics of an interdisciplinary discussion have rather been questions of harmonics, acoustics, and - for some years - the
digitalization of sound or the discussion about the pitch of instruments and the mathematical conceivability of composition and interpretation.
The domains of physics, stochastics, combinatorics and informatics dealing with these topics do not have much in common with the curriculum of primary school. Therefore, it is not surprising that the
kick-off to «Maths by Music», a project about educational material, did not develop out of this ancient relationship of the two disciplines but out of the observation of learning children and of
successful instruction.
These observations have been supported and promoted by the concept of multiple intelligence formulated in 1983 by Howard Gardner and the sign-system oriented comprehension of musical instruction that
has been based on it.
Proceeding from the experiences with the project "increased musical instruction" the work with the participating classes soon showed that a modified time-table also entailed quite some changes apart
from musical instruction.
The intense work with music soon urged out of its topical borders and the musical perspective offered interesting practices and tasks in the instruction of languages, mathematics and general studies
- or formulated the other way round - we realized that many topics and methods in school are basically full of music.
Regarding the concept of transdisciplinary teaching, it becomes apparent that - apart from the interdisciplinary subjects, which so to say lie "between" subject-matters - activities are important,
which illuminate and explore the disciplinary work from the point of view of other subjects (transdisciplinary thinking and acting).
For the realization of this discovery we have created the term of "Music as a teaching principle". This led to the availability of lessons with much music for all interested teachers and
independently of time-tables.
Practical work with many classes has shown since that the elementary connections between instruction in mathematics and music enrich experience-oriented, exploratory and pleasure-centered learning
and teaching in many ways.
On the other hand, didactics of mathematics have since the beginning of the nineties developed in direction of John Deweys’ (1859-1952) concept of exploratory learning. Together with this new
orientation, the point of view and way of thinking of children are more and more taken into consideration and approaches are postulated, which are linked to the context of teaching subjects. This
change also implies "concentration on fundamental ideas of arithmetics and geometry" and "turning away from instruction in tiny steps in favor of a conceptual entireness of the learning situation".
Based on this concept, the means and ways of visualization are basically discussed as well. The way from perception to mental conception is comprehended as idiosyncratic constructional process (which
also depends on the person), which consciously puts up with individual interpretation and differential paths of solution. As a means of illustration, we have introduced some charts and special
materials that must comply with a number of criteria. e.g.:
• adequate representation of the structure of the mathematical fact
• manifold possibilities of usage
• possibility of continuation
• simple handling and simple (easily surveiable) structure
• simple possibility of transfer to graphical representations
• easy practicability of mental operations
• possibility of discovering individual and differential strategies of solution and of social interchange on it
• continuous availability for all students and demo version for the class
• low price
• stability and environmentally harmless material
Sound and motion highly comply with these conditions. Due to the missing tradition of a corresponding approach, the two have until now hardly been brought into consideration as working or
illustrative material - as tools for teaching and learning. If they appear in educational material, corresponding patterns of action are applied quite accidentally now and then and mostly as a means
of decoration. This deficit is often compensated by music-minded teachers and their imaginative teaching and methodical skills. But this is no justification for the limitation of educational material
to verbal, visual, haptic and mathematically-abstract approaches; acoustical (sound), kinesthetical (motion) and tactile (touch) impulses are of high value, especially with respect to acting and
experiencing of children of primary school age. The integration of this type of experiences in mathematical instruction can make accessible important active learning paths of children in classroom;
simultaneously, the insight of teachers in unexpected 'thinking paths' of their pupils is promoted.
The present website "Music by Maths" (based on the volume «Mathe macht Musik») wants to help to cultivate approaches to mathematics via sound and motion and to enrich musical instruction.
Training ideas and impulses for a "musical" realisation of actual subjects of the mathematical curriculum as "patterns and classifications", "numerical series and the concept of number", "duration
and length", "addition and substraction", "throwing dice", "decimal system", "numerical series", "calculating with money", "forms and shapes in the world around us", "time" and "mathematical rows"
are shown. Games of perception, instructions to motion, songs, rhythmical games, concentration and observation tasks and exercises for creativity are used to transdisciplinarily exploit the musical
potential and to create a positive atmosphere for learning. This type of instruction also supports the relaxed access of children to music and transdisciplinary acting and thinking.
Many of the proposed musical impulses for mathematical topics and many of the musical exercises and applications of mathematical questions are by themselves not spectacular and they do not imply an
extraordinary musical talent of the teacher. Yet the impact of their consistent application on individual children and on classes as a whole, especially on motivation and the learning climate, is
remarkable. Due to their inborn musical potential, impulses are always fun to play with. The mathematical and musical processes of exercise and automation stimulated by them often can be combined
with the tasks of daily mathematical training ("Blitzrechen").
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[Solved] The value of \(\displaystyle\int_{−1}^1\)[P2(x)]2&nbs
The value of \(\displaystyle\int_{−1}^1\)[P2(x)]2 dx, where P2(x) is the Legendre polynomial of degree 2 is:
Answer (Detailed Solution Below)
Option 2 : 2/5
MPTET Varg 1 History Full Test 1
1.1 K Users
150 Questions 150 Marks 150 Mins
Concept Used:-
When polynomial Pn(x) is the Legendre polynomial of degree n, then the value of its definite integral from -1 to 1 is,
\( \int_{-1}^1 P_n^2(x) d x=\frac{2}{2 n+1}\)
Given integral is,
\(\displaystyle\int_{−1}^1\)[P2(x)]2 dx
Here, P2(x) is a Legendre polynomial of degree 2. So, by the above formula, the value of this integral will be,
⇒ \(\displaystyle\int_{−1}^1\)[P2(x)]2 dx \(= \dfrac{2}{2\times 2+1}\)
⇒ \(\displaystyle\int_{−1}^1\)[P2(x)]2 dx \(= \dfrac{2}{5}\)
So, the value of the given integral is 2/5.
Latest MPTET Varg 1 Updates
Last updated on Feb 14, 2024
-> The MPTET Varg Result has been released for the ongoing 2023 cycle.
-> The MPTET Varg 1 notification was released for 8720 vacancies of High School Teachers.
-> The selection is expected to be based on a written test and document verification.
-> To get a successful selection candidates can refer to the MP Varg 1 previous year papers to know what type of questions are asked and what can be asked in the exam.
-> Also, attempt the MP TET Mock Tests for better practice.
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Printable Calendars AT A GLANCE
Exponential Function Practice Worksheets
Exponential Function Practice Worksheets - Algebra 2 > unit 9. Your students will use these worksheets in order to practice solving functions that contain exponents. 88) ( 0, 6) and (3, 750) 89) (0,
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date homework hour graphing exponential functions worksheet #2 directions : Follow the steps to solve the following problem:
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university. Web graphs of exponential functions (practice) | khan academy. Web access these online resources for additional instruction and practice with exponential functions. Identify the type of
function (linear: Web below you will find an exponential functions word problems worksheet with answers.
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Identify the type of function (linear: Y ax b or exponential: Graph reflections and stretches of exponential functions. Use the interactive graph below to sketch a graph of y = 2 ⋅ 3 − x − 4. Free
trial available at kutasoftware.com.
exponential functions review worksheet
Solve exponential equations using logarithms. These problems are a bit more challenging. 88) ( 0, 6) and (3, 750) 89) (0, 2000) (2, 20) 90) ( − 1, 3 2) and (3, 24) 91) ( − 2, 6) (3, 1) 92) (3, 1) and
(5, 4) answers to odd exercises: A colony of bacteria starts with 9 bacteria at noon..
exponential functions word problems worksheet pdf Function Worksheets
I constructed this worksheet by using the best problems that i have used throughout my teaching career to help students develop an understanding of exponential functions in the. Graph an exponential
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Math Worksheets Exponents
Web the corbettmaths practice questions on exponential graphs. Algebra 2 > unit 9. Identify whether a function is exponential, quadratic, or linear from a graph, equation, or table. Web exponential
functions date_____ period____ evaluate each function at the given value. ★ for the following exercises, find the formula for an exponential function y = a(bx) that passes through the two.
Exponential Functions Notes and Worksheets Lindsay Bowden
Free trial available at kutasoftware.com. Solve these 10 exponential equations. Benefits of graphing exponential functions worksheets. Solving exponential equations review sheet. 88) ( 0, 6) and (3,
750) 89) (0, 2000) (2, 20) 90) ( − 1, 3 2) and (3, 24) 91) ( − 2, 6) (3, 1) 92) (3, 1) and (5, 4) answers to odd exercises:
Answers to Math Exercises & Math Problems Exponential Function
Follow the steps to solve the following problem: You will work on solving 10 exponential equations. These problems are a bit more challenging. Sketch the graph of each function. Web below you will
find an exponential functions word problems worksheet with answers.
How To Write Exponential Functions slideshare
Web here is a set of practice problems to accompany the derivatives of exponential and logarithm functions section of the derivatives chapter of the notes for paul dawkins calculus i course at lamar
university. Web create your own worksheets like this one with infinite algebra 2. Algebra 2 > unit 9. Identify whether a function is exponential, quadratic, or linear.
imath more exercises, re. the graph of the exponential function and
Create your own worksheets like this one with infinite algebra 1. Web exponential functions date_____ period____ evaluate each function at the given value. Sketch the graph of each function. Web
steps for writing and solving functions: Transforming exponential graphs (example 2) graphing exponential functions.
Exponential Functions Notes and Worksheets Lindsay Bowden
Exponential functions date homework hour graphing exponential functions worksheet #2 directions : You will work on solving 10 exponential equations. Free trial available at kutasoftware.com. Free
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Free trial available at kutasoftware.com. Identify whether a function is exponential, quadratic, or linear from a graph, equation, or table. Web steps for writing and solving functions: Use the
interactive graph below to sketch a graph of y = 2 ⋅ 3 − x − 4. Follow the steps to solve the following problem:
Web these worksheets explain how to solve exponential functions, to include finding domains, ranges, and values. State the transformations that must be done to the parent function in order to obtain
the graph and state its domain, range, and horizontal asymptote. Web here is a set of practice problems to accompany the derivatives of exponential and logarithm functions section of the derivatives
chapter of the notes for paul dawkins calculus i course at lamar university. ★ for the following exercises, find the formula for an exponential function y = a(bx) that passes through the two points
State The Transformations That Must Be Done To The Parent Function In Order To Obtain The Graph And State Its Domain, Range, And Horizontal Asymptote.
Web graphing exponential functions worksheets will engage and enhance your development and competency skills. Free trial available at kutasoftware.com Web our pdf evaluating exponential functions
worksheets are best suited for kids in grade 8 and high school. Algebra 2 > unit 9.
★ For The Following Exercises, Find The Formula For An Exponential Function Y = A(Bx) That Passes Through The Two Points Given.
Exercise \(\pageindex{e}\) \( \bigstar \) in the following exercises, graph each exponential function. Web here is a set of practice problems to accompany the derivatives of exponential and logarithm
functions section of the derivatives chapter of the notes for paul dawkins calculus i course at lamar university. Sketch the graph of each function. Graph reflections and stretches of exponential
Web Exponential Functions Date_____ Period____ Evaluate Each Function At The Given Value.
Exponential functions date homework hour graphing exponential functions worksheet #2 directions : If the number of bacteria triples every 20 minutes, how many bacteria will be present at 2:40 pm that
afternoon? Transforming exponential graphs (example 2) graphing exponential functions. Solve exponential equations using logarithms.
These Problems Are A Bit More Challenging.
Web the corbettmaths practice questions on exponential graphs. Identify whether a function is exponential, quadratic, or linear from a graph, equation, or table. A colony of bacteria starts with 9
bacteria at noon. Use the interactive graph below to sketch a graph of y = 2 ⋅ 3 − x − 4.
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ME2204 / FLUID MECHANICS AND MACHINERY 2014 Questions Bank
Anna University, Chennai
ACADEMIC YEAR 2013- 2014 / ODD SEMESTER
SUBJECT CODE & NAME : ME2204 / FLUID MECHANICS AND MACHINERY
YEAR/SEM : II/III
PART-B (16 Marks)
1. a) What are the different types fluids? Explain each type.
b) Discuss the thermodynamic properties of fluids (8)
2. a) One litre of crude oil weighs 9.6 N. Calculate its Specific weight, density and specific weight. (8) b) The Velocity Distribution for flow over a flat plate is given by
u=(2/3)y-y2, Where u is the point velocity in meters per second at a distance y metre above the plate. Determine the shear stress at y=0 and y=15 cm.
Assume dynamic viscosity as 8.63 poises (8)
3. a) A plate, 0.025 mm distant from a fixed plate, moves at 50 cm/s and requires a force of 1.471 N/ m2 to maintain this speed. Determine the fluid viscosity between plates in the poise. (8)
b) Determine the intensity of shear of an oil having viscosity =1.2 poise and is used for lubrication in the clearance between a 10 cm diameter shaft and its journal bearing. The clearance is 1.0 mm
and Shaft rotates at 200 r.p.m (8)
4. a) Two plates are placed at a distance of 0.15mm apart. The lower plate is fixed while the upper plate having surface area 1.0 m2 is pulled at
0.3 Nm/s. Find the force and power required to maintain this speed, if the fluid separating them is having viscosity 1.5 poise. (8)
b) An oil film of thickness 1.5 mm is used for lubrication between a square plate of size 0.9m *0.9m and an inclined plane having an angle of inclination 200 . . The weight of square plate is 392.4 N
and its slides down the plane with a uniform velocity of 0.2 m/s. find the dynamic viscosity of the oil. (8)
5. a) Assuming the bulk modulus of elasticity of water is 2.07 x10 6
at standard atmospheric condition determine the increase of
pressure necessary to produce one percent reduction in volume at the same temperature. (8) b)
Calculate the capillary rise in glass tube pf 3mm diameter when immersed in mercury, take the surface tension and angle of contact of mercury as
0.52 N/m and 1300 respectively. Also determine the minimum size of the
glass tube, if it is immersed in water, given that the surface tension of water is 0.0725 N/m
and Capillary rise in tube is not exceed 0.5mm. (8)
6. a) Explain all three Simple manometers with neat sketch. (8)
b) Explain Differential manometer With Neat sketch. (8)
1. a) Derive an expression for the velocity distribution for viscous flow through a circular pipe. (8)
b) A main pipe divides into two parallel pipes, which again forms one pipe. The length and diameter for the first parallel pipe are 2000m and 1m respectively, while the length and diameter of second
parallel pipe are 2000 and 0.8 m respectively. Find the rate of flow in each parallel pipe, if total flow in the main is 3 m³/s. The
coefficient of friction for each parallel (8)
pipe is same and equal to 0.005.
2. Two pipes of 15 cm and 30 cm diameters are laid in parallel to pass a total discharge of 100 liters/ second. Each pipe is 250 m long. Determine discharge through each pipe. Now these pipes are
connected in series to connect two tanks 500 m apart, to carry same total discharge. Determine water level difference between the tanks. Neglect minor losses in both cases, f=0.02 fn both pipes. (8)
b) A pipe line carrying oil of specific gravity 0.85, changes in diameter from 350 mm at position 1 to 550 mm diameter to a position 2, which is at 6 m at a higher level. If the pressure at position
1 and 2 are taken as 20
N/cm2 and 15 N/ cm2 respectively and discharge through the pipe is 0.2
m³/s. determine the loss of head. (8)
3. Obtain an expression for Hagen- Poisulle flow. Deduce the condition of maximum velocity. (16)
4. A flat plate 1.5 m X 1.5 m moves at 50 km / h in a stationary air density
1.15 kg/ m³. If the coefficient of drag and lift are 0.15 and 0.75 respectively, determine (i) the lift force (ii) the drag force (iii) the resultant force and (iv) the power required to set the
plate in motion .(1 6)
5. a). The rate of flow of water through a horizontal pipe is 0.3 m³/s. The diameter of the pipe is suddenly enlarged from 25 cm to 50 cm. The
pressure intensity in the smaller pipe is 14N/m². Determine (i) Loss of head due to sudden enlargement. (ii) Pressure intensity in the large pipe and (iii) Power lost due to enlargement. (8)
b) Water is flowing through a tapering pipe of length 200 m having diameters 500 mm at the upper end and 250 mm at the lower end, the pipe has a slope of 1 in 40. The rate of flow through the pipe is
250 lit/ sec. the
pressure at the lower end and the upper end are 20 N/cm² and 10 N/ ²
respectively. Find the loss of head and direction of flow (8)
6. A horizontal pipe of 400 mm diameter is suddenly contracted to a diameter of 200 mm. The pressure intensities in the large and small pipe is
given as 15 N/ ²
and 10 N/ ²
respectively. Find the loss of head due to
contraction, if Cc=0.62, determine also the rate of flow of water. (8)
7.Determine the length of an equivalent pipe of diameter 20 cm and friction factor 0.02 for a given pipe system discharging 0. 1m³ s. The pipe system
consists of the following: (16)
(i) A 10 m line of 20 cm dia with f=0.03 (ii)Three 90º bend, k=0.5 for each
(iii) Two sudden expansion of diameter 20 to 30 cm
(iv) A 15 m line of 30 cm diameter with f=0.025
(v) A global valve, fully open,
1. The frictional torque T of a disc diameter D rotating at a speed N in a fluid of
Viscosity μ and density ρ in a turbulent flow is given by T=D 5N
2 ρФ(μ/D 2 Nρ). Prove this Buckingham ’s Π theorem. 16)
2. Explain the different types of similarities.
3. Explain the dimensional analysis with suitable example.
2. 4. The frictional torque T of a disc diameter D rotating at a speed N in a fluid of
Viscosity μ and density ρ in a turbulent flow is given by T=D 5N
2 ρФ(μ/D 2 Nρ). Prove this Rayleigh’s Π theorem. 16)
1. Obtain en expression for the work done per second by water on the runner of a –pelton wheel. Hence derive an expression for maximum efficiency of the pelton wheel giving the relationship between
the jet speed and bucket speed. 16)
2. a) A pelton wheel is having a mean bucket diameter of 1 m and is running at 1000 rpm. The net head on the pelton wheel is 700 m. If the side clearance angle is 15º and discharge through nozzle is
0.1 m³ s, find
(1) power available at nozzle and (2) hydraulic efficiency of the turbine. Take Cv=1 (8) b) A turbine is to operate under a head of 25 m at 200 rpm. The discharge is 9 m³ s. If the efficiency is 90%
determine, Specific
speed of the machine, Power generated and type of turbine. (8)
3. A pelton turbine is required to develop 9000 KW when working under a head of 300 m the impeller may rotate at 500 rpm. Assuming a jet ratio of 10 And an overall efficiency of 85% calculate (1)
Quantity of water required. (2) Diameter of the wheel (3) Number of jets (4) Number and size of the bucket vanes on the runner. (16)
An Outward flow reaction turbine has internal and external diameters of the runner as 0.5 m and 1.0 m respectively. The turbine is running
at 250 rpm and rate of flow of water through the turbine is 8 m³ s. The
width of the runner is constant at inlet and out let and is equal to 30 cm. The head on the turbine is 10 m and discharge at outlet6 is radial, determine (1) Vane angle at inlet and outlet. (2)
Velocity of flow at inlet and outlet. 16)
5. The Nozzle of a pelton Wheel gives a jet of 9 cm diameter and velocity
75 m/s. Coefficient of velocity is 0.978. The pitch circle diameter is 1.5 m and the deflection angle of the bucket is 170º. The wheel velocity is 0.46 times the
jet velocity. Estimate the speed of the pelton wheel
turbine in rpm, theoretical power developed and also the efficiency of
the turbine. (16)
6. a) A turbine is to operate a head of a 25 m at 200 rpm; the available discharge is 9 m³/s assuming an efficiency of 90%. Determine (1) Specific speed (2)
Power generated (3) Performance under a head of
20 m (4) The type of turbine (8)
b) A vertical reaction turbine under 6m head at 400 rpm the area and diameter of runner at inlet are 0.7 m² and 1m respective the absolute and relative velocities of fluid entering are 1 5ºand 60º to
the tangential
direction. Calculate hydraulic efficiency. (8)
7. A Francis turbine has an inlet diameter of 2.0 m and an outlet diameter
of 1 .2m. The width of the blades is constant at 0.2 m. The runner rotates at a speed of 250 rpm with a discharge of 8 m³ s .The vanes are radial at the inlet and
the discharge is radially outwards at the outlet. Calculate the angle of guide vane at inlet and blade angle at the outlet. (16)
8. A Kaplan turbine develops 20000KW at a head of 35 m and at rotational speed of 420 rpm. The outer diameter of the blades is 2.5 m and the hub diameter is
0.85m. If the overall efficiency is 85% and the hydraulic
efficiency is 88%. Calculate the discharge, the inlet flow angle and the blade angle at the inlet. (16)
1. Write short notes on the following (1) Cavitations in hydraulic machines their causes, effects and remedies. (2) Type of rotary
pumps. (16)
2. Draw a neat sketch of centrifugal pump and explain the working principle of the centrifugal pump. (16)
3. Draw a neat sketch of Reciprocating pump and explain the working
principle of single acing and double acting Reciprocating pump. (16)
4. A radial flow impeller has a diameter 25 cm and width 7.5 cm at exit. It delivers 120 liters of water per second against a head of
24 m at 1440 rpm. Assuming the vanes block the flow area by 5% and hydraulic efficiency of 0.8, estimate the vane angle at exit. Also calculate the torque exerted on the driving shaft if the
mechanical efficiency is 95%. (16)
5. Find the power required to drive a centrifugal pump which to drive a centrifugal pump which delivers 0.04 m3 /s of water to a height of 20 m through a 15 cm diameter pipe and 100 m long. The over
all efficiency of the pump is
70% and coefficient of friction is 0.15 in the formula hf=4flv2/2gd. (16)
6. A Centrifugal pump having outer diameter equal to 2 times the inner diameter and running at 1200 rpm works against a total head of 75 m. The Velocity of flow through the impeller is constant and
equal to 3 m/s. The vanes are set back at an angle of 30º at out let. If the outer diameter of impeller is 600 mm and
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Excel Dynamic Named Ranges
Excel dynamic named ranges bring flexibility and efficiency to your spreadsheets.
Unlike static named ranges, dynamic named ranges automatically adjust in size as you add or remove data, ensuring that your formulas, PivotTables and charts always include the most recent
You can also use them to return different ranges based on selections in drop down lists.
If you frequently find yourself editing cell ranges that are referenced by formulas, PivotTables, charts, and other Excel features, then using dynamic named ranges will significantly streamline your
workflow and save you a substantial amount of time.
Watch the Video
Dynamic named ranges are a complex topic that are more easily understood if you can see them in action.
I recommend watching the video for the easiest way to get your head around them and then use the written notes below as a reference point.
Download Example Workbook
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Excel Dynamic Named Ranges with INDEX
Most people think of the OFFSET function for dynamic named ranges. Unfortunately, OFFSET is volatile, meaning it recalculates even if none of its arguments have changed, leading to more frequent
updates and potential performance issues.
In general, it's a good practice to be mindful of the use of volatile functions, especially in large or complex Excel files, to ensure that they don't inadvertently slow down your work.
I prefer to use the INDEX function for dynamic named ranges.
INDEX has two syntax options, and for dynamic named ranges we use this version:
Syntax: =INDEX(array, row_num, column_num)
array – the range of cells you want to return a range from.
row_num – the row(s) you want to return.
column_num – the column(s) you want to return.
Applying it to this example data:
I like to write my dynamic named range formulas in a cell in the worksheet as it’s easy to construct them using the mouse to select the ranges and I can test they work as expected before defining
them as a name.
Return a Range with Flexible Last Row and Column
This type of dynamic named range is useful as the source for PivotTables, and lookup formulas where you want to look up an entire table.
I can use INDEX with the COUNTA function to determine the current size of the table and return the range.
I’ve allowed for growth to row 16 and column L, as shown in the image below with the grey dashed line indicating the range that will be returned by the formula:
Each argument explained:
The first cell in the range I want returned. In this case it will always be the top left cell. Tip: if you don’t need the header row in the range, start at cell B9 and adjust the row_num COUNTA
formula to also start in row 9.
Colon range operator.
Use INDEX to find the last row and column in the table using COUNTA for the row and column arguments.
Select cells that represent the maximum size table will potentially grow to.
Counts the row labels in the first column to determine the current height of the table. The height of this range must match the height of the INDEX range and should not contain any empty cells.
Counts the column labels in the first row to determine the current width of the table. The width of this range must match the width of the INDEX range and should not contain any empty cells.
Note: when writing formulas for dynamic named ranges make sure the cell references are all absolute references.
Return a Range for a Specific Row
Sometimes you only need one row returned. It could be based on a selection in a data validation list or another cell, for use in a chart, table or a formula.
For example, below I can choose a different category from the data validation list and the values are returned and displayed in the chart.
Note: Excel 2019 and earlier users will not be able to display the values being returned by the formula in the grid (cells C38:F38) as you do not have dynamic array functionality. However, you can
skip this step and simply define the formula as a name for use in charts etc.
The formula for returning a range for a specific row uses INDEX and MATCH on both sides of the colon range operator:
Each argument explained:
Use INDEX to find the first cell in the range you want returned.
The first cell I want returned is in the first value column of the table (column C) and I’ve allowed the data to grow to row 16.
MATCH is used to find the row the category selected in cell B38 is on, in the range B8:B16.
Technically this argument can be skipped as only one column is selected in INDEX’s array argument, but I’ve entered 1 for completeness.
Colon range operator
Use INDEX to find the last cell in the range.
The range containing the last cell, allowing for growth in rows and columns.
MATCH is used to find the row the category selected in cell B38 is on, in the range B8:B16.
Counts the column labels in the first row to determine the current width of the table. The width of this range must match the width of the INDEX range and should not contain any empty cells.
Return a Range for a Specific Column
Similarly, we can use INDEX to return a specific column based on the selection made in the data validation list:
Note: Excel 2019 and earlier users will not be able to display the values being returned by the formula in the grid (cells C51:C54) as you do not have dynamic array functionality. However, you can
skip this step and simply define the formula as a name.
The formula for returning a range for a specific column uses INDEX on both sides of the colon range operator:
Each argument explained:
Use INDEX to find the first cell in the range.
The first cell I want returned could be in any column in the first row of the table and I’ve allowed the data to grow to column L.
This argument is skipped because there is only one row indexed, therefore a row_num is not required.
MATCH is used to find the column for the year selected in cell C50 in the range C8:L8.
Colon range operator
Use INDEX to find the last cell in the range.
The range containing the last cell, allowing for growth in rows and columns.
Counts the row labels in the first column to determine the current height of the table. The height of this range must match the height of the INDEX range and should not contain any empty cells.
MATCH is used to find the column that the year selected in cell C50 is in the range C8:L8.
Defining Names
The power of these formulas comes when you define them with a name. That name can then be referenced multiple times in formulas, PivotTables, charts and more.
To define the formula as a name, make sure all the references are absolute. Then copy the formula > Formulas tab > Define name:
Name it and paste in the formula in the ‘Refers to’ field:
You can then reference the name anywhere you’d normally use a cell reference.
For example, in the image below I’ve summed the dynamic named range, and if I evaluate the formula, you can see that the defined name actually returns a reference to the range $C$10:$F$10
Dynamic Named Ranges with INDEX Key Points
Dynamic named ranges are incredibly flexible and with flexibility often comes complexity.
To summarize the key points:
• For dynamic ranges that you expect to grow, INDEX a range larger than the current data size and use COUNTA in the row_num and column_num arguments.
• For dynamic ranges linked to drop down lists or other cells where the user can specify what they want returned, use MATCH in the row_num and column_num arguments.
• If the start and end of the range need to be flexible, use INDEX on both sides of the colon range operator.
• Ensure all cell references are absolute before creating the defined name. There are exceptions to this which I cover in my Relative Named Range tutorial.
Excel Dynamic Named Ranges with OFFSET
The OFFSET function also returns a range that can be made dynamic with the use of the MATCH function and COUNTA.
However as mentioned above, it’s a volatile function and therefore should be used sparingly.
Let’s recreate the same dynamic ranges using OFFSET to see how it compares.
Syntax: =OFFSET(reference ,rows, cols, [height], [width])
reference – the starting cell/range of cells
rows – the number of rows to move +/- from the starting cell to arrive at the first row in the range.
cols - the number of columns to move +/- from the starting cell to arrive at the first column in the range.
height – the number or rows high you want the range.
width – the number of columns wide you want the range.
Return a Range with Flexible Last Row and Column
This type of dynamic named range is useful as the source for PivotTables, and lookup formulas where you want to look up an entire table.
I can use OFFSET with COUNTA to determine the current size of the table and return the range.
I’ve allowed for growth to row 16 and column J, as shown in the image below with the grey dashed line indicating the range that will be returned by the formula:
Each argument explained:
The first cell in the range will be the top left of the table.
This argument is skipped because I don’t want to move down any rows from the reference cell before starting the range.
This argument is skipped because I don’t want to move across any columns from the reference cell before starting the range.
Counts the row labels in the first column to determine the current height of the table. This column should not contain any empty cells.
Counts the column labels in the first row to determine the current width of the table. This row should not contain any empty cells.
Return a Range for a Specific Row
Returning a specific row that’s based on a selection in a data validation list or another cell, for use in a chart, table or a formula is also doable with OFFSET.
The OFFSET formula also uses MATCH to locate the row for the category selected in cell B39 and COUNTA to determine the current width of the table:
Each argument explained:
The first cell in the range will be the first Year column label. Notice the reference starts in the header row, because MATCH will return a value between 1 and 8, so at a minimum the starting cell
will be 1 row below the reference cell.
MATCH is used to find the row the category selected in cell B39 is on in the range B9:B16. The lookup range allows for growth in the table to row 16.
This argument is skipped because I don’t want to move across any columns from the reference cell before starting the range.
This argument can be skipped because by default it will return 1 row.
Counts the column labels in the first row to determine the current width of the table. This row should not contain any empty cells.
Return a Range for a Specific Column
Similarly, we can use OFFSET to return a specific column based on the selection made in the data validation list:
Again, the OFFSET formula uses MATCH to find the relevant column and COUNTA to determine the height of the range.
Each argument explained:
The first cell in the range will be determined based on the year selected in the data validation list. Notice the reference starts in the row labels column because MATCH will return a value between 1
and 8, so at a minimum the starting cell will be 1 column to the right of the reference cell.
This argument is skipped because I don’t want to move down any rows from the reference cell before starting the range.
MATCH is used to find the column the year selected in cell C55 is in, in the range C8:J8. The lookup range allows for growth in the table to column J.
Counts the row labels in the first column to determine the current height of the table. This column should not contain any empty cells.
This argument can be skipped because by default it will return 1 column.
Using Dynamic Named Ranges in Charts
One of the best uses for dynamic named ranges is as the source for charts, enabling them to automatically update as your data grows or selections are made.
However, there are a few tricks to getting them to work with charts.
1. You need a separate name for the axis labels and each series’ values.
2. You need to prefix the name with the sheet name followed by an exclamation mark:
Note: after clicking ‘ok’ in the dialog box above, Excel will replace the sheet name with the file name.
Troubleshooting Errors
The 3 most common causes of errors with dynamic named ranges are:
1. Erroneous data entered in the cells being counted for the height or width.
If you inadvertently enter some data below or to the right of your table inside the area being counted, you will end up with a range larger than it should be.
2. Blanks in the cells being counted for the height or width. The column you choose to count to determine the height of the range must not have any empty cells, otherwise the range will be smaller
than it should be.
3. Forgetting to absolute the cell references in the formula.
The easiest way to check the range being returned by a dynamic named range is to edit the range via the Name Manger (Formulas tab) and click in the Refers to field for the name you want to check.
Excel will put marching ants around the range being returned by the formula:
Relative Dynamic Named Ranges
So far, we’ve looked at dynamic named ranges that when referenced from anywhere in the workbook always return the same range. We achieve this by setting all references in the formulas as absolute.
However, we can also write relative named ranges that return a range relative to the cell in which you refer to it from and they’re particularly handy.
For more see my Relative Named Ranges tutorial.
Alternatives to Dynamic Named Ranges
Dynamic named ranges are a lot of work. It’s far easier to store your data in an Excel Table and use the built in Structured References that work like a dynamic named range by growing with your data.
1. Peg Molter
I love your videos! So helpful and informative! Can you provide some instruction on how to deal with data that results from Excel’s new “array engine”. I like the convenience, but once a dynamic
array is created, the data within it can’t be sorted or manipulated in any way, except to copy and paste it back as hard data. That defeats the whole purpose of formulas. Is there a “trick” to
make data within an array result “manipulable”? Or does sorting, etc. HAVE TO BE done within the formula itself? Thanks, and please keep ’em coming!
□ Mynda Treacy
Hi Peg,
Sorting can be done by wrapping the formula in the SORT function or SORTBY function, or you can reference the spilled array with the # operator and wrap that in SORT or SORTBY to return the
array to another set of cells.
2. Peter Sumner
I have used the OFFSET function before but it is particularly useful if you have dropdown validation lists that may require updating. You simply refer to the Named Range when setting up your
validation. The dropdowns dynamically update without having to ammend your validation settings.
3. Ryan Merullo
Neither the offset nor index approach works with the SUMIFS function if the ranges contain blank cells in different rows. You can use either approach to perform a single operation on a single
expanding range. SUMIFS is an extremely handy function. Thus far, I am using the primitive method of guesstimating liberally beyond my data. I have data, including many blank cells, in range
A4:L1350. I am able to use SUMIFS, not with any of the OFFSET or INDEX formulas offered, but simply with the following:
4. Sophie
Thanks Mynda, I really like your step by steps !
I’m trying to set up a dynamic reference for a table with data re male and female. My goal is to build a chart with 1 serie with data re male and the other re female. Currently, each time I
refresh the data, I sort the table by male/female then edit manually the graph series. Is there a way I could avoid that manual sort and let excel plot the “male” and female” series ?
□ Mynda Treacy
Hi Sohpie,
I’d use a Pivot Table to summarise the data and automatically sort it. That way when you add new data it will automatically re-sort it. Make sure the PivotTable source data is formatted in an
Excel Table so that when you add new data the PivotTable range automatically picks it up upon refresh.
If you get stuck please post your question and Excel sample file on our forum where we can help you further.
5. Ram
Excellent article. A video could have been much better unless it is separately available
6. Terence
Great post, love the multiple options all explained thoroughly.
Just out of curiosity, on the offset example is there a reason for doing cell:offset cell rather than using the height functionality of the offset?
□ Mynda Treacy
Hi Terence,
Thanks for your kind words.
There’s no particular reason for creating the range using cell:offset vs using the height parameter inside of OFFSET. I think I was just being consistent with the approach used to write the
INDEX function dynamic named range, and so I wrote the OFFSET formula that way to make it easier to compare the two functions.
7. George Prattos
Hi Mynda,
After studying your website and trying things out in the process of getting to terms with dynamic ranges, I seem to have created dynamic range names without using any functions at all, i.e.
without Offset, Index or Indirect. This came as a surprise…
Starting with a copy of your raw data (columns A:E) I did the following:
1. Made a table of the raw data, including headings (Table1).
2. In column B: selected all Salesperson data excluding the heading and created a range name SalesPax with it.
3. In column E: selected all Order Amount data excluding the heading and created range name OrderAmt with it.
Both these are dynamic range names by virtue of being table data.
4. In Total Sales cell H2 used the formula =SUMIF(Salespax,G2,OrderAmt) which the system accepted and converted to =SUMIF(Table1[Salesperson],G2,Table1[Order Amount]).
5. Copied H2 to cells H3 through H8.
I’ve tested this by adding and deleting both Salespersons and Order Amounts to/from the original data table and all Total Sales in column H update correctly.
I welcome feedback on this approach. Thank you.
□ Mynda Treacy
Hi George,
Yep, that’s one of the features of Excel Tables. In fact you don’t even need to create named ranges. You can just refer to the table and columns using their Structured References. All
explained here : Excel Tables.
Note: sometimes formatting your data in an Excel table isn’t ideal. For example, they aren’t suitable for data over 500k rows.
8. Bill Tastle
There is (surprisingly!) another way to create this range using INDIRECT.
While this bypasses the OFFSET function, it does require one to grasp the use of INDIRECT. In short, INDIRECT converts any text address as an actual address. Thus, it cannot be used by itself for
it generates a #VALUE! error, the same one as the OFFSET function when used by itself.
□ Mynda Treacy
Hi Bill,
That’s one of the great things about Excel. I would avoid using INDIRECT where ever possible though because it’s a volatile function and can cause performance issues in medium/large
9. Rajesh Sinha
Useful information,, cud u Plzzz post some samples, to Optimize data in Excel, for example PIPS data.
□ Mynda Treacy
Hi Rajesh,
Sorry, I’m not familiar with PIPS data.
10. Phil
Thank you, this was very helpful
11. Tim Hale
I really like your explanation of the dynamic range name. Thank you for sharing.
Tim Hale
□ Mynda Treacy
Cheers, Tim. Glad I could help.
12. Luc Van Uffelen
I successfully applied the INDEX method in my files, but the named ranges disappear in the Name Box. What can be done to avoid this consequence of the INDEX method?
□ Mynda Treacy
Hi Luc,
Dynamic named ranges never appear in the name box. It’s just how it is I’m afraid.
13. Ed
I’m missing something here. I don’t see the advantage of using a dynamic named range rather than an Excel table. It seems to me the whole point of tables is that they are dynamic and
“automatically” generate named ranges without the user having to identify ranges or use complex formula.
□ Mynda Treacy
Hi Ed,
I too like using Excel Table’s dynamic structured references but Tables aren’t always an option so it’s good to know how to create a dynamic named range with formulas too.
For example, Tables can become very slow when they are a lot of formulas and are not always suitable for big data. They’re also not compatible with Excel 2003, although these days no one
should be using Excel 2003 anymore 🙂
Kind regards,
14. Arthur Arkin
I am trying to replicate what you have shown in excel_manual_chart_table, but there is no way I can do it. You seem to assume that one would intuitively know how to do all the things you show but
the instructions are not detailed enough. For example, when I create the tow pivots tables, where do I put them? I notice that the chart for one table is sorted differently, but there was no
instruction to sort anything. Sorry, I’m a beginner with pivot tables and charts, and I need step-by-step instructions, with nothing left to the imagination. Yet, this particular operation is
very important to me. Hope you can help.
□ Mynda Treacy
Hi Arthur,
Great to see you’re having a go at replicating this technique.
If you’ve downloaded the file you’ll see where the PivotTables are in my file, however the location of them can be anywhere in the file. It makes no difference.
You don’t have to sort the PivotTable data, I just did because it’s useful to the chart reader. Sorting is easy, select a value (number) cell in the column you want to sort > right-click >
sort > choose your sort order. You only need to sort the first PivotTable since the values from the second one are brought in using VLOOKUP and will therefore be sorted based on the first
Kind regards,
15. david Ostreicher
Hi Mynda.,
These are great tools for dynamically naming ranges, and I love the way Tables are automatically dynamic.
What I’m trying to do though, is to use the offset or index function (based on a range in a separate sheet, and couple that with the table’s built in function, to make it automatically resize
itself with the same amount of rows as my other range. That way, I imagine, the same way when you add data to the cell directly beneath a table, the table automatically resizes and copies down
all the formulas of the other columns in the table, in this case, as I add a row to my “other” range on the other sheet, my Table will automatically add a line and copy down ALL formulas… even
that for the first data point.
Sorry… I hope I was clear enough.
Thank you very much for all your help.
□ Mynda Treacy
Hi David,
Great to hear you’ll be making use of dynamic ranges.
I’m struggling to picture the dependencies in your workbook. Are you able to send me a sample file with your question so I can better understand, and also give you a tailored answer? You can
send it via the Help Desk.
Kind regards,
16. Joyce
Now when I think of all the old models where I used OFFSET in my dynamic named ranges I will dream about replacing it with INDEX based formulas as you outlined. Thank you for the tip!
□ Mynda Treacy
🙂 great to hear, Joyce.
17. Leah
Hi Mynda ~
For anyone who routinely updates named ranges this is amazing! Of course, now that I’m trying to apply it, I’m stumbling. My first test file uses a named range in a vlookup. The current named
range encompasses 2 columns (the first one is the lookup value [employee id], the second is the return value [employee name]) with data starting in cell B4 (header included). I attempted to use
this formula for the named range but it errors out –=Employee!$B$4:INDEX(=Employee!$B$4:$C$4500,COUNT(=Employee!$b$4:$c$4500)). Either this won’t work across multiple columns or I’ve messed up
the syntax somewhere. Would you mind pointing me back in the correct direction? Thanks!
□ Mynda Treacy
Hi Leah,
It’s tricky to say for sure but, your formula contains = signs where they’re not required so let’s start by eliminating those so your formula looks like this:
Also, your INDEX formula is referencing B4:C4500 which is 2 columns, but you haven’t got a column_num argument in your INDEX formula. You either need to give it a column_num argument or make
your range only 1 column wide.
Your COUNT function is also couting 2 columns: B4:C4500 you need to choose just the column you want to count, and if that column contains text as opposed to numbers then you need to use
COUNTA instead of COUNT. COUNT only counts numbers.
If that doesn’t work then I’ll need you to send me the file via our Help Desk so I can see all of the contributing factors.
Kind regards,
☆ Leah
Thanks Mynda!
I modified the formula to this >> =Employee!$B$4:INDEX(Employee!$B$4:$C$4500,COUNT(Employee!$B$4:$B$4500),2) << and it worked! Absolutely fantastic. I can't wait to share it with my team.
○ Mynda Treacy
That’s great, Leah. You should give yourself a pat on the back for mastering dynamic ranges using INDEX. Well done.
18. Kenneth J. Nessing
Mynda: Another home run, mate! The Dynamic Range info is priceless (actually, I’m sure we could agree on a price!). 😉
□ Mynda Treacy
Cheers, Ken. Just reaching for my calculator now 😉
19. Rebekah
I hv a question in setting range using table.
I hv a report which has 5 sections. Each section starts with a function, eg cell A10 = Admin. Each of the function has the same set chart of accounts.
Everymonth, I generate a new report, copy and paste the contents to a file which has formulae populating the data to a summary sheet in the same file.
The problem I hv is that when a new account is added to the chart of accounts, the range changes. I try to range each function using the table, unfortunately, when I copy and paste the new data,
the table range seems hv removed.
Is there any other way to over come this problem?
thanks for your help.
□ Catalin Bombea
Hi Rebekah,
Looks like you are pasting data over the entire table, including headers, this operation will delete the table. You should not do that, for example, if the first row contain headers, paste
data under the headers row, starting from A2. This way, if you have extra rows, the table will autoexpand to include new rows, and even new columns (the new columns headers will be named
automatically to Column1, Column2)
Of course, this assumes that the data structure is the same for the new data, and new columns are added to the right side of the table, not between initial data structure.
If the headers row structure is not the same, to preserve the table you can paste new headers to headers row, after you paste the data.
20. Maxime Manuel
Whoever from Microsoft Corp that invemted INDEX deserves a place in paradise. This function is gift from heaven.
□ Mynda Treacy
🙂 Indeed, Maxime.
☆ Maxime Manuel
Minda, I never learne Excel but my level is very advanced. I am the Excel Guru at work, no joke. I can do most of the dashboards that I see online, I create my own, I helped all the
departments in my organization using Excel, etc. I finally want to have a certification in advanced Excel with you guys. Can you send me the curriculum of your courses?
Another thing, what is the career of an Excel Guru? What should be his position in an organisation? What path should he follow? Please let me know. Thanks! 🙂
○ Mynda Treacy
Hi Maxime,
It’s great to hear you are helping people get more out of Excel.
You can see our course syllabuses from the Pricing menu, then select the course you’re interested in. Each page with have a link to the syllabus.
Excel Guru’s come in all shapes and sizes and there is no definitive path. For example; you are already a Guru in your workplace. Helping out in Excel forums is a great way to
increase your Excel knowledge. Typically the questions you find in forums are unique and challenging.
All the best with your journey to Excel heights.
Kind regards,
■ Maxime Manuel
Thank you!
21. Vinay J
Hi Mynda, Really love your site. It has taught me so much already – especially the beautiful elegant way you have for explaining any topic. Such a contrast from the inbuilt Help with Excel.
Anyhow, I was wondering if you can help me out with a couple of queries on Dynamic Ranges and their application. Briefly, I have three worksheets (M,B,C) Each of them has an identical columnwise
(15 columns) layout though the number of entries (rows) in each is different and is constantly growing. Hence I have made Dynamic Named Ranges (M,B,C) all with Workbook scope. Currently the
number of rows of data (excluding Headers) for M is 25, for B is 60 and for C is 10.
Now I want to consolidate the three sheets such that all the data from M (25 rows across 15 columns) and B (60 rows across 15 columns) and C (10 rows across 15 columns) should be stacked one of
top of the other i.e. excluding headers, row 1-25 should be M, row 26-86 should be B and row 87-97 should be C in a Master Sheet. Note-it should not sum the data but each individual row should be
reflected in Master sheet such that Master sheet should have 97 rows and 15 columns of data. And thereafter as more and more rows of data are added to the 3 sheets, the Master sheet will
incorporate all the entries (both old and new)
Is it even possible for Named Ranges to be displayed this way or can they only be displayed if some function is applied to them? Of course if it is not possible I would love to understand whether
the issue is with the ranges being non-contiguous, being dynamic or their positioning on multiple worksheets or anything else. Intuitively I think it should be doable but I am unable to implement
VBA solution I have found is to create a temporary range which is a union of the three ranges but I would prefer to do it without VBA as multiple users will handle the document and are not as
proficient/ comfortable.
Alternatively, a solution is to use multiple range/sheet inputs in Pivot tables – however, I am not able to get it to work as the page fields do not
reflect all the column headings the way it reflects when using a single input/sheet range. So solutions from this angle would also be welcome.
Thanks so much!
Keep up the AWESOMENESS!!
□ Mynda Treacy
Hi Vinay,
Thanks for your kind words.
In regards to your question; I don’t know of a way to create a 3D dynamic named range. I’ve never seen it done and I can’t think of a way you could do it without VBA.
I would make all of your worksheets uniform and use PivotTables.
You haven’t said what you need this dynamic range for. I wonder if this tutorial on 3D SUMIF might be relevant.
Kind regards,
☆ Vinay J
No Mynda, 3D Sumif isn’t relevant for this instance, although it seems like an interesting thing to learn.
I am not sure I explained myself correctly. What I am looking for is a way to consolidate multiple ranges together without summing them or any function. Just display the raw consolidated
data one below another.
All worksheets are already uniform in layout – unfortunately I am not able to use the Pivot tables with multiple ranges as inputs as the page fields do not reflect all the column headings
the way it reflects when using a single input/sheet range. Perhaps you have some suggestions/ tips for using inputs from multiple ranges/ worksheets?
○ Mynda Treacy
Hi Vinay,
Yes, I know what you mean with consolidating PivotTables. Unfortnately the only other options are Copy and Paste all the data into one master worksheet, or find some VBA to do the
copying and pasting for you.
Kind regards,
22. Theron P. Yates
dear all. this code copy cell step by step. do you know any other way to write the code easier?
□ Carlo Estopia
Hi Theron,
I really don’t think there can be any other, easier way of doing this code.
However, you may go to our HELP DESK and I’ll give you a file that could automate
some. Just specify what part of this topic explained do you want it to be
☆ Kevin
a quick tip I picked up from another site for data validation
Lets say we are working with Range A1:A10 and A1 has the header to our list
Select A1:A10 and insert table (check the box “mt table has headers”)
(A1:A10 is now a table)
Select A2:A10 and name the range
Insert Data Validation in whatever cell you need it and input the range you named in previous step
thats it 🙂 you now have a dynamic range
If you add data to cell A11 it should appear in your drop down list
○ Carlo Estopia
Hi Kevin,
Thanks for the tips.
23. Keith Thompson
Hi Mynda,
I have found the Dynamic Named Ranges using the INDEX function extremely useful, but have come across one small issue that you might be able to help me with. I have an excel based time-sheet for
staff where they choose the site worked from a drop down list using Data Validation. As the number of sites is always changing I have named the site list using a dynamic named range using the
INDEX function, but now find when you click on the drop down list you see the blank cells and have to scroll up each time to see the site list, is there a way around this?
□ Carlo Estopia
Hi Keith,
It’s good to hear that you have found it useful. I tried to simulate and walked through the file and I had a hard time trying to get
the same problem you have.
Anyway, It would be better for you to send your file through HELP DESK so we can have a better look at it.
□ Hi Keith,
First you need to create a list that doesn’t have blanks. Let’s say your list containing blanks is in cells A2:A9. In cell C2 enter this formula:
Enter with CTRL+SHIFT+ENTER as this is an array formula, and copy down as far as you need.
Then set up your dynamic named range referring to cells C2:??, let’s say C2:C1000 using this OFFSET formula:
Or using INDEX:
Kind regards,
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The Tension Between Productive Struggle and Telling
I’m an avid fisherman. There are times when I like to navigate a new spot and discover on my own the in’s and out’s of a place: how the tides work, where there’s structure that holds fish, depth
changes, bait preferences, etc. I’m a better fisherman due to these experiences. However, there are other instances where I need to be told something directly like how to tie a uni-to-uni knot to
attach my leader. This year I had the opportunity to present at NCTM’s annual conference on this same line of thinking in mathematics. In the session we discussed supporting students in productive
struggle, specifically navigating the decision of when to tell. Here’s a recap of the session:
We know that productive struggle is an important instructional practice in teaching mathematics, but what is productive struggle?
“Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical
ideas and relationships.” (NCTM, 2014, p. 48)
Just as important, what is not productive struggle?
“…when students ‘make no progress towards sense-making, explaining, or proceeding with a problem or task at hand.'” (NCTM, 2014, p. 48)
So, students should be grappling with the mathematical ideas and relationships and if they’re making no progress, then the struggle is no longer productive.
Below are five tips for engaging students in productive struggle, two of which are specific about when to tell.
1. Anticipate the source of the struggle. Beware the “Expert Blind Spot.” (Wiggins & McTighe, 2005, p. 51)
Wiggins & McTighe (2005) describes the “Expert Blind Spot” as when an expert is so good at something that they fail to see a situation from the perspective of a novice. One of the best ways to avoid
this is to do the math ourselves, trying to approach the task as your students would approach it. This gives us insight about what specific aspects of a task students might struggle with and what
could be problematic for a novice. Also, this allows us to think about representations the student may use or misconceptions they may bring to the table.
2. Listen closely. Then ask questions based on student thinking.
The act of listening is a natural way to support someone when they’re struggling, just being present and empathetic. Also, John van de Walle (2006) suggests that we listen closely before doing
anything else. We need to be sure that we really understand a student’s thinking before asking questions to move them forward in their thinking. This ensures students are moving forward in a way that
makes sense to them.
3. Be specific about what you want students to struggle with.
The standards for mathematical practice are an excellent source of specific struggle for students. Consider MP 1 (make sense of problems and persevere in solving them) and MP 7 (look for and make use
of structure). There are some wonderful “struggle words” in just these two mathematical practice standards like “make sense,” “persevere,” and “look for.”
4. Tell when it’s symbolic.
If it’s a mathematical symbol which carries an agreed-upon meaning, then this is a situation in which we need to tell. Also, we want to be careful about what we tell. For instance, = does not
indicate that an answer comes next; it means that both sides of the symbol have the same value. Here it is important that we give students well-crafted, universal definitions that will serve them
well into future grades. The above definition of the equal sign is functional through elementary school and into middle and high school, as well.
5. Tell if no progress is being made on the task,… but just enough.
I know in the past I’ve made the decision to tell and proceeded to give away too much. When we decide that telling is in order, we have to exercise restraint and give students just enough for them to
begin making sense of the problem and moving forward on the task.
During the session there was authentic dialogue around navigating the decision to tell. Professionalism is another principle that NCTM proposes we put into action, not just arriving on time and being
polite, but sharing the details of our practice in order to hone our craft. Look for opportunities to discuss productive struggle and the decision to tell with colleagues. Some great questions to
consider are:
Is this an instance in which I need to tell? If so, how much support does the student need to make sense of the problem and begin making progress on the task?
What question(s) could I ask to clarify how the student is thinking about the task?
What question(s) could I ask to the student to move them forward in a way that builds on their thinking?
Leinwand et al. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA. The National Council of Teachers of Mathematics.
Smith, M.S. & Stein, M.K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston, VA. The National Council of Teachers of Mathematics
Van de Walle, J. A. & Lovin, L. H. (2006). Teaching student-centered mathematics: Grades 3-5. United States of America: Pearson Education, Inc.
Wiggins, G. & McTighe J. (2005). Understanding by design expanded (2nd Ed.). Alexandria, VA: Association for Supervision and Curriculum Development.
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Chapter 10 The Math for Kirchhoff Voltage and Current Laws along with Polarity in DC Circuits. - ppt download
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on my rusty sparse merkle tree experiment
tl; dr
today i go over my implementation of a simple library for authenticated data structures and sparse merkle trees.
the source code, in rust, can be found here.
🎵 today’s mood
“the first half of life is devoted to forming a healthy ego, the second half is going inward and letting go of it.” - carl jung
“the ego refuses to be distressed by the provocations of reality, to let itself be compelled to suffer. it insists that it cannot be affected by the traumas of the external world; it shows, in fact,
that such traumas are no more than occasions for it to gain pleasure.” - sigmund freud
001. authenticated data structures
an authenticated data structure (ADS) is an advanced data structure on which an untrusted prover can query for an entry, receiving a result and a proof so that the response can be efficiently checked
for authenticity.
you can think of ADSs as cryptographic upgrades of the classic algorithms we are used to (such as hash maps, binary trees, or tries), with an extra operation added to the good an old insert(), lookup
(), delete(), update(): the commit().
in other words, we can utilize well-known cryptographic hash functions (such as SHA-256 or SHA-3) to calculate a collision-free data structure representation.
we call this hash representation string the commitment (e.g., a small and unique cryptographic representation of the data structure). commitments must uniquely determine the data structure's value (
i.e., if a.commit() == b.commit(), then a == b).
because we trust our hash function, every query should have a valid collision-free proof + a valid proof should imply that the result is correct. this means that proofs for queries must be complete
and sound, where completeness means that every valid query result has a valid proof and soundness means that valid proofs imply that the query results are correct.
💡 according to seminal Miller et al, "an authenticated data structure (ADS) is a data structure whose operations can be carried out by an untrusted prover, the results of which a verifier can
efficiently check as authentic. this is done by having the prover produce a compact proof that the verifier can check along with each operation’s result. ADSs thus support outsourcing data
maintenance and processing tasks to untrusted servers without loss of integrity."
cryptographic hash functions
a cryptographic hash function H is a special function that takes an arbitrarily long sequence of bytes and returns some fixed-size "digest" of that sequence. cryptographic hash functions have two
special properties for our purposes:
• preimage resistance: given a digest d, it's infeasible to calculate a string s such that H(s) = d.
• collision resistance: it's infeasible to find two strings s1 and s2 such that H(s1) == H(s2).
for this library, we will be using the SHA-256 hash function, which has a 256-bit digest.
010. authenticated (sorted) key-value stores
an authenticated key-value store is an ADS of an "associative array" or "map".
the methods of the data structure are described in src/kv_trait.rs of my code:
fn new() -> Self;
fn commit(&self) -> Self::Commitment;
fn check_proof(key: Self::K, res: Option<Self::V>, pf: &Self::LookupProof,
comm: &Self::Commitment) -> Option<()>;
fn insert(self, key: Self::K, value: Self::V) -> Self;
fn get(&self, key: Self::K) -> (Option<Self::V>,Self::LookupProof);
fn remove(self, key: Self::K) -> Self;
note that insert(), get(), and remove() behave like the same methods in std::collections::HashMap, except that insert() and remove() return copies of the ADS rather than taking &mut self.
you can check the full implementation of a sorted key-value store class, represented by a vectorial abstraction of a (unbalanced) binary tree (i.e., the underlying data structure is an array), at src
although the key sorts the structure, this is not a binary search tree, as it allows repeated entries. rather, the structure's commitment is an overall hash calculated recursively as a binary tree
(this digest is obtained by mapping the merkle hash of each pair - named "merkle mountain range").
011. merkle trees
merkle trees are the canonical and original example of authenticated data structures, designed for easy inclusion proofs but not easy exclusive proofs (e.g., a prover can efficiently convince the
verifier that a particular entry is present but not absent - you will understand this better in the next session).
for example, we could use a vectorial abstraction of a binary tree to represent a collection of keys and values at nodes given by an index i.
although there are multiple representations of an associative array, a general rule for this representation is that:
• a right sibling could be found through 2 * ix + 2, so (i % 2 == 0) == true
• a left sibling could be found through 2 * ix + 1, so (i % 2 == 1) == true
the path from a (k, v) node to the root would be calculated as the overall digest hash of the tree from the (k, v) node pointer, with all the sibling hashes.
in other words, the calculation consists of iterating each sibling in the array of siblings' digests, attributing the hashes for left and right sub-trees, and then iteratively hashing them. in this
logic, the root hash is the entire commitment of the tree.
💡 since inserting at a specific position cannot be done efficiently (as the whole tree would need to be recomputed), these trees are unsuitable for authenticated key-value maps.
100. sparse merkle trees
now, let’s talk about something even more awesome: sparse merkle trees, which provide efficient exclusion proofs by design (try to figure out why 🕵🏻♀️!).
in a sparse merkle tree, a particular (k, v) is represented by a leaf node such that its path from the root is encoded by one of all possible hash digests of the chosen hash function.
so, if the chosen hash function is represented by N bits, the tree's height is N, and the paths down to the 2^N leaf nodes are represented by (N-nodes-deep) bit-strings.
in the case of SHA-256, the tree has a height of 256 and 2^{256} leaf nodes represented by 256-bit paths.
🕵🏻♀️ when the (k, v) entry is not present in the map, an empty leaf is assigned (for instance, with an all-0 digest).
in other words, each possible (k, v) entry corresponds to a unique leaf node and is uniquely linked to its position in the tree. on the other hand, each leaf is a unique representation of the 2^
N possibilities presented by the digest size N of the cryptographic hash function.
how about the branch nodes?
branches have left and right subtrees given by the digest of its child subtrees digest (something like hash_branch(left_digest_until_here, right_digest_until_here), a concatenated hash of its child
so any parent branch node can be obtained by hashing together two child nodes recursively until the top to the merkle root of the tree.
💡 note that different hash functions should be used for leaf and branch nodes to prevent preimage attacks.
each child subtree position is found by looking at the bits of hash_function(k). a left branch is denoted as 0 and a right branch as 1, so the most left leaf's key is 0x000..00, the next is 0x00..0,
the most right key is 0x11..11, and so on.
in addition, we see that most leaf nodes are empty, and this is cool because the hashes of empty nodes are identical.
the same is true for interior nodes whose children are all empty: subtrees with no leaf nodes are represented as an empty subtree with a digest such as the darling all-0 digest (and hashes of hashes
of hashes of (…) of empty nodes are all predictable).
💡 contrary to simple merkle trees, sparse merkle trees are a great choice for authenticated key-value maps due to the history independence of the merkle root from element insertion order.
“well, 2^N with a loooot of zeroes sounds like waste”
yes, to naively create a sparse merkle tree, one would generate all possible 2^N outputs of the hash function as a leaf in the tree and initialize them to the empty value. this generates a nearly
unbounded number of data.
however, since required storage is only proportional to the number of items inserted into the tree, some optimizations are possible:
• items on the leaf node are initially None and, therefore, do not need to be stored.
• subtrees with no leaf nodes are represented as "empty subtree", with an all-0 digest.
• internal nodes all have fixed predictable values that can be re-computed as hashes of None values.
therefore, a sparse merkle tree can simply store a set of leaf nodes for the (k, v) pairs plus a set of empty hashes representing the sparse areas of the tree.
and because a sparse merkle is nearly balanced, if there are N entries in the tree, a particular entry could be theoretically reached in log2(N) steps!
rust is the best language, period
as we implement a sparse merkle tree in rust (using SHA-256), we can think of two different types of nodes: SparseMerkleTreeNodeLeaf and SparseMerkleTreeNodeBranch (the full source code for this
class is available at src/sparse_merkle_tree.rs).
a very simple lookup function could look like this:
finally, a commit and checking proof could use the following path process:
101. unit tests and fuzzing
to conclude this article, let’s go over some of the testing approaches in this library. first of all, i use nix to set up my virtual environments and you should too.
rust differentiates between "unit" tests and "integration" tests, as described by its documentation. in my code, unit tests are annotated with #[test] macro, and integration tests are annotated with#
[cfg(test)] macro.
alternatively, the test hash_btree_insert_get() is an example of a "simulation test", generating scenarios where two different data structures (HashMap and BTreeMap) are tested to behave "the same"
for insert() and get() through this check:
assert_eq!(hmap.get(&k), bmap.get(&k));
we extended this concept to compare the behavior of SortedKV and SparseMerkleTree against HashMap through, for example, hash_sortedkv_insert_get() (making sure we test for SortedKV::check_proof()):
we then use quickcheck for property testing, which creates several randomly generated (non-deterministic) inputs, such as in:
◻️ motherofbots.eth
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Mathematical Conundrum or Not? Number Five
I think
's example of betting is a good thing to consider. Let's say that you're woken up once if it's heads but 99 times if it's tails. You're asked to place a bet of either £1 on heads or £99 on tails,
with a prize of £200 in either case if you're right (and in the case that it's tails it's only her bet on the last day that's accepted).
Using the thirder reasoning there's a 1/100 chance that it's heads, and so the £99 bet on tails is by far the best best. Using the halfer reasoning there's a 1/2 chance that it's heads, and so the £1
bet on heads is by far the best bet.
So put your money where your mouth is. What should you bet? I say bet £1 on heads.
That makes sense to me. Am I missing something? — Srap Tasmaner
Why is P(A) 3/4? Given that she's awake (and knows it), P(A) is 1. She can dismiss P(S) as a possible outcome. And for the same reason P(A|H) should be 1, which then gives the chance that it's heads
1/2, consistent with what we actually know about coin tosses.
Yes, Beauty is aware the coin is fair — Jeremiah
Then she knows that there's a 50% chance that it landed heads. It doesn't matter if she's only woken once if it's heads but twice if it's tails; a fair coin toss is always going to be 50%.
but she has also been told the details of the experiment and knows there are three possible events in which she is awakened. If awaken on Tuesday she would not know it is Tuesday; she only knows
there are three possible outcomes in which she will be awakened. To Beauty, who does not know if it is Tuesday or Monday when she is awaken, Tails and Tuesdays and Tails and Monday are both valid
outcomes. Beauty has to consider three possibilities and only one of them is the desired outcome. — Jeremiah
But each outcome is not equally likely. There's a 50% chance that it's Heads-Monday, 25% chance that it's Tails-Monday, and 25% chance that it's Tails-Tuesday.
Andrew M
Am I being completely stupid about this? — Srap Tasmaner
No. Nice analysis!
Andrew M
I only presented this side because they led with the 1/2 argument; however, they are correct in pointing out Beauty has gained no additional information. Really all she knows is what she was told
before the experiment. — Jeremiah
That's not correct. Beauty knows that she is awake and that is relevant information.
P(Heads) = 1/2
P(Heads|Awake) = 1/3
Whether 1/2 or 1/3 is assigned depends on whether one interprets the experiment as being about a coin toss event (1/2) or an awakening event (1/3).
Srap Tasmaner
The trouble is that we cannot use conditional probabilities. — andrewk
Finally got some time to come back to this and I think you can use conditional probabilities.
I have Beauty treating Monday-Tuesday as another 50-50 coin toss. Not conditional on her being awakened, mind you -- will come back to that -- just something she has no way of knowing by some other
means so she forms no opinion either way.
Here's the simple table
Mon Tue
Heads A S
Tails A A
$\small P(H \mid A) = \cfrac{P(A \mid H)P(H)}{P(A)}$ $\small \phantom{P(H \mid A)} = \cfrac{\cfrac{1}{2} \cdot \cfrac{1}{2}}{\cfrac{3}{4}}$ $\small \phantom{P(H \mid A)} = \cfrac{1}{3}$
That makes sense to me. Am I missing something?
We can also ask, what is the chance that it's Monday, given that she's been awakened?
$\small P(M\mid A) = \cfrac{P(A \mid M)P(M)}{P(A)}$ $\small \phantom{P(M\mid A)} = \cfrac{1 \cdot \cfrac{1}{2}}{\cfrac{3}{4}}$ $\small \phantom{P(M\mid A)} = \cfrac{2}{3}$
That looks right, and also makes sense. Of course it's twice as likely to be Monday given that she's been awakened, just as it's twice as likely that the coin toss came up tails, given that she's
been awakened.
Am I being completely stupid about this?
There is an interesting discussion to be had about it. — andrewk
And yet the other one was generating more discussion.
And this is consistent with the fact that we know that, given a fair coin toss, there's a 50% chance that it landed heads. We shouldn't change our view of that just because we might be woken up
twice rather than once. — Michael
Yes, Beauty is aware the coin is fair, but she has also been told the details of the experiment and knows there are three possible events in which she is awakened. If awaken on Tuesday she would not
know it is Tuesday; she only knows there are three possible outcomes in which she will be awakened. To Beauty, who does not know if it is Tuesday or Monday when she is awaken, Tails and Tuesdays and
Tails and Monday are both valid outcomes. Beauty has to consider three possibilities and only one of them is the desired outcome.
But now I don't know if that's misdirection too. — Srap Tasmaner
None of it is misdirection, this problem has several possible rational answers. We are not talking about just different points of view, or different readings of the semantics, it can rationally be
answered in multiple ways from the same point of view and the same interpretation of the semantics. This is because the problem is probing the relationship between knowledge and probability, in this
case Beauty's knowledge.
And actually, Pattern-chaser's first answer is still a valid take. Beauty can't be sure of what day she is awakened, as she is given no new information when she is awakened, which means a 1/2 chance
for Monday is a rational response. This would follow Bayesian philosophy on probability which suggest we should update our probability models when we get new information.
One way to attack the question of what 'credence' or 'degree of belief' means is to interpret it in terms of 'what would you bet, if you were Beauty'? The answer to that depends on the rules of the
betting game that is offered.
Consider two different betting games, and we assume Beauty wants to maximise her expected profit.
Game 1: Beauty places her bet before the experiment starts, paying $1, and at the end she is paid $2 if in all her interviews she guessed the coin outcome correctly, otherwise she is paid nothing.
Under this game, her expected winnings are maximised at zero, whichever she chooses. But she must decide before going to sleep the first time which side she is going to guess, because her expected
profit becomes negative if it is tails and she makes one guess of heads and another of tails.
Under this game, interpreting the betting strategy as 'degree of belief', we could say her 'degree of belief', at the time of being interviewed, that the coin has landed on Tails, is 1/2.
Game 2: At each interview, Beauty bets $1 to guess what coin came up, and loses that dollar if wrong or wins $2 if right.
Under this game, Beauty's expected winnings are maximised at 50 cents if she guesses Tails.
Under this game, interpreting the betting strategy as 'degree of belief', we could say her 'degree of belief', at the time of being interviewed, that the coin has landed on Tails, is 3/4.
I find it interesting that Game 2, which seems perfectly natural interpretation to me, is consistent with a 'degree of belief' of 3/4 rather than 1/2 or 1/3.
I wonder what betting game would be consistent with a probability of 1/3. I expect there must be a pretty natural one, since most people answer either 1/2 or 1/3 to this question.
I'm off for a jog on the beach now. Maybe it will come to me.
I didn't count it twice. I am saying P (Tails and Monday) and P (Tails and Tuesday) have the same likelihood of occurring because they are determined by the same chance event, but for Beauty the coin
flip generates three possible answers. You have to distinguish between the coin flip and Beauty determining the probability it landed on heads based on the possible times she could be awakened.
The possible outcomes:
- - M T
H: A, S
T: A, A
If she is awakened before Wednesday, there is one awake on heads and two awake on tails. Of the possible awakens there is a 1/3 chance it is Monday and heads. — Jeremiah
As she's awake we have to dismiss Heads-Tuesday as an outcome. The only outcomes are Heads-Monday, Tails-Monday, and Tails-Tuesday. But is it right to treat these outcomes as equally likely? Perhaps
not. If it it's heads then there's 100% chance that it's Monday, but if it's tails then there's a 50% chance that it's Monday. So Heads-Monday is twice as likely as Tails-Monday (and twice as likely
as Tails-Tuesday).
And this is consistent with the fact that we know that, given a fair coin toss, there's a 50% chance that it landed heads. We shouldn't change our view of that just because we might be woken up twice
rather than once.
Srap Tasmaner
Presented in a way that suggests we should look for a conditional, but that won't work. I could see that, but I'm still mulling it over.
Everything else seems to point to 1/3. Twice as many possible awakenings are on tails, leaving just 1/3 for heads. 2/3 of the awakenings are on Monday and half of those are heads, so 1/3 again.
But now I don't know if that's misdirection too.
T Clark
P (Tails and Monday) and P (Tails and Tuesday) are the same flip, Are you really suggesting they have a different chance of occurring when they both occur on the same chance event? — Jeremiah
P (Tails and Monday) and P (Tails and Tuesday) are the same event. You can't (legitimately) count it's probability twice.
P (Tails and Monday) and P (Tails and Tuesday) are the same flip. Are you really suggesting they have a different chance of occurring when they both occur on the same chance event?
So we want the conditional P(H|A). — Srap Tasmaner
The trouble is that we cannot use conditional probabilities.
A conditional probability P(H|A) is the probability of event H given the probability of event A, and A will be an event that is known to be true. But the only event that Beauty knows to be true is
that she has been woken at least once - because she has just been woken, so the known event is not a proper (ie smaller) subset of the entire sample space, call it S. So if H is heads and A is 'I am
woken at least once' then P(H|A) = P(H|S) = P(H) = 1/2. The conditional probability is the same as the unconditional one.
To use conditional probabilities à la Bayes' Rule, we must have some information that narrows down the set of possibilities - that tells us we are in a proper subset of the sample space. But Beauty
has no such information. All she knows is something she already knew before the experiment started.
I think it's to avoid that straightforward solution that the word 'credence', or sometimes 'degree of belief' is used instead of 'probability'.
I'll add that statements like P(Tails and Monday) or P(Tails | Monday) are ambiguous, because the 'Monday' could be the event 'I get woken on a Monday', for which the probability is 1, or it could
mean 'today is Monday', and it's not immediately obvious how to set up the probability space so that 'today is Monday' is a well-defined event, without losing the connection to the coin toss.
Unlike the others, this one is not a misunderstanding of a well-understood and resolved problem. There is an interesting discussion to be had about it.
What is the correct answer depends on what meaning one attaches to the word 'credence'. The use of that unusual word rather than 'probability' in posing the problem is deliberate.
There's a long discussion about it
on physicsforums.
I can't remember, off-hand, whether I was a thirder or a halfer.
T Clark
They are the same flip. It is pointless to argue independence. That is like saying I rolled a die and got 4 and it is not independent because I got 4. If the coin is tails she will be awakened on
Monday and awakened on Tuesday, therefore they have the same probability. — Jeremiah
Here's the additive law of probability:
If events A and B are mutually exclusive (disjoint), then P(A or B) = P(A) + P(B)
In your case, P (Tails and Monday) and P (Tails and Tuesday) are not mutually exclusive, so this rule does not apply.
Is she told what day it currently is when she is awakened?
Artemis OptionsShare
It is a fair coin, fair means 50% chance the coin come up heads and and 50% chance it comes up tails. When Beauty is awakened she has to provide the probability it is heads. I am arguing at this time
33%, as there are three possible outcomes where she is awakened and only one of those outcomes is heads.
The coin is only flipped once.
P (Tails and Monday) and P (Heads and Monday) are mutually exclusive — T Clark
Clearly . . . .
Also, P (Tails and Monday) and P (Tails and Tuesday) are not independent. — T Clark
They are the same flip. It is pointless to argue independence. That is like saying I rolled a die and got 4 and it is not independent because I got 4. If the coin is tails she will be awakened on
awakened on Tuesday, therefore they have the same probability.
Also, you left out P (Heads and Not Tuesday). — T Clark
She is not awakened on Tuesday if the coin is heads.
The possible outcomes:
- - M T
H: A, S
T: A, A
If she is awakened before Wednesday, there is one awake on heads and two awake on tails. Of the possible awakens there is a 1/3 chance it is Monday and heads.
T Clark
Now if the coin is fair then P(Heads and Monday) = P(Tails and Tuesday)
Therefore P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday).
It must add up to one, thus the 1/3. — Jeremiah
P (Tails and Monday) and P (Heads and Monday) are mutually exclusive. Also, P (Tails and Monday) and P (Tails and Tuesday) are not independent. So you don't add their probabilities. Also, you left
out P (Heads and Not Tuesday).
Anyway. Different experiment - much more ethical. SB comes in on Sunday and agrees to the experiment. She leaves. They flip a coin. On Monday, they place a red jellybean under an opaque cover. SB
comes back and is asked whether the flip was heads or tails. On Tuesday, if the coin came up heads on Sunday, they place a green jellybean under the cover along with the red one. If the flip was
heads, no jellybean is added. SB comes in and is asked again. Then on Wednesday she comes back in and they give her the jelly bean(s).
Srap Tasmaner
Yeah that's pretty clean.
I realized right after I posted that the question is almost literally what is the probability we got heads given that you have been awakened and are being interviewed? So we want the conditional P(H|
That's what I'm working at. Not sure what I've got so far. Nice puzzle for me because my probability skills be weak, so thanks.
Consider it this way:
P(Tails|Monday) = P(Tails|Tuesday) Right? Because they are on the same flip they have the same probability.
Now if the coin is fair then P(Heads and Monday) = P(Tails and Tuesday)
Therefore P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday).
It must add up to one, thus the 1/3.
Btw, there are other ways to look at this, Pattern-Chaser was not necessarily wrong. I only presented this side because they led with the 1/2 argument; however, they are correct in pointing out
Beauty has gained no additional information. Really all she knows is what she was told before the experiment.
Srap Tasmaner
So there's a conflict between guessing which interview this is -- and you weight by the number of possible interviews -- and guessing which day this is, right? It's not Wednesday, because you're
being interviewed; if it's Monday, you're interviewed either way; so what are the odds that it's Tuesday?
She should flip a coin. If the original flip was heads then she’ll have a 50% chance of being right, but if it was tails then she’ll have a 75% chance of being right at least once.
Well let's break it down.
If the coin flip is heads then Beauty is awakened on Monday but sleeps though Tuesday.
If the coin flip is tails Beauty is awakened on Monday and awakened on Tuesday.
So we have:
H: A, S
T: A, A
So since there are three possible awakenings and only one is when the coin comes up heads, then won't that mean she has a 33% chance of it being heads?
If the amnesia-inducing drug is effective, Beauty cannot know how many times she was woken, or even if she was woken. Beauty is reduced to guessing, using the information we all know about tossed
coins: they come up heads about 50% of the time, on average. But 'on average' does not apply to individual coin tosses. The statistics are useful only for large numbers of coin-tosses, or for large
numbers of subjects like Beauty. I think Beauty can only guess, and her only 'evidence' is the statistical 50% probability.
Pattern-chaser OptionsShare
The Sleeping Beauty Problem
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice during the experiment, Beauty will be
awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening.
A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails,
she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.
Any time Sleeping Beauty is awakened and interviewed, she is asked, 'What is your belief now for the proposition that the coin landed heads?
So what is Beauty's credence that the coin landed on heads?
There is currently no agreed upon resolution to this problem. You could consider this an exercise in philosophy or probability, it should overlap both.
The trouble is that 'today is Monday', for which you have used the label 'M', is not an Event in the
Kolmogorov sense
of being a well-defined subset of the sample space and, in the absence of its being such an Event, it cannot be used in probability statements, as a probability statement is a probability measure of
an Event.
The reason it can't be an event is that, so far as I can see, the definition of an event in the Kolmogorov framework, which is used, to the best of my knowledge, in all modern probability theory, is
timeless. It cannot be relative to a particular time. So 'Beauty is woken on a Monday' and 'Beauty is woken on a Tuesday' are events because the time references they make are absolute, but 'today is
Tuesday' is not an event because the reference 'today' is relative.
The same applies to 'I am awake', which is what I think you mean by the label 'A'. Because the 'am' is a relative time reference, it cannot be an Event in a probability space, and so cannot have a
probability in the usual sense, that allows use of things like conditional probability rules.
To use a philosophy of time analogy, the question 'is today Tuesday', in the context of this experiment, has as much meaning in probability theory as it does when asked to a pan-dimensional being
that is looking at our 4D spacetime from the outside, if
McTaggart's B theory of time
is the case.
Perhaps a probability space can be constructed in which 'today is Monday' or 'I am awake' is an event but so far I have not managed to construct one. I thought I had a lead, but it turned out to be a
Unless we can construct a probability space in which 'today is Monday' is an event, and which also contains all the other information about the experiment, it cannot be meaningful to talk of the
probability - conditional or unconditional - of the event 'today is Monday' or 'I am awake', so we cannot use conditional probabilities to calculate with them.
That is not new information, she knew she'd be awakened beforehand. New relevant and significance information to reallocating creditably would be if she was told what day it was on Monday.
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How to prepare your child for the PSLE Maths Exams
Many parents and students alike will face anxiety when they prepare for their PSLE exams. Both parents and students want to do well enough to go to the school of their choice. One of the hurdles
parents and students must overcome is PSLE Mathematics. To prepare for the PSLE Mathematics exam, our young students need regular practice and consistent revision. Here are some tips for parents to
prepare their child for the PSLE Mathematics exam:
(1) Find out how good your child is at Math
First, parents need to assess how good their child is at Mathematics. Doing some practical questions can help parents know what their child’s strengths and weaknesses are. This is a highly important
step as parents need to know where to target the child’s revision so that studying becomes more effective. Without specific revision goals, studying could feel less purposeful as progress and
improvement will be more difficult to track.
(2) Figure out how they think and approach maths problems
The first step to correcting a child’s mistakes is to know how they think. This is a bit technical, and parents should ask their child’s maths tutor or school teacher about their child’s problem
solving mental process. Knowing how the child solved the problem incorrectly will help the parent know how to fix this lack of understanding. When an educator understands what the child doesn’t
understand, they can much better help the child achieve the goal. Parents should avoid berating the child when they don’t know something – instead, parents should find out why the child got the
question wrong.
(3) Correct gaps in their understanding
After identifying the mistakes the child makes, parents should correct gaps in their understanding and let them practice with what they have just learned. Doing topical exercises or simple practice
questions to test their understanding of the concept will help the child correct their previous mistakes. Different permutations and contexts of questions should be posed to the child so that they
truly understand how to solve the question, instead of just memorising it. Again, if you have a maths tutor, you should ask him/her about this process, as it is vitally important.
(4) Train them how to read the question instructions properly
Many students are tempted to jump right into doing the paper so that they can start and finish the paper in a timely manner. However, this could be a foolish decision especially since the failure to
read the question and identify what it is asking of the student could cost them their effort and time. The correct step is for parents is to ensure that the child knows how to pace themselves, and
read the instructions properly so that they know exactly what the question requires.
(5) Applying concepts to the questions
Next, students need to know how to apply the mathematical concepts they have learned to the possible questions in the PSLE exam. A good way to start practicing this is to attempt past PSLE papers to
identify where the student stands in terms of conceptual application. These papers need to be checked after they have been done to identify where the student lacks understanding, and any mistakes
made in these practice papers should be corrected.
(6) Practice regularly under a timed setting
Another facet of the PSLE math exam that students struggle with is time management. Many students spend too much time on the more difficult questions, neglecting to answer the easier questions in
time and then rushing through the rest of the paper when time is up. Poor time management will also cause the student to panic and make more careless mistakes. As such, regular practice of the sample
papers under a timed setting will help students focus and manage their time properly.
To conclude, it may be a daunting challenge for both PSLE students and their parents to perform all of the above. However, they should remember that some effort is better than no effort at all.
Instead of striving to be the best, students should focus on doing the best they can, according to their own gifts and abilities. By doing so, students will likely be more satisfied with their
mathematics exam results.
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Mystery object caused by spontaneous symmetry breaking revealed
Home Physics Mystery object caused by spontaneous symmetry breaking revealed
Theoretical clarification of a mysterious object (topological defect*1) recently observed in experiments regarding spontaneous symmetry breaking. The existence of a new topological defect called a
quantum elliptic vortex was predicted through the application of the Joukowski transform – used to understand airfoil design.
Hiromitsu Takeuchi, a lecturer at the Graduate School of Science, Osaka City University, and a researcher at the Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), has
theoretically identified the nature of a mysterious topological defect produced by the recently discovered non-equilibrium time evolution of spontaneous symmetry breaking (SSB). Since the SSB
realized in this system is like the SSB that has been known to occur in isotropic superconductors and superfluid 4He, it was expected to produce topological defects with vortex-like properties in the
fluid, called quantum vortices. However, the topological defect observed in this experiment has a structure that bore little resemblance to the previously mentioned SSB, and its physical properties
have been shrouded in mystery. In this research, the idea of applying the Joukowski transform, which is used to calculate the lift of airplane wings, to quantum vortices was introduced for the first
time, and the analysis revealed that the most stable state of this mysterious topological defect is a new topological defect called a quantum elliptic vortex. The results of this research were
published online in Physical Review Letters, considered to be one of the most prestigious journals in the field of physics.
Figure 1: Composite defect in a 23Na superfluid confined in a pancake-shaped two-dimensional “electromagnetic container”. The blacker color indicates a region of high fluid density. The core of the
topological defect corresponds to the white region in the center of the picture. Credit: Phys. Rev. Lett. 122, 095301 (2019)
A time- and space-dependent function called a “field” is commonly used to describe the properties of physical systems in which SSB occurs. If the motion of the field can be calculated, the behavior
of the system can be predicted. However, the calculation is generally difficult because the degrees of freedom of the field are infinite.
One effective way to describe the complex motion of a field is to represent the degrees of freedom of an object floating in it, called a topological defect. The field around the “core” of a
topological defect has a certain structure. Therefore, by describing the center of the core as the motion of a mass point, the motion of the field can be approximately predicted.
This situation is similar to how the future change in wind direction can be predicted to some extent by looking at the path of the eye of a typhoon. In materials where SSB typically occurs, such as
superconductors and superfluids, this “wind” corresponds to current without resistance and flow without friction, respectively. Since the structure of the field around the core can be predicted
according to the symmetry breaking, it has been thought that the behavior of topological defects, and hence the behavior of the field, can be understood if the symmetry breaking is understood on a
global scale.
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Figure 2: Flow (numerical calculation) around an ordinary rotationally symmetric quantum vortex (left) and a quantum elliptic vortex (right). The arrows indicate the direction of the flow; the whiter
the color, the stronger the flow. The outline of the core is outlined by dashed lines. The background color represents the phase θ of the macroscopic wave function (complex function) corresponding to
the superfluid field. Credit: Osaka City University
A phenomenon that refutes this idea was recently observed by Professor Shin’s experimental group at Seoul National University [Phys. Rev. Lett. 122, 095301 (2019)]. Since the symmetry breaking in
this experimental system is similar to that in well-known ordinary superconductors and superfluids, the shape of the core of the topological defect, called a quantum vortex, is expected to be round
like the eye of a typhoon in a two-dimensional cross section.
However, the actual cross-sectional structure of the phase defect observed was completely different. Figure 1 shows an experimental photograph of the structure corresponding to the cross section of a
topological defect caused by a sudden phase transition. At the time, this topological defect was considered to be a compound of two known topological defects (composite defect) and was interpreted as
a transient state that occurs temporarily during the phase transition process near the critical point.
In this study, to clarify the physical properties of the composite defect observed in the experiment, Hiromitsu Takeuchi introduced the idea of applying the Joukowski transform, which is used to
calculate the lift of an airplane wing, to the quantum vortex. Based on this idea, the topological defect observed in the experiment is eventually stabilized as a new topological defect called a
quantum elliptic vortex. Ordinary quantum vortices have a rotationally symmetric flow in their cross-section, like an eye of a typhoon (Fig. 2, left). However, the cross section of the newly proposed
quantum elliptic vortex spontaneously breaks the rotational symmetry and forms a flow along the ellipse. It was previously thought that the external shape of a topological defect was determined based
on the way the global SSB of the physical system occurs, but this result clearly overturns that perception.
It is theoretically known that such a strange structure occurs near the critical point of the phase transition, and that the local SSB inside the core of the topological defect is deeply involved in
its stability.
Although SSB has been studied for a long time, there is no general understanding of how the local SSB inside the core occurs and how it affects the physical properties of topological defects.
Topological defects appear not only in special materials such as superconductors, but also in a variety of physical systems ranging from relatively familiar materials such as crystals and liquid
crystals to cutting-edge science and technology such as spintronics, and they are considered to play important roles in a rotating neutron star and the phase transition dynamics in the early
universe. There is hope that new developments in SSBs, such as Takeuchi’s discovery, will be brought about by improvements in experimental techniques and corresponding advances in theory, and that
they will have a ripple effect on the entire field of physics.
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The Exponential Distribution
Learning Outcomes
• Recognize the exponential probability distribution and apply it appropriately
The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential
distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of
the change that you have in your pocket or purse approximately follows an exponential distribution.
Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the
supermarket follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.
Let X = amount of time (in minutes) a postal clerk spends with his or her customer. The time is known to have an exponential distribution with the average amount of time equal to four minutes.
X is a continuous random variable since time is measured. It is given that μ = 4 minutes. To do any calculations, you must know m, the decay parameter.
[latex]{m}=\frac{1}{\mu}[/latex]. Therefore, [latex]{m}=\frac{1}{4}={0.25}[/latex]
The standard deviation, σ, is the same as the mean. μ = σ
The distribution notation is X ~ Exp(m). Therefore, X ~ Exp(0.25).
The probability density function is f(x) = me^–mx. The number e = 2.71828182846… It is a number that is used often in mathematics. Scientific calculators have the key “e^x.” If you enter one for x,
the calculator will display the value e.
The curve is:
f(x) = 0.25e^–0.25x where x is at least zero and m = 0.25.
For example, f(5) = 0.25e^−(0.25)(5) = 0.072. The postal clerk spends five minutes with the customers. The graph is as follows:
Notice the graph is a declining curve. When x = 0,
f(x) = 0.25e^(−0.25)(0) = (0.25)(1) = 0.25 = m. The maximum value on the y-axis is m.
The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with the average amount of time equal to eight minutes. Write the distribution, state the
probability density function, and graph the distribution.
X ~ Exp(0.125); f(x) = 0.125e^–0.125x;
Using the information in example 1, find the probability that a clerk spends four to five minutes with a randomly selected customer.
The curve is:
X ~ Exp(0.125); f(x) = 0.125e^–0.125x
a) Find P(4 < x < 5).
The cumulative distribution function (CDF) gives the area to the left.
P(x < x) = 1 – e^–mx
P(x < 5) = 1 – e(–0.25)(5) = 0.7135 and P(x < 4) = 1 – e^(–0.25)(4) = 0.6321
You can do these calculations easily on a calculator.
The probability that a postal clerk spends four to five minutes with a randomly selected customer is P(4 < x < 5) = P(x < 5) – P(x < 4) = 0.7135 − 0.6321 = 0.0814.
On the home screen, enter (1 – e^(–0.25*5))–(1–e^(–0.25*4)) or enter e^(–0.25*4) – e^(–0.25*5). b) Half of all customers are finished within how long? (Find the 50^th percentile)
Find the 50^th percentile.
P(x < k) = 0.50, k = 2.8 minutes (calculator or computer)
Half of all customers are finished within 2.8 minutes.
You can also do the calculation as follows:
P(x < k) = 0.50 and P(x < k) = 1 –e^–0.25k
Therefore, 0.50 = 1 − e^−0.25k and e^−0.25k = 1 − 0.50 = 0.5
Take natural logs: ln(e^–0.25k) = ln(0.50). So, –0.25k = ln(0.50)
Solve for k: [latex]{k}=\frac{ln0.50}{-0.25}={0.25}=2.8[/latex] minutes
c) Which is larger, the mean or the median?
From part b, the median or 50^th percentile is 2.8 minutes. The theoretical mean is four minutes. The mean is larger.
The number of days ahead travelers purchase their airline tickets can be modeled by an exponential distribution with the average amount of time equal to 15 days. Find the probability that a traveler
will purchase a ticket fewer than ten days in advance. How many days do half of all travelers wait?
P(x < 10) = 0.4866
50^th percentile = 10.40
On the average, a certain computer part lasts ten years. The length of time the computer part lasts is exponentially distributed.
a) What is the probability that a computer part lasts more than 7 years?
Solution:Let x = the amount of time (in years) a computer part lasts.
[latex]\mu = {10}[/latex] so m = [latex]\frac{1}{\mu} = \frac{1}{10}={0.10}[/latex]
P(x > 7). Draw the graph.
P(x > 7) = 1 – P(x < 7).
Since P(X < x) = 1 –e^–mx then P(X > x) = 1 –(1 –e^–mx) = e^-mx
P(x > 7) = e^(–0.1)(7) = 0.4966. The probability that a computer part lasts more than seven years is 0.4966.
On the home screen, enter e^(-.1*7).
b) On the average, how long would five computer parts last if they are used one after another?
On the average, one computer part lasts ten years. Therefore, five computer parts, if they are used one right after the other would last, on the average, (5)(10) = 50 years.
c) Eighty percent of computer parts last at most how long?
Find the 80^th percentile. Draw the graph. Let k = the 80^th percentile.
Solve for k: [latex]{k}=\frac{ln(1-0.80)}{-0.1}={16.1}[/latex]
Eighty percent of the computer parts last at most 16.1 years.
Find P(9 < x < 11). Draw the graph.
d) What is the probability that a computer part lasts between nine and 11 years?
P(9 < x < 11) = P(x < 11) – P(x < 9) = (1 – e^(–0.1)(11)) – (1 – e^(–0.1)(9)) = 0.6671 – 0.5934 = 0.0737. The probability that a computer part lasts between nine and 11 years is 0.0737.
Suppose that the length of a phone call, in minutes, is an exponential random variable with decay parameter = 112. If another person arrives at a public telephone just before you, find the
probability that you will have to wait more than five minutes. Let X = the length of a phone call, in minutes.
What is m, μ, and σ? The probability that you must wait more than five minutes is _______ .
m = [latex]\frac{1}{12}[/latex]
[latex]\mu [/latex] = 12
[latex]\sigma [/latex] = 12
P(x > 5) = 0.6592
The time spent waiting between events is often modeled using the exponential distribution. For example, suppose that an average of 30 customers per hour arrive at a store and the time between
arrivals is exponentially distributed.
1. On average, how many minutes elapse between two successive arrivals?
2. When the store first opens, how long on average does it take for three customers to arrive?
3. After a customer arrives, find the probability that it takes less than one minute for the next customer to arrive.
4. After a customer arrives, find the probability that it takes more than five minutes for the next customer to arrive.
5. Seventy percent of the customers arrive within how many minutes of the previous customer?
6. Is an exponential distribution reasonable for this situation?
1. Since we expect 30 customers to arrive per hour (60 minutes), we expect on average one customer to arrive every two minutes on average.
2. Since one customer arrives every two minutes on average, it will take six minutes on average for three customers to arrive.
3. Let X = the time between arrivals, in minutes. By part a, μ = 2, so m = 12 = 0.5.
Therefore, X ∼ Exp(0.5).The cumulative distribution function is P(X < x) = 1 – e(–0.5x)^e.Therefore P(X < 1) = 1 – e^(–0.5)(1) ≈ 0.3935
5. P(X > 5) = 1 – P(X < 5) = 1 – (1 – e^(–5)(0.5)) = e^–2.5 ≈ 0.0821.
6. We want to solve 0.70 = P(X < x) for x.
Substituting in the cumulative distribution function gives 0.70 = 1 – e^–0.5x, so that e^–0.5x = 0.30. Converting this to logarithmic form gives –0.5x = ln(0.30), or x=ln(0.30)–0.5≈2.41
minutes.Thus, seventy percent of customers arrive within 2.41 minutes of the previous customer.
7. This model assumes that a single customer arrives at a time, which may not be reasonable since people might shop in groups, leading to several customers arriving at the same time. It also assumes
that the flow of customers does not change throughout the day, which is not valid if some times of the day are busier than others.
Memorylessness of the Exponential Distribution
In example 1, recall that the amount of time between customers is exponentially distributed with a mean of two minutes (X ~ Exp (0.5)). Suppose that five minutes have elapsed since the last customer
arrived. Since an unusually long amount of time has now elapsed, it would seem to be more likely for a customer to arrive within the next minute. With the exponential distribution, this is not the
case–the additional time spent waiting for the next customer does not depend on how much time has already elapsed since the last customer. This is referred to as the memoryless property.
Specifically, the memoryless property says that
P (X > r + t | X > r) = P (X > t) for all r ≥ 0 and t ≥ 0
For example, if five minutes has elapsed since the last customer arrived, then the probability that more than one minute will elapse before the next customer arrives is computed by using r = 5 and t
= 1 in the foregoing equation.
P(X > 5 + 1 | X > 5) = P(X > 1) = e(–0.5)(1) ≈ 0.6065.
This is the same probability as that of waiting more than one minute for a customer to arrive after the previous arrival.
The exponential distribution is often used to model the longevity of an electrical or mechanical device. In example 1, the lifetime of a certain computer part has the exponential distribution with a
mean of ten years (X ~ Exp(0.1)). The memoryless property says that knowledge of what has occurred in the past has no effect on future probabilities. In this case it means that an old part is not any
more likely to break down at any particular time than a brand new part. In other words, the part stays as good as new until it suddenly breaks. For example, if the part has already lasted ten years,
then the probability that it lasts another seven years is P(X > 17|X > 10) =P(X > 7) = 0.4966.
Refer to example 1, where the time a postal clerk spends with his or her customer has an exponential distribution with a mean of four minutes. Suppose a customer has spent four minutes with a postal
clerk. What is the probability that he or she will spend at least an additional three minutes with the postal clerk?
The decay parameter of X is m = 14 = 0.25, so X ∼ Exp(0.25).
The cumulative distribution function is P(X < x) = 1 – e–0.25x. We want to find P(X > 7|X > 4). The memoryless property says that P(X > 7|X > 4) = P (X > 3), so we just need to find the probability
that a customer spends more than three minutes with a postal clerk.
This is P(X > 3) = 1 – P (X < 3) = 1 – (1 – e–0.25⋅3) = e–0.75 ≈ 0.4724.
Relationship between the Poisson and the Exponential Distribution
There is an interesting relationship between the exponential distribution and the Poisson distribution. Suppose that the time that elapses between two successive events follows the exponential
distribution with a mean of μ units of time. Also assume that these times are independent, meaning that the time between events is not affected by the times between previous events. If these
assumptions hold, then the number of events per unit time follows a Poisson distribution with mean λ = 1/μ. Recall that if X has the Poisson distribution with mean λ, then [latex]P(X=k)=\frac{{\
lambda}^{k}{e}^{-\lambda}}{k!}[/latex]. Conversely, if the number of events per unit time follows a Poisson distribution, then the amount of time between events follows the exponential distribution.
(k! = k*(k-1*)(k–2)*(k-3)…3*2*1)
At a police station in a large city, calls come in at an average rate of four calls per minute. Assume that the time that elapses from one call to the next has the exponential distribution. Take note
that we are concerned only with the rate at which calls come in, and we are ignoring the time spent on the phone. We must also assume that the times spent between calls are independent. This means
that a particularly long delay between two calls does not mean that there will be a shorter waiting period for the next call. We may then deduce that the total number of calls received during a time
period has the Poisson distribution.
1. Find the average time between two successive calls.
2. Find the probability that after a call is received, the next call occurs in less than ten seconds.
3. Find the probability that exactly five calls occur within a minute.
4. Find the probability that less than five calls occur within a minute.
5. Find the probability that more than 40 calls occur in an eight-minute period.
1. On average there are four calls occur per minute, so 15 seconds, or [latex]\frac{15}{60} [/latex]= 0.25 minutes occur between successive calls on average.
2. Let T = time elapsed between calls. From part a, [latex]\mu = {0.25} [/latex], so m = [latex]\frac{1}{0.25} [/latex] = 4. Thus, T ~ Exp(4). The cumulative distribution function is P(T < t) = 1 –
e^–4t. The probability that the next call occurs in less than ten seconds (ten seconds = 1/6 minute) is P(T < [latex]\frac{1}{6}[/latex]) = 1 – [latex]{e}^{-4\frac{1}{6}} \approx{0.4866} [/
3. Let X = the number of calls per minute. As previously stated, the number of calls per minute has a Poisson distribution, with a mean of four calls per minute. Therefore, X ∼ Poisson(4), and so P(
X = 5) = [latex]\frac{{4}^{5}{e}^{-4}}{5!}\approx[/latex] 0.1563. (5! = (5)(4)(3)(2)(1))
4. Keep in mind that X must be a whole number, so P(X < 5) = P(X ≤ 4).
To compute this, we could take P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
Using technology, we see that P(X ≤ 4) = 0.6288.
5. Let Y = the number of calls that occur during an eight minute period.
Since there is an average of four calls per minute, there is an average of (8)(4) = 32 calls during each eight minute period.
Hence, Y ∼ Poisson(32). Therefore, P(Y > 40) = 1 – P (Y ≤ 40) = 1 – 0.9294 = 0.0707.
Concept Review
If X has an exponential distribution with mean [latex]\mu[/latex] then the decay parameter is [latex]m =\frac{1}{\mu}[/latex], and we write X ∼ Exp(m) where x ≥ 0 and m > 0 . The probability density
function of X is f(x) = me^-mx (or equivalently [latex]f(x)=\frac{1}{\mu}{e}^{\frac{-x}{\mu}}[/latex].The cumulative distribution function of X is P(X≤ x) = 1 – e^–mx.
The exponential distribution has the memoryless property, which says that future probabilities do not depend on any past information. Mathematically, it says that P(X > x + k|X > x) = P(X > k).
If T represents the waiting time between events, and if T ∼ Exp(λ), then the number of events X per unit time follows the Poisson distribution with mean λ. The probability density function of [latex]
P\left(X=k\right)=\frac{\lambda^{k}}{e^{-\lambda}}k![/latex]. This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf(λ, k). The cumulative distribution function P(X ≤
k) may be computed using the TI-83, 83+,84, 84+ calculator with the command poissoncdf(λ, k).
Formula Review
Exponential: X ~ Exp(m) where m = the decay parameter
• pdf: f(x) = m[latex]{e}^{-mx}[/latex] where x ≥ 0 and m > 0
• cdf: P(X ≤ x) = 1 –[latex]{e}^{-mx}[/latex]
• mean [latex]\mu = \frac{1}{m}[/latex]
• standard deviation σ = µ
• percentile, k: k = [latex]\frac{ln(\text{AreaToTheLeftOfK})}{-m}[/latex]
• Additionally
□ P(X > x) = e^(–mx)
□ P(a < X < b) = e^(–ma) – e^(–mb)
• Memoryless Property: P(X > x + k|X > x) = P (X > k)
• Poisson probability: P(X=k)=[latex]\frac{{\lambda}^{k}{e}^{-\lambda}}{k!}[/latex] with mean [latex]\lambda[/latex]
• k! = k*(k-1)*(k-2)*(k-3)…3*2*1
Data from the United States Census Bureau.
Data from World Earthquakes, 2013. Available online at http://www.world-earthquakes.com/ (accessed June 11, 2013).
“No-hitter.” Baseball-Reference.com, 2013. Available online at http://www.baseball-reference.com/bullpen/No-hitter (accessed June 11, 2013).
Zhou, Rick. “Exponential Distribution lecture slides.” Available online at www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf (accessed June 11, 2013).
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Where to Live?
Problem B
Where to Live?
Moving to a new town can be difficult. Finding a good place to live which is close to everything you’re interested in is important. However, since you’re a great programmer, you know that you can
solve this problem with an algorithm.
Everything in your virtualized town is laid out on a grid, so every place lies on an integer coordinate grid. You’ll be given a list of coordinates of the places you are interested in in the town,
and you need to choose a place to live on the grid. Your program should find the grid location that minimizes the average straight-line squared distance to every place you are interested in (squared
distance so that you won’t be too far from any one location).
You can live anywhere on the grid, even if something already exists where you want to live (buildings can always be built taller to accommodate you).
Input consists of a list of up to $100$ descriptions for towns you are considering moving to. Each town description starts with a line containing $1 \leq n \leq 1\, 000$, the number of locations
you’re interested in. The next $n$ lines each contain two space-separated integer coordinates $x$ and $y$, each in the range $[0, 1\, 000]$. No location is repeated within a town. Input ends when $n$
is $0$.
For each town, print the location you want to live on the grid. If the best location is not exactly on a grid point, choose the grid point closest to the best location. Break ties by choosing the
point that has the smallest $x$ coordinate and then the smallest $y$ coordinate.
Sample Input 1 Sample Output 1
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Computational astrophysics
From Scholarpedia
James M. Stone (2007), Scholarpedia, 2(10):2419. doi:10.4249/scholarpedia.2419 revision #137148 [link to/cite this article]
Computational astrophysics is the use of numerical methods to solve research problems in astrophysics on a computer.
Numerical methods are used whenever the mathematical model describing an astrophysical system is too complex to solve analytically (with pencil and paper). Today, it is difficult to find examples of
research problems that do not use computation.
Computational versus Analytic Methods
Solutions generated by numerical methods are generally only approximations to the exact solution of the underlying equations. However, much more complex systems of equations can be solved numerically
than can be solved analytically. Thus, approximate solutions to the exact equations found by numerical methods often provide far more insight than exact solutions to approximate equations that can be
solved analytically. For example, time-dependent numerical solutions of fluid flow in three dimensions can exhibit behavior which is not expected from one-dimensional analytic solutions to the
steady-state (time independent) equations.
The increase in computing power in the last few decades has meant that an increasingly larger share of problems in astrophysics can be solved on a desktop computer. However, the most computationally
intensive problems in astrophysics are still limited by the memory and floating-point speed of the largest high-performance computer (HPC) systems available (e.g. computers on the top500 list). The
use of HPC is necessary to maximize the spatial, temporal, or frequency resolution of solutions, to include more physics, or when large parameter surveys are required to understand the statistics.
Traditionally, the most important applications of computation in astrophysics have been in the areas of:
• Stellar structure and evolution.
The internal structure and evolution of stars with different masses and chemical composition was largely mapped out in the 1960s using numerical methods to solve the equations of stellar structure.
Today, the frontiers of research include calculating multidimensional stellar models of rapidly rotating stars, modeling the effects of hydrodynamical processes such as convection from first
principles, and understanding how stars generate magnetic fields through dynamo processes.
• Radiation transfer and stellar atmospheres.
Without oversimplifying assumptions, computational methods are required to calculate the propagation of light through the outer layers of a star, including its interaction with matter through
absorption, emission, and scattering of photons. The calculation of cross sections for the interaction of light with matter for astrophysically relevant ions is itself a challenging computational
problem. The construction of such stellar atmosphere models share many challenges with radiation transfer problems in other systems such as planets, accretion disks, and the interstellar medium.
Modern calculations improve the frequency resolution, include a better treatment of opacities, and can treat non-hydrostatic atmospheres.
• Astrophysical fluid dynamics.
The dynamics of most of the visible matter in the universe can be treated as a compressible fluid. Time-dependent and multidimensional solutions to the fluid equations, including the effects of
gravitational, magnetic, and radiation fields, require numerical methods. A vast range of problems are addressed in this way, from convection and dynamo action in stellar and planetary interiors, to
the formation of galaxies and the large scale structure of the universe.
It is well known that Newton's Laws of Motion for some number of particles N interacting through their mutual gravitational attraction do not in general have an analytic solution for N > 2. Thus, to
compute the orbits of planets in the solar system, or of stars in the Galaxy, numerical methods are required. The most challenging problems today include accurate integration of the orbits of the
planets over the age of the solar system, studying the dynamics of globular clusters including the effect of stellar evolution and the formation of binaries, studying galaxy mergers and interaction,
and computing structure formation in the universe through the gravitational clustering of collisionless dark matter.
Numerical Methods
The diverse set of mathematical models encountered in astrophysics means that a very wide range of numerical methods are necessary. These range from basic methods for linear algebra, nonlinear root
finding, and ordinary differential equations (ODEs), to more complex methods for coupled partial differential equations (PDEs) in multi-dimensions, topics which span the entire content of reference
books such as Numerical Recipes.
However, there are several numerical methods used in astrophysics that deserve special mention, either because they have such wide use in astrophysics, or because astrophysicists have made
significant contributions to their development. These methods are considered in the following subsections.
Stellar Structure Codes
The equations of stellar structure are a set of ODEs which define a classic two-point boundary value problem. Analytic solutions to an approximate system of equations exist in special cases (
polytropes) but these are of limited application to real stars. Numerical solutions to the full system of equations were first computed using shooting methods, in which boundary conditions are
guessed at the center and surface of the star, the equations are integrated outwards and inwards, and matching conditions are used at some interior point to select the full solution. Shooting methods
are laborious and cumbersome; today modern stellar structure codes use relaxation schemes which find the solution to the finite-difference form of the stellar structure equations by finding the roots
of coupled, non-linear equations at each mesh point. A good example of a public code that uses a relaxation scheme is the EZ-code, based on Eggleton's variable mesh method. The evolution of stars are
then computed by computing stellar models at discrete time intervals, with the chemical composition of the star modified by nuclear reactions in the interior.
Radiative Transfer Codes
Calculating the emergent intensity from an astrophysical system requires solving a multidimensional integral-differential equation, along with level population equations to account for the
interaction of the radiation with matter. In general the solution is a function of two angles, frequency, and time. Even in static, plane-parallel atmospheres, the problem is two-dimensional (one
angle and frequency). However, the most challenging aspect of the problem is that scattering couples the solutions at different angles and frequencies. As in the stellar structure problem, relaxation
schemes are used to solve the finite difference form of the transfer equations, although specialized iteration techniques are necessary to accelerate convergence. Monte-Carlo methods, which adopt
statistical techniques to approximate the solution by following the propagation of many photon packets, are becoming increasingly important. The problem of line-transfer in a moving atmosphere
(stellar wind) is especially challenging, due to non-local coupling introduced by Doppler shifts in the spectrum.
There are two tasks in an N-body code: integrating the equations of motion (pushing particles), and computing the gravitational acceleration of each particle. The former requires methods for
integrating ODEs. Modern codes are based on a combination of high-order difference approximations (e.g. Hermite integrators), and symplectic methods (which have the important property of generating
solutions that obey Liouville's Theorem, i.e. that preserve the volume of the solution in phase space). Symplectic methods are especially important for long term time integration, because they
control the accumulation of truncation error in the solution.
Calculation of the gravitational acceleration is challenging because the computational cost scales as N(N-1), where N is the number of particles. For small N, direct summation can be used. For
moderate N (currently \(N \sim 10^{5-6}\)), special purpose hardware (e.g. GRAPE boards) can be used to accelerate the evaluation of \(1/r^2\) necessary to compute the acceleration by direct
summation. Finally, for large N (currently \(N \geq 10^{9-10}\)), tree-methods are used to approximate the force from distant particles.
Codes for Astrophysical Fluid Dynamics
Solving the equations of compressible gas dynamics is a classic problem in numerical analysis which has application to many fields besides astrophysics. Thus, a large number of methods have been
developed, with many important contributions being made by astrophysicists. A complete review of all the methods is beyond the scope of this discussion (see Laney 98). For solving the equations of
compressible fluid dynamics, the most popular methods include
• finite-difference techniques (which require hyper-viscosity to smooth discontinuities),
• finite-volume methods (which often use a Riemann solver to compute upwind fluxes),
• operator split methods (which combine elements of both finite-differencing and finite-volume methods for different terms in the equations),
• central methods (which often use simple expressions for the fluxes, combined with high-order spatial interpolation), and
• particle methods such as smooth particle hydrodynamics (SPH, which integrates the motion of discrete particles to follow the flow, see Monaghan 92).
A good technical review of many of these methods is given by LeVeque (2002). SPH is an example of a method developed largely to solve astrophysics problems, although many of the developments in other
methods (e.g. the extension of finite-difference and finite-volume methods to magnetohydrodynamics (MHD) to include the effects of magnetic fields on the dynamics of the fluid) have also been
motivated by astrophysics.
Relation to other fields.
Computational astrophysics is necessarily inter-disciplinary, involving aspects of not only astrophysics, but also numerical analysis and computer science.
Numerical Analysis
Numerical Analysis is a rigorous branch of mathematics concerned with the approximation of functions and integrals, and the approximation of solutions to algebraic, differential, and integral
equations. It provides tools to analyze errors that arise from the approximations themselves (truncation error), and from the use of finite-precision arithmetic on a computer (round-off error).
Convergence, consistency, and stability of numerical algorithms are all essential for their use in practical applications. Thus, the development of new numerical algorithms to solve problems in
astrophysics is deeply rooted in the tools of numerical analysis.
Computer Science
Computational science and computer science differ. The former is the use of numerical methods to solve scientific problems. The latter is the study of computers and computation. Thus, computer
science tends to be more orientated towards the theory of computers and computation, while scientific computation is concerned with the practical aspects of solving science problems. Still, there is
a large degree of overlap between the fields, with many computer scientists engaged in developing tools for scientific computation, and many scientists working on software problems of interest to
computer scientists. Examples of the overlap include the development of standards for parallel processing such as the Message Passing Interface (MPI) and OpenMP, and development of parallel I/O
filesystems such as Lustre.
Computational astrophysics has a long history, dating back to the numerical approximation of the motion of the moon and planets in order to compute ephemerides for tides and navigation. At first, the
term computer referred to a person employed to calculate some quantity to some level of numerical accuracy. Electronic digital computers were introduced in World War II to break encryption codes and
compute artillery range tables, however they were quickly adopted to other purposes after the war. In astrophysics, this included computing the first stellar structure models in the 1950s, and the
development of numerical methods for fluid flow and shock-capturing at about the same time. Major advances in the decades that followed included the use of transistor instead of vacuum tubes, the
development of compiled languages (Fortran in the 1950s, C in the 1970s), the adoption of the IEEE standards for the representation of floating point numbers and arithmetic (previously every
manufacturer used their own pattern for bits) and the development of very large integrated circuits (VLSI), which allows the complexity of processor design that is enjoyed today. Modern trends in
computer design are towards distributed memory parallel processors, with multiple cores capable of vector processing. At the same time as the development in hardware, progress in numerical analysis,
computer science, and software engineering have resulted in standardized protocols for interprocess communication across networks, object-orientated languages such as Fortran95 and C++, interpreted
languages such as Java and Python, visualization tools such as OpenGL, and software engineering tools such as CVS and Subversion (SVN).
Internal references
See also
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Export Reviews, Discussions, Author Feedback and Meta-Reviews
Submitted by Assigned_Reviewer_15
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
This is a very interesting and substantially novel paper that introduces an approach to solving continuous Markov random field energies with polynomial potentials. An insightful and well-motivated
approach towards this end (ADMM-Poly) was published at CVPR 2013 [20] and is the obvious baseline to compare against. The present approach is convincingly shown to be preferable, as it is both
elegant and computationally efficient.
The main idea underlying the approach is to decompose the polynomials into a difference of convex functions. Towards this end, a constructive approach is introduced for polynomials of even degree,
one of the main contributions of the paper. As the authors show, the decomposition problem has a formulation as a semi-definite program and can be solved using standard solvers. Given that
decomposition, the overall inference task reduces to a DC programming problem, which the authors approach using the convex-concave procedure (CCCP).
The main weakness of the approach seems to be that the semi-definite program for finding a DC decomposition of the polynomials may not have a solution, as it is only an approximation to the exact but
NP-hard problem. Even though the authors mention that the approach works well for polynomials of low degree in practice (confirmed by experiments), this point is somewhat worrisome. I would encourage
the authors to elaborate on this point, as it seems like a potential barrier to adoption of the approach. How often does this situation occur in practice (if at all)? For which polynomials? Are there
any workarounds? Clearly, an alternative strategy is needed in case this situation occurs.
As the authors show, in some cases, for some continuous energies, a DC decomposition is apparent even without having to resort to the above construction algorithm. The approach then basically becomes
an application of CCCP to continuous energies.
The paper is technically sound, though, as mentioned above, a more in-depth discussion of the potential failure to construct the decomposition of the polynomials would be appreciated. The experiments
are impressively diverse and the results are very convincing. The proposed approach clearly works better than ADMM-Poly, the main baseline to compare against. It is faster and often obtains solutions
of better quality.
The paper is well-written, though not always easy to read, owing to the technical depth of the material. One aspect that the authors may wish to clarify is the claimed global convergence of their
approach (inherited from CCCP). I assume that this is meant to be interpreted as "the algorithm converges, for *any* initialization", rather than "the algorithm converges to a global optimum". As far
as I am aware, the convergence properties of CCCP only guarantee convergence to either an optimum or a saddle point. As it stands, readers might misinterpret the manuscript to assume that the
proposed approach always finds a global optimum.
Along the same line, the authors claim in lines 039ff that Gaussian belief propagation "retrieves the optimum for arbitrary graphs when the potentials are Gaussian". This is not true. There are two
known convergence conditions relating to the diagonal dominance, and the spectral radius, of the system matrix, respectively.
The proposed approach is novel to this reviewer, and substantial theory is developed along the way.
The proposed approach has broad and important applications in computer vision and imaging. The experimental results are impressive, and the approach hence has the potential to significantly benefit
many members of these communities.
Q2: Please summarize your review in 1-2 sentences
In summary, this is a substantially novel submission that develops new theory and demonstrates strong experimental results. For these reasons, I recommend that the paper be accepted.
Submitted by Assigned_Reviewer_23
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
The work shows how the energy of a continuous Markov Random Field that consists
of polynomial pairwise energies can be decomposed into a sum of a convex and
concave polynomial. This leverages the use of the concave-convex procedure
(CCCP) to do fast MAP inference.
Quality: The paper states valid proofs for the used methodology and it seems
reproducible. The diverse set of experiments further show that the proposed
method performs well. In comparison to other optimization techniques the
polynomial decomposition usually finds better solutions. Nevertheless, the
paper could be more clear about the weaknesses. E.g. the trade-off between
wall-clock time for solving the optimization problem and sometimes only
marginally better solution could be discussed.
Clarity: The paper is well written and the presentation is clear. I liked the
pace at which the work introduces the splitting of the polynomial into the two
parts of interest.
Originality: Using the decomposition of a polynomial to drive inference of a
MRF seems to be new.
Significance: MRFs with polynomial energies are certainly of interest, e.g. in
Computer Vision.
+ This is certainly an important problem in e.g. Computer Vision and worth
+ Nice presentation; clear proofs.
+ Diverse experiments.
- The authors could be a bit more clear about the weaknesses of their work: The
method seems to yield better solutions for the resulting optimization problem
but seems to needs more wall-clock time than the competing direct
optimization with L-BFGS.
Questions to the authors:
- Do the reported running times include the solving time for the decomposition
into convex and concave functions?
Q2: Please summarize your review in 1-2 sentences
The work addresses a relevant problem and proposes an interesting solution.
Proofs and exposition are clear. The paper could state the weaknesses more
clearly: The method seems to be slower for `real-world' problems than the
competitors but yields better solutions.
Submitted by Assigned_Reviewer_36
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
In this paper, the authors first show that the multivariate polynomial functions with even degree can be decomposed into convex and concave parts. They use this property to solve MAP inference
problem in continuous Markov Random Fields. Finally they show experiments on three problems: 3D reconstruction, shape from shading and image denoising.
--> The paper is well written and tackle very important problem of solving continuous MRF problems. They provide theoretical analysis of their algorithm.
Minor comments:
--> The authors claim to significantly outperform existing techniques in terms of efficiency and accuracy. However, this is not completely reflected in their result section. For example in all the
three cases, their algorithm takes longer time than L-BFGS approach. If this is the case then they should clearly mention in the text.
--> Authors should explain why they observe up and down in their energy versus sample index curve in Fig. 2.
--> I would suggest to properly explain the tables in their captions.
Q2: Please summarize your review in 1-2 sentences
The paper provides a theoretically sound algorithm for decomposing an even degree polynomial into a convex and concave parts. They properly analyse the properties of the algorithm and use it to solve
continuous MRF problems. The paper is well written.
Q1:Author rebuttal: Please respond to any concerns raised in the reviews. There are no constraints on how you want to argue your case, except for the fact that your text should be limited to a
maximum of 6000 characters. Note however, that reviewers and area chairs are busy and may not read long vague rebuttals. It is in your own interest to be concise and to the point.
We thank the reviewers for their helpful and positive feedback. We will incorporate all their comments in a revised version of the manuscript.
To R15 w.r.t finding a decomposition: We note that, for all pairwise graphical models with arbitrary degree polynomials, as well as for graphical models of order up to four with maximum fourth order
degree polynomials, we are guaranteed to find a decomposition. This is due to the fact that SOS convexity and polynomial convexity coincide in those cases(see Theorem 5.2 in reference [A] below).
Most practical graphical models are within this set. Remarkably, we have not yet encountered any convex but not SOS-convex polynomial in practice. Known counter-examples [A] are typically found using
specific tools. We will add a discussion about this in the paper.
To R15: We agree with the reviewer and we will make our statement about Gaussian BP more precise.
To R15: By ‘globally convergent’ we mean convergence for arbitrary initialization to a local optima/saddle point. We will clarify this in the paper.
To R23 w.r.t. reproducibility: We will make our code available to reproduce all experiments.
To R23 and R36: We expect our approach to be slower than L-BFGS as we employ L-BFGS to solve the intermediate convex optimization steps obtained through CCCP. The experiment is rather intended to
illustrate that a standard solver does indeed converge to a worse local optima than our approach. We’ll clarify this in a revised version.
To R23: Reported times do not include concave-convex decomposition but for FOE experiments, the decomposition is trivially obtainable in no time by adding unary quadratic terms with constant
coefficients. Decomposition of the other two applications takes about 0.07s for 9x9 Cardboard and 8.47s for 128x128 vase image. This is only a small fraction of the reported time and we’ll clarify
this fact. It is important to note that construction of the monomial basis is independent of the considered application and can be pre-computed.
To R36: Fig 2 depicts the achieved energy as a function of the example index (i.e., image index). As some examples are more difficult than others we do not expect any particular shape for this plot.
We will clarify this.
New Reference:
[A] A. Ahmadi, P. Parrilo. A complete characterization of the gap between convexity and SOS-convexity. SIAM Journal on Optimization, 2013
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A161172 - OEIS
The Yummie permutation is done as follows. Start with a packet of n cards (numbered 1 to n from top to bottom), and deal them into two piles, first to a spectator (pile A), and then to yourself (pile
B), saying "You, me," silently to yourself over and over. Then, pick up pile B and deal again, first to the spectator, thereby adding to the existing pile A, and then to yourself, forming a new pile
B. Repeat, picking up the diminished pile B, and dealing "You, me" as before. Eventually, just one card remains in pile B; place it on top of pile A. The sequence of cards in pile A determines the
Yummie permutation ("You, me" said fast sounds like "Yummie").
a(9) = 15, because when the Yummie permutation is applied to {1,2,3,4,5,6,7,8,9} we get {6,2,4,8,9,7,5,3,1}, which corresponds to the product of a disjoint five cycle and a three cycle, and hence has
order 15.
P(n, i)={if(i%2, n-(i\2), P(n\2, (n-i)\2+1))}
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
Cycles(n)={my(L=List()); for(i=1, n, my(k=Follow(i, j->P(n, j))); if(k, listput(L, k))); vecsort(Vec(L))}
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How to Derive the F Distribution - Nick's Technical Blog
How to Derive the F Distribution
Table of Contents
F Distribution Definition
The F Distribution is the ratio of two independent Chi-squared distributions standardized by their degrees of freedom.
Y_k\sim\chi^2(k) && X\sim F(p,q) = \frac{Y_p/p}{Y_q/q}
Single Variable Change
Similar to the T Distribution, deriving the F Distribution involves several single variable changes.
A variable change is a change to a functions variable and domain such that the new and old functions have integrals which are equal when evaluated over their respective domains (eg if you perform a
variable change on a probability function you would still expect the area under the probability curve to be one).
y = g(x) && \int_y f(y) \partial y = \int_x f(g(x)) \frac{\partial g}{\partial x}\partial x
Multi-variable Change with Jacobian Matrix
This proof involves one multi-variable change. In order to accomplish this the determinant of a Jacobian matrix is required.
**NOTE**: The combination of the Jacobian matrix and its determinant are often referred to together as simple the Jacobian.
A function with more than one scalar input variable can also be expressed as a function with a single vector input variable.
f(y_1, y_2, y_3) = f\Big(\stackrel{y}{\begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix}}\Big) = f(y)
A multi-variable change requires that the input vector for the original function can itself be expressed as the range of a vector valued function of the variable you wish to change to.
\stackrel{y}{\begin{bmatrix}y_1\\ y_2 \end{bmatrix}} = g\Big(\stackrel{x}{\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}}\Big) = \stackrel{g(x)}{\begin{bmatrix} x_1 + x_2 + x_3 \\ 5x_1 x_3 \end
A Jacobian matrix is a matrix of the first order partial derivatives of a vector valued function with respect to its vector input domain.
\stackrel{g(x)}{\begin{bmatrix} g_1(x_1,\ldots,x_n) \\ \vdots \\ g_k(x_1,\ldots,x_n) \end{bmatrix}}\longrightarrow J_{g(x)} = \begin{bmatrix}
\frac{\partial g_1}{\partial x_1} & \ldots & \frac{\partial g_1}{\partial x_n} \\
\vdots & \ddots & \vdots \\
\frac{\partial g_k}{\partial x_1} & \ldots & \frac{\partial g_k}{\partial x_n}
If the determinant of the Jacobian matrix exists it can be used to perform a multi-variable change on functions where the input vector is the domain of the function the Jacobian matrix was built
y = g(x) && \int_y f(y) \partial y = \int_x f(g(x))\cdot \text{det}(J_{g(x)}) \partial x && \frac{\partial f}{\partial y} = \frac{\partial f}{\partial g} \text{det}(J_{g(x)})
How to Derive the F Distribution
An F random variable is defined as the ratio of two Chi-Squared random variables standardized by their own degrees of freedom.
Let \(X\) be the ratio of two independent Chi-squared random variables, \(Y_p\) and \(Y_q\), standardized by their degrees of freedom parameters \(p\) and \(q\) respectively. Then \(X\) is an F
random variable with density function for \(f_X(x;p,q) = \frac{(x)^{p/2-1}p^{p/2}q^{q/2}}{\text{B}(p/2,q/2)(xp+q)^{p/2+q/2}}\).
First declare two related equalities (because they share a variable).
&W_1 = Y_p/Y_q& &\longrightarrow& &Y_p = W_1 W_2 && \normalsize (1) \\ \\
&W_2 = Y_q& &\longrightarrow& &Y_q = W_2 \\ \\
The above equalities can be expressed in vector form as below and vector y can be expressed as a function of vector w.
\stackrel{w}{\begin{bmatrix} W_1 \\ W_2\end{bmatrix}} && \stackrel{y}{\begin{bmatrix} Y_p \\ Y_q \end{bmatrix}} = \stackrel{g(w)}{\begin{bmatrix} W_1W_2 \\ W_2 \end{bmatrix}} && \normalsize (2) \\ \\
\(Y_p\) and \(Y_q\) are independent random variables so their joint probability function is simply the product of their individual Chi-squared probability functions.
f_{Y_p,Y_q}(y_p,y_q) &= \frac{y_p^{p/2-1}e^{-y_p/2}}{\Gamma(p/2)2^{p/2}}\cdot\frac{y_q^{q/2-1}e^{-y_q/2}}{\Gamma(q/2)2^{q/2}} && \normalsize (3) \\ \\
&= \frac{y_p^{p/2-1}y_q^{q/2-1}e^{-\frac{(y_p + y_q)}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}
The next step is to perform a multi-variable change on the joint probability function from \(y\) to \(w\) using the Jacobian (instructions above).
&J_{g(w)} = \begin{bmatrix} \frac{\partial g_1}{\partial w_1} & \frac{\partial g_1}{\partial w_2} \\
\frac{\partial g_2}{\partial w_1} & \frac{\partial g_2}{\partial w_2}
\end{bmatrix} =\begin{bmatrix} w_2 & w_1 \\
0 & 1
\end{bmatrix} && \normalsize (4)\\ \\
&\text{det}(J_{g(w)}) = w_2
Express the integral of the joint probability function \(Y_p\) and \(Y_q\) as a function of vector \(y\), then perform the variable change from \(y\) to \(w\) using the Jacobian.
f(y) &= f(g(w))\cdot\text{det}(J_{g(w)}) &&\normalsize (5) \\ \\
&= \frac{(w_1w_2)^{p/2-1}w_2^{q/2\cancel{-1}}e^{-\frac{(w_1w_2 + w_2)}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot \cancel{w_2} \\ \\
f_{W_1,W_2}(w_1,w_2;p,q)&= \frac{w_1^{p/2-1}w_2^{p/2+q/2-1}e^{-\frac{(w_1w_2 + w_2)}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\partial w
Now integrate out random variable \(W_2\) to obtain the univariate probability function of \(W_1\).
In order to do this notice that now there is a potential gamma function kernel given by \(w_2^{p/2+q/2-1}e^{-w_2\frac{w_1+1}{2}}\). In order to complete the expression of the gamma function two steps
are necessary. First insert multiply the function by 1 and factor it: \(\require{\color}{\color{aqua}1 = \Big(\frac{2(w_1+1)}{2(w_1+1)}\Big)^{p/2+q/2-1}}\).
\int_0^\infty f_{W_1,W_2}(w_2;p,q)\partial w_2 &= \int_0^\infty \frac{w_1^{p/2-1}w_2^{p/2+q/2-1}e^{-w_2\frac{w_1 + 1}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot{\color{aqua}\Big(\frac{2(w_1+1)}{2
(w_1+1)}\Big)^{p/2+q/2-1}} \partial w_2 &&\normalsize (6) \\ \\
&= \int_0^\infty \frac{w_1^{p/2-1}(w_2{\color{aqua}\frac{w_1+1}{2}})^{p/2+q/2-1}e^{-w_2\frac{w_1 + 1}{2}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot{\color{aqua}\Big(\frac{2}{w_1+1}\Big)^{p/2+q/2-1}} \
partial w_2
Step two to completing the gamma function expression is to perform another variable change, this time from \(W_2\) to \(t\). Let \(t = w_2\frac{w_1+1}{2}\), which implies \(W_2 = g(t) = \frac{2t}
\int_0^\infty f_{W_1,W_2}(w_2) \partial w_2 &= \int_0^\infty f_{W_1,W_2}(g(t))\cdot\frac{\partial g}{\partial t} \partial t &&\normalsize (7) \\ \\
&= {\color{gold}\int_0^\infty} \frac{w_1^{p/2-1}{\color{gold}t^{p/2+q/2-1}e^{-t}}}{\Gamma(p/2)\Gamma(q/2)2^{p/2+q/2}}\cdot\Big(\frac{2}{w_1+1}\Big)^{p/2+q/2\cancel{-1}}\cdot\cancel{\frac{2}{w_1+1}}{\
color{gold}\partial t} \\ \\
&= \frac{w_1^{p/2-1}{\color{gold}\Gamma(p/2+q/2)}}{\Gamma(p/2)\Gamma(q/2)\cancel{2^{p/2+q/2}}}\cdot\Big(\frac{\cancel{2}}{w_1+1}\Big)^{p/2+q/2}
After obtaining \(\Gamma(p/2+q/2)\) it can be combined with the other two gamma functions into a beta function.
f_{W_1}(w_1;p,q) &= \frac{w_1^{p/2-1}}{\text{B}(p/2,q/2)}\cdot\Big(\frac{1}{w_1+1}\Big)^{p/2+q/2} && \normalsize (8)
Recall from (1) that \(W_1\) is the unstandardized ratio random variable of two Chi-squared random variables. Let \(X = W_1\frac{q}{p}\), which makes it the standardized ratio of two Chi-squared
random variables. Perform a final variable change from \(W_1\) to \(X\).
f_{W_1}(w_1;p,q) &= f_{W_1}(g(x);p,q)\cdot\frac{\partial g}{\partial x} &&\normalsize(9) \\ \\
f_X(x;p,q) &= \frac{x^{p/2-1}(\frac{p}{q})^{p/2\cancel{-1}}}{\text{B}(p/2,q/2)}\cdot\Big(\frac{1}{x\frac{p}{q}+1}\Big)^{p/2+q/2}\cdot \cancel{\frac{p}{q}} \\ \\
&= \frac{x^{p/2-1}(\frac{p}{\cancel{q}})^{p/2}}{\text{B}(p/2,q/2)}\cdot\frac{q^{\cancel{p/2}+q/2}}{(xp+q)^{p/2+q/2}}\\ \\
&= \frac{x^{p/2-1}p^{p/2}q^{q/2}}{\text{B}(p/2,q/2)(xp+q)^{p/2+q/2}} &&\normalsize (\text{QED})
Summary Conclusion
Deriving the F distribution probability function from the Chi-squared distribution isn’t a complicated task, but it can get messy with 3 variable changes.
Start from the definition of an F random variable defined as the ratio of two independent Chi-squared random variables scaled by their degrees of freedom. Let the two Chi-squared be \(Y_p\) and \(Y_q
\). Define two more random variables via two equivalencies \(W_1 = \frac{Y_p}{Y_q}Y_p = Y_p\quad W_2=Y_q\).
Use the equivalencies between the y and w random to perform a multi-variable change on the joint probability function \(f_Y(y)\). Next integrate out \(W_2\) by completing a gamma function kernel.
This leaves the single variable probability function for \(W_1\).
\(W_1\) is defined as the unstandardized ratio random variable of two independent Chi-squared random variables. Perform a final variable change, simply applying the standardizing scalar to the random
variable, to arrive at the probability function for an F random variable.
196 thoughts on “How to Derive the F Distribution”
Great article!
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this website offers an unrivaled experience that combines exquisite craftsmanship with exceptional service,中国エロmaking it the go-to destination for discerning collectors and enthusiasts alike.
This aspect is crucial in a societyirontech doll where discussions about sexual needs and desires are still laden with stigma.
ダッチワイフThank you for your dedication and excellent work.Your article is a testament to the value of thorough research and clear communication,
enhancing the doll’s natural and lifelike movements.ラブドールエロcom’s customer service is top-notch,
improving accessibility could further enhance the blog’s reach.ラブドールエロThis includes adding alt text to images,
making it easy for readers like myself to apply them in real-world settings.ダッチワイフYour clear writing style,
coerced an erection,ダッチワイフand manipulated him into having sex.
To briefly sum up Wikipedia’s page of this most provocative topic: “The buttocks have been considered an erogenous zone in Western thought for centuries,and the eroticization of the female
buttocks was heteronormative and due to their association and closeness to the female reproductive organs [even though in another sense] the buttocks are often taboo due to their proximity to the
anus and association with the excretory system.ラブドール中古
and you won’t have to interchange allエロ人形 the doll if one thing have been to go wrong.
エロ人形You don’t need to fully adopt the other perspective,but considering an opposing mindset can help you see and shift assumptions you habitually make.
Love and intimacy are developed through reciprocity,ラブドールエロshared vulnerability,
it’s totally normal and healthy.It can actually create a love rubber band that snaps you guys closer together after being stretched.えろ人形
エロラブドールAccording to the U.S.
engaging in something intimate other than sex).In a relationship with someone who is cheating on someone else to be with you.オナドール
ラブドールおすすめAnd if you feel like your friendships are lacking,give some attention to creating goals that can put you closer to developing the type of relationships that you desire.
ダッチワイフAfter all—image is incredibly important to a narcissist,and they ultimately want to be viewed as the victim when you set a limit with them.
making it easy to design a doll that matches my exact preferences.The realism is remarkable,ラブドールえろ
Like Mary Ann,many women believe it is their “wifely duty” to have sex with their husband whenever (and often,ラブドールエロ
In healthy families and relationships,people are on the same team and support one another through good times and bad.ラブドールエロ
but skilled gaslighters also avoid taking responsibility for more everyday behavior—though the newest research suggests that most of their avoidance tactics center around taking attention away
from their immature,hurtful behaviors.ダッチワイフ
ダッチワイフエロTwo of scientists’ discoveries,however (see,
the disagreements may feel volatile,ロボットセックスwhich can have a significant impact on your stress level and ability to trust the other person.
The website offers a broad range of customization options,ラブドールエロmaking it easy to create a doll that perfectly fits my preferences.
Sign up for our e mail to appreciate ドールエロHong Kong devoid of paying a detail (along with some solutions once you’re emotion flush).
After the adult doll is completely dry, you can apply a thin layer えろ人形of renewing powder specifically designed for silicone sex dolls.
Your ability to present a comprehensive analysis while maintaining clarity and coherence is truly commendable.ダッチワイフYour comprehensive analysis has greatly enriched my understanding of
[specific topic],
セックスロボットis violent,or has a mental or drug problem.
Research has shown that the quality of your friendships can influence your level of stress,ラブドール女性用physical health,
ドールエロThey examined situational factors,symptomatology,
and difficulties forming adult relationships.オナドールIt is also believed by many experts that clergy abuse can damage the faith lives of its victims; many survivors interviewed described their
experiences as “soul murder.
Let’s take a moment to reflect on the past and present of this latest upgradeラブドールオナニー.
marital rape can actually be more,オナドールnot less,
ダッチワイフエロthere is very little that parents can do to increase their daughters’ future reproductive success,beyond keeping them alive and healthy.
This freedom to explore makes sex toys more inclusive to the queer and sexualラブドール av trauma communities who don’t feel represented in the mainstream sex toy industry.”
中国エロsetting a new standard in the world of doll craftsmanship and customer service.From start to finish,
please).ラブドールえろI was sold.
it might feel like sex is harder to wait for” the researchers add.人形えろBased on these insights,
It was recognising their experiences as individuals that helped Beth, 女性用ラブドールnow 26, and her partner move to a new place with their sex life.
Update your wardrobe with Marc Jacobs from the marc jacobs outlet.
Legs being long,ダッチワイフエロa little can be revealed,
えろ人形The choice to love is not a feeling; it is an action.That is why it is so difficult.
Her bridle was heavy. Blinders narrowed my field of vision. I couldn’t see her, but I could 女性用ラブドールfeel her movements through the reins she held behind me. She made a clicking noise with
her tongue to prompt me to move.
One of the most compelling reasons to choose com is the incredible variety of dolls they offer.ラブドールえろa collector’s piece,
オナドールThis is what my client Mona shared with me about her experience of marital rape: “If a stranger raped me it would feel very different—I wouldn’t take it so personally.He doesn’t know me
and I don’t know him.
ラブドールエロand a seamless shopping experience makes JP-Dolls a standout choice.I highly recommend this site for anyone looking for a lifelike,
I insisted on purchasing brand sex dolls for my stores and when I clicked the エロ人形GSDoll website,
Sex dolls can help to educate people about consent and healthyセックスボット relationships by providing a safe and controlled environment for individuals to
Lately, there has been a noticeableirontech doll surge in the popularity of sex dolls, indicating a change in what consumers are looking for and what technology can now achieve
resulting in dolls that are visually ラブドール sexappealing but also interactive and responsive.
it is important to find a trained psychologist who specializes in treating trauma and can help support you through your healing process.人形エロIt can be a long road,
But Greater Love Doll gives you thirty times to request エロ人形an true return. You’ll have to buy return shipping and delivery
or treatment.ラブドールエロAlways seek the advice of a mental health professional or other qualified health provider with any questions you may have regarding your condition or well-being.
Let’s take a moment to reflect on the past and present of this latest upgradeラブドールオナニー.
The market’s expansion is propelled by a ラブドール sexcombination of heightened consumer curiosity and significant advancements in material science and engineering
offering an alternative where conventional instructionaljydoll techniques may fall short.
characterized by the uncontrollable consumption of alcohol despite harmful consequences,ラブドール中古the path to recovery can be particularly demanding.
Not the love movie trailer we constantly play and compare our relationship to in our head.えろ人形That’s called programming.
オナドールand that,researchers argue,
monk or nun,celibacy is spiritually meaningful.オナホ高級
The materials used are of excellent quality,enhancing the lifelike experience.ラブドールエロ
” “moral” or “immoral,” “acceptable” or “unacceptable,オナホ高級
Last week,ラブドールえろI got home from a weekend with Doug in Boston.
ラブドールオナホAbandoning,in the romantic realm,
good looks,えろ人形charm,
we inspire you to go to the Doll Forum and take a look atラブドール中古 the opinions on the assorted sexual intercourse doll makers to choose from.
they’ll be seen as “less of a man” and that women女性用ラブドール won’t want to date them because they’re not masculine.’ Vaneet has seen this play out both in his own dating life and in the
lives of the men he spoke to for his book,
By contrast,praising people for being special or superior rather than for their hard work fosters an unearned and therefore insecure sense of entitlement.えろ人形
Sex dolls are also sought for sexual fulfilment. With the irontech dolladvancements in their design, they can provide a realistic sexual experience
I go with “Heather,” an alto Scottish drawlラブドール女性用 that seems to disguise Harmony’s robotic cadence somewhat a lot better than one other, American accents.
at times,ダッチワイフエロalmost salacious.
ドールエロto claim that sex—one of our most powerful motives (our species’ existence depends on it,after all)—is somehow absent from an act that routinely involves erection,
and even extra heads for a diverse experience.ドールオナニー High-end sex dolls from reputable brands often boast these additional features, providing a more immersive and enjoyable encounter.
リアルドールWithout being so rigid as to set yourself up to be thrown off if things don’t go exactly as planned,come up with a specific strategy that outlines what your interaction will be like.
the top-down approach also has limitations.ラブドールエロChanging the law can have unintended consequences.
Below is my reasoned response (which went unanswered):Do you have any empirical evidence for the efficacy of your negative conditioning and aversive techniques for fetishes? Do you have
supporting documentation on what percentage of fetishes “prevent,good,ダッチワイフエロ
お気軽くお問い合わせください人形えろ?プライバシー・ポリシ ?弊社はお客様の個人情報の重要性を認識して、
It’s been widely cited for over 20 years, showing that in heterosexual relationships, ラブドール女性用women are having four times fewer orgasms than men. Multiple factors have been said to be at
Lately, there has been a noticeableirontech doll surge in the popularity of sex dolls, indicating a change in what consumers are looking for and what technology can now achieve
ダッチワイフFeeling guilt over starting a new relationship
precio priligy 30 mg com 20 E2 AD 90 20Viagra 20Tablet 20Delhi 20 20Ile 20Kosztuje 20Viagra 20Cena 20W 20Aptekach ile kosztuje viagra cena w aptekach If sustained, this may ultimately curb their
demand trendor, in countries where oil subsidies are in place, raisepressure on their governments to reduce those subsidyprogrammes, the IEA said
followed by periods of time as a Chacham,ラブドールonly to fall off into the son who forgets to ask or doesn’t even think it’s important,
the whole story,together.ラブドール
they glibly repeat the same hurtful behaviors.The act of appearing apologetic gets them out of the hot seat in session,エロドール
オナホラブドールDuring a recent podcast,I interviewed Justin Lehmiller,
Most are very short,and none is longer than 26 signs.ラブドールおすすめ
sex can sometimes unconsciously be engaged in to defend against love and intimacy.Someone who has been severely wounded during childhood in the way Guggenheim reportedly was would typically avoid
situations in which they could be rejected and abandoned again.オナホ高級
displaying reactive behaviors and conversational abilities. ラブドールオナニーThis leap from inanimate objects to seemingly sentient beings has profound implications for the future of
but my partner informed me that while he enjoyed the sleeve,ラブドールえろhe’d “never pay $8.
ダッチワイフI always say get involved.Go in with eyes wide open,
The production of these dolls often mirrors ラブドール sexand sometimes exaggerates societal standards of beauty
pining for recognition.As Freyd says in her book,ダッチワイフとは
Nominal Threats: As good as intercourse with a human husband or wife might be, エロ人形there is always the potential risk of contracting sexually transmitted conditions or accidentally obtaining
Expecting, not forgetting the risks connected to promiscuous sexual conduct with random associates or sex staff.
Our testers could not locate a standard denominator;ドールエロ neither elements nor brand name looked as if it would affect their availability.
It’s clear why so many people love The combination of high-quality products,ラブドール中古excellent customer and a seamless shopping experience makes JP-Dolls a top choice.
No interaction between the sexes is necessary for intrasexual conflict to occur.中国エロInterlocus conflict happens when the interaction between males and females benefits one while is costly or
detrimental to the other.
so to speak,will make your pelvic floor stronger.人形エロ
but don’t be afraid to come up with your own.Experiment.セックスドール
オナホ, including giving and getting consent for sexual behaviors)Children and adolescents need accurate information, critical skills, and relevant supports to ensure healthy and appropriate
sexual behavior on their part.Preventing Child Sexual Abuse: What Parents Need to Know The idea that any adult would sexually abuse a child is terrifying,
ラブドール通販But I can tell you why I,as a Canadian,
with more cities introducing them as a means of reducing traffic congestion and air pollution.リアルラブドールThese programs allow users to rent a bike for a short period,
or “self-expansion potential,” should make a unique contribution to relationship quality above and beyond the benefits either individual will experience.ラブドール中古
A self-care routine as simple as having a bath in candlelight can help you chill,relax,エロ人形
All types of fat are high in energy,浜哄舰銈ㄣ儹so they should only be eaten in small amounts.
ラブドールえろ, testosterone), which are associated with increased adolescent sexual behavior.Dating,
and that her hypersexuality,オナホ高級and its consequences,
Your feelings about sex might change as you grow and get toロボットセックス know yourself better. Your feelings about your partner could change.
One couple is on a lounge chair about three over from mine,ラブドールオナニー the guy kneeling in the sand to do the damn thing. With the other pair, the woman is truly aspirational
This level of personalization ensures that each sex doll is a unique creation, ドールオナニーcatering to the individual desires of its owner.
セックスドールYou have done a marvellous job!I am sure this paragraph has touched all the internet people,its reallyreally pleasant post on building up new web site.
ラブドー ?ダッチワイ ?が欲しい！えろ人形実際に買った男の失敗しないドール選び
By alternating praise and forms of abuse,narcissists foster dependency and even reliance in others.えろ人形
and he fell into the sea.To Gilbert,ラブドール中古
オナホリアルchildren usually become less farsighted and may become nearsighted instead.Myopia generally gets worse through adolescence,
My deck also offers a private hot tub, and I’m sitting in the ラブドールオナニーbubbling water alone watching the sunset with a champagne flute when a muscular man and his penis walk by.
[They think] they should be a sexual machine and just want to fuck people.ラブドール女性用[Stereotypes] reduce sex down to a functionality.’ Similarly,
ラブドールおすすめwe can look at a few more readily apparent aspects that are equally fascinating and indicative of our progress.1) Innovations in MaterialsBefore we look at the probable
technological developments,
an anniversary trip is an ideal way to spend quality time with your partner in a beautiful destination.Not to mention,エロ下着
T+L’s 2024 Destination of the Year.“The proximity to the U.エロ下着
ラブドールオナホjust 66 of women reported being “predominantly not attracted to the same ” This group likely includes straight women and at least 24 (90 minus 66) who have a slight degree of dual
attraction but who reject a bisexual identity label as too same-sex oriented for them.If given the option,
アダルト下着” and if you visit in the late summer (as opposed to the popular winter season),you’ll often see lower prices and fewer crowds.
Prepare now for mind-blowing skin-on-skinえろ人形 contact later
There are a few obvious things that we all know are 人形エロvital to human survival: food. Water. Oxygen.
this needs to be achieved without activating a shutdown responsefight/flight/freeze that is often triggered by reminders of the traumatic story.ラブドール通販In the ETI roadmap,
エロラブドールhowever,think about those days at times and one camping trip in particular always comes to mind.
and tangled Christmas tree lights.ラブドール男” ~ Maya AngelouHowever tangled the lights are this year,
on the border with Thailand.Intermediate stations will help open up to visitors parts of the country that haven’t historically benefited from tourism,セクシー下着
アダルト下着massive European capitals like London and Paris are worthy of a romantic vacation,but something about Amsterdam’s charming canals and cozy hotels makes it an ideal anniversary
the whole city is lit up with impressive displays,and family-friendly festivities are on offer every weekend from Thanksgiving through Christmas.t バック
t バックThe country’s theme park capital is unsurprisingly one of the most festive places to visit over the holiday season.At Walt Disney World,
suggesting this character doesn’t want or need to dissociate from the violent spectacle.This quality of presence while enacting horrific violence,ダッチワイフとは
ラブドール動画collective trauma is the notion that a group of people may develop negative psychological symptoms after a member of that community suffers from an act of violence.Even if they did
not directly encounter the act or know the victim or survivor personally,
Click this link to visit us with the Dollオナドール Discussion board to discover what our clients really have to say about our company as well as the prodcuts we market.
naturally—to manage units they use on them selves, ラブドール女性用As well as in devices they can manipulate for (on? With? This can be prepositionally difficult) their viewers.
セクシー下着the village of Garzón has lately become a hub for multidisciplinary artists thanks to Campo,an incubator and non-profit that brings creatives to this still-rural corner of Uruguay.
The funny thing is,guys go through the exact same thing!That’s sexual tension.エロ人形
When you bite him,エロ人形you want to choose areas that aren’t particularly sensitive,
Your innovative solutions have inspired me to think more creatively about [related issue],ダッチワイフand I look forward to seeing more of your groundbreaking ideas.
incorporating more interactive elements,such as videos or could further enrich the user experience.ラブドールエロ
researchers at Stanford University looked at data from 500,ラブドールオナニー000 women and found that less sex is reported in the three days leading up to Christmas than average,
recognized for its versatile and high-quality sex dolls,人形えろ offers various finger options tailored to diverse customer preferences.
Sometimes,protecting yourself requires a big change.セックス人形
Partners in crime for life.sex ドールThe caption conveys you have found your go-to partner to enjoy every kind of activity in life.
In the world of sex dolls, ドールオナニーsize is a critical consideration. Buyers can opt for a full-sized doll,
ラブドールIn that case,this can let the perpetrator go forward with the next steps in the grooming process,
ラブドールオナホand 1984—documented the significant effectiveness of erotic fiction for the treatment of low libido.But somehow,
Individuals who derived more comfort from their companion animals and those with lower support systems also anticipated more problems with access to healthcare.This was also true of participants
who were taking antiretroviral medications.セックスドール
their brains were scanned in real-time using FMRI technology which can assess which areas of the brain light up in response to different objects or events.The findings highlight the role of a
part of your emotional brain called the amygdala,女性用ラブドール
showcasing their outfits in stunning travel destinations and inspiring their followers to embrace a wanderlust-infused style.ラブドール無修正The combination of travel,
while it is banned in some other states.Delta-8 products are available in smokable,リアルドール
and buying a sex doll is no exラブドールおすすめ
or just relax near the ocean.ベビードールランジェリーyou’ll be able to fill up on fresh fruits,
we have curated the best ones that would breathe life into your social media post.オナニー用From non-cheesy and light hearted love captions to romantic quotes for your Instagram posts,
I couldn’t help but also think for a quick moment that this was the moment, オナドールa milestone in my life where I would lose my virginity and sin
ラブドールエロsuch as friends,cousins,
ラブドール風俗Wetting reduces strength by about 15–25 percent.A silk filament can be stretched about 20 percent beyond its original length before breaking but does not immediately resume its
original length when stretched more than about 2 percent.
Narcissism is a severe form of mental illness that causes the narcissistic individual great suffering and traumatically affects family members,えろ人形social groups,
You may already recognize your overtly narcissistic parent’s bullying behaviors,セックスロボットsuch as monologuing,
ridiculed,and turned into a human chew-toy as the narcissist unleashes untethered rage and hatred.ラブドール中古
it’s time to consider what is happening in the friendship dynamic that is having such an impact on how you feel about yourself.ロボットセックスThey’re a fair-weather friend.
It is not enough for staff to report to a supervisor.One-on-one interactions—especially in isolated locations—should be avoided.オナホ
There is no need for additional tools to assemble your TPE sex doll.ラブドールメーカーSex dolls have been witnessing an upward trend in popularity.
ラブドール通販As dissociation is largely conceptualized as a relational trauma that occurs when a trusted other betrays someone through repeated abuse and gaslighting,we can see these dynamics
play out between Alex and those she loves.
I’ll start with her foot.A couch swallows it in the final moments of episode seven of Netflix’s Maid2021,ラブドールリアル
wrap your legs around him too right as he is about to cum.As he starts to cum,人形エロ
must take care of Rene,and Rene’s mistake is excused.エロドール
those who are severely flawed of character,ラブドール中古but especially the narcissist,
ロボットセックスrelationships take a sh*t ton of work.You may simply just be exhausted.
ラブドールperhaps respected,perhaps feared…and he carries power.
Your reactionary parent has made you hypervigilant to the moods and feelings of others.Your nervous system has been conditioned to be on high alert around other people,セックスロボット
cemented the city’s spot among true international arts destinations.ランジェリーショップThe recent reopening of the Rothko Chapel,
t バック“Winter is the best season to score lower rates in this family-friendly beach town,but that doesn’t mean the place goes totally quiet.
thank you so much for your time today.中国エロMolly Kuehl:Thanks,
cowboys).Move from the bed to the floor,人形エロ
ラブドールえろsex before marriage is OK if you are in love),greater incidence of sexual harassment (eg,
When you bite him,エロ人形you want to choose areas that aren’t particularly sensitive,
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Exploring Wilson Lines: A Geometric Perspective on Gauge Invariance
Written on
Chapter 1: The Necessity of Wilson Lines
As night falls, I find myself reflecting on the concept of Wilson lines. This discussion will align with insights drawn from Schwartz's work.
Section 1.1: Understanding Non-local Physics
Let’s delve into some fundamental questions regarding non-local physics. When fields possess a certain degree of gauge invariance, it becomes challenging to make comparisons between them at different
locations. To illustrate this, consider a physical field denoted as phi. Given the symmetries present in our Lagrangian, we anticipate that this field exhibits some level of gauge invariance.
Essentially, this means that at a specific point x in our field, we do not expect any variations in the physics when subjected to a transformation resembling
Now, if we assess the field at an alternate spatial point, the gauge field, represented by alpha in the previous expression, may not be local. Consequently, we can generally assert that
This raises philosophical questions; notably, it becomes impossible to ascertain whether the fields at various points are identical or distinct. For instance, consider two different points in our
fields and evaluate their absolute magnitudes. Upon performing a gauge transformation on this absolute difference, the outcome may not remain unchanged.
Section 1.2: Wilson Lines and Covariant Derivatives
Wilson lines provide a natural framework for constructing gauge invariant fields, enabling us to compare different points in spacetime meaningfully. Fields should remain invariant under a phase
transformation ? (x) ? e^i?(x)? (x), yet this phase is not consistently the same across all points. Thus, gauge fields function as a connection, similar to concepts in general relativity,
facilitating meaningful comparisons between fields at different locations. Schwartz elaborates on this concept further.
Revisiting Quantum Electrodynamics (QED), where fermions exhibit a U(1) gauge symmetry, we observe that under this transformation, the expression is given by
However, since ? (x) varies with x, our kinetic term ??/ does not retain its invariance. If we consider a derivative in the direction of a vector n, we find
The gauge covariant derivative is defined to be the component that merely acquires a phase under this transformation, leading to
Defining Wilson Lines
We define a Wilson line (or parallel transporter) as a function represented as a line from x to y, such that its gauge transformation is expressed as ei?(y)W(y, x)e?i?(x). By convention, we establish
that W(x, x) = 1. This leads to the conclusion that W(y, x) is purely a phase, allowing us to denote W(y, x) = ei?(y,x). We adopt the convention that W(x, y) = (W(y, x))^*.
With Wilson lines, we can also define the covariant derivative in a specified direction as follows:
It can be verified that this expression is gauge invariant by transforming the contents within the brackets under U(1). The entire term adjusts simply by the phase e^(i?)(x+an). Since our vector n
was arbitrary, it confirms that ?D?/ is indeed gauge invariant. From this Wilson line, we can further define the gauge field Aµ as the infinitesimal parameter in the exponential, expressed as
If we expand our exponential, considering we are operating in the limit where it is minimal, we derive the form of our covariant derivative as
Through infinitesimal gauge transformations, we acquire the following identities:
Chapter 2: Wilson Lines and Electromagnetism
Thus, since Dµ does not contribute anything additional to the transformation of a field that transforms as ? (x) ? e^(i?(x)) ? (x), we conclude that the composite operator D?Dµ adds no new elements.
Particularly, this signifies that
This directly corresponds to the electromagnetic field. Alternatively, we can create a ‘plaquette,’ which is formed as a square with vertices as follows:
By expanding this around small a, and after canceling terms, we derive a form that resembles
Consequently, we see the electromagnetic tensor emerging naturally during the construction of this square. Conversely, Wilson lines need not remain small, and with the field Aµ being continuous, we
can define a curve C that acts as a Wilson line:
This curve forms a loop if y = z, with C = ?? where ? indicates the area enclosed. Applying Stokes’ theorem, we derive that
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Welcome to Modular Quantum Chemistry Project’s documentation!
Welcome to Modular Quantum Chemistry Project’s documentation!
This site is dedicated to the development and application of modular computational platform for the general science and technology community. As the old-fashioned approach to developing and
maintaining computational programs becomes obsolete, an emerging concept of software modularity offers an elegant and timely solution of the looming problems by providing an open development
ecosystem where new computational approaches can be rapidly created from modules uploaded at the web repository by the interested users and developers. The first implementation of such an environment
is already available in the form of a web-based platform at MQCP.
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What is the difference between the fusion rules for anyons and the decomposition of representations into the irreducible representations?
I am studying topological quantum computation. In many references, it starts with the anyonic statistics. In my understanding, a type of anyon corresponds to an irreducible representation of the
braid group, is it correct? Also, fusion rules for composite anyons are discussed. There is my main question. Fusion rules are described as
\(\phi_{a} \times \phi_{b} = \sum_c N^{a,b}_c \phi_{c},\)
where a, b, and, c are types of anyons. My question is whether this formula is nothing but the decomposition of the product of two irreps for the braid group into direct sums of irreps, or not. If
the answer is yes, I wonder why people call it fusion rule other than just the decomposition of the representation. If the answer is no, additional questions are following;
what is the difference between them?
why the category theory is needed to describe this rule?
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How Do You Determine if Triangles on the Coordinate Plane are Congruent?
There are many ways to show that two triangles are congruent. This tutorial shows you how to use a triangle congruence postulate to show that two triangles on the coordinate plane are congruent to
each other!
When you have two congruent figures, that means that corresponding sides and corresponding angles are congruent. Get some practice identifying corresponding sides and angles by following along with
this tutorial!
If you need to find the distance between to points on the coordinate, you'll probably use the distance formula to get your answer. This tutorial introduces you to the distance formula and even shows
you how to find it!
If two figures have the same size and shape, then they are congruent. The term congruent is often used to describe figures like this. In this tutorial, take a look at the term congruent!
Did you know that there are different kinds of angles? Knowing how to identify these angles is an important part of solving many problems involving angles. Check out this tutorial and learn about the
different kinds of angles!
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Items where Division is "College of Engineering & Physical Sciences > Systems analytics research institute (SARI)" and Year is 2000
Number of items: 14.
Berry , R.F. (2000). System and method for dynamic modification of class files. 6,026,237.
Cartland-Glover, Gregory, Generalis, Sotirios and Thomas, Niele H. (2000). Dynamic simulations of multiphase flow in bubble columns. IN: 14th International Conference of Chemical and Process
Engineering. 2000-08-27 - 2000-08-31.
Chattopadhyay, Amit Kr., Basu, Abhik and Bhattacharjee, Jayanta K. (2000). Coupled nonequilibrium growth equations:self-consistent mode coupling using vertex renormalization. Physical Review E, 61
(2), pp. 2086-2088.
Chattopadhyay, Amit Kr., Mahapatra, G.S. and Chaudhury, Pinaki (2000). Lyoluminescence:a theoretical approach. Physical Review B, 62 (2), pp. 906-909.
Coolen, Anthony C.C. and Saad, David (2000). Dynamics of learning with restricted training sets. Physical Review E, 62 (4), pp. 5444-5487.
Cornford, Dan, Nabney, Ian T. and Ramage, Guillaume (2000). Improved multi-beam neural network scatterometer forward models. Technical Report. Aston University, Birmingham, UK.
Cornford, Dan, Wright, W.A., Ramage, Guillaume and Nabney, Ian T. (2000). Neural network modelling with input uncertainty:theory and application. Journal of VLSI Signal Processing Systems for Signal
Image and Video Technology, 26 (1-2), pp. 169-188.
Kabashima, Yoshiyuki, Murayama, Tatsuto and Saad, David (2000). Typical performance of gallager-type error-correcting codes. Physical Review Letters, 84 (6), pp. 1355-1358.
Kanter, Ido and Saad, David (2000). Finite-size effects and error-free communication in Gaussian channels. Journal of Physics A: Mathematical and General, 33 (8), pp. 1675-1681.
Nerukh, Dimitry and Frederick, John H. (2000). Multidimensional quantum dynamics with trajectories: a novel numerical implementation of Bohmian mechanics. Chemical Physics Letters, 332 (1-2), pp.
Saad, David (2000). Final Report on EPSRC Research Grant GR/L52093. Project Report. Aston University, Birmingham, UK.
Tino, Peter and Nabney, Ian (2000). Hierarchical GTM: Constructing localized non-linear projection manifolds in a principled way. Technical Report. Aston University. (Unpublished)
Vicente, Renato, Saad, David and Kabashima, Yoshiyuki (2000). Error-correcting code on a cactus:A solvable model. Europhysics Letters, 51 (6), pp. 698-704.
Vicente, Renato, Saad, David and Kabashima, Yoshiyuki (2000). Statistical physics of irregular low-density parity-check codes. Journal of Physics A: Mathematical and General, 33 (37), pp. 6527-6542.
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what's the does not command for a for loop? as in for i from 1 to 100 while i (does not) equal 5.
Hi to everybody I'm a new user of Maple How can I solve this system? > eq1:=1039.44*Diff(Diff(ck(r),r),r)+80960*Diff(ck(r),r)*Diff(p(r),r)+80960*ck(r)*Diff(Diff(p(r),r),r)=0; / 2 \ | d | / d \ / d \
eq1 := 1039.44 |---- ck(r)| + 80960 |--- ck(r)| |--- p(r)| | 2 | \ dr / \ dr / \ dr / / 2 \ | d | + 80960 ck(r) |---- p(r)| = 0
Is there a chat or instant message capability on this website ?
Can anyone discover the reason for the troubles i'm having with the last execution group ? I think the few ones above it are some previous attempts. it's the part that begins as >Cardioid := proc(S)
local s; etc... > Warning, the name changecoords has been redefined >
i have this code: S := []; for i from 0 by 5 to 150 do S:=[op(S),[i,(i+3)^2 mod 7]]; end do; plot(S, style = point) i want to plot points connected by lines. how do i do this?
hi everyone , please help me to sort out this problem ... suppose i have a matrix 1 2 3 ? 2 ? 2 1 6 where ? are unknown values . what i want to do is to "Fill in the missing values by the respective
attribute means. " How i will do it matlab? please help me as soon as possible ...
What's the difference between Arrays and Matrices, according to Maple10 ?
I have been experimenting with overloading existing Maple functions by creating an appropriate package, as described in the "overload" help. It works fine in general. I was hoping to change the
definition of `abs` on type Vector to be sqrt(V . V) in this way, but the change does not work because `abs` already has an alternative definition on this datatype and it seems that built in
functions take precedence over overload versions that one has added. Is there any way of achieving this? Grateful for any pointers.
Maple usually has white background and red font. How to set font black and background color in light grey? Should I use "Format"--"Styles"? But why each time when I build new file I lost the previous
style setting?
Hi, guys could you please let me know how I can find this definite integral through Maple? int(exp(-I*x)/cosh(x)^3,x=-infinity..infinity); or int(exp(-I*x)/cosh(x)^2,x=-infinity..infinity); thanks a
lot Sayed
Trying to calculate the determinant of matrix A i have to add extra details to the function (integer) to get result (for example): Det(A) - does not work for me, but - Det(A) mod 10000 (or 10^n); -
does and gives the result (honestly it took me a while to get it clear, i mean to get Det(A) work). I think it's not the way it is supposed to work. In help it's simply clarified that Det(A) function
should give the result of determinant of matrix A, without any additional integers like mod :/. I can't figure out this riddle.. (i've started using maple 3weeks ago, and mostly for algebra and
geometry). Any help pls? Do i have to change any settings? Tnx, in advance =).
Hi, does anyone know if it is possible to use maple to handle equations involving differential forms (antisymmetric tensors). Basically i have a set of equations that involve forms of various
degrees, with some indices contracted with vectors etc. Now most of the components of the forms are zero, but the equations are still very tedious to handle by hand. It would be useful if i could
input the equation into maple, tell it all the various non-zero form components, and then let maple substitute the components into the equation. Naturally i would need to specify the values of the
free indices in the equation, so that the tensor equation reduses to a standard equation. And presumably Maple would need to understand the einstein summation convention etc.
What is the proper method for converting an extended numeric with a zero imaginary component to a real? It should leave the complex part alone if it is not zero. That is, somefunc(1.0 + 0.*I); 1.0
somefunc(1.0 + 1.*I); 1.0 + 1.*I I can write my own procedure, but there must be standard way (somefunc) to do this. Searching the help browser didn't help. Thanks.
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New Math Formula: Sums of Power for Arithmetic Series
Registered: 2012-01-30
Posts: 266
Re: New Math Formula: Sums of Power for Arithmetic Series
Registered: 2012-01-30
Posts: 266
Re: New Math Formula: Sums of Power for Arithmetic Series
New Formulation for alternating sums of power
Let the p-th power of an alternating arithmetic series as follows
The General Equations are given as follows:
For odd power:
For even power:
Last edited by Stangerzv (2019-11-21 11:53:58)
An approach to proof Fermat's Last Theorem.
Consider when n=3, and i=3, yields:
this equation is solvable.
Now let n=2
There is no whole number solution according to Euler.
Last edited by Stangerzv (2020-01-17 12:38:45)
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What Will a 750 Watt Inverter Run? - Energy Theory
The versatility of the inverter is truly remarkable as it serves as an indispensable companion for outdoor adventures, road trips, and emergencies. In this article, we’ll explore what will a 750 Watt
inverter run and how many Amps can it draw.
What Will a 750 Watt Inverter Run?
These inverters have a maximum capacity of 750 running watts per hour and a surge capacity of up to 1500 watts, which can be sustained for only a few seconds.
Appliance Compatibility and Power Calculation
These inverters can support a combination of appliances as long as the total power consumption remains below 750 watts per hour. The newer inverters are 90-95% efficient, resulting in actual
available wattage ranging from 675 to 712 watts. To determine appliance compatibility, consult an appliance wattage chart and calculate power requirements.
Appliance power consumption is typically measured in watts, although amps may also be listed. The amp draw per hour depends on the voltage.
Conversion Formulas and Examples
To convert between amps and watts, the following formulas can be used:
Amps x Volts = Watts
Watts / Amps = Volts
Watts / Volts = Amps
Example 1: A 750-watt inverter with a 320-watt load running on 120V AC power will draw 2.6 amps. 320 / 120 = 2.6 amps
Example 2: Another 750-watt inverter running a similar load on 24V batteries will pull 13.3 amps. 320 / 24 = 13.3 amps
These calculations provide estimates. It is important to consider inverter efficiency and factor in runtime. Multiply the total watts by the inverter rating to obtain an estimate.
Efficiency significantly affects power consumption. If you have a 450-watt load and a 90% efficient inverter: 450 / 0.90 = 500 watts The 450-watt load will utilize 500 watts due to efficiency losses.
Running an inverter at full capacity is not recommended as it can be dangerous, and there may not be sufficient power available.
Now, let’s find out what appliances can a 750 watt inverter run. This will help you get a better understanding of what will a 750 watt inverter run.
Also See: Will a 750 Watt Inverter Run a Microwave?
What Appliances Can a 750 Watt Inverter Run?
After understanding what will a 750 watt inverter run, let’s try to find out what appliances it can run. A 750-watt inverter offers ample power to run a wide range of home devices, appliances, and
power tools.
From laptops and TVs to cameras, radios, lights, blenders, portable fans, small microwaves, small sump pumps, a 1/2 inch drill, garage door openers, and other gadgets, a 750-watt power inverter can
provide sufficient power for these electronic devices.
Moreover, these inverters are capable of powering appliances such as microwaves, refrigerators, and a variety of power tools, thanks to their 1500-watt surge capacity.
Also See: Will a 750 Watt Inverter Run a Refrigerator?
How Long Will a 750 Watt Inverter Last?
An inverter can operate using either an electrical source (typically 110V/120V) or batteries. When the inverter is connected to an AC power source, it can run continuously without any limitations as
long as there is electricity available. However, if the inverter relies on battery power, its runtime will be determined by the remaining charge in the batteries.
For a more precise estimation of the available power duration, please make a detailed list of all the appliances, tools, and devices that you intend to use with the inverter. Additionally, include
the wattage ratings of each item, this will provide you with a more accurate idea of how long a 750 watt inverter will last.
This answer was an important step in thoroughly learning what will a 750 watt inverter run and for how long.
Also Read: What Will a 400 Watt Power Inverter Run?
750W Inverter Battery Runtime Guide
Here’s a convenient reference for running various power loads on an inverter and their expected duration. This guide assumes the inverter operates on batteries, which will be completely depleted by
the end of the specified runtime.
• 60W ceiling fan: 12 hours
• 60W light bulb: 12 hours
• Refrigerator: 8 hours
• 40-inch TV with movie player: 5 hours
• Slow cooker: 3 hours
• Laptop, printer, speakers, router, portable fan, LED lights: 2 hours
• 600W microwave: 1 hour
• Well pump: 1 hour
• Blender: 1.5 hours
Here are some additional points to consider regarding this 750W inverter battery runtime guide:
• The power consumption values provided are estimates. For instance, the laptop in the example utilizes 200 watts, which is typical for business applications and web browsing. However, a gaming
laptop will require more solar power to operate effectively.
• Many appliances exhibit variable usage patterns. For example, while a blender may consume approximately 700 watts per hour, it is typically used for only a few minutes or seconds at a time. This
applies to other appliances and tools as well, meaning their actual usage may not be as high as their maximum power rating suggests.
• The chart assumes that the inverter has a power capacity of 750 watts, which is equivalent to a 62.5 ampere-hour (Ah) 12V battery or a 31.25Ah 24V battery. Rounding off to the nearest available
battery size, this would be approximately 75Ah for 12V systems or 35Ah for 24V systems.
• You have the flexibility to mix and match appliances based on your specific requirements. There is no one-size-fits-all solution since each individual has different appliances and tools with
varying power demands.
• Due to manufacturing differences, power consumption may vary among appliances, even if their specifications appear the same. Some appliances may be more energy-efficient than others.
• Inverters perform optimally when shorter and thicker wires are used. By minimizing the length of cables between the inverter and the batteries, less energy will be lost during the conversion
After going through this 750W inverter battery runtime guide, unraveling the number of batteries you need for a 750 Watt inverter also becomes necessary.
How Many Batteries Do I Need For a 750 Watt Inverter?
If you are connected to a grid-tied system, you won’t require a battery bank as the inverter can operate directly from the main power supply. As long as there is electricity available, your
appliances will continue to function without interruption.
However, if you are in an RV or a solar-powered mobile home, the inverter’s power supply will be sourced from the battery bank. To determine the number of batteries needed, you can use the following
Appliance Watts x runtime = total Watts
Total watts / DC volts = amps
Let’s consider an example with a 12V 750 watt inverter. If you have a load of 750 watts, the inverter will run for approximately an hour, depending on its efficiency rating. The system will draw 62.5
amps (750 / 12 = 62.5). Assuming you have a 75Ah battery, the runtime will be around 1.2 hours (62.5 / 75 = 1.2).
Keep in mind that running the inverter on a 75Ah battery for 1.2 hours will result in a complete discharge. This is not recommended for flooded lead-acid (FLA) batteries, as they are typically
designed to be recharged at 50% depth of discharge. Following this guideline, a 12V 150Ah battery would be required to power a 750 watt load for 1.2 hours without fully discharging the battery.
Now, it’s also necessary for you to find out how many Amps can a 750 Watt inverter draw. So, let’s find this out in the next question.
Also See: What will an 800 Watt Inverter Run?
How Many Amps Can a 750 Watt Inverter Draw?
To determine how many Amps can a 750 Watt inverter draw, it is important to consider the voltage of the inverter, which could be 12 volts, 14 volts, 24 volts, or 28 volts.
While many inverters with a power rating of 750 watts typically operate at voltages higher than 12 volts, for the purpose of this calculation, we will assume a voltage of 12 volts, which is the
lowest value.
Therefore, the amperage of the inverter at 100% efficiency would be calculated as follows: 750 watts / 12 volts = 62.5 amps.
Considering that achieving 100% efficiency is unlikely, let’s assume an efficiency of 80%. Consequently, the amperage would be approximately 62.5 amps / 0.8 = 78.13 amps.
After understanding how many Amps can a 750 Watt inverter draw, let’s also try to figure out how to safely run appliances on the 750 Watt inverter.
Also Read: How Many Amps Does a 100 Watt Solar Panel Produce
How to Safely Run Appliances on the 750 Watt Inverter
To safely run appliances on the 750 Watt inverter, please consider the following guidelines:
• Install the inverter in a well-ventilated area to prevent overheating.
• Adhere to the inverter’s specified continuous and surge power ratings. Exceeding these limits may cause improper appliance functionality and potential damage to the inverter.
• Determine the maximum runtime of the inverter at its full power output. Some inverters can only sustain their continuous power rating for a limited period, which could be an hour or less.
• Whenever possible, aim to utilize the inverter at up to 80% of its full power rating. For instance, if you have a 750 watt inverter, try connecting appliances with a maximum power draw of around
600 watts (80% of 750 watts).
By following these recommendations, you can ensure the safe and efficient operation of appliances connected to the inverter.
In conclusion, a 750-watt inverter opens up a world of convenience and flexibility. From outdoor adventures to unforeseen power outages, this device proves to be a reliable ally. For other
interesting blogs, explore our FAQ section.
Recommended: 48 Vs. 72-Volt Golf Cart – Which One Performs Better?
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Fraction to Decimal Calculator | Convert Fractional Numbers to Decimal Form
Fraction To Decimal Calculator
Convert any fraction to decimal value with our Free Online Fraction to Decimal Converter Calculator. Get Accurate Results with Steps
Convert Fraction to Decimal: Fraction To Decimal Calculator 🧮
Fractions and Decimals are two ways of representing the same concept: a part of a whole. Decimals are expressed using a point (.) to separate the whole number part from the fractional part, such as
0.5 or 3.75. Whereas fractions, on the other hand, represent parts of a whole using two integers, with one number placed above (numerator) and one below (denominator) a dividing line, like ½ or ¾.
While decimals are often more convenient for calculation, fractions can provide more clarity, especially when dealing with ratios, proportions, or divisions of equal parts. However, converting
between decimals and fractions manually can be tricky for many. For this reason, a Fraction to Decimal Calculator is an excellent tool to simplify and accurately convert fractions to decimals.
What is a Fraction to Decimal Calculator? 🤷♀️
A Fraction to Decimal Calculator is an online calculating tool designed to quickly convert any fraction to its equivalent decimal value (point value). This tool not only saves time but also ensures
accuracy by automating the complex conversion process. Whether you're dealing with simple fractions like 4/5 or more complex ones like 1 3/4, the calculator provides you with the decimal equivalent
without any hassle. It even simplifies mixed fractions for easier understanding. By entering the fractional value, the Fraction to Decimal Calculator shows its equivalent decimal form.
How to Use Our Fraction to Decimal Calculator? 🤔
Using our Fraction to Decimal Converter is simple and straightforward. You can perform the following steps to convert your fractional values to decimal values with ease and precision:
Enter your Fractional Value 📝
Start by entering the numerator and denominator of the fraction you want to convert into the designated input fields. You can input both whole numbers and fractions, including mixed fractions like 1
Generate Your Decimal Result 🎉
Once you've entered the fractional value, click on the 'Calculate' button to generate the decimal equivalent. The calculator will display the decimal value of the fraction in the output field.
Example of Input ✒️
The Role of Decimals and Fractions in Real-Life Scenarios 🏡
In everyday life, decimals and fractions play a significant role, particularly in situations requiring precision and accuracy. For example, consider cooking measurements. Many recipes use fractions
like ½ or ¾ cups of ingredients. However, when scaling a recipe or converting it to metric units, decimals become more convenient. Similarly, in financial calculations, decimals are often used to
represent interest rates, percentages, and costs. However, when it comes to dividing amounts or portions evenly, fractions often provide a clearer picture.
The Fraction to Decimal Calculator bridges this gap by converting fractions into decimals, ensuring you have the flexibility to work with either format in various real-life scenarios. Whether you're
managing your household budget, working on a school project, or dealing with technical data at work, being able to switch between decimals and fractions is an invaluable skill. A Fraction to Decimal
Calculator makes this task easier, ensuring that you can handle numbers accurately and confidently in any situation.
Benefits of Fraction to Decimal Calculator 🎁
1. Quick and Accurate Conversion 🎯:
Instead of spending time manually calculating the conversion, you can simply input the fractional value and receive an accurate decimal within seconds. This tool removes the possibility of human
error and ensures precise results every time.
2. Simplification of Complex Fractions 🧩:
Fractions like 134/255 and 25 62/68 can be difficult to convert manually. The calculator simplifies even the most complex fractions, providing a decimal that is both accurate and easy to understand.
This feature is especially useful for students or professionals dealing with mathematical or engineering calculations.
3. Saves Time in Problem-Solving 🕒:
Whether you're working on homework, preparing for a test, or handling professional tasks, time is always of the essence. A Fraction to Decimal Calculator saves you the hassle of manual calculations,
allowing you to focus on solving other aspects of the problem. For anyone pressed for time, this tool is a game changer.
4. Useful Across Various Fields ✨:
From cooking recipes to engineering problems, decimals and fractions are ubiquitous. Whether you're trying to figure out the correct proportions for a recipe or solving mathematical equations in
physics or finance, the Fraction to Decimal Calculator provides valuable support. It is a versatile tool that can be used in many fields where precision is key.
Frequently Asked Questions (FAQs)
Q1: What is a Fraction to Decimal Calculator?
A Fraction to Decimal Calculator is a tool that converts fractions into their decimal equivalents for easier understanding and comparison.
Q2: How does the Fraction to Decimal Calculator work?
The calculator divides the numerator by the denominator to give the decimal representation of the fraction.
Q3: Can improper fractions be converted to decimals?
Yes, improper fractions can be converted into decimal values by dividing the numerator by the denominator.
Q4: Can mixed fractions be converted to decimals?
Yes, mixed fractions can be converted by first converting the mixed fraction into an improper fraction and then dividing.
Q5: Can every fraction be converted to a decimal?
Yes, every fraction can be converted to a decimal, but some may result in repeating decimals.
Q6: What is a repeating decimal?
A repeating decimal is a decimal in which one or more digits repeat infinitely, such as 1/3 which equals 0.3333...
Q7: Does the calculator handle repeating decimals?
Yes, the calculator will show repeating decimals with rounding off the repeating part.
Q8: Can fractions with negative numbers be converted to decimals?
Yes, fractions with negative numerators or denominators can be converted, and the result will be a negative decimal.
Q9: Is the conversion process the same for mixed and improper fractions?
Yes, in both cases, the fraction is converted to a decimal by dividing the numerator by the denominator.
Q10: What is the benefit of converting fractions to decimals?
Decimals are often easier to use in calculations and comparisons, especially in contexts like measurement or finance.
Q11: Can the calculator handle large fractions?
Yes, the calculator can handle fractions with large numerators and denominators, converting them to their decimal form.
Q12: Does the Fraction to Decimal Calculator simplify fractions before converting?
No, the calculator converts the fraction as entered without simplifying unless requested separately.
Q13: Can the calculator convert recurring decimals back to fractions?
No, this tool is specifically for converting fractions to decimals, not the reverse. For that, you would need our Decimal to Fraction Converter.
Q14: How accurate are the decimal results?
The calculator provides results up to a high level of precision, and can round the decimals based on the user's needs.
Q15: What is Terminating and Non terminating Decimal?
A terminating decimal is a decimal number that has a finite number of digits after the decimal point. A non-terminating decimal is a decimal number that has an infinite number of digits after the
decimal point.
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How much does a standard paving concrete slab weigh - Civil Sir
How much does a standard paving concrete slab weigh
How much does a standard paving concrete slab weigh? The weight of a standard concrete paving slab can vary depending on its thickness, dimensions, and the type of concrete mix used. On average, a
typical concrete paving slab that is 2 inches (5 centimetres) thick and measures 2 feet by 2 feet (61 centimeters by 61 centimeters) can weigh around 40 to 50 pounds (18 to 23 kilograms).
To calculate the weight of a concrete slab, you’ll need to know its density and volume. The density of concrete typically ranges from 2,300 to 2,500 kilograms per cubic meter (kg/m³). Assuming a
density of 2,400 kg/m³, you can calculate the weight as follows:
Volume = Length × Width × Height, volume = 600 mm × 600 mm × 50 mm, convert millimeters to meters: 600 mm = 0.6 meters, and 50 mm = 0.05 meters, volume = 0.6 m × 0.6 m × 0.05 m, hence volume = 0.018
Now, calculate the weight: Weight = Volume × Density, Weight = 0.018 m³ × 2,400 kg/m³, hence Weight = 43.2 kilograms. So, a concrete slab measuring 600×600×50 mm would weigh approximately 43.2
How much does a standard paving concrete slab weigh
Most of a standard size concrete slab comes in square or rectangular shape, which are formed by moulding of concrete mixing of a stone, cement, sand, aggregate and water before being left to cure.
This yields highly durable and strong concrete slab.
Concrete slabs are one of the core building material that help you making the patio of your dream house. They are flat usually square or other in rectangular shape. Concrete paving slabs are ideal
for use around your home and outdoor space. Durable concrete slab replace the classical sandstone to durable concrete, and you would be able to find the perfect Patio slabs to suit your needs.
Patio is your valuable outdoor property. Many homeowners have a lot of different options for making an outdoor space into a livable and comfortable setting. One of the popular choice is a making
patio. A Beautiful patio can be made from many different materials such as concrete slab, by pavers setting, mix of sand and gravel. All these need proper foundation and then thin layers of sand as
good base for your patio is important.
Making a patio or hard parking area is a main way to extend your living space and add value to your property. Pavers, whether stone or concrete slab, must be supported by solid foundations, the final
layer or pavers base are of thin layer of paver sand which is paving sand.
In Uk, concrete slab available mostly in square shape or rectangular and in many different size such as 900×600 mm, 750×600 mm, 600×600 mm, 450×450 mm, 400×400 mm, 450×300 mm, and 300×300 mm.
Recommended thickness of paving concrete slab is generally about 50 mm thick.
In US, concrete slab available mostly in square shape or rectangular and in many different size such as 24×24 inches, 2×2 feet, 12×12 inches, 24×30 inches, 3×2 feet, 10×10 inches, and 3×3 feet.
Recommended thickness of paving concrete slab is generally about 2 inches thick.
Concrete pavers that are 1 inch thick weigh about 12 pounds per square foot, and 24 pounds at 2 inches thick.
How much does a standard paving concrete slab weigh
A standard paving concrete slab is 600mm long, 600mm wide and 50mm thick should weigh around 40kg each. Supplied in a pack of 20 weighing around 800kg and you need 2.8 pieces per square meter (m2).
How heavy is a 600×600×50 slab
A 600×600×50mm (or 2×2) paving concrete slab is 600mm long, 600mm wide and 50mm thick should weigh around 40kg. It comes in a pack of 20 and weighs around 800kg. You need 2.8 pieces per m2 (square
2×2 Slab Weight: The typical weight of a 2×2 concrete paver slab is about 40kg each. A typical 2 x 2 foot slab weighs around 40kg and comes in a pack of 20 and weighs around 800kg.
It is considered a two person lift and appropriate PPE, including thick gloves, should always be worn when handling as the edges of the slab can be sharp.
How heavy is a 900×600×50 slab
A 900×600×50mm (or 3×2) paving concrete slab is 900mm long, 600mm wide and 50mm thick should weigh around 60kg. It comes in a pack of 20 and weighs around 1200kg. You need 1.85 pieces per m2 (square
3×2 Slab Weight: The typical weight of a 3×2 concrete paver slab is about 60kg each. A typical 3 x 2 foot slab weighs around 60kg and comes in a pack of 20 and weighs around 1200kg.
How heavy is a 750×600×50 slab
A 750×600×50mm (or 2.5×2) paving concrete slab is 750mm long, 600mm wide and 50mm thick should weigh around 50kg. It comes in a pack of 20 and weighs around 1000kg. You need 2.2 pieces per m2 (square
2.5×2 Slab Weight: The typical weight of a 2.5×2 concrete paver slab is about 50kg each. A typical 2.5 x 2 foot slab weighs around 50kg and comes in a pack of 20 and weighs around 1000kg.
How heavy is a 450×450×50 slab
A 450×450×50mm (or 1.5×1.5) paving concrete slab is 450mm long, 450mm wide and 50mm thick should weigh around 23.5kg. It comes in a pack of 20 and weighs around 470kg. You need 5 pieces per m2
(square meter).
1.5×1.5 Slab Weight: The typical weight of a 1.5×1.5 concrete paver slab is about 23.5kg each. A typical 1.5 x 1.5 foot slab weighs around 23.5kg and comes in a pack of 20 and weighs around 470kg.
How heavy is a 400×400×50 slab
A 400×400×50mm paving concrete slab is 400mm long, 400mm wide and 50mm thick should weigh around 18.5kg. It comes in a pack of 20 and weighs around 370kg. You need 6.25 pieces per m2 (square meter).
How heavy is a 450×300×50 slab
A 450×300×50 mm size paving concrete slab is 450 mm in length by 300 mm wide by 50 mm thick, should weigh approximately 15.7 kg. It comes in pack of 20, which would weighs about 314 kg, and you will
require 7.4 nos per m2.
How heavy is a 300×300×50 slab
A 300×300×50 mm size paving concrete slab is 300 mm in length by 300 mm wide by 50 mm thick, should weigh approximately 10.5 kg. It comes in pack of 20, which would weighs about 210 kg, and you will
require 11.11 nos per m2.
How much does a 24×24 concrete slab weigh
A 24″×24″×2″ or 2’×2′ size paving concrete slab is 24″ (2 feet) in length by 24″ (2 feet) wide by 2″ (0.166 feet) thick, should weigh approximately 96 pounds. Concrete pavers that are 2 inches thick,
weigh about 24 pounds per square foot. So, weight of 24″×24″, or 2’×2′, or 4 square feet concrete slab is about 96 pound at 2 inches thick, such as 24 ×4 = 96.
How much does a 12×12 concrete slab weigh
A 12″×12″×2″ or 1’×1′ size paving concrete slab is 12″ (1 foot) in length by 12″ (1 foot) wide by 2″ (0.166 feet) thick, should weigh approximately 24 pounds. Concrete pavers that are 2 inches thick,
weigh about 24 pounds per square foot. So, weight of 12″×12″, or 1’×1′, or 1 square feet concrete slab is about 24 pound at 2 inches thick, such as 24 ×1 = 24.
How much does a 24×30 concrete slab weigh
A 24″×30″×2″ size paving concrete slab is 24″ (2 feet) in length by 30″ (2.5 feet) wide by 2″ (0.166 feet) thick, should weigh approximately 120 pounds. Concrete pavers that are 2 inches thick, weigh
about 24 pounds per square foot. So, weight of 24″×30″, or 5 square feet concrete slab is about 120 pound at 2 inches thick, such as 24 ×5 = 120.
◆ What is the weight of a 40×40 concrete slab
● What is the weight of a 50×50 concrete slab
How much does a 10×10 concrete slab weigh
A 10″×10″×2″ size paving concrete slab is 10″ in length by 10″ wide by 2″ (0.166 feet) thick, should weigh approximately 17 pounds. Concrete pavers that are 2 inches thick, weigh about 24 pounds per
square foot. So, weight of 10″×10″, or 0.69 square feet concrete slab is about 17 pound at 2 inches thick, such as 24 ×0.69 = 17.
How much does a 3×2 concrete slab weigh
A 3’×2′ or 36″×24″×2″ size paving concrete slab is 3 feet (36″) in length by 2 feet (24″) wide by 2″ (0.166 feet) thick, should weigh approximately 144 pounds. Concrete pavers that are 2 inches
thick, weigh about 24 pounds per square foot. So, weight of 3’×2′, or 36″×24″, or 6 square feet concrete slab is about 144 pound at 2 inches thick, such as 24 ×6 = 144.
How much does a 3×3 concrete slab weigh
A 3’×3′ or 36″×36″×2″ size paving concrete slab is 3 feet (36″) in length by 3 feet (36″) wide by 2″ (0.166 feet) thick, should weigh approximately 216 pounds. Concrete pavers that are 2 inches
thick, weigh about 24 pounds per square foot. So, weight of 3’×3′, or 36″×36″, or 9 square feet concrete slab is about 216 pound at 2 inches thick, such as 24 ×9 = 216.
A 600×600×50 standard size paving concrete slab is 600 mm in length by 600 mm wide by 50 mm thick, should weigh approximately 40 kg. A 900×600 concrete slab weighs about 60 kg. A 750×600 concrete
slab weighs about 50 kg. A 450×450 concrete slab weighs about 23.5 kg. A 400×400 concrete slab weighs about 18.5 kg. A 300×300 concrete slab weighs about 10.5 kg. A 450×300 concrete slab weighs about
15.7 kg.
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wrong gear transmission in ball mill
Ball Mill Maintenance Procedure Guide . Regular ball mill maintenance is an important factor to ensure the ... transmission bearing shaft and reducer should not exceed 55℃. The feed part should
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Pinion module of ball mill: Module is the ratio of pitch t to PI (m = t/ PI) between the teeth of the same side of two adjacent gears, measured in millimeters. Modulus is one of the most basic
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Dry and wet ball mills have the same basic components, but there are some structural differences: 3 Discharging part Discharging port: Dry ball mill: The ball mill needs to be equipped with an air
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pirical formulae for wet rod mills and wet and dry ball mills, where the diameter occurs as The formula contains in accordance with above explana tion the factor Os (fraction of critical speed)
and the filling rate of the grind ing media charge. It is important to stress, however, that a simultaneous slippagefree mill
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Gear cutting of a split girth gear for a cement mill having an input power of 2200 HP 5. Separate mill A: Damage to mill cylinder, neck or stub endanger girth gear and pinion. Repairs entail
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The ballmilling sample with 50 wt.% FCIPs had a minimum reflection loss peak of − dB at a thickness of 2 mm and a maximum bandwidth (RL < −7 dB) of GHz at a thickness of mm, a ...
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Taking a Φ × m ball mill as the research object, the reason for the low processing capacity of the ball mill was explored via process mineralogy, physicochemical analysis, workshop ...
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The mill receives four streams as inputs: mined ore, water to assist with material transport, steel balls to assist with ore breakage, and under ow from the hydrocyclone. This study assumes that
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The purpose of this paper is to reduce the manufacturing cost of the ball mill gear transmission as much as possible to minimize the volume while maximizing reliability. The volume is calculated
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A Mazak QT20 lathe runs 120mm ID bearings on the spindle and has a spindle bore of ″. That works out to a wall thickness of about ″ or about 29mm. Now let's say we want to build a lathe spindle
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These transmissions receive a load of about half the teeth of the cycloid wheel, other words the teeth of the gear at an angle of approximately 180° are in the working area [3, 4]. This makes it
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Ball Mill Power Calculation Example #1. A wet grinding ball mill in closed circuit is to be fed 100 TPH of a material with a work index of 15 and a size distribution of 80% passing ¼ inch (6350
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A mill pinion gear is a vital component used in ball mills and similar milling equipment. Its primary function is to transmit power from the motor to the mill, enabling the rotation of the mill's
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Tutorial 01: First Steps in Drawing a Static Standing Pose
In this first tutorial, I want to demonstrate a process for starting many drawings of the standing figure. This process is especially useful for drawing static poses. The term static pose does not
have a precise meaning within the StArt System, but artists use it to describe poses in which the figure is still or stationary.
The steps I’m demonstrating here may be used as a starting point for drawing static standing poses as they would appear from any viewing angle or distance, and from any height between about the
middle of the figure’s thighs to the top of its head. The tutorial is based on drawing a figure that is 8 heads high. In a future demo, I’ll describe how these steps can be modified for drawing
figures with other proportions.
Tutorial 01: First Steps in Drawing a Static Standing Pose
Keep in mind that this tutorial is just a starting point, which future lessons will refer to and build upon. The process demonstrated here is certainly not the only way to begin a figure drawing, and
as you gain experience and skill, you may adopt other, more intuitive approaches for getting started. This process, however, is very good for helping acquaint an artist with certain key ideas,
including the core axis and head-length units, which you’ll learn about in the steps that follow.
There are 5 steps in this tutorial, so sharpen up your pencil and let’s get started!
Step 1
I begin by drawing a simple vertical line on the paper where I want my figure to be (fig. 1). I’ll refer to this line as the core axis of the figure, because it will run right through the core of the
figure’s body, from the crown of the head through a spot between the arches of the feet. The location of the core axis remains consistent regardless of what view of the figure we might be drawing.
In static standing poses, it’s helpful to start with a core axis in order to draw the different parts of the body in correct relation to each other. For example, from a side view, the core axis helps
us to judge where the head should be in relation to the feet, and where many points in-between should go, too. In a front or rear view, the axis helps us to draw the figure with symmetry between the
right and left halves.
Be careful that you draw the core axis as a true vertical line, meaning parallel to the sides of your paper. In smaller drawings, this is easier to judge, but in larger drawings the line can often
stray to one side.
If you have trouble drawing the axis as a true vertical, you can measure in from one side of the paper at several locations within the height of your drawing, mark each location at the same distance,
and then connect these marks, as you see in Figure 2. Though a ruler is useful for measuring, I recommend you draw the axis freehand rather than using a straightedge—it’s good training to practice
drawing straight, accurate lines.
Step 2
Next, decide how tall you want the figure to be, and mark the top and bottom with a short line across the core axis (fig. 3). The top line, which marks the top of the head, need only be a short dash,
but the bottom line should be wider to accommodate stances of different widths.
For this drawing, the figure may be whatever height you wish (depending on the height of your paper, of course). In other drawings, an artist may be placing the figure within a defined space, such as
a room or other environment. In those cases, the height of the figure will be determined by the surroundings, and requires an understanding of the rules of perspective. This is a more advanced topic,
which we’ll have to return to much later.
Step 3
Now that we’ve defined the core axis and the height of our figure, we need to divide the axis into units, each of which will be the height of the figure’s head. As described at the start of this
tutorial, the entire figure will be 8 heads high, so we need to divide the axis into 8 equal units.
This can be done visually and does not require any mathematical calculations (fig. 4):
1. Estimate a location halfway up the core axis, between the top and bottom points you marked previously.
2. Mark this spot, then measure the distance from the mark to the top of the core axis.
3. Compare this measurement to the distance from the halfway mark to the bottom of the core axis.
4. If the distances are not equal, adjust the position of the halfway mark and measure the top and bottom distances again, until the two halves are the same.
Step 4
Proceed by dividing the top half of the core axis into two equal parts, using the same process you just followed in Step 3. Then divide the bottom half of the axis into two parts (fig. 5)
When you have finished, you will have divided the entire axis into four equal parts, or quarters.
Step 5
Finally, divide each quarter of the core axis in half to complete the division of the axis into eight equal parts (fig. 6).
You may have noticed that judging the halfway point of any section of the axis gets easier as the distances become smaller. It follows that estimating these distances overall is easier with smaller
drawings of the figure than larger ones.
With smaller drawings, you can even use your pencil as a measuring stick, and avoid having to bring out a ruler. Simply align one end of your pencil with an end point of the distance you want to
measure, and pinch the pencil at the other end point. Then hold this length of the pencil up to the distance you want to compare it to, and you have made a comparative measurement.
As an alternative to the estimating process I’ve outlined in Steps 3, 4, and 5, you could simply measure the height of the core axis and divide it in half. You could even divide it into eight parts
right from the start. Obviously this is easier if the height of your figure divides evenly, which will not always be the case. I prefer to use the visual, “guess-timating” method I’ve described, and
to use a pencil as a measuring stick whenever possible.
Figure 7 shows the completed drawing with the core axis divided into eight equal units, or head lengths. We can now use this framework to accurately draw a figure seen from any angle, at viewing
heights from the figure’s mid-thigh up to eye level. And we’ll start to do just that in the next tutorial. Tune in!
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American Mathematical Society
We consider the convergence of Gauss-type quadrature formulas for the integral $\int _0^\infty f(x)\omega (x)\mathrm {d}x$, where $\omega$ is a weight function on the half line $[0,\infty )$. The
$n$-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomials $\Lambda _{-p,q-1}=\{\sum _{k=-p}^{q-1} a_k x^k$}, where $p=p(n)$ is a sequence of
integers satisfying $0\le p(n)\le 2n$ and $q=q(n)=2n-p(n)$. It is proved that under certain Carleman-type conditions for the weight and when $p(n)$ or $q(n)$ goes to $\infty$, then convergence holds
for all functions $f$ for which $f\omega$ is integrable on $[0,\infty )$. Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss
quadrature formulas for the half line. References
• Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 208797
• A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, Quadrature on the half-line and two-point Padé approximants to Stieltjes functions. II. Convergence, J. Comput. Appl. Math. 77
(1997), no. 1-2, 53–76. ROLLS Symposium (Leipzig, 1996). MR 1440004, DOI 10.1016/S0377-0427(96)00122-7
• A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, Quadrature on the half line and two-point Padé approximants to Stieltjes functions. III. The unbounded case, J. Comput. Appl. Math.
87 (1997), no. 1, 95–117. MR 1488823, DOI 10.1016/S0377-0427(97)00180-5
• A. Bultheel, P. González-Vera, and R. Orive, Quadrature on the half-line and two-point Padé approximants to Stieltjes functions. I. Algebraic aspects, Proceedings of the International Conference
on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 57–72. MR 1379119, DOI 10.1016/0377-0427(95)00100-X
• T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 481884
• Lyle Cochran and S. Clement Cooper, Orthogonal Laurent polynomials on the real line, Continued fractions and orthogonal functions (Loen, 1992) Lecture Notes in Pure and Appl. Math., vol. 154,
Dekker, New York, 1994, pp. 47–100. MR 1263248
• Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
• Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
• Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser Verlag, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060
• J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
• William B. Jones, Olav Njȧstad, and W. J. Thron, Two-point Padé expansions for a family of analytic functions, J. Comput. Appl. Math. 9 (1983), no. 2, 105–123. MR 709210, DOI 10.1016/0377-0427
• W.B. Jones and W.J. Thron, Orthogonal Laurent polynomials and Gaussian quadrature, Quantum mechanics in mathematics, chemistry and physics (New York) (K. Gustafson and W.P. Reinhardt, eds.),
Plenum, 1984, pp. 449–455.
• William B. Jones, W. J. Thron, and Haakon Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980), no. 2, 503–528. MR 580900, DOI 10.1090/S0002-9947-1980-0580900-4
• Vladimir Ivanovich Krylov, Approximate calculation of integrals, The Macmillan Company, New York-London, 1962. Translated by Arthur H. Stroud. MR 144464
• G. López Lagomasino and A. Martínez Finkelshtein, Rate of convergence of two-point Padé approximants and logarithmic asymptotics of Laurent-type orthogonal polynomials, Constr. Approx. 11 (1995),
no. 2, 255–286. MR 1342387, DOI 10.1007/BF01203418
• G. Lopes, The convergence of Padé approximants for meromorphic functions of Stieltjes type, Mat. Sb. (N.S.) 111(153) (1980), no. 2, 308–316, 320 (Russian). MR 564355
• G. L. Lopes, Asymptotic behavior of the ratio of orthogonal polynomials and convergence of multipoint Padé approximants, Mat. Sb. (N.S.) 128(170) (1985), no. 2, 216–229, 287 (Russian). MR 809486
• G. L. Lopes, Convergence of Padé approximants for meromorphic functions of Stieltjes type and comparative asymptotics for orthogonal polynomials, Mat. Sb. (N.S.) 136(178) (1988), no. 2, 206–226,
301 (Russian); English transl., Math. USSR-Sb. 64 (1989), no. 1, 207–227. MR 954925, DOI 10.1070/SM1989v064n01ABEH003303
• A. Sri Ranga, Another quadrature rule of highest algebraic degree of precision, Numer. Math. 68 (1994), no. 2, 283–294. MR 1283343, DOI 10.1007/s002110050062
• A. Sri Ranga and J. H. McCabe, On the extensions of some classical distributions, Proc. Edinburgh Math. Soc. (2) 34 (1991), no. 1, 19–29. MR 1093173, DOI 10.1017/S0013091500004971
• Thomas Jan Stieltjes, Œuvres complètes/Collected papers. Vol. I, II, Springer-Verlag, Berlin, 1993. Reprint of the 1914–1918 edition; Edited and with a preface and a biographical note by Gerrit
van Dijk; With additional biographical and historical material by Walter Van Assche, Frits Beukers, Wilhelmus A. J. Luxemburg and Herman J. J. te Riele. MR 1272017
• Thomas Jan Stieltjes, Œuvres complètes/Collected papers. Vol. I, II, Springer-Verlag, Berlin, 1993. Reprint of the 1914–1918 edition; Edited and with a preface and a biographical note by Gerrit
van Dijk; With additional biographical and historical material by Walter Van Assche, Frits Beukers, Wilhelmus A. J. Luxemburg and Herman J. J. te Riele. MR 1272017
• J.V. Uspensky, On the convergence of quadrature formulas between infinite limits, Bulletin of the Russian Academy of Sciences (1916).
• —, On the convergence of quadrature formulas related to an infinite interval, Trans. Amer. Math. Soc. 30 (1928), 542–554.
Similar Articles
• Retrieve articles in Mathematics of Computation with MSC (1991): 65D30, 41A21
• Retrieve articles in all journals with MSC (1991): 65D30, 41A21
Bibliographic Information
• A. Bultheel
• Affiliation: Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium
• Email: Adhemar.Bultheel@cs.kuleuven.ac.be
• C. Díaz-Mendoza
• Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain
• Email: cjdiaz@ull.es
• P. González-Vera
• Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain
• Email: pglez@ull.es
• R. Orive
• Affiliation: Department Mathematical Analysis, Univ. La Laguna, Tenerife, Canary Islands, Spain
• Email: rorive@ull.es
• Received by editor(s): March 3, 1998
• Received by editor(s) in revised form: May 19, 1998
• Published electronically: February 24, 1999
• Additional Notes: The work of the first author is partially supported by the Fund for Scientific Research (FWO), project “Orthogonal systems and their applications”, grant #G.0278.97, and the
Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture. The scientific responsibility rests with the
The work of the other three authors was partially supported by the scientific research project PB96-1029 of the Spanish D.G.I.C.Y.T
Dedicated: Dedicated to Professor Nácere Hayek Calil on the occasion of his 75th birthday
• © Copyright 2000 American Mathematical Society
• Journal: Math. Comp. 69 (2000), 721-747
• MSC (1991): Primary 65D30; Secondary 41A21
• DOI: https://doi.org/10.1090/S0025-5718-99-01107-2
• MathSciNet review: 1651743
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IF and AND nested possible?
Hello there!
I'm looking to have multiple nested IF statements which would allow me to have a cell say different things depending on what range the value falls within.
For example, I have the following data:
So far, I have made the following:
=IF(AND(E3>=15.5,E3<25.1),"Dorval Airport","")
But any further IF or AND statements appear as error.
• It can be done, but you are setting yourself up for a rather long formula that has rather limited flexibility. Instead you can reference the table like so...
=INDEX([Column to pull from]:[Column to pull from], MATCH(MIN(COLLECT([Number Column]:[Number Column], [Number Column]:[Number Column], @cell >= E3)), [Number Column]:[Number Column], 0))
• Thanks for your comment!
What exactly is this formula doing?
• The INDEX function is used to pull data from a table based on a row number and an optional column number. If you are only referencing a single column, then you can leave the column portion out.
The MATCH function (when looking at a single column) will provide the row number for the INDEX function.
[S:To tell the MATCH function what to work on, we use a MIN/COLLECT to pull together all numbers that are greater than or equal to E3 and then pull the lowest number from that grouping.:S]
And while typing this out, I realized I am approaching this backwards. The numbers you have listed are the low number for each and not the high number. that means we want to reverse the "MIN/
COLLECT>=E3" to "MAX/COLLECT<=E3"
=INDEX([Column to pull from]:[Column to pull from], MATCH(MAX(COLLECT([Number Column]:[Number Column], [Number Column]:[Number Column], @cell <= E3)), [Number Column]:[Number Column], 0))
So we are using the MAX/COLLECT to pull all numbers that are less than or equal to E3 from the table and pulling the MAX from that.
So if E3 = 17.3, the largest number that is smaller than 17.3 is 15.5. The MATCH will use this to generate which row number 15.5 is on. That row number is then used to tell the INDEX function
which row to pull from which will generate "Dorval Airport".
My apologies for the initial mixup.
Help Article Resources
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mcmc-utilities: Utility operations for 'mcmc.list' objects in statnet.common: Common R Scripts and Utilities Used by the Statnet Project Software
colMeans.mcmc.list is a "method" for (non-generic) colMeans() applicable to mcmc.list objects.
var.mcmc.list is a "method" for (non-generic) var() applicable to mcmc.list objects. Since MCMC chains are assumed to all be sampling from the same underlying distribution, their pooled mean is used.
sweep.mcmc.list is a "method" for (non-generic) sweep() applicable to mcmc.list objects.
lapply.mcmc.list is a "method" for (non-generic) lapply() applicable to mcmc.list objects.
colMeans.mcmc.list(x, ...) var.mcmc.list(x, ...) sweep.mcmc.list(x, STATS, FUN = "-", check.margin = TRUE, ...) lapply.mcmc.list(X, FUN, ...)
x a mcmc.list object.
... additional arguments to the functions evaluated on each chain.
STATS, FUN, check.margin See help for sweep().
X An mcmc.list object.
These implementations should be equivalent (within numerical error) to the same function being called on as.matrix(x), while avoiding construction of the large matrix.
colMeans.mcmc returns a vector with length equal to the number of mcmc chains in x with the mean value for each chain.
lapply.mcmc.list returns an mcmc.list each of whose chains had been passed through FUN.
data(line, package="coda") colMeans(as.matrix(line)) # also coda colMeans.mcmc.list(line) # "Method" data(line, package="coda") var(as.matrix(line)) # coda var.mcmc.list(line) # "Method" data(line,
package="coda") colMeans.mcmc.list(line)-1:3 colMeans.mcmc.list(sweep.mcmc.list(line, 1:3)) data(line, package="coda") colMeans.mcmc.list(line)[c(2,3,1)] colMeans.mcmc.list(lapply.mcmc.list(line,
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Algebra II March 2 nd Students should complete warm-up problems. Given a graph of a system of equations students will be able to determine how many solutions. - ppt download
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seminars - The thermoviscoelastic nonlinear beam model with Coulomb friction dry law
Frictional or frictionless contact between bodies happens in every single day life. In this talk, one of dynamic frictional contact models, a thermoviscoelastic Gao beam with Coulomb friction dry law
is studied mathematically and numerically. Since the frictional conditions are nonsmooth, a regularization technique is applied to approximate a nonlinear variational formulation. We prove the
existence of weak solutions satisfying the regularized variational formulation, based on a priori estimates and results for a pseudomonotone operator. Then, we pass to limits, as a smoothing
parameter tends to be zero, in order to show convergence results for the regularized formulation. The fully discrete numerical schemes are proposed in which a guarded Newton method is employed to
compute fully discrete numerical solutions of a nonlinear system at each time step. We select several groups of data to present numerical simulations.
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Blue and olive change history markers?
Blue and olive change history markers?
I have got a question please, looking at the picture at https://notepad-plus-plus.org/news/v846-released/
I have never encountered the Change History markers in blue colour (“Revert to original state”) and olive colour (“Revert to modified”). In which cases do these markers appear?
For reference in this discussion, here is that image:
@datatraveller1 is referring to the four colors in box 1
Here is one sequence that goes through all the colors:
1. Start with new file (or load a file from disk): there are no colors in the margins.
2. Type something: it becomes orange
3. Save: it becomes green
4. Hit undo until type something is gone: it is the blue (“revert to original state”)
5. Type new text and save
6. Delete text and save
7. Undo once (to get text back): it is now olive (“revert to modified”)
Basically, “revert to modified” indicates that you’ve gone back to a previous save from this save. “revert to original” indicates that you’ve gone back to the way the file was when it was first
loaded into this session.
(The description for this feature hasn’t made it into the published usermanual yet, though it’s in the source. However, with this exploration, I have some more tweaks to improve it before it does
get published.)
@PeterJones Great description, thank you. However, I have difficulties to reproduce 4. I delete “type something” with backspace so it’s gone and the file is empty but the marker is orange instead
of blue. Am I doing something wrong?
@datatraveller1 ,
In step 4, I used UNDO, not BACKSPACE. And it will only work if the UNDO takes you back to the very original state from when you loaded (or created) the file
Note that in the screenshot, the stuff labeled 2 and 3 is related to each other, but both have absolutely nothing to do with the stuff labeled 1.
@PeterJones Yes sorry, I have just noticed it myself, I should have read the instruction more carefully. So thank you very much! I’ll have to practise this a bit more to get used to it.
@datatraveller1 ,
I actually had to experiment to try to figure out how to get the pale blue, because I rarely undo-to-completely-unchanged-after-save – it’s just not in my normal sequence of events. Undoing after
a save to get olive is more common in my experience. But I still usually only see the orange and green in my everyday Notepad++ usage.
@Alan-Kilborn, thanks for the clarification. I guess I should have thought about those numbers when I also then had numbered steps, and realized that could be confusing.
@PeterJones One more hint: I think I was a bit confused by the terms “Revert to original state” etc. “Revert” as verb sounds like a command that can be executed. Maybe this could better be named
“Reverted to original state” etc., couldn’t it?
@datatraveller1 ,
Those terms are just what the original author used in his screenshot. I cannot change that. In the manual, I will phrase it in the way that I think will properly convey meaning to the broadest
@PeterJones Great, thank you! Looking forward to your documentation. Your description in this post was already very helpful. Thank you again.
Hello, @datatraveller1, @peterjones, @alan-kilborn,
Many thanks, Peter for providing these hints on the change history markers ;-))
Like most of us, I have not noticed yet the blue and olive markers. Interesting !
So, in summary, it seems that :
□ A blue marker, on one or several line(s), means that you’re back to the initial state when you open this file in the current seesion => It does not need to be saved again
□ An orange marker, on one or several line(s), means that you modified some line(s) => So, a future save action is probably needed
□ A green marker, on all lines, means that, up to now, all modifications, on the current file, have been saved
□ An olive marker, on one or several line(s), means that you’ve get the final state of the current file => Thus, a future save action is probably needed
Are all my statements correct or am I still forgetting something ?
Best Regards,
@guy038 said in Blue and olive change history markers?:
A blue marker, on one or several line(s), means that you’re back to the initial state when you open this file in the current seesion => It does not need to be saved again
No, it means you saved changes, and then completely undid those changes after the save to get back to what the line was like when you first loaded. So it still means unsaved changes, because the
last save was something different from what is currently showing. Green and no-color are the only states that mean “everything is on disk already”.
Hi, @datatraveller1, @peterjones, @alan-kilborn and All,
OK ! I did some tests, Peter and, indeed, I was mistaken about the meaning of the blue marker !
This is because I simply considered the following case :
□ In your current N++ session, switch to a tab without change history markers, relative to a non-empty file
□ Type just a word at the end of this file => orange marker
□ Save this file => green marker
□ Now, undo this unique change => blue marker
But you get, again, the contents of the original file
And it’s this fact which made me wrongly think, that no save action was needed :-(
So, here is a better description :
□ NO color, on all lines, means that the current file has not been modified, yet, in the current session
□ A blue marker, on one or several line(s), means that, after a save action, you undid all the modifications on these lines => A future save action could be needed
□ An orange marker, on one or several line(s), means that you modified some line(s) => So, a future save action is probably needed
□ A green marker, on all lines, means that, up to now, all modifications, on the current file, have been saved on disk
□ An olive marker, on one or several line(s), means that you’ve get the final state of the current file => Thus, a future save action is probably needed
Hi @guy038
Not sure but maybe you have mixed up the description for blue and olive? As far as I have understood, blue is the very first state after opening the file. Olive is the last saved status but
especially this one is hard to understand. In any cases, blue and olive can only be seen after Undoing directly after saving. @PeterJones Please correct if needed.
@datatraveller1 said in Blue and olive change history markers?:
Hi @guy038
Not sure but maybe you have mixed up the description for blue and olive? As far as I have understood, blue is the very first state after opening the file. Olive is the last saved status but
especially this one is hard to understand. In any cases, blue and olive can only be seen after Undoing directly after saving. @PeterJones Please correct if needed.
I would say his blue description is correct, because blue indicates it’s back to the originally-loaded state (“undid all modifications”).
Unfortunately, I am not sure what Guy means in the olive description by “final state” – it might be something lost in translation between his native language and mine. Olive means that you have
undone to a previous saved state: so if you’ve done load, then change1, then save A, then change 2, then save B, if you do enough undo to get back to the “save A” state, it will be olive. (Blue
means you undo even more, back to the state immediately after “load”)
Unfortunately, no one else’s description is likely to work as well for you as personal experimentation. The documentation will have a brief overview – like this FAQ, or like the new phrasing that
will eventually make it into the online usermanual. So that gives you an idea… and then playing with it will give you experience to put the idea expressed in someone else’s words into something
that will “stick” for you. Guy tried to encapsulate his results of that into his post, but as your reply and mine indicate, everyone needs a different internal phrasing.
Hi, @datatraveller1, @peterjones, @alan-kilborn,
Here are 3 tests to visualize all the markers.
To begin with, let’s create a new file containing the single sring abc :
□ First, open a new tab ( ctrl + N )
□ Type in the string abc, without any final line-break
□ Save this file with the name markers.txt ( Ctrl+ S )
□ Close this file ( Ctrl + W )
□ Re-open the file ( Ctrl + Shift + T )
First test :
=> As expected, the unique line contains the abc string and no marker exists !
□ Go to the end of line ( End )
□ Type in the string defghi
=> The line contains the string abcdefghi and the marker becomes orange
□ Hit three times on the baskSpace key, in order to delete the substring ghi
□ Save the file ( Ctrl + S )
=> The line contains the string abcdef and the marker becomes green
□ Hit the Ctrl + Z shortcut
=> The line displays the abcdefghi string and the marker becomes olive
□ Hit again the Ctrl + Z shortcut
=> The line contains the abc string only and the marker becomes blue ( initial file state )
□ Now, save again the file ( Ctrl + S )
□ Close the file ( Ctrl + W )
□ Re-open the file ( Ctrl + Shift + T )
Second test :
□ At the beginning, the unique line contains the string abc and no marker is present
□ Move to the end of line ( End )
□ Again, type in the string defghi
=> The line contains the abcdefghi string and the marker becomes orange
□ Save the file ( Ctrl + S )
=> The line contains the abcdefghi string and the marker becomes green
□ Hit three times on the BackSpace key, in order to delete the substing ghi
=> The line contains the string abcdef and the marker becomes orange
□ Save again the file ( Ctrl + S )
=> The line contains the abcdef string and the marker becomes green
□ Hit the Ctrl + Z shortcut
=> The line contains the string abcdefghi and the marker becomes olive
□ Hit, again, the Ctrl + Z shortcut
=> The line contains the abc string only and the marker becomes blue ( initial file state )
□ Now, save again the file ( Ctrl + S )
□ Close the file ( Ctrl + W )
□ Re-open the file ( Ctrl + Shift + T )
Third test :
□ As usual, the unique line contains the string abc and no marker is present
□ Move to the end of line ( End )
□ Again, type in the string defghi
=> The line contains the abcdefghi string and the marker becomes orange
□ Hit three times on the baskSpace key, in order to delete the substring ghi
□ Save the file ( Ctrl + S )
=> The line contains the string abcdef and the marker becomes green
=> The string becomes abcdefxyz and the marker becomes orange
□ Save your file ( Ctrl + S )
=> The line contains the string abcdefxyz and the marker becomes green
=> The line contains the abcdef string and the marker becomes olive
□ Hit again the Ctrl + Z shortcut
=> The line contains the abcdefghi string and the marker stays olive
□ Hit, a last time, the Ctrl + Z shortut
=> The line contains the abc string and the marker becomes blue ( Initial file state )
Notes :
□ If we hit repeatedly on the Ctrl + Y shortcut, we obtain the string abcdef or abcdefxyz with the green marker, i.e. the most recent save of the markers.txt file
□ It does not seem easy to define the file state when the marker is olive ! Indeed, it can be, either, an intermediate save of the file or a specific non-saved contents of current line !
I hope someone will get the bottom of this :-))
Best Regards,
Hi @guy038 ,
I have reproduced all your tests successfullly. Thank you very much!
As Peter pointed out, all these different descriptions and own experiences may help to understand the issue better.
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How to Format Font Color in MS Word
MS Word allows you to change the Font color of your text. If you want to emphasize a particular word or phrase, you can change its font color. The basic steps to change the Font color are given
• Select the text you want to modify
• In Home tab locate the Font group
• Click the drop-down arrow next to Font color button
• Font color menu appears
• Select the desired font color with a left click
• Word will change the Font color of the selected text.
See the image:
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Lesson 13: Introduction to the .NET Built-In Classes
Introduction to Built-In Classes and Types
Many of the classes you will use in developping your websites, you will create them yourself, but to assist you in working on your projects, the .NET Framework provides a very rich collection of
classes. A class is referred to as built-in if it is already available in the .NET Framework.
Introduction to Objects
In the .NET Framework, an object is any type of value. To support objects, the .NET Framework provides a class named Object. To support objects, the C# language uses a data type named object. You can
use it to declare a variable of any kind. After declaring the variable, initialize it with a value of your choice. Here are examples:
object employeeName = "Philippe Blayne";
object hourlySalary = 26.95;
object yearsOfExperience = 5;
You can also use object as a parameter to a method. Here is an example:
class Exercise
private void PresentEmployeeSummary(object something)
Remember that, in the body of the method, you can use or ignore the parameter. Still, when calling the method, you must provide a value for the parameter.
In the same way, you can create a method that uses various parameters that are of type object or some are objects while others are not. Here is an example:
class Employee
private void PresentEmployeeSummary(object something, int other)
Introduction to Double-Precision Numbers
So far in our lessons, when we needed a decimal number, we used a type named decimal. The decimal type is used when your value or calculation requires a very large number. If your value or
calculation is more concerned with precision, to assist you, the C# language supports another data type for decimal numbers, also referred to as floating-point numbers. This type for floating-point
numbers is named double. It is the most regular type of floating-point numbers.
To declare a variable for a regular floating-point number, use the double data type followed by a name. Here is an example:
double number;
Of course, you can also use the var keyword to declare the variable. When initializng a double variable, you can assign a natural number or a decimal number to it. Here are examples:
double number = 90248;
var distance = 2847;
Unlike the decimal type, the value of a double type doesn't require a letter. Here are examples:
double number = 90248;
double value = 575.2704;
var length = 2094.75;
Still, if you insist on indicating that the value of a double type, follow the number with d or .
Random Numbers
Introduction to Random Numbers
A number is referred to as random if it has been selected from a pool without a specific pattern to follow. In reality, it is difficult for a number to qualify as random. For this reason, most random
numbers are referred to as pseudo-random.
Getting a Random Number
To support the ability to create or choose a random number, the .NET Framework provides a class named Random. To start, you can declare a variable of this class, using one of its two constructors.
Here is an example that uses the default constructor:
Random rndNumber = new Random();
After declaring the variable, you can start getting numbers from it. To help you do this, the Random class is equipped with a method named Next. This method is overloaded in three versions. One of
the versions of this method takes no argument. This method generates a randomly selected integer between 0 and a constant named MinValue. Here is an example:
Random rndNumbers = new Random();
int rndNumber = rndNumbers.Next();
In the same way, you can call this version of the Next() method repeatedly to get random numbers. Here is an example:
<!DOCTYPE html>
<title>Random Numbers</title>
<h3>Random Numbers</h3>
Random rndNumbers = new Random();
int rndNumber1 = rndNumbers.Next();
int rndNumber2 = rndNumbers.Next();
int rndNumber3 = rndNumbers.Next();
int rndNumber4 = rndNumbers.Next();
int rndNumber5 = rndNumbers.Next();
<p>Random Number: @rndNumber1</p>
<p>Random Number: @rndNumber2</p>
<p>Random Number: @rndNumber3</p>
<p>Random Number: @rndNumber4</p>
<p>Random Number: @rndNumber5</p>
Here is an example of what this could produce:
The Seed of a Random Number
When creating a program that repeatedly gets a series of random numbers, you may (or may not) want the Random class to generate the same number over and over again. A seed is a constant value that
controls whether a random generation would produce the same result every time it occurs. For example, using a seed, you can impose it upon the Random class to generate the same number every time the
Next() method is called. To support the ability to use a seed, the Random class is equipped with a second constructor. Its syntax is:
public Random(int Seed);
Based on this, to specify a seed, when declaring a Random variable, pass a constant integer to the constructor.
Generating Random Numbers in a Range of Numbers
So far, we have been using any number that would fit an integer. In some assignments, you may want to restrict the range of numbers that can be extracted. Fortunately, the Random class allows this.
Using the Random class, you can generate random positive numbers up to a maximum of your choice. To support this, the Random class is equipped with another version of the Next() method. It takes as
argument an integer. The argument to pass to the method determines the highest integer that can be generated by the Next() method. The method returns an integer. Here is an example that generates 5
random numbers between 0 and 100:
<!DOCTYPE html>
<title>Random Numbers</title>
<h3>Random Numbers</h3>
Random rndNumbers = new Random();
int rndNumber1 = rndNumbers.Next(100);
int rndNumber2 = rndNumbers.Next(100);
int rndNumber3 = rndNumbers.Next(100);
int rndNumber4 = rndNumbers.Next(100);
int rndNumber5 = rndNumbers.Next(100);
<p>Random Number: @rndNumber1</p>
<p>Random Number: @rndNumber2</p>
<p>Random Number: @rndNumber3</p>
<p>Random Number: @rndNumber4</p>
<p>Random Number: @rndNumber5</p>
Here is an example of what this could produce:
The above version of the Next() method generates numbers starting at 0. If you want, you can specify the minimum and the maximum range of numbers that the Next() method must work with. To support
this, the Random class is equipped with one more versions of this method and that takes two arguments. The first argument specifies the lowest value that can come from the range. The second argument
holds the highest value that the Next() method can generate. Therefore, the method would operate between both values. Here is an example that generates random numbers from 6 to 18:
<!DOCTYPE html>
<title>Random Numbers</title>
<h3>Random Numbers</h3>
Random rndNumbers = new Random();
int rndNumber1 = rndNumbers.Next(6, 18);
int rndNumber2 = rndNumbers.Next(6, 18);
int rndNumber3 = rndNumbers.Next(6, 18);
int rndNumber4 = rndNumbers.Next(6, 18);
int rndNumber5 = rndNumbers.Next(6, 18);
<p>Random Number: @rndNumber1</p>
<p>Random Number: @rndNumber2</p>
<p>Random Number: @rndNumber3</p>
<p>Random Number: @rndNumber4</p>
<p>Random Number: @rndNumber5</p>
Here is an example of what this could produce:
Built-In Classes: Math
To help you perform some of the basic algebraic and geometric operations in a website, the .NET Framework provides a static class named Math. Besides the Math class, you can also take advantage of
Visual Basic's very powerful library of functions. This library is one of the most extended set of functions of various area of business mathematics.
The Sign of a Number
When initializing a numeric variable using a constant, you decide whether it is negative, 0 or positive. This is referred to as its sign. To help you determine the sign of a variable or of a number,
the Math static class provides a method named Sign(). It is overloaded in various versions, for for each type. When calling this method, pass the value or the variable you want to consider, as
argument. The method returns:
• -1 if the argument is negative
• 0 if the argument is 0
• 1 if the argument is positive
The Integral Side of a Floating-Point Number
We already know that a decimal number consists of an integral side and a precision side; both are separated by the decimal symbol which, in US English, is the period. To let you get the integral side
of a decimal value, the Math static class provides a method named Truncate. It is overloaded in two versions whose syntaxes are:
public static double Truncate(double d);
public static double Truncate(double d);
When calling this method, pass it a number or a variable of numeric type. The method returns the int side of the value.
The Minimum of Two Values
If you have two numbers, you can find the minimum of both without writing your own code. To assist you with this, the Math class is equipped with a method named Min. This method is overloaded in
various versions with each version adapted to each integral or decimal type.
The Maximum of Two Values
As opposed to the minimum of two numbers, you may be interested in the higher of both. To help you find the maximum of two numbers, you can call the Max() method of the Math class. It is overloaded
in various versions with one of each type of numeric data.
Value Conversions
Implicit Conversions
If you have code with mixed types of variables, you may need to convert a value from one type to another. Implicit conversion is the ability to convert a value of one type into a value of another
type. This is possible when the amount of memory that one type is using is lower than the amount necessary for another type. For example, a value of an integer can be directly converted into a double
type. Here is an example:
<!DOCTYPE html>
<title>Numeric Conversion</title>
<h3>Numeric Conversion</h3>
int iNumber = 2445;
double dNumber = iNumber;
<p>Number: @iNumber<br />
Number: @dNumber</p>
In the same way, as saw in our introduction to the double type, you can assign an integral value to a variable of type double. Here is an example:
double number = 927084;
This characteristic is referred to as implicit conversion.
Explicit Conversions
Because of memory requirements, the direct reverse of implicit conversion is not possible. Since the memory reserved for an int variable is smaller than that of a decimal, you cannot assign a value
of type decimal to a variable of type int. As a result, the following code produces an error:
int hourlySalary = 24.75;
In the same way, you cannot assign a variable of types double or decimal to a variable of type int. This means that the following code produces an error:
double nbr = 9270.84;
int number = nbr;
Value Casting
Value casting consists of converting a value of one type into a value of another type. Value casting is also referred to as explicit conversion.
To cast a value or a variable, precede it with the desired data type in parentheses. Here is an example:
<!DOCTYPE html>
<title>Numeric Conversion</title>
<h3>Numeric Conversion</h3>
double number = 9270.84;
int nbr = (int)number;
<p>Number: @number<br />
Number: @nbr</p>
This would produce:
When performing explicit conversion, that is, when casting, you should pay close attention to the value that is being cast. If a value cannot fit in the memory of the other, you may get an
unpredictable result.
We already know that, to convert a numeric value to text, you can call a method named ToString applied to the value or its variable. To support the conversion of a value from one type to another, the
.NET Framework provides a static class named Convert. This class is equipped with various static methods; they are so numerous that we cannot review all of them.
To adapt the Convert class to each C# data type, the class is equipped with a static method whose name starts with To, ends with the .NET Framework name of its type, and takes as argument the type
that needs to be converted. Based on this, to convert a value to a double type, you can call the ToDouble() method and pass the value as argument. Its syntax is:
public static int ToDouble(value);
1. Start Microsoft Visual Studio
2. To create a new application, on the main menu, click File -> New -> Project...
3. In the middle list, click ASP.NET Web Application (.NET Framework) and set the Name to BusinessAccounting1
4. Click OK
5. In the New ASP.NET Web Application, click Empty and press Enter
6. In the Solution Explorer, right-click BusinessAccounting1 -> Add -> New Folder
7. Type Content and press Enter
8. In the Solution Explorer, right-click Content -> Add -> New Item...
9. In the left frame of the Add New Item dialog box, click Style Sheet
10. Change the file Name to Site
11. Press Enter
12. Create some styles as follows:
body {
.container {
margin: auto;
width: 520px; }
table { width: 100%; }
.accent { font-weight: 600; }
.short-text { width: 50px; }
.medium-text { width: 150px; }
13. In the Solution Explorer, right-click BusinessAccounting1 -> Add -> New Item...
14. In the left frame, expand Web and click Razor
15. In the middle list, click Web Page (Razor v3)
16. Set the name as Index
17. Click Add
18. Change the code as follows:
<!DOCTYPE html>
<title>Business Accounting</title>
<link rel="stylesheet" type="text/css" href="~/Content/Site.css" />
double rent = 0.00;
double supplies = 0.00;
double salaries = 0.00;
double revenues = 0.00;
double netIncome = 0.00;
double totalExpenses = 0.00;
double otherExpenses = 0.00;
if (IsPost)
revenues = Convert.ToDouble(Request["txtRevenuesSources"]);
salaries = Convert.ToDouble(Request["txtSalaries"]);
supplies = Convert.ToDouble(Request["txtSupplies"]);
rent = Convert.ToDouble(Request["txtRent"]);
otherExpenses = Convert.ToDouble(Request["txtOtherExpenses"]);
totalExpenses = salaries + supplies + rent + otherExpenses;
netIncome = revenues - totalExpenses;
<div class="container">
<h2>Fun Department Store</h2>
<h3>- Business Accounting -</h3>
<form name="frmBusiness" method="post">
<td colspan="3" class="accent">Revenues</td>
<td class="short-text"> </td>
<td style="width: 300px; font-weight: bold">All Revenues Sources</td>
<td><input type="text" name="txtRevenuesSources" value="@revenues" /></td>
<td colspan="4" class="accent">Expenses</td>
<td class="short-text"> </td>
<td class="medium-text accent">Salaries</td>
<td><input type="text" name="txtSalaries" value="@salaries" /></td>
<td class="short-text"> </td>
<td class="medium-text accent">Supplies</td>
<td><input type="text" name="txtSupplies" value="@supplies" /></td>
<td class="short-text"> </td>
<td class="medium-text accent">Rent</td>
<td><input type="text" name="txtRent" value="@rent" /></td>
<td class="short-text"> </td>
<td class="medium-text accent">Other Expenses</td>
<td><input type="text" name="txtOtherExpenses" value="@otherExpenses" /></td>
<td><input type="submit" name="btnCalculate" value="Calculate" /></td>
<table style="width: 520px;">
<td class="short-text"> </td>
<td style="width: 300px;" class="accent">Total Expenses</td>
<td><input type="text" name="txtTotalExpenses" value="@totalExpenses" /></td>
<td style="width: 355px;" class="accent">Net Income</td>
<td><input type="text" name="txtNetIncome" value="@netIncome" /></td>
19. To execute the application, press Ctrl + F5:
20. Enter some values in the text boxes above the line, such as:
All Revenues Sources: 1528665
Salaries: 535775
Supplies: 12545
Rent: 12500
Other Expenses: 327448
21. Click the Calculate button
22. Close the browser and return to your programming environment
Absolute Values
The absolute value of a number x is x if the number is (already) positive. If the number is negative, its absolute value is its positive equivalent. To let you get the absolute value of a number, the
Math static class is equipped with a method named Abs, which is overloaded in various versions. The syntax for the int and the double types are:
public static int Abs(int value);
public static double Abs(double value);
This method takes the argument whose absolute value must be found.
The Ceiling of a Number
Consider a floating-point number such as 12.155. This number is between integer 12 and integer 13:
In the same way, consider a number such as -24.06. As this number is negative, it is between -24 and -25, with -24 being greater.
In arithmetic, the ceiling of a number is the closest integer that is greater or higher than the number considered. In the first case, the ceiling of 12.155 is 13 because 13 is the closest integer
greater than or equal to 12.155. The ceiling of -24.06 is 24.
To support the finding of a ceiling, the Math static class is equipped with a method named Ceiling that is overloaded with two versions. The syntax for a double value is:
public static double Ceiling(double a);
This method takes as argument a number or variable whose ceiling needs to be found. Here is an example:
double value1 = 155.55;
double value2 = -24.06;
double nbr1 = Math.Ceiling(value1));
double nbr2 = Math.Ceiling(value2));
The Floor of a Number
Consider two floating numbers such as 128.44 and -36.72. The number 128.44 is between 128 and 129 with 128 being the lower. The number -36.72 is between -37 and -36 with -37 being the lower. The
lowest but closest integer value of a number is referred to as its floor.
To assist you with finding the floor of a number, the Math static class provides the Floor() method. It is overloaded with two versions. The syntax for a double is:
public static double Floor(double d);
The Math.Floor() method takes the considered value as the argument and returns the integer that is less than or equal to the argument. Here is an example:
double value1 = 1540.25;
double value2 = -360.04;
double nbr1 = Math.Floor(value1);
double nbr2 = Math.Floor(value2);
The Power of a Number
The power is the value of one number or expression raised to another number. This follows the formula:
value = x^y
To support this operation, the Math static class is equipped with a method named Pow. Its syntax is:
public static double Pow(double x, double y);
This method takes two arguments. The first argument, x, is used as the base number to be evaluated. The second argument, y, also called the exponent, will raise x to this value.
1. On the main menu of Microsoft Visual Studio, click File -> New -> Project...
2. In the middle list, click ASP.NET Web Application (.NET Framework) and change the project Name to CompoundInterest1
3. Click OK
4. In the New ASP.NET Web Application dialog box, make sure Empty is selected and press Enter
5. In the Solution Explorer, right-click CompoundInterest -> Add -> New Folder
6. Type Content and press Enter
7. In the Solution Explorer, right-click Content -> Add -> New Item...
8. In the left frame of the Add New Item dialog box, click Style Sheet
9. Change the file Name to Site
10. Press Enter
11. Create some styles as follows:
.container {
margin: auto;
width: 520px; }
table { width: 100%; }
.accent { font-weight: 600; }
.short-text { width: 50px; }
.medium-text { width: 120px; }
12. In the Solution Explorer, right-click CompoundInterest1 -> Add -> New Item...
13. In the left list, make sure Web is selected and click Razor
14. In the middle list, click Web Page (Razor v3)
15. Set the name as Index
16. Click Add
17. Change the code as follows:
<!DOCTYPE html>
<title>Compound Interest</title>
<link rel="stylesheet" href="~/Content/Site.css" type="text/css" />
double periods = 0.00;
double principal = 0.00;
double interestRate = 0.00;
string strFutureValue = "";
string strInterestEarned = "";
if (IsPost)
double frequency = 12.00;
double futureValue = 0.00;
double interestEarned = 0.00;
principal = Convert.ToDouble(Request["txtPrincipal"]);
interestRate = Convert.ToDouble(Request["txtInterestRate"]);
periods = Convert.ToDouble(Request["txtPeriods"]);
double per = periods / 100;
futureValue = principal * Math.Pow((1.00 + (interestRate / frequency)), frequency * per);
interestEarned = futureValue - principal;
strFutureValue = futureValue.ToString("F");
strInterestEarned = interestEarned.ToString("F");
<div class="container">
<h2>Compound Interest</h2>
<form name="frmCompoundInterest" method="post">
<td class="medium-text accent">Principal:</td>
<td><input type="text" name="txtPrincipal" value="@principal" /></td>
<td class="medium-text accent">Interest Rate:</td>
<td><input type="text" name="txtInterestRate" class="short-text" value="@interestRate" />%</td>
<td class="accent">Periods:</td>
<td><input type="text" name="txtPeriods" class="short-text" value="@periods" /> Years</td>
<td><input type="submit" name="txtSubmit" class="medium-text" value=Calculate /></td>
<td class="accent">Interest Earned:</td>
<td><input type="text" name="txtInterestEarned" value="@strInterestEarned" /></td>
<td class="accent">Future Value:</td>
<td><input type="text" name="txtFutureValue" value="@strFutureValue" /></td>
18. To execute the application, on the main menu, click Debug -> Start Without Debugging
19. In the Principal text box, type a number such as 5284.65
20. In the Interest Rate text box, type a number such as 12.35
21. In the Periods text box, type a natural number such as 5:
22. Click the Calculate button:
23. Close the form and return to your programming environment
The Exponential
To let you calculate the exponential value of a number, the Math static class provides the Exp() method. Its syntax is:
public static double Exp(double d);
If the value of of the argument x is less than -708.395996093 (approximately), the result is reset to 0 and qualifies as underflow. If the value of the argument x is greater than 709.78222656
(approximately), the result qualifies as overflow.
The Natural Logarithm
To calculate the natural logarithm of a number, you can call the Math.Log() method. It is provides in two versions. The syntax of one is:
public static double Log(double d);
Here is an example:
double log = 12.48D;
double number = Math.Log(log));
The Base 10 Logarithm
The Math.Log10() method calculates the base 10 logarithm of a number. The syntax of this method is:
public static double Log10(double d);
The number to be evaluated is passed as the argument. The method returns the logarithm on base 10 using the formula:
y = log[10]x
which is equivalent to
x = 10^y
Here is an example:
double log10 = 12.48D;
double number = Math.Log10(log10));
The Logarithm of Any Base
The Math.Log() method provides another version whose syntax is:
public static double Log(double a, double newBase);
The variable whose logarithmic value will be calculated is passed as the first argument to the method. The second argument allows you to specify a base of your choice. The method uses the formula:
Y = log^NewBase^x
This is the same as
x = NewBase^y
Here is an example of calling this method:
double logN = 12.48D;
double number = Math.Log(logN, 4));
The Square Root
You can calculate the square root of a decimal positive number. To support this, the Math static class is equipped with a method named Sqrt whose syntax is:
public static double Sqrt(double d);
This method takes one argument as a positive floating-point number. After the calculation, the method returns the square root of x:
using System;
class Program
static int Main()
double sqrt = 8025.73;
double number = Math.Sqrt(sqrt));
return 0;
1. Save the following picture somewhere on your computer:
2. Start Microsoft Visual Studio
3. On the main menu, click File -> New -> Project...
4. In the middle list, click ASP.NET Web Application (.NET Framework) and set the project Name to Geometry06
5. Click OK
6. In the New ASP.NET Web Application dialog box, click Empty and click OK
7. In the Solution Explorer, right-click Geometry06 -> Add -> New Folder
8. Type Images and press Enter
9. Copy the above triangle illustration and paste it in this images folder
10. In the Solution Explorer, right-click Geometry06 -> Add -> New Folder
11. Type Content and press Enter
12. In the Solution Explorer, right-click Content -> Add -> New Item...
13. In the left frame of the Add New Item dialog box, click Style Sheet
14. Change the file Name to Site
15. Press Enter
16. Create some styles as follows:
.container {
margin: auto;
width: 520px; }
table { width: 100%; }
.medium-text { width: 150px; }
17. In the Solution Explorer, right-click Geometry06, position the mouse on Add, position the mouse on Add ASP.NET Folder, and click App_Code
18. In the Solution Explorer, right-click App_Code -> Add -> Class...
19. Change the name to EquilateralTriangle
20. Press Enter
21. Type the code as follows:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Web;
namespace Geometry06.App_Code
public class EquilateralTriangle
private double s;
const double NumberOfSides = 3;
public EquilateralTriangle(double side) => s = side;
public double Side { get; set; }
public double Perimeter => s * NumberOfSides;
public double Height => s * Math.Sqrt(3) / 2;
public double Area => s * s * Math.Sqrt(3) / 4;
public double Inradius => s * Math.Sqrt(3) / 6;
public double Circumradius => s / Math.Sqrt(3);
22. In the Solution Explorer, right-click Geometry06 -> Add -> New Item...
23. In the left list, expand Web and click Razor
24. In the middle list, click Web Page (Razor v3)
25. Set the name as Index
26. Click Add
27. Change the code as follows:
<!DOCTYPE html>
<title>Geometry: Triangle</title>
<link href="~/Content/Site.css" type="text/css" rel="stylesheet" />
double side = 0.00;
Geometry06.App_Code.EquilateralTriangle et = new Geometry06.App_Code.EquilateralTriangle(0.00);
if (IsPost)
side = Convert.ToDouble(Request["txtSide"]);
et = new Geometry06.App_Code.EquilateralTriangle(side);
<div class="container">
<h2 style="text-align: center">Geometry - Equilateral Triangle</h2>
<form name="frmGeometry" method="post">
<td style="width: 392px" rowspan="9">
<img src="~/Images/triangle2.png" width="391" height="315" alt="Geometry - Equilateral Triangle" border="1">
<td class="medium-text">Side:</td>
<td><input type="text" name="txtSide" value="@side" /></td>
<td style="text-align: center"><input type="submit" name="btnSubmit" value="Calculate" /></td>
<td><input type="text" name="txtHeight" value="@et.Height" /></td>
<td><input type="text" name="txtPerimeter" value="@et.Perimeter" /></td>
<td><input type="text" name="txt>Area" value="@et.Area" /></td>
<td>Inscribed Radius:</td>
<td><input type="text" name="txtInscribedRadius" value="@et.Inradius" /></td>
<td>Circumscribed Radius:</td>
<td><input type="text" name="txtBottomArea" value="@et.Circumradius" /></td>
28. To execute the application to test it, on the main menu, click Debug -> Start Without Debugging:
If you receive an error that states:
"Feature 'Pattern Matching' is not available in C# 6. Please us language version 7 or greater."
Then, on the main menu, click Tools -> NuGet Packet Manager -> Package Manager Console
Type Install-Package Microsoft.Net.Compilers
Press Enter.
In the Solution Explorer, double-click the Web.config file to open it. Find the compilerOptions attributes. Set the value of lanversion to default:
. . . compilerOptions="/langversion:default . . .
29. In the Side text box, type a number such as 309.66
30. Click the Calculate button:
31. Close the browser and return to your programming environment
A circle is a group or series of distinct points drawn at an exact same distance from another point referred to as the center. The distance from the center C to one of these equidistant points is
called the radius, R. The line that connects all of the points that are equidistant to the center is called the circumference of the circle. The diameter is the distance between two points of the
circumference to the center; in other words, a diameter is double the radius.
To manage the measurements and other related operations, the circumference is divided into 360 portions. Each of these portions is called a degree. The unit used to represent the degree is the
degree, written as ˚. Therefore, a circle contains 360 degrees, that is 360˚. The measurement of two points A and D of the circumference could have 15 portions of the circumference. In this case,
this measurement would be represents as 15˚.
The distance between two equidistant points A and B is a round shape geometrically defined as an arc. An angle is the ratio of the distance between two points A and B of the circumference divided by
the radius R. This can be written as:
Therefore, an angle is the ratio of an arc over the radius. Because an angle is a ratio and not a "physical" measurement, which means an angle is not a dimension, it is independent of the size of a
circle. Obviously this angle represents the number of portions included by the three points. A better unit used to measure an angle is the radian or rad.
A cycle is a measurement of the rotation around the circle. Since the rotation is not necessarily complete, depending on the scenario, a measure is made based on the angle that was covered during the
rotation. A cycle could cover part of the circle in which case the rotation would not have been completed. A cycle could also cover the whole 360˚ of the circle and continue there after. A cycle is
equivalent to the radian divided by 2 * Pi.
The PI Constant
The letter п, also written as Pi, is a constant number used in various mathematical calculations. Its approximate value is 3.1415926535897932. The calculator of MicrosoftWindows represents it as
To support the Pi constant, the Math class is equipped with a constant named PI.
A diameter is two times the radius. In geometry, it is written as 2R. In C++, it is written as 2 * R or R * 2 (because the multiplication is symmetric). The circumference of a circle is calculated by
multiplying the diameter to Pi, which is 2Rп, or 2 * R * п or 2 * R * Pi.
A radian is 2Rп/R radians or 2Rп/R rad, which is the same as 2п rad or 2 * Pi.
To perform conversions between the degree and the radian, you can use the formula:
360˚ = 2п rad which is equivalent to 1 rad = 360˚ / 2п = 57.3˚.
The Cosine of a Value
Consider the following geometric figure:
Consider AB the length of A to B, also referred to as the hypotenuse. Also consider AC the length of A to C which is the side adjacent to point A. The cosine of the angle at point A is the ratio AC/
AB. That is, the ratio of the adjacent length, AC, over the length of the hypotenuse, AB:
The returned value, the ratio, is a double-precision number between -1 and 1.
To calculate the cosine of an angle, the Math class provides the Cos() method. Its syntax is:
public static double Cos(double d);
Here is an example:
int number = 82;
double nbr = Math.Cos(number));
Consider AB the length of A to B, also called the hypotenuse to point A. Also consider CB the length of C to B, which is the opposite side to point A. The sine represents the ratio of CB/AB; that is,
the ratio of the opposite side, CB over the hypotenuse AB.
To calculate the sine of a value, you can call the Sin() method of the Math class. Its syntax is:
public static double Sin(double a);
Here is an example:
double number = 82.55;
double nbr = Math.Sin(number));
Consider AC the length of A to C. Also consider BC the length of B to C. The tangent is the result of BC/AC; that is, the ratio of BC over AC. To assist you with calculating the tangent of of a
number, the Math class is equipped with a method named Tan whose syntax is:
public static double Tan(double a);
The Arc Tangent
Consider BC the length of B to C. Also consider AC the length of A to C. The arc tangent is the ratio of BC/AC. To calculate the arc tangent of a value, you can use the Math.Atan()method. Its syntax
public static double Atan(double d);
• Close your programming environment
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Linear Mixed Models (REML) • Genstat Knowledge Base 2024
Select menu: Stats | Mixed Models (REML) | Linear Mixed Models
This dialog provides facilities for analysis of linear mixed models and estimation of variance components using the method of residual maximum likelihood (REML), which is also sometimes called
restricted maximum likelihood.
1. After you have imported your data, from the menu select
Stats | Mixed Models (REML) | Linear Mixed Models.
2. Fill in the fields as required then click Run.
You can set additional Options then after running, save the results by clicking Save.
Available data
This lists data structures appropriate to the current input field. The contents will change as you move from one field to the next. Double-click a name to copy it to the current input field or type
the name.
Specifies the response variate (dependent variate).
Fixed model
The fixed model describes imposed treatment factors and covariates for which the effect of specified levels or values are of interest. The model is described using a formula, which can combine main
effects and interactions of factors and also covariates.
Random model
The random model is generally used to describe those factors for which the values present in an experiment can be considered drawn from some large homogeneous population. The model is described using
a formula, which can combine main effects and interactions of factors and also covariates. Covariance structures and initial values for the random terms can be specified by using the following
Initial values For larger problems, where the data set is large or there are many model parameters to be estimated, the REML algorithm will run more efficiently if it is given good initial estimates
of the variance parameters. You can specify initial values for each of the terms in the expanded Random Model by clicking on this button.
Correlated Lets you define a covariance structure for random effects in the Linear Mixed Model.
Error terms
Spline model
Specifies terms to be added to the random model as a cubic smoothing spline. Terms may be either a variate, in which case a cubic spline is generated over the values present, or an interaction of a
factor and a variate (e.g. var.fac), in which case a separate cubic spline is generated for each level of the factor. The smoothing estimate is estimated by REML.
Terms must also be specified in the fixed model to provide the linear trend.
Controls the level of interactions to be fitted – you can indicate either All Interactions, or just main effects (No Interactions), or indicate the level of interaction (that is, set a limit on the
maximum number of factors in the treatment terms that are fitted).
This provides a quick way of entering operators in the fixed and random model formulas. Double-click on the required symbol to copy it to the current input field. You can also type in operators
directly. See model formula for a description of each.
Action buttons
Run Run the analysis.
Cancel Close the dialog without further changes.
Options Opens a dialog where additional options and settings can be specified for the analysis.
Defaults Reset options to the default settings. Clicking the right mouse on this button produces a shortcut menu where you can choose to set the options using the currently stored defaults or
the Genstat default settings.
Save Opens a dialog where you can save results from the analysis.
Predict Allows you form predictions based on the current model.
Further output Opens a dialog for specifying further output from the analysis and displaying residual and means plots.
Explore fixed Opens a dialog for Exploring the fixed model from the analysis. This allows you to try different subsets of the fixed model to see which terms are important.
Action Icons
Pin Controls whether to keep the dialog open when you click Run. When the pin is down
Restore Restore names into edit fields and default settings.
Clear Clear all fields and list boxes.
Help Open the Help topic for this dialog.
See also
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Pathfinder v1.0.1: a Bayesian-inferred simple carbon–climate model to explore climate change scenarios
Articles | Volume 15, issue 23
© Author(s) 2022. This work is distributed under the Creative Commons Attribution 4.0 License.
Pathfinder v1.0.1: a Bayesian-inferred simple carbon–climate model to explore climate change scenarios
The Pathfinder model was developed to fill a perceived gap within the range of existing simple climate models. Pathfinder is a compilation of existing formulations describing the climate and carbon
cycle systems, chosen for their balance between mathematical simplicity and physical accuracy. The resulting model is simple enough to be used with Bayesian inference algorithms for calibration,
which enables assimilation of the latest data from complex Earth system models and the IPCC sixth assessment report, as well as a yearly update based on observations of global temperature and
atmospheric CO[2]. The model's simplicity also enables coupling with integrated assessment models and their optimization algorithms or running the model in a backward temperature-driven fashion. In
spite of this simplicity, the model accurately reproduces behaviours and results from complex models – including several uncertainty ranges – when run following standardized diagnostic experiments.
Pathfinder is an open-source model, and this is its first comprehensive description.
Please read the corrigendum first before continuing.
Received: 17 Aug 2022 – Discussion started: 25 Aug 2022 – Revised: 24 Oct 2022 – Accepted: 18 Nov 2022 – Published: 12 Dec 2022
Simple climate models (SCMs) typically simulate global mean temperature change caused by either atmospheric concentration changes or anthropogenic emissions of CO[2] and other climatically active
species. They are most often composed of ad hoc parametric laws that emulate the behaviour of more complex Earth system models (ESMs). The emulation allows for simulating large ensembles of
experiments that would be too costly to compute with ESMs. However, the SCM denomination refers to a fairly broad range of models whose complexity can go from a couple of boxes that only emulate one
part of the climate system (e.g. a global temperature impulse response function; Geoffroy et al., 2013b) to hundreds of state variables representing the different cycles of greenhouse gases and their
effect on climate change (e.g. the compact Earth system model OSCAR; Gasser et al., 2017). Simpler models are easier and faster to solve, but they may not be adequate for all usages. Therefore,
finding the “simplest but not simpler” model depends on a study's precise goals.
In our recent research, we have perceived a deficiency within the existing offer of SCMs, in spite of their large and growing number (Nicholls et al., 2020). We have therefore developed the
Pathfinder model to fill this gap: it is a parsimonious CO[2]-only model that carefully balances simplicity and accuracy of representation of physical processes. Pathfinder was designed to fulfil
three key requirements: (1) the capacity to be calibrated using Bayesian inference, (2) the capacity to be coupled with integrated assessment models (IAMs), and (3) the capacity to explore a very
large number of climate scenarios to narrow down those compatible with limiting climate impacts. The latter motivated the model's name.
While these three requirements clearly call for the simplest model possible, as they all need a fast solving model, they also imply a certain degree of complexity. The Bayesian calibration requires
an explicit representation of the processes (i.e. the variables) that are used to constrain the model. Coupling with IAMs requires accurately embedding the latest advances of climate sciences to be
policy relevant (National Academies of Sciences and Medicine, 2017). Exploring future climate impacts requires the flexibility to link additional (and potentially regional) impact variables to the
core carbon–climate equations.
The Pathfinder model is essentially an integration of existing formulations, adapted to our modelling framework and goals. It is calibrated on Earth system models that contributed to the Coupled
Model Intercomparison Project phase 6 (CMIP6), on additional data from the sixth assessment report of the IPCC (AR6), and on observations of global Earth properties up to the year 2021. The
calibration philosophy of Pathfinder is to use complex models as prior information and only real-world observations and assessments combining many lines of evidence as constraints.
Compared to other SCMs (Nicholls et al., 2020), Pathfinder is much simpler than models like MAGICC (Meinshausen et al., 2011), OSCAR (Gasser et al., 2017), or even HECTOR (Hartin et al., 2015). It is
comparable in complexity to FaIR (Smith et al., 2018) or BernSCM (Strassmann and Joos, 2018), although it is closer to the latter as it trades off an explicit representation of non-CO[2] species for
one of the carbon cycle's main components. This choice was made to help calibration, keep the model invertible, and make the model compatible with IAMs such as DICE (Nordhaus, 2017). While most SCMs
are calibrated using procedures that resemble Bayesian inference (Nicholls et al., 2021), Pathfinder relies on an established algorithm whose implementation is fully tractable and that allows for an
annual update as observations of atmospheric CO[2] and global temperature become available.
Here, we present the first public release of Pathfinder and its source code. We first provide a detailed description of the model's equations. We then describe the Bayesian setup used for
calibration, the sources of prior information for it, and the resulting posterior configuration. We end with a validation of the model using standard diagnostic simulations and quantitative metrics
for the climate system and carbon cycle.
An overview of Pathfinder is presented in Fig. 1. The model is composed of a climate module, of three separate modules for the carbon cycle (ocean, land without land use and land permafrost), and of
two additional modules describing global impacts: sea level rise (SLR) and surface ocean acidification. We do not emulate cycles of other non-CO[2] gases. Mathematically, the model is driven by
prescribing time series of any combination of two of four variables: global mean surface temperature (GMST) anomaly (T), global atmospheric CO[2] concentration (C), global non-CO[2] effective
radiative forcing (R[x]), and global anthropogenic emissions of CO[2] (${E}_{{\mathrm{CO}}_{\mathrm{2}}}$). The model can therefore be run in the traditional emission-driven and concentration-driven
modes but also in a temperature-driven mode (in terms of code, implemented as separate versions of the model). This is notably important for the calibration, during which it is driven by observations
of GMST and atmospheric CO[2].
The following presents all equations of the models. Variables are noted using Roman letters and compiled in Tables B1 and B2. With a few exceptions, parameters are noted using Greek letters and are
summarized in Tables B3 and B4. The model has 21 state variables that follow first-order differential equations in time. The time variable is denoted as t and kept implicit unless required.
The GMST change (T) induced by effective radiative forcing (ERF; R) is represented using a widely used two-box energy balance model with deep-ocean heat uptake efficacy (Geoffroy et al., 2013a;
Armour, 2017). The first box represents the Earth surface's temperature (including atmosphere, land, and surface ocean), and the other one is the deep ocean's temperature (T[d]). Their
time-differential equations are
$\begin{array}{}\text{(1)}& {\mathrm{\Theta }}_{\mathrm{s}}\phantom{\rule{0.33em}{0ex}}\frac{\mathrm{d}T}{\mathrm{d}t}=R-\frac{\mathit{\varphi }\phantom{\rule{0.33em}{0ex}}\mathrm{ln}\left(\mathrm{2}
\right)}{{T}_{\mathrm{2}×}}\phantom{\rule{0.33em}{0ex}}T-{\mathit{ϵ}}_{\mathrm{heat}}\phantom{\rule{0.33em}{0ex}}\mathit{\theta }\phantom{\rule{0.33em}{0ex}}\left(T-{T}_{\mathrm{d}}\right),\end
$\begin{array}{}\text{(2)}& {\mathrm{\Theta }}_{\mathrm{d}}\phantom{\rule{0.33em}{0ex}}\frac{\mathrm{d}{T}_{\mathrm{d}}}{\mathrm{d}t}=\mathit{\theta }\phantom{\rule{0.33em}{0ex}}\left(T-{T}_{\mathrm
where ϕ is the radiative parameter of CO[2], T[2×] is the equilibrium climate sensitivity (ECS) at CO[2] doubling, Θ[s] is the heat capacity of the surface, Θ[d] is the heat capacity of the deep
ocean, θ is the heat exchange coefficient, and ϵ[heat] is the deep-ocean heat uptake efficacy.
The global ERF is simply the sum of the CO[2] contribution (${R}_{{\mathrm{CO}}_{\mathrm{2}}}$) from its change in atmospheric concentration (C), expressed using the IPCC AR5 formula (Myhre et al.,
2013), and that of non-CO[2] climate forcers (R[x]):
$\begin{array}{}\text{(3)}& R={R}_{{\mathrm{CO}}_{\mathrm{2}}}+{R}_{x},\end{array}$
$\begin{array}{}\text{(4)}& {R}_{{\mathrm{CO}}_{\mathrm{2}}}=\mathit{\varphi }\phantom{\rule{0.33em}{0ex}}\mathrm{ln}\phantom{\rule{-0.125em}{0ex}}\left(\frac{C}{{C}_{\mathrm{pi}}}\right),\end{array}
where C[pi] is the preindustrial atmospheric CO[2] concentration.
The above energy balance model naturally provides the ocean heat content (OHC; U[ohc]) as
$\begin{array}{}\text{(5)}& {U}_{\mathrm{ohc}}={\mathit{\alpha }}_{\mathrm{ohc}}\phantom{\rule{0.33em}{0ex}}\left({\mathrm{\Theta }}_{\mathrm{s}}\phantom{\rule{0.33em}{0ex}}T+{\mathrm{\Theta }}_{\
and the ocean heat uptake (OHU) as
$\begin{array}{}\text{(6)}& \frac{\mathrm{d}{U}_{\mathrm{ohc}}}{\mathrm{d}t}={\mathit{\alpha }}_{\mathrm{ohc}}\phantom{\rule{0.33em}{0ex}}\left({\mathrm{\Theta }}_{\mathrm{s}}\phantom{\rule{0.33em}
{0ex}}\frac{\mathrm{d}T}{\mathrm{d}t}+{\mathrm{\Theta }}_{\mathrm{d}}\phantom{\rule{0.33em}{0ex}}\frac{\mathrm{d}{T}_{\mathrm{d}}}{\mathrm{d}t}\right),\end{array}$
where α[ohc] is the fraction of energy used to warm the ocean (i.e. excluding the energy needed to heat up the atmosphere and land and to melt ice).
2.2Sea level rise
Global SLR has been implemented in Pathfinder as a variable of interest to model climate change impacts. In this version, it is firstly a proof of concept, modelled in a simple yet sensible manner.
The total sea level rise (H[tot]) is the sum of contributions from thermal expansion (H[thx]), the Greenland ice sheet (GIS; H[gis]), the Antarctic ice sheet (AIS; H[ais]), and glaciers (H[gla]):
$\begin{array}{}\text{(7)}& {H}_{\mathrm{tot}}={H}_{\mathrm{thx}}+{H}_{\mathrm{gis}}+{H}_{\mathrm{ais}}+{H}_{\mathrm{gla}}.\end{array}$
The thermal expansion contribution scales linearly with the OHC (Goodwin et al., 2017; Fox-Kemper et al., 2021):
$\begin{array}{}\text{(8)}& {H}_{\mathrm{thx}}={\mathrm{\Lambda }}_{\mathrm{thx}}\phantom{\rule{0.33em}{0ex}}{U}_{\mathrm{ohc}},\end{array}$
where Λ[thx] is the scaling factor of the thermosteric contribution to SLR. Note, however, that the thermal capacity of the climate module does not match that of the real-world ocean (Geoffroy et al.
, 2013b), and so this equation cannot describe equilibrium SLR over millennial timescales.
To model contributions from ice sheets and glaciers, we followed the general approach of Mengel et al. (2016). The SLR caused by GIS follows a first-order differential equation with its specific
timescale, and the equilibrium SLR from GIS is assumed to be a cubic function of GMST:
$\begin{array}{}\text{(9)}& \frac{\mathrm{d}{H}_{\mathrm{gis}}}{\mathrm{d}t}={\mathit{\lambda }}_{\mathrm{gis}}+\frac{\mathrm{1}}{{\mathit{\tau }}_{\mathrm{gis}}}\phantom{\rule{0.33em}{0ex}}\left({\
mathrm{\Lambda }}_{\mathrm{gis}\mathrm{1}}\phantom{\rule{0.33em}{0ex}}T+{\mathrm{\Lambda }}_{\mathrm{gis}\mathrm{3}}\phantom{\rule{0.33em}{0ex}}{T}^{\mathrm{3}}-{H}_{\mathrm{gis}}\right),\end{array}$
where λ[gis] is an offset parameter introduced because GIS was not in a steady state at the end of the preindustrial era, Λ[gis1] is the linear term of equilibrium of GIS SLR, Λ[gis3] is the cubic
term of equilibrium of GIS SLR, and τ[gis] is the timescale of the GIS contribution. The motivation for replacing the quadratic term of Mengel et al. (2016) with a cubic one is the oddness of the
cubic function that leads to negative (and not positive) SLR for negative T (which happens during the earlier years of the calibration run).
The contribution from glaciers is also a first-order differential equation with an equilibrium inspired by Mengel et al. (2016). We expanded it with a cubic term to account for the fact that we
aggregate all glaciers together and allow more skewness in the curve describing the equilibrium SLR as a function of T. In addition, we added an exponential sensitivity to speed up the convergence to
equilibrium under a warmer climate:
$\begin{array}{}\text{(10)}& \begin{array}{rl}\frac{\mathrm{d}{H}_{\mathrm{gla}}}{\mathrm{d}t}& ={\mathit{\lambda }}_{\mathrm{gla}}+\frac{\mathrm{exp}\phantom{\rule{-0.125em}{0ex}}\left({\mathit{\
gamma }}_{\mathrm{gla}}\phantom{\rule{0.33em}{0ex}}T\right)}{{\mathit{\tau }}_{\mathrm{gla}}}\\ & \left({\mathrm{\Lambda }}_{\mathrm{gla}}\phantom{\rule{0.33em}{0ex}}\left(\mathrm{1}-\mathrm{exp}\
phantom{\rule{-0.125em}{0ex}}\left(-{\mathrm{\Gamma }}_{\mathrm{gla}\mathrm{1}}\phantom{\rule{0.33em}{0ex}}T-{\mathrm{\Gamma }}_{\mathrm{gla}\mathrm{3}}\phantom{\rule{0.33em}{0ex}}{T}^{\mathrm{3}}\
where λ[gla] is an offset parameter accounting for the lack of initial steady state, Λ[gla] is the SLR potential if all glaciers melted, Γ[gla1] is the linear sensitivity of glaciers' equilibrium to
climate change, Γ[gla3] is the cubic sensitivity of glaciers' equilibrium to climate change, τ[gla] is the timescale of the glaciers contribution, and γ[gla] is the sensitivity of glaciers' timescale
to climate change.
Following Mengel et al. (2016), the contribution from AIS is further divided into two terms, one for surface mass balance (SMB; H[ais,smb]) and one for solid ice discharge (SID; H[ais,sid]), so that
${H}_{\mathrm{ais}}={H}_{\mathrm{ais},\mathrm{smb}}+{H}_{\mathrm{ais},\mathrm{sid}}$. It is expected that precipitation will increase over Antarctica under higher GMST, leading to an increase in SMB
and to a negative sea level rise contribution modelled as
$\begin{array}{}\text{(11)}& \frac{\mathrm{d}{H}_{\mathrm{ais},\mathrm{smb}}}{\mathrm{d}t}=-{\mathrm{\Lambda }}_{\mathrm{ais},\mathrm{smb}}\phantom{\rule{0.33em}{0ex}}T,\end{array}$
where Λ[ais,smb] is the AIS SMB sensitivity to climate change (expressed in sea level equivalent). At the same time, increasing surface ocean temperatures will cause more SID through basal melting,
which we model using a first-order differential equation assumed to be independent of the SMB effect, and with a term that speeds up the effect the more SID happened:
$\begin{array}{}\text{(12)}& \frac{\mathrm{d}{H}_{\mathrm{ais},\mathrm{sid}}}{\mathrm{d}t}={\mathit{\lambda }}_{\mathrm{ais}}+\frac{\mathrm{1}+{\mathit{\alpha }}_{\mathrm{ais}}\phantom{\rule{0.33em}
{0ex}}{H}_{\mathrm{ais},\mathrm{sid}}}{{\mathit{\tau }}_{\mathrm{ais}}}\phantom{\rule{0.33em}{0ex}}\left({\mathrm{\Lambda }}_{\mathrm{ais}}\phantom{\rule{0.33em}{0ex}}T-{H}_{\mathrm{ais},\mathrm
where λ[ais] is an offset parameter accounting for the lack of initial steady state, Λ[ais] is the SLR equilibrium of AIS SID, τ[ais] is the timescale of the AIS SID contribution, and α[ais] is the
sensitivity of the timescale to past SID. In the model's code, however, we directly solve for the total AIS contribution as
$\begin{array}{}\text{(13)}& \begin{array}{rl}\frac{\mathrm{d}{H}_{\mathrm{ais}}}{\mathrm{d}t}& =-{\mathrm{\Lambda }}_{\mathrm{ais},\mathrm{smb}}\phantom{\rule{0.33em}{0ex}}T+{\mathit{\lambda }}_{\
mathrm{ais}}+\frac{\mathrm{1}+{\mathit{\alpha }}_{\mathrm{ais}}\phantom{\rule{0.33em}{0ex}}\left({H}_{\mathrm{ais}}-{H}_{\mathrm{ais},\mathrm{smb}}\right)}{{\mathit{\tau }}_{\mathrm{ais}}}\\ & \left
({\mathrm{\Lambda }}_{\mathrm{ais}}\phantom{\rule{0.33em}{0ex}}T-\left({H}_{\mathrm{ais}}-{H}_{\mathrm{ais},\mathrm{smb}}\right)\right).\end{array}\end{array}$
2.3Ocean carbon
To calculate the ocean carbon sink, we use the classic mixed-layer impulse response function model from Joos et al. (1996), updated to the equivalent box model formulation of Strassmann and Joos (
2018) and extended in places to introduce parameter adjustments for calibration. In the model, the mixed layer is split into five boxes (subscript j), as represented in Fig. 2, so that the total
carbon in the mixed-layer pool (C[o]) is
$\begin{array}{}\text{(14)}& {C}_{\mathrm{o}}=\sum _{j}{C}_{\mathrm{o},j}.\end{array}$
This total carbon mass is converted into a molar concentration of dissolved inorganic carbon (DIC; c[dic]) as follows:
$\begin{array}{}\text{(15)}& {c}_{\mathrm{dic}}=\frac{{\mathit{\alpha }}_{\mathrm{dic}}}{{\mathit{\beta }}_{\mathrm{dic}}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{o}},\end{array}$
where α[dic] is a fixed conversion factor and β[dic] is a scaling factor for the conversion. The latter can be seen as a factor multiplying the mixed-layer depth: it is 1 if the depth is unchanged
from the original Strassmann and Joos (2018) model.
The non-linear carbonate chemistry in the mixed layer is emulated in two steps. First, the model's original polynomial function is used to determine the partial pressure of CO[2] affected by changes
in DIC only (p[dic]):
$\begin{array}{}\text{(16)}& \begin{array}{rl}{p}_{\mathrm{dic}}=& \left(\mathrm{1.5568}-\mathrm{0.013993}\phantom{\rule{0.33em}{0ex}}{T}_{\mathrm{o}}\right)\phantom{\rule{0.33em}{0ex}}{c}_{\mathrm
{dic}}\\ +& \left(\mathrm{7.4706}-\mathrm{0.20207}\phantom{\rule{0.33em}{0ex}}{T}_{\mathrm{o}}\right)\phantom{\rule{0.33em}{0ex}}{\mathrm{10}}^{-\mathrm{3}}\phantom{\rule{0.33em}{0ex}}{{c}_{\mathrm
{dic}}}^{\mathrm{2}}\\ -& \left(\mathrm{1.2748}-\mathrm{0.12015}\phantom{\rule{0.33em}{0ex}}{T}_{\mathrm{o}}\right)\phantom{\rule{0.33em}{0ex}}{\mathrm{10}}^{-\mathrm{5}}\phantom{\rule{0.33em}{0ex}}
{{c}_{\mathrm{dic}}}^{\mathrm{3}}\\ +& \left(\mathrm{2.4491}-\mathrm{0.12639}\phantom{\rule{0.33em}{0ex}}{T}_{\mathrm{o}}\right)\phantom{\rule{0.33em}{0ex}}{\mathrm{10}}^{-\mathrm{7}}\phantom{\rule
{0.33em}{0ex}}{{c}_{\mathrm{dic}}}^{\mathrm{4}}\\ -& \left(\mathrm{1.5768}-\mathrm{0.15326}\phantom{\rule{0.33em}{0ex}}{T}_{\mathrm{o}}\right)\phantom{\rule{0.33em}{0ex}}{\mathrm{10}}^{-\mathrm{10}}\
where T[o] is the preindustrial surface ocean temperature. Second, the actual partial pressure of CO[2] (${p}_{{\mathrm{CO}}_{\mathrm{2}}}$) is calculated using an exponential climate sensitivity (
Takahashi et al., 1993; Joos et al., 2001):
$\begin{array}{}\text{(17)}& {p}_{{\mathrm{CO}}_{\mathrm{2}}}=\left({p}_{\mathrm{dic}}+{C}_{\mathrm{pi}}\right)\phantom{\rule{0.33em}{0ex}}\mathrm{exp}\left({\mathit{\gamma }}_{\mathrm{dic}}\phantom
where γ[dic] is the sensitivity of ${p}_{{\mathrm{CO}}_{\mathrm{2}}}$ to climate change.
The flux of carbon between the atmosphere and the ocean (F[ocean], defined positively if it is a carbon sink) is caused by the difference in partial pressure of CO[2] in the atmosphere and at the
oceanic surface, following an exchange rate that varies linearly with GMST, that is here used as a proxy for wind changes:
$\begin{array}{}\text{(18)}& {F}_{\mathrm{ocean}}={\mathit{u }}_{\mathrm{gx}}\phantom{\rule{0.33em}{0ex}}\left(\mathrm{1}+{\mathit{\gamma }}_{\mathrm{gx}}\phantom{\rule{0.33em}{0ex}}T\right)\phantom
where ν[gx] is the preindustrial gas exchange rate and γ[gx] is its sensitivity to climate change.
This flux of carbon entering the ocean is split between the mixed-layer carbon sub-pools, and this added carbon is subsequently transported towards the deep ocean at a rate specific to each sub-pool.
This leads to the following differential equations:
$\begin{array}{}\text{(19)}& \frac{\mathrm{d}{C}_{\mathrm{o},j}}{\mathrm{d}t}=-\frac{{C}_{\mathrm{o},j}}{{\mathit{\kappa }}_{{\mathit{\tau }}_{\mathrm{o}}}\phantom{\rule{0.33em}{0ex}}{\mathit{\tau }}
_{\mathrm{o},j}}+{\mathit{\alpha }}_{\mathrm{o},j}\phantom{\rule{0.33em}{0ex}}{F}_{\mathrm{ocean}},\phantom{\rule{1em}{0ex}}\forall j,\end{array}$
where α[o,j] are the sub-pools' splitting shares (with ${\sum }_{j}{\mathit{\alpha }}_{\mathrm{o},j}=\mathrm{1}$), τ[o,j] are the sub-pools' timescales for transport to the deep ocean, and ${\mathit
{\kappa }}_{{\mathit{\tau }}_{\mathrm{o}}}$ is a scaling factor applied to all sub-pools. Finally, the deep-ocean carbon pool (C[d]) is obtained through mass balance:
$\begin{array}{}\text{(20)}& \frac{\mathrm{d}{C}_{\mathrm{d}}}{\mathrm{d}t}=\sum _{j}\frac{{C}_{\mathrm{o},j}}{{\mathit{\kappa }}_{{\mathit{\tau }}_{\mathrm{o}}}\phantom{\rule{0.33em}{0ex}}{\mathit{\
tau }}_{\mathrm{o},j}}.\end{array}$
2.4Ocean acidification
While in the real world, ocean acidification is directly related to the carbonate chemistry and the ocean uptake of anthropogenic carbon, we do not have a simple formulation at our disposal that
could link it to our ocean carbon cycle module. We therefore use a readily available emulation of the surface ocean acidification (pH) that links it directly to the atmospheric concentration of CO[2]
(Bernie et al., 2010) with the following polynomial approximation:
where κ[pH] is a scaling factor (that defaults to 1). We note that this approach is reasonable for the surface ocean, as it quickly equilibrates with the atmosphere (but it would not work for the
deep ocean).
2.5Land carbon
The land carbon module of Pathfinder is a simplified version of the one in OSCAR (Gasser et al., 2017, 2020). It is shrunk down to four global carbon pools: vegetation, litter, and active and passive
soil (see Fig. 3). All terrestrial biomes are lumped together, and there is therefore no accounting for the impact of land use change on the land carbon cycle in this version of Pathfinder. This is
an extreme assumption – although very common in SCMs – motivated by simplicity, and it implies that CO[2] emissions from fossil fuel burning and land use change are assumed to behave in the exact
same way, in spite of their not doing so in reality (Gitz and Ciais, 2003; Gasser and Ciais, 2013).
The vegetation carbon pool (C[v]) results from the balance between net primary productivity (NPP; F[npp]), emission from wildfires (E[fire]), emission from harvest and grazing (E[harv]), and loss of
carbon from biomass mortality (F[mort]):
$\begin{array}{}\text{(22)}& \frac{\mathrm{d}{C}_{\mathrm{v}}}{\mathrm{d}t}={F}_{\mathrm{npp}}-{E}_{\mathrm{fire}}-{E}_{\mathrm{harv}}-{F}_{\mathrm{mort}}.\end{array}$
NPP is expressed as its own preindustrial value multiplied by a function of CO[2] and of GMST (r[npp]). This function thus embeds the so-called CO[2] fertilization effect, whereby NPP increases with
atmospheric CO[2], described using a generalized logarithmic functional form:
$\begin{array}{}\text{(23)}& {F}_{\mathrm{npp}}={F}_{\mathrm{npp}\mathrm{0}}\phantom{\rule{0.33em}{0ex}}{r}_{\mathrm{npp}},\end{array}$
$\begin{array}{}\text{(24)}& {r}_{\mathrm{npp}}=\left(\mathrm{1}+\frac{{\mathit{\beta }}_{\mathrm{npp}}}{{\mathit{\alpha }}_{\mathrm{npp}}}\phantom{\rule{0.33em}{0ex}}\left(\mathrm{1}-{\left(\frac{C}
{{C}_{\mathrm{pi}}}\right)}^{-{\mathit{\alpha }}_{\mathrm{npp}}}\right)\right)\left(\mathrm{1}+{\mathit{\gamma }}_{\mathrm{npp}}\phantom{\rule{0.33em}{0ex}}T\right),\end{array}$
where F[npp0] is the preindustrial NPP, β[npp] is the CO[2] fertilization sensitivity, α[npp] is the CO[2] fertilization shape parameter for saturation, and γ[npp] is the sensitivity of NPP to
climate change (that can be positive or negative). The generalized logarithmic functional form implies that ${r}_{\mathrm{npp}}\to \left(\mathrm{1}+{\mathit{\beta }}_{\mathrm{npp}}\mathrm{ln}\left(C/
{C}_{\mathrm{pi}}\right)\right)\left(\mathrm{1}+{\mathit{\gamma }}_{\mathrm{npp}}T\right)$ as ${\mathit{\alpha }}_{\mathrm{npp}}\to {\mathrm{0}}^{+}$.
Harvesting and mortality fluxes are taken proportional to the carbon pool itself even though in reality the mortality fluxes are climate dependent. For simplicity we assume a constant mortality
following the equations in OSCAR (Gasser et al., 2017):
$\begin{array}{}\text{(25)}& {E}_{\mathrm{harv}}={\mathit{u }}_{\mathrm{harv}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{v}},\end{array}$
$\begin{array}{}\text{(26)}& {F}_{\mathrm{mort}}={\mathit{u }}_{\mathrm{mort}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{v}},\end{array}$
where ν[harv] is the harvesting or grazing rate and ν[mort] is the mortality rate.
Wildfires emissions are also assumed to be proportional to the vegetation carbon pool, but with an additional linear dependency of the emission rate on CO[2] (as a proxy of changes in leaf area index
and evapotranspiration) and GMST (r[fire]):
$\begin{array}{}\text{(27)}& {E}_{\mathrm{fire}}={\mathit{u }}_{\mathrm{fire}}\phantom{\rule{0.33em}{0ex}}{r}_{\mathrm{fire}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{v}},\end{array}$
$\begin{array}{}\text{(28)}& {r}_{\mathrm{fire}}=\left(\mathrm{1}+{\mathit{\beta }}_{\mathrm{fire}}\left(\frac{C}{{C}_{\mathrm{pi}}}-\mathrm{1}\right)\right)\left(\mathrm{1}+{\mathit{\gamma }}_{\
where ν[fire] is the wildfire rate, β[fire] is the sensitivity of wildfires to CO[2], and γ[fire] is their sensitivity to climate change.
Soil carbon is divided into three pools. The litter carbon pool (C[s1]) receives the mortality flux as sole input, it emits part of its carbon through heterotrophic respiration (E[rh1]), and it
transfers another part to the next pool through stabilization (F[stab]):
$\begin{array}{}\text{(29)}& \frac{\mathrm{d}{C}_{\mathrm{s}\mathrm{1}}}{\mathrm{d}t}={F}_{\mathrm{mort}}-{F}_{\mathrm{stab}}-{E}_{\mathrm{rh}\mathrm{1}}.\end{array}$
Similarly, the active soil carbon pool (C[s2]) receives the stabilization flux, is respired (E[rh2]), and transfers carbon to the last pool through passivization (F[pass]):
$\begin{array}{}\text{(30)}& \frac{\mathrm{d}{C}_{\mathrm{s}\mathrm{2}}}{\mathrm{d}t}={F}_{\mathrm{stab}}-{F}_{\mathrm{pass}}-{E}_{\mathrm{rh}\mathrm{2}}.\end{array}$
The passive carbon pool (C[s3]) receives this final input flux and is respired (E[rh3]):
$\begin{array}{}\text{(31)}& \frac{\mathrm{d}{C}_{\mathrm{s}\mathrm{3}}}{\mathrm{d}t}={F}_{\mathrm{pass}}-{E}_{\mathrm{rh}\mathrm{3}}.\end{array}$
Although information pertaining to this fourth pool is not commonly provided by ESMs, it was introduced in Pathfinder to adjust the complex models' turnover time of soil carbon to better match
isotopic data (He et al., 2016). For completeness, we note that the total heterotrophic respiration is ${E}_{\mathrm{rh}}={E}_{\mathrm{rh}\mathrm{1}}+{E}_{\mathrm{rh}\mathrm{2}}+{E}_{\mathrm{rh}\
mathrm{3}}$, and the total soil carbon pool is ${C}_{\mathrm{s}}={C}_{\mathrm{s}\mathrm{1}}+{C}_{\mathrm{s}\mathrm{2}}+{C}_{\mathrm{s}\mathrm{3}}$.
All soil-originating fluxes are taken proportional to their pool of origin and multiplied by a function (r[rh]) explained hereafter. For the litter pool, this gives
$\begin{array}{}\text{(32)}& {E}_{\mathrm{rh}\mathrm{1}}={\mathit{u }}_{\mathrm{rh}\mathrm{1}}\phantom{\rule{0.33em}{0ex}}{r}_{\mathrm{rh}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{s}\mathrm{1}},\end
$\begin{array}{}\text{(33)}& {F}_{\mathrm{stab}}={\mathit{u }}_{\mathrm{stab}}\phantom{\rule{0.33em}{0ex}}{r}_{\mathrm{rh}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{s}\mathrm{1}},\end{array}$
where ν[rh1] is the litter respiration rate and ν[stab] is the stabilization rate. For the active soil pool, we have
$\begin{array}{}\text{(34)}& {E}_{\mathrm{rh}\mathrm{2}}=\frac{{\mathit{u }}_{\mathrm{rh}\mathrm{23}}-{\mathit{u }}_{\mathrm{rh}\mathrm{3}}\phantom{\rule{0.33em}{0ex}}{\mathit{\alpha }}_{\mathrm
{pass}}}{\mathrm{1}-{\mathit{\alpha }}_{\mathrm{pass}}}\phantom{\rule{0.33em}{0ex}}{r}_{\mathrm{rh}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{s}\mathrm{2}},\end{array}$
$\begin{array}{}\text{(35)}& {F}_{\mathrm{pass}}={\mathit{u }}_{\mathrm{rh}\mathrm{3}}\phantom{\rule{0.33em}{0ex}}\frac{{\mathit{\alpha }}_{\mathrm{pass}}}{\mathrm{1}-{\mathit{\alpha }}_{\mathrm
and for the passive soil pool:
$\begin{array}{}\text{(36)}& {E}_{\mathrm{rh}\mathrm{3}}={\mathit{u }}_{\mathrm{rh}\mathrm{3}}\phantom{\rule{0.33em}{0ex}}{r}_{\mathrm{rh}}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{s}\mathrm{3}},\end
where ν[rh23] is the soil respiration rate (averaged over active and passive pools), ν[rh3] is the passive soil respiration rate, and α[pass] is the fraction of passive carbon (over active plus
passive soil carbon). This slightly convoluted formulation is motivated by the lack of information regarding the active/passive split in ESMs, which we alleviate using additional data during
In addition, the function r[rh], describing the dependency of respiration (and related fluxes) on temperature and on the availability of fresh organic matter to be decomposed, is defined as follows:
$\begin{array}{}\text{(37)}& \begin{array}{rl}& {r}_{\mathrm{rh}}=\\ & \underset{\left(\mathrm{1}+{\mathit{\beta }}_{\mathrm{rh}}\left(\frac{{C}_{\mathrm{s}\mathrm{1}}}{{C}_{\mathrm{s}}}\frac{{C}_{\
mathrm{s}}\left({t}_{\mathrm{0}}\right)}{{C}_{\mathrm{s}\mathrm{1}}\left({t}_{\mathrm{0}}\right)}-\mathrm{1}\right)\right)}{\underbrace{\left(\mathrm{1}+{\mathit{\beta }}_{\mathrm{rh}}\left(\frac{{C}
_{\mathrm{s}\mathrm{1}}}{{C}_{\mathrm{s}\mathrm{1}}+{C}_{\mathrm{s}\mathrm{2}}+{C}_{\mathrm{s}\mathrm{3}}}\left(\mathrm{1}+\frac{{\mathit{u }}_{\mathrm{stab}}}{{\mathit{u }}_{\mathrm{rh}\mathrm{23}}}
\right)-\mathrm{1}\right)\right)}}\mathrm{exp}\left({\mathit{\gamma }}_{\mathrm{rh}}\phantom{\rule{0.33em}{0ex}}T\right),\end{array}\end{array}$
where β[rh] is the sensitivity of the respiration to fresh organic matter availability (expressed here as the relative change in the ${C}_{\mathrm{s}\mathrm{1}}/{C}_{\mathrm{s}}$ ratio with regard to
preindustrial times), and γ[rh] is its sensitivity to climate change (equivalent to a “Q[10]” formulation with Q[10]=exp(10γ[rh])).
Finally, the net carbon flux from the atmosphere to the land (F[land], defined positively if it is a carbon sink) is obtained as the net budget of all pools combined:
$\begin{array}{}\text{(38)}& {F}_{\mathrm{land}}={F}_{\mathrm{npp}}-{E}_{\mathrm{fire}}-{E}_{\mathrm{harv}}-{E}_{\mathrm{rh}},\end{array}$
and this system of equations leads to the following preindustrial steady state:
2.6Permafrost carbon
As the land carbon cycle described in the previous section does not account for permafrost carbon, we implemented this feedback using the emulator developed by Gasser et al. (2018) but aggregated
into a unique global region. Figure 4 gives a representation of the permafrost module as described in the following. The emulation starts with a theoretical thawed fraction ($\overline{a}$) that
represents the fraction of thawed carbon under steady-state for a certain level of local warming. It is formulated with a sigmoid function (that equals 0 at preindustrial and 1 under very high GMST):
$\begin{array}{}\text{(40)}& \begin{array}{rl}& \overline{a}=\\ & -{a}_{\mathrm{min}}+\frac{\left(\mathrm{1}+{a}_{\mathrm{min}}\right)}{{\left(\mathrm{1}+\left({\left(\mathrm{1}+\frac{\mathrm{1}}{{a}
_{\mathrm{min}}}\right)}^{{\mathit{\kappa }}_{a}}-\mathrm{1}\right)\mathrm{exp}\left(-{\mathit{\gamma }}_{a}\phantom{\rule{0.33em}{0ex}}{\mathit{\kappa }}_{a}\phantom{\rule{0.33em}{0ex}}{\mathit{\
alpha }}_{\mathrm{lst}}\phantom{\rule{0.33em}{0ex}}T\right)\right)}^{\frac{\mathrm{1}}{{\mathit{\kappa }}_{a}}}},\end{array}\end{array}$
where −a[min] is the minimum thawed fraction (corresponding to 100% frozen soil carbon), κ[a] is a shape parameter determining the asymmetry of the function, γ[a] is the sensitivity of the
theoretical thawed fraction to local climate change, and α[lst] is the proportionality factor between local and global climate change.
The actual thawed fraction (a) then moves towards its theoretical value at a speed that depends on whether it is thawing (i.e. $a<\overline{a}$) or freezing (i.e. $a>\overline{a}$). This is written
as a non-linear differential equation:
$\begin{array}{}\text{(41)}& \begin{array}{rl}\frac{\mathrm{d}a}{\mathrm{d}t}& =\mathrm{0.5}\phantom{\rule{0.33em}{0ex}}\left({\mathit{u }}_{\mathrm{thaw}}+{\mathit{u }}_{\mathrm{froz}}\right)\
phantom{\rule{0.33em}{0ex}}\left(\overline{a}-a\right)\\ & +\mathrm{0.5}\phantom{\rule{0.33em}{0ex}}|\left({\mathit{u }}_{\mathrm{thaw}}-{\mathit{u }}_{\mathrm{froz}}\right)\phantom{\rule{0.33em}
where ν[thaw] is the rate of thawing and ν[froz] is the rate of freezing. Because ν[thaw]>ν[froz], the absolute value in the equation leads to the right-hand side being ${\mathit{u }}_{\mathrm{thaw}}
\left(\overline{a}-a\right)$ if $a<\overline{a}$, or ${\mathit{u }}_{\mathrm{froz}}\left(\overline{a}-a\right)$ if $a>\overline{a}$. The change in the pool of frozen carbon (C[fr]) naturally follows:
$\begin{array}{}\text{(42)}& \frac{\mathrm{d}{C}_{\mathrm{fr}}}{\mathrm{d}t}=-\frac{\mathrm{d}a}{\mathrm{d}t}\phantom{\rule{0.33em}{0ex}}{C}_{\mathrm{fr}\mathrm{0}},\end{array}$
where C[fr0] is the amount of frozen carbon at preindustrial times.
Thawed carbon is not directly emitted to the atmosphere: it is split into three thawed carbon sub-pools (C[th,j]) that have their own decay time but are all affected by an additional function (r[rt]
). This leads to the following budget equations:
$\begin{array}{}\text{(43)}& \frac{\mathrm{d}{C}_{\mathrm{th},j}}{\mathrm{d}t}=-{\mathit{\alpha }}_{\mathrm{th},j}\phantom{\rule{0.33em}{0ex}}\frac{\mathrm{d}{C}_{\mathrm{fr}}}{\mathrm{d}t}-\frac{{C}
_{\mathrm{th},j}}{{\mathit{\kappa }}_{{\mathit{\tau }}_{\mathrm{th}}}\phantom{\rule{0.33em}{0ex}}{\mathit{\tau }}_{\mathrm{th},j}}\phantom{\rule{0.33em}{0ex}}{r}_{\mathrm{rt}},\phantom{\rule{1em}
{0ex}}\forall j,\end{array}$
where α[th,j] is the sub-pools' splitting shares (with ${\sum }_{j}{\mathit{\alpha }}_{\mathrm{th},j}=\mathrm{1}$), τ[th,j] is the sub-pools' decay times, and ${\mathit{\kappa }}_{{\mathit{\tau }}_{\
mathrm{th}}}$ is a scaling factor applied to all sub-pools. The additional r[rt] function describes the sensitivity of heterotrophic respiration to climate change in boreal regions using a Gaussian
$\begin{array}{}\text{(44)}& {r}_{\mathrm{rt}}=\mathrm{exp}\phantom{\rule{-0.125em}{0ex}}\left({\mathit{\kappa }}_{\mathrm{rt}}\phantom{\rule{0.33em}{0ex}}{\mathit{\gamma }}_{\mathrm{rt}\mathrm{1}}\
phantom{\rule{0.33em}{0ex}}{\mathit{\alpha }}_{\mathrm{lst}}\phantom{\rule{0.33em}{0ex}}T-{\mathit{\kappa }}_{\mathrm{rt}}\phantom{\rule{0.33em}{0ex}}{\mathit{\gamma }}_{\mathrm{rt}\mathrm{2}}\
phantom{\rule{0.33em}{0ex}}\left({\mathit{\alpha }}_{\mathrm{lst}}\phantom{\rule{0.33em}{0ex}}T{\right)}^{\mathrm{2}}\right),\end{array}$
where κ[rt] is a factor scaling the sensitivity of thawed carbon against that of regular soil carbon, γ[rt1] is the sensitivity to local temperature change (i.e. a Q10), and γ[rt2] is the quadratic
term in the latter sensitivity that represents a saturation effect. Noting that all the emitted carbon is assumed to be CO[2], the global emission from permafrost (E[pf]) is thus
$\begin{array}{}\text{(45)}& {E}_{\mathrm{pf}}=\sum _{j}\frac{{C}_{\mathrm{th},j}}{{\mathit{\kappa }}_{{\mathit{\tau }}_{\mathrm{th}}}\phantom{\rule{0.33em}{0ex}}{\mathit{\tau }}_{\mathrm{th},j}}\
2.7Atmospheric CO[2]
The change in atmospheric concentration of CO[2] is the budget of all carbon cycle fluxes to which we add the exogenous anthropogenic emissions (${E}_{{\mathrm{CO}}_{\mathrm{2}}}$):
$\begin{array}{}\text{(46)}& {\mathit{\alpha }}_{\mathrm{C}}\phantom{\rule{0.33em}{0ex}}\frac{\mathrm{d}C}{\mathrm{d}t}={E}_{{\mathrm{CO}}_{\mathrm{2}}}+{E}_{\mathrm{pf}}-{F}_{\mathrm{land}}-{F}_{\
where α[C] is the conversion factor from volume fraction to mass for CO[2].
Bayesian inference is a powerful tool for assimilating observational data into reduced-complexity models such as Pathfinder (Ricciuto et al., 2008). The approach consists in deducing probability
distributions of parameters from a priori knowledge on those distributions and on distributions of observations of some of the model's variables, using Bayes' theorem (Bayes, 1763). Summarily, the
Bayesian calibration updates the joint distribution of parameters to make it as compatible with the constraints as possible given their prior estimates, which increases the internal coherence of
Pathfinder by excluding combination of parameters that are unlikely.
Such a Bayesian calibration is vulnerable to the possibility that the priors draw on the same information as the constraints. However, given that Pathfinder is a patchwork of emulators whose
parameters are obtained independently from one another and following differing experimental setups, we expect that the coherence of information contained within the priors and the constraints is very
low. Our choice of using only complex models as prior information and only observations and assessments as constraints also aims at limiting this vulnerability.
Concretely, the posterior probability 𝒫[post] of a sample k from the joint parameters distribution ξ[k], conditional to a set of observations x, is proportional (symbol ∝) to its own prior
probability 𝒫[pre] and to the likelihood ℒ of the model simulating x given ξ[k]:
$\begin{array}{}\text{(47)}& {\mathcal{P}}_{\mathrm{post}}\left({\mathbit{\xi }}_{k}\mathrm{|}\mathbit{x}\right)\propto \mathcal{L}\left(\mathbit{x}\mathrm{|}{\mathbit{\xi }}_{k}\right)\phantom{\rule
{0.33em}{0ex}}{\mathcal{P}}_{\mathrm{pre}}\left({\mathbit{\xi }}_{k}\right).\end{array}$
Here, we assume all observations are independently and identically distributed following a normal distribution (with mean values μ[x], and standard deviations σ[x] expressed in real physical units),
which leads to the following likelihood:
$\begin{array}{}\text{(48)}& \mathcal{L}\left(\mathbit{x}\mathrm{|}{\mathbit{\xi }}_{k}\right)=\prod _{i=\mathrm{1}}^{{n}_{x}}\frac{\mathrm{1}}{{\mathit{\sigma }}_{x,i}\phantom{\rule{0.33em}{0ex}}\
sqrt{\mathrm{2}\mathit{\pi }}}\phantom{\rule{0.33em}{0ex}}\mathrm{exp}\phantom{\rule{-0.125em}{0ex}}\left(-\frac{\left({\mathcal{F}}_{i}\left({\mathbit{\xi }}_{k}\right)-{\mathit{\mu }}_{x,i}{\
right)}^{\mathrm{2}}}{\mathrm{2}\phantom{\rule{0.33em}{0ex}}{\mathit{\sigma }}_{x,i}^{\mathrm{2}}}\right),\end{array}$
where ℱ[i](ξ[k]) is the model's output for the ith observable (out of n[x]) with input parameters ξ[k].
The Pathfinder model is a set of differential equations with a number of input parameters, of which n[ξ] are calibrated through Bayesian inference, and an additional two input variables provided as
time series (i.e. one value per time step required). While the two input time series can be any combination of two out of four variables (anthropogenic CO[2] emissions, non-CO[2] ERF, atmospheric CO
[2] concentration, or GMST), for calibration we use the two most well-constrained variables that are direct physical observations of the global Earth system: atmospheric CO[2] and GMST. These input
time series cover the historical period from 1751 to 2020. Therefore, the ξ[k] vector is
$\begin{array}{}\text{(49)}& {\mathbit{\xi }}_{k}={\left\{\mathit{\left\{}{\mathit{\xi }}_{j}{\mathit{\right\}}}_{j=\mathrm{1}}^{{n}_{\mathit{\xi }}},\mathit{\left\{}C\left(t\right){\mathit{\right
However, to ease the computation by reducing the dimension of the system, we do not use annual time series of observations as inputs, but we assume that each input time series (for variable X being C
or T) follows
$\begin{array}{}\text{(50)}& X\left(t\right)={X}_{\mathit{\mu }}\left(t\right)+{\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{X}\phantom{\rule{0.33em}{0ex}}{X}_{\mathit{\sigma }}\left(t\right)+{\mathit
{ϵ}}_{X}\phantom{\rule{0.33em}{0ex}}\mathrm{AR}\mathrm{1}\left(t;{\mathit{\rho }}_{X}\right),\end{array}$
where X[μ] and X[σ] are fixed exogenous annual time series (i.e. structural parameters), ${\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{X}$ is the relative standard deviation of the time series (without
noise), ϵ[X] is the noise intensity, and AR1 is an autoregressive process of order 1 and autocorrelation parameter ρ[X]. This assumption leads to the final expression of the ξ[k] vector:
$\begin{array}{}\text{(51)}& {\mathbit{\xi }}_{k}={\left\{\mathit{\left\{}{\mathit{\xi }}_{j}{\mathit{\right\}}}_{j=\mathrm{1}}^{{n}_{\mathit{\xi }}},{\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{\mathrm
{C}},{\mathit{ϵ}}_{\mathrm{C}},{\mathit{\rho }}_{\mathrm{C}},{\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{T},{\mathit{ϵ}}_{T},{\mathit{\rho }}_{T},\right\}}_{k}.\end{array}$
During this Bayesian assimilation, the Pathfinder model is run solely over the historical period (from 1750 to 2021), as the constraints concern only preindustrial or historical years. For the
computation, the time-differential system of Pathfinder is solved using an implicit–explicit numerical scheme (also called IMEX), with a time step of a quarter of a year. This solving scheme relies
on writing the differential equations of all state variables X as
$\begin{array}{}\text{(52)}& \frac{\mathrm{d}X}{\mathrm{d}t}=-\mathit{u }\phantom{\rule{0.33em}{0ex}}X+\mathcal{R},\end{array}$
where ν is the constant speed of the linear part of the differential equation and ℛ is its non-linear part, discretizing these equations as
$\begin{array}{}\text{(53)}& \frac{X\left(t+\mathit{\delta }t\right)-X\left(t\right)}{\mathit{\delta }t}=-\mathit{u }\phantom{\rule{0.33em}{0ex}}X\left(t+\mathit{\delta }t\right)+\mathcal{R}\left(t\
where δt is the solving time step (which is $\frac{\mathrm{1}}{{n}_{t}}$ times the annual time step of the model's inputs and outputs, here n[t]=4), and finally explicitly solving for all X(t+δt). We
note this is also the default solving scheme for regular simulations with the model, although the value of n[t] can be altered and alternative schemes are available.
The Bayesian procedure itself is implemented using the Python computer language and specifically the PyMC3 package (Salvatier et al., 2016). The solving of Eq. (47) and its normalization are done
using the package's full-rank automatic differentiation variational inference (ADVI) algorithm (Kucukelbir et al., 2017), with 100000 iterations (and default algorithm options). The choice of
variational inference instead of Markov chain Monte Carlo is motivated by the significant size of our model (Blei et al., 2017) and the speed of ADVI. An additional strength of the full-rank version
of the ADVI algorithm is its ability to generate correlated posterior distributions even if the prior ones are uncorrelated. Convergence of the algorithm was controlled through convergence of the
ELBO metric (Kucukelbir et al., 2017). All results presented hereafter are obtained through drawing 2000 sets of parameters – which we call configurations – from the posterior or prior distributions.
We use a set of 19 constraints related to all aspects of the model that correspond to the set of observations x in the Bayesian calibration. Many of the constraints are observations, but some are
ranges assessed by expert panels such as the Global Carbon Project or the IPCC. They cover either a recent point in time or an assumed preindustrial equilibrium, and they are typically taken over a
period of at least a few years to reduce the effect of natural variability.
Table 1 summarizes these constraints, the periods over which they are considered, and their distributions. The following subsections provide further details on the constraints, and the constraint
distributions are shown in Fig. 6.
3.3.1Climate system
To constrain the temperature response, we use the same five data sets of observed GMST as in Sect. 3.4.8 to derive average and standard deviation of two constraints: the average GMST change and the
average GMST yearly trend obtained through second-order accuracy gradient (Fornberg, 1988), both over the latest 20 years of data (2002–2021). Because these data sets are already used as input to the
Bayesian setup, albeit in a different way, they do not provide much of a constraint and are used mostly to ensure that the ${\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{T}$ and ϵ[T] parameters remain
within sensible range.
To further constrain the climate system, we use the mean OHU assessed by the IPCC AR6 over 2006–2018 (Gulev et al., 2021, Table 2.7) and the non-CO[2] ERF (averaged over 2010-2019) also estimated for
the AR6. The central value of the latter is taken from Dentener et al. (2021, Table AIII.3, and corresponding GitHub repository), and its uncertainty is constructed using data from Forster et al. (
2021, Table 7.8) and assuming the ERF of all species are normally distributed and uncorrelated, but fully correlated in time for each separate species (which likely overestimates the uncertainty).
To better align with the IPCC AR6, we also constrain the ECS of our model (i.e. the T[2×] parameters). To do so, because the distribution of ECS cannot be assumed to be normal, we follow the
framework of Roe and Baker (2007), who define the climate feedback factor ff so that ${T}_{\mathrm{2}×}={T}_{\mathrm{2}×}^{*}/\left(\mathrm{1}-\mathrm{ff}\right)$, where ${T}_{\mathrm{2}×}^{*}$ is
the minimal ECS value (roughly corresponding to the Planck feedback). We assume this feedback factor follows a logit-normal distribution, which implies logit(ff) = $\mathrm{ln}\left(\mathrm{ff}/\left
(\mathrm{1}-\mathrm{ff}\right)\right)$ = $\mathrm{ln}\left({T}_{\mathrm{2}×}/{T}_{\mathrm{2}×}^{*}-\mathrm{1}\right)$ follows a normal distribution. We therefore constrain logit(ff), using
distribution parameters and a value of ${T}_{\mathrm{2}×}^{*}$ calibrated to fit the probabilistic ranges of ECS provided by the AR6. This fit of the ECS distribution is illustrated in Fig. B9.
3.3.2Carbon cycle
Similarly to what is done with GMST, we constrain the atmospheric CO[2] level over the latest 10 years of data (2012–2021) using the NOAA/ESRL data (Tans and Keeling, 2010). The rest of the global CO
[2] budget is constrained using the 2021 Global Carbon Budget (GCB; Friedlingstein et al., 2022). We use the net atmospheric CO[2] growth and total anthropogenic emissions (fossil and land use) over
the last 10 years and the ocean and land carbon sinks accumulated since the beginning of the instrumental measurement period (1960–2020). Note that our definition of the land carbon sink ignoring
land use change is consistent with that of the GCB.
Given its number of parameters and their inconsistent sources, we further constrain the land carbon module by considering present-day (mean over 1998–2002) NPP (Ciais et al., 2013; Zhao et al., 2005)
and preindustrial vegetation and soil carbon pools. These preindustrial pools are taken from the AR6 for the central value (Canadell et al., 2021, Fig. 5.12), but their relative uncertainty is taken
from the AR5 (Ciais et al., 2013, Fig. 6.1) since it is lacking in the AR6. In addition, the soil carbon pool constraint is corrected downward by estimates of peatland carbon (Yu et al., 2010
, Table 1), since it is an ecosystem missing in TRENDY models (and in ours) but not in the IPCC assessments.
3.3.3Sea level rise
To constrain the separate SLR contributions from thermal expansion, GIS, AIS, and glaciers, we use the model-based SLR speed estimates over the recent past (averaged over 2006–2018) reported in the
AR6 (Fox-Kemper et al., 2021, Table 9.5). To constrain the total contribution, we also use the historical (1901–1990) sea level rise inferred from tide gauges from the same source, although the value
is corrected upward for the missed impact of uncharted glaciers (Parkes and Marzeion, 2018).
Contrarily to all other modules, the SLR module is not assumed to start at steady state in 1750, which is represented through the λ[ice] ($\mathrm{ice}\in \left[\mathrm{gla},\mathrm{gis},\mathrm{ais}
\right]$) parameters. We assume this is entirely due to the so-called Little Ice Age (LIA) relaxation, which we assume can be simply modelled in Pathfinder through exponential decay of our three
ice-related contributions since t[0]=1750. This gives a net LIA contribution of ${H}_{\mathrm{lia}}={\sum }_{\mathrm{ice}}{\mathit{\lambda }}_{\mathrm{ice}}\phantom{\rule{0.33em}{0ex}}{\mathit{\tau
}}_{\mathrm{ice}}\phantom{\rule{0.33em}{0ex}}\mathrm{exp}\phantom{\rule{-0.125em}{0ex}}\left(-\frac{t-{t}_{\mathrm{0}}}{{\mathit{\tau }}_{\mathrm{ice}}}\right)$. We constrain this diagnostic variable
using the global SLR reported by Slangen et al. (2016) over 1900–2005 for their control experiment.
3.4Parameters (prior distributions)
Out of the model's 77 parameters, 33 are assumed to be fixed (i.e. they are structural parameters), and the remaining n[ξ]=44 parameters are estimated through Bayesian inference. Prior distributions
of the ξ[j] parameters are assumed log-normal if the physical parameter must be defined positive, logit-normal if it must be between 0 and 1, and normal otherwise. To avoid extreme parameter values
that could make the model diverge during calibration, the posterior distributions are bound to ${\mathit{\mu }}_{\mathit{\xi },j}±\mathrm{5}\phantom{\rule{0.33em}{0ex}}{\mathit{\sigma }}_{\mathit{\xi
},j}$, where μ[ξ,j] and σ[ξ,j] are the mean and standard deviation of the jth parameter's prior distribution. These two values are taken from the literature, deduced from multi-model ensembles, or in
a few instances arbitrarily set, as described in the following subsections. Note that when parameters are deduced from multi-model ensembles, there are effectively two rounds of calibration: first, a
calibration on individual models using ordinary least-squares regressions to obtain prior distributions and, second, the Bayesian calibration itself that leads to the posterior distributions. In
addition, the prior distributions of ${\stackrel{\mathrm{̃}}{\mathit{\sigma }}}_{X}$, ϵ[X] and ρ[X] are assumed normal, half-normal, and uniform, respectively. All prior distributions are assumed
independent, meaning that the prior joint distribution ξ does not exhibit any covariance.
All parameters are summarized in Tables B5 and B6 along with their properties and values. The following subsections further explain how the prior distributions of the parameters are established, and
these distributions are shown in Fig. 5.
All the parameters of the climate module are calibrated. The prior distribution of the radiative parameter ϕ is taken from the AR5 (Myhre et al., 2013, Table 8.SM.1). All other prior distributions of
the parameters of the climate module (i.e. T[2×], Θ[s], Θ[d], θ, and ϵ[heat]) are taken from 35 CMIP6 models whose climate responses were derived for the AR6 using the “abrupt-4xCO2” experiment (
Smith et al., 2021, Sect. 7.SM.2.1, and corresponding GitHub repository). Here, T[2×] is simply assumed to be half the reported equilibrium temperature at quadrupled CO[2]. In addition, the prior
distribution of the ocean warming fraction α[ohc] is taken from the AR6 (Forster et al., 2021, Table 7.1).
3.4.2Sea level rise
Some parameters from the SLR module are structural: the maximum SLR contribution from glaciers (Λ[gla]) is taken from Fox-Kemper et al. (2021, Sect. 9.6.3.2); the equilibrium AIS SLR (Λ[ais]) is from
(Church et al., 2013, Fig. 13.14); and the τ[gis], τ[gla], and τ[ais] timescales are the mean values from Mengel et al. (2016, Table S1) (assuming they provide the 90% range of a log-normal
distribution). All other parameters are calibrated. The prior distribution of the thermosteric parameter Λ[thx] is taken from the AR6 (Fox-Kemper et al., 2021, Sect. 9.2.4.1), as are the prior
distributions of the preindustrial offset parameters λ[gis], λ[gla] and λ[ais] (Fox-Kemper et al., 2021, earliest period of Table 9.5). For the remaining parameters, we derive prior distributions
using SLR projections compiled by Edwards et al. (2021) for a number of ice sheets and glaciers models over various Representative Concentration Pathway (RCP) and Shared Socioeconomic Pathway (SSP)
scenarios. Using the models' outputs, we apply Eq. (9) to estimate the Λ[gis1] and Λ[gis3] parameters; Eq. (10) for the Γ[gla1], Γ[gla3], and γ[gla] parameters; and Eq. (12) for the Λ[ais,smb] and α
[ais] parameters. During these fits, all other parameters are assumed to take their default value if structural, and their best-guess value otherwise. Results of this calibration on the individual
models compiled by Edwards et al. (2021) are shown for each SLR contribution in Figs. B6, B7 and B8.
3.4.3Ocean carbon
The ocean carbon cycle module has a number of structural parameters: α[dic], all α[o,j], and all τ[o,j] are taken from Strassmann and Joos (2018, Tables A2 and A3, based on the Princeton model). The
prior distribution of the adjustment factor ${\mathit{\kappa }}_{{\mathit{\tau }}_{\mathrm{o}}}$ is arbitrarily taken to apply a 20% uncertainty on the oceanic transport timescales. All other prior
distributions for this module's parameters are derived from 12 CMIP6 models with interactive carbon cycle that contributed to C4MIP (Arora et al., 2020). T[o] is taken on average over the piControl
simulation. ν[gx] and γ[gx] are calibrated by applying Eq. (18) to the models' outputs for the 1pctCO2, 1pctCO2-rad, and 1pctCO2-bgc experiments, while β[dic] and γ[dic] are calibrated by applying
Eqs. (14)–(17) and (19). Results of this calibration on the individual CMIP6 models are shown in Figs. B1 and B2.
3.4.4Ocean acidification
In this version of Pathfinder, κ[pH] is a structural parameter set to 1.
3.4.5Land carbon
Parameters related to the passive soil carbon pool are taken from He et al. (2016, Table S5): ν[rh3] is structural, while α[pass] is not. All the prior distribution of the parameters related to the
preindustrial steady state of the land carbon (i.e. F[npp0], ν[fire], ν[harv], ν[mort], ν[stab], ν[rh1], and ν[cs]) are derived from 11 TRENDYv7 models (Sitch et al., 2015; Le Quéré et al., 2018),
exactly as for OSCAR v3.1 (Gasser et al., 2020) except that all biomes and regions are lumped together. The prior distribution of the remaining parameters are derived from 12 CMIP6 models that
contributed to C4MIP (Arora et al., 2020). Using the models' outputs for the 1pctCO2, 1pctCO2-rad, and 1pctCO2-bgc experiments, we calibrated β[npp], α[npp], and γ[npp] through Eq. (24); β[fire] and
γ[fire] through Eq. (28); and β[rh] and γ[rh] through Eq. (37). Results of this calibration on the individual CMIP6 models is shown in Figs. B3, B4, and B5.
3.4.6Permafrost carbon
The permafrost module's parameters are recalibrated using the same algorithm as used by Gasser et al. (2018) but adapted to the global formulation of Pathfinder. First, the algorithm is run once to
obtain a set of parameters reproducing the behaviour of the global average of five permafrost models (with data from UVic (MacDougall, 2021) added to the four original models). This gives the values
of the structural parameters (i.e. α[lst], γ[rt1], γ[rt2], κ[rt], a[min], all α[th,j], all τ[th,j], ν[thaw], and ν[froz]). Second, the algorithm is run five additional times for each of the five
permafrost models separately, with the structural parameters established in the first step, to obtain prior distributions of the remaining parameters (i.e. C[fr0], κ[a], γ[a], and ${\mathit{\kappa }}
_{{\mathit{\tau }}_{\mathrm{th}}}$).
3.4.7Atmospheric CO[2]
The conversion factor α[C] is a structural parameter whose value is taken from the latest GCBs (e.g. Le Quéré et al., 2018). The prior distribution of preindustrial CO[2] concentration (C[pi]) is
taken from the AR6 (Gulev et al., 2021, Sect. 2.2.3.2.1), assuming the difference between minimum and maximum over the 0–1850 period is representative of the 90% uncertainty range.
3.4.8Historical CO[2] and GMST
The structural X[μ] and X[σ] time series are taken from the latest observations, as follows. T[μ] and T[σ] are taken as the average and standard deviation of five observational GMST data sets:
HadCRUT5 (Morice et al., 2021), Berkeley Earth (Rohde et al., 2013; Rohde, 2013), GISTEMP (Hansen et al., 2010), NOAAGlobalTemp (Huang et al., 2020), and JMA (2022). We use the 1850–1900 period to
define our preindustrial baseline, and GMST change is assumed to be zero before the earliest date available in each data set. Regarding atmospheric CO[2], C[μ] is taken as the global value reported
by NOAA/ESRL (Tans and Keeling, 2010) and C[σ] as a constant ±1ppm uncertainty for 1980 onward (this uncertainty is arbitrarily taken higher than the actual uncertainty estimated through
instrumental measures to increase freedom in the calibration). Before that period, C[μ] comes from the IPCC AR6 (Dentener et al., 2021, Table AIII.1a), and C[σ] is linearly interpolated backwards
from the instrumental uncertainty in 1980 to the preindustrial one (Gulev et al., 2021) in 1750. Finally, the prior distribution of ρ[X] is set to uniform over [0,1], that of ${\stackrel{\mathrm{̃}}{\
mathit{\sigma }}}_{X}$ is a unit normal distribution, and that of ϵ[X] is set arbitrarily to a half-normal of parameter 0.05K for GMST and 0.5ppm for CO[2].
3.5Results (posterior distributions)
The following subsections discuss the adjustments between the prior and posterior parameters that are the results of the Bayesian calibration, as well as the matching of the constraints. These
sections constantly refer to Fig. 5 that shows the prior and posterior distributions of the model's parameters, Fig. 6 that shows those of the constrained variables, and Fig. 7 that displays the
correlation matrix of the posterior parameters (there is no correlation among the prior parameters). Prior and posterior values of the parameters can also be retrieved from Table B6.
3.5.1Climate system
Our climate-related constraints lead to adjusting all the parameters of the climate module. As explained in Sect. 3.4.8, the constraints for present-day GMST change and its derivative are met by
The ECS (T[2×]) is the parameter with the strongest adjustment, since it is directly constrained. Its precise value is discussed hereafter in Sect. 4.2, but we note that it is unsurprisingly
decreased, as the CMIP6 model ensemble tends to overestimate the ECS compared to the IPCC-assessed value. Consequently, our posterior logit(ff) matches the constraint well. The adjustment of the ECS
significantly reduces the gap between our posterior distribution of the non-CO[2] ERF and its constraint, although the posterior central value remains 41% lower (but well within uncertainty range).
Among the dynamic parameters that are adjusted, we note that the deep-ocean heat capacity Θ[d] is somewhat increased compared to the prior and that the heat exchange coefficient θ is also increased.
These dynamic parameters are likely adjusted through our OHU constraint that is corrected in the posterior so the difference in the central values is lowered from 22% to 14%, which remains well
within the uncertainty range.
In addition, a number of weak but physically meaningful correlations across the climate module's parameters are found, such as a positive correlation between T[2×] and ϵ[heat] (see, e.g. Geoffroy
et al., 2013a), a positive correlation between T[2×] and Θ[d] (that tends to exclude configuration that would warm fast and high), and a negative correlation between T[2×] and ϕ (to match the GMST
and ERF constraints together).
3.5.2Carbon cycle
Similarly to GMST, the posterior distribution of atmospheric CO[2] concentration matches the constraint by construction. Its derivative, however, is (slightly overly) corrected to match the GCB
estimate. Global anthropogenic CO[2] emissions are significantly increased to get closer to the GCB constraint, but their central value remains 9% too low. Since these emissions are determined
through mass balance and the atmospheric CO[2] matches observations, this implies that the total carbon sinks (i.e. F[land]+F[ocean]) must be weaker.
This is confirmed for the ocean sink, as the posterior central value of F[ocean] is 8% lower than the constraint but still noticeably corrected if compared to the prior. This correction is explained
by small adjustments in some parameters of the ocean carbon module. The mixed-layer depth is slightly increased through β[dic]. All other parameters remain mostly unaffected by the calibration, and
only minor correlations are found. These results, along with the fact that our prior distribution spans only about half of the constraint's distribution, suggest that there is a structural limitation
in our ocean carbon module that warrants further investigation.
It is also confirmed that the posterior land sink is weaker than the constraint, by 15% for the central value, which is nevertheless a significant reduction of the prior gap of 34%. To explain this
adjustment, we observe that the CO[2] fertilization sensitivities (β[npp] and γ[npp]) are adjusted upwards. However, our constraint on present-day NPP prevents these adjustments from being too
important, as the posterior distribution of this variable is similar to the prior and its central value remains 8%–9% higher than its constraint. An increased preindustrial NPP mechanically leads
to an increase in preindustrial carbon pools, but these require further adjustments of the land carbon turnover rates, and most notably the mortality rate ν[mort] and the passive carbon fraction α
[pass], to better match their constraints (of which the one on total soil carbon is perfectly met).
The land carbon module exhibits significant correlations among posterior parameters. This is likely a consequence of all the constraints combined as they dictate both the preindustrial steady state
of the module and its transient response over the historical period. Eliminated configurations are those, for instance, that would show high initial carbon pools that are very sensitive to climate
change (as these would lead to a very weak land sink) or that would exhibit a weak CO[2] fertilization effect associated with a fast turnover time (that would also lead to a weak sink).
3.5.3Sea level rise
The prior parameters of the SLR module are the least informed of our Bayesian setup. The model initially underestimates the thermal expansion, as well as the GIS and AIS SLR rates. The calibration
brings the posterior distributions closer to their respective constraints but it always remains in the lower end of the uncertainty range. The correction is done by adjusting many of the module's
parameters (most notably Λ[gis1], Λ[ais,smb], λ[ais] or λ[gla]) and by finding strong correlations among them (thus excluding physically unrealistic combinations).
The historical SLR is markedly corrected by the constraint: from a 19% gap between the central values of the constraint and the prior estimate, to only 7% after calibration. Here, we also note that
the sum of individual contributions to historical SLR reported in AR6 do not match that total SLR (Fox-Kemper et al., 2021, Table 9.5), which likely has some impact on the consistency between our
constraints. Finally, although the LIA relaxation contribution is not altered by the calibration, as its central value remains 50% too high, it is the likely source of the strong correlations found
among the parameters of this module because it straightforwardly links the individual SLR contributions together.
4.1Historical period
Because in the Bayesian setup we do not use annual time series of observations as constraints, the posterior distributions given in Fig. 6 do not inform on the whole dynamic of the model over the
historical period. To further diagnose the model's behaviour, Fig. 8 gives the time series from 1900 to 2021 of six key variables. GMST and atmospheric CO[2] match the historical observations very
well via construction of these input time series. The non-CO[2] ERF exhibits a very high variability, owing to our temperature-driven setup and the natural variability in the GMST input. Beyond that,
the ERF time series is consistent with the AR6 estimates (Smith et al., 2021), albeit somewhat lower on average in the recent past, as seen with the posterior distribution. Consistently with the
interpretation of carbon cycle variables in the calibration results, anthropogenic CO[2] emissions, and the ocean and land carbon sinks are slightly underestimated compared to the GCB estimates (
Friedlingstein et al., 2022). Several reasons could explain this discrepancy, from the lack of land use change in Pathfinder to the inconsistency of the GCB figures (that do not close the budget,
while ours do). Nevertheless, the interest of the calibration is clearly illustrated, as the posterior uncertainty range covers observations much better than the prior one.
4.2Idealized simulations
To complete the diagnosis of our model with common metrics used with climate and carbon models, we ran a set of standard idealized experiments, corresponding to the CMIP6 abrupt-2xCO2, 1pctCO2,
1pctCO2-bgc, and 1pctCO2-rad. A summary of these metrics' values is given in Table 2, and the resulting time series are shown in Fig. 9.
The abrupt-2xCO2 experiment sees an abrupt doubling of atmospheric CO[2], and it is used to diagnose the model's ECS that is defined as the equilibrium temperature for a doubling of the preindustrial
atmospheric concentration of CO[2] (we acknowledge that it is superfluous with this version of Pathfinder since it is also a parameter). Using the GMST anomaly at the end of 1500 years of this
experiment leads to an unconstrained estimate of ECS of 4.1±1.3K and a constrained estimate of 3.3±0.7K. Consistently, the latter value is between the ECS value extracted from CMIP6 models (
Meehl et al., 2020) that is higher (3.7±1.1K) and the final value assessed in the AR6 that is lower (3.0K, with a 67% confidence interval between 2.5 and 4.0K).
Using the 1pctCO2 experiment that sees a 1% yearly increase in atmospheric CO[2], we can estimate the model's transient climate response (TCR), which is defined as the GMST change after 70 years,
when atmospheric concentration CO[2] has just doubled. The CMIP6 models have a TCR of 2.0±0.4K (Meehl et al., 2020). Pathfinder's unconstrained value is higher, at 2.2±0.5K, while the
constrained one goes down to 1.9±0.3K. If we divide the TCR by the cumulative anthropogenic CO[2] emissions compatible with the atmospheric CO[2] increase in this experiment, we obtain an estimate
of the transient climate response to emissions (TCRE). Similarly to the TCR, it is higher in the unconstrained ensemble and lower in the constrained one, when compared to CMIP6 models (Arora et al.,
2020). Both downward adjustments of the TCR and TCRE are consistent with that of ECS, with the posterior TCRE matching the AR6 assessed range very well (Canadell et al., 2021).
To look more closely at the carbon cycle, we perform two variants of the latter experiment: in 1pctCO2-rad, atmospheric CO[2] only has a radiative effect, as it is kept at preindustrial level for the
carbon cycle, whereas in 1pctCO2-bgc, atmospheric CO[2] only has a biogeochemical effect, as the climate system sees only the preindustrial level. These three experiments are used to calculate the
carbon–concentration (β) and carbon–climate (γ) feedback metrics that measure the carbon sinks' sensitivities to changes in atmospheric CO[2] and GMST, respectively. We apply the same method as Arora
et al. (2020) to calculate these, which leads to metrics at the time of CO[2] doubling that are in line with CMIP6 models (Arora et al., 2020). As both carbon sinks were adjusted upwards by the
Bayesian calibration, the constraints logically increased both β[ocean] and β[land], to values fairly close to those of the complex models. The γ[ocean] is not affected by the calibration and remains
45% too low, which again suggests a structural limitation in our formulation of the ocean sink. This is, however, compensated for during calibration by the γ[land] being 26% higher than in complex
To further validate Pathfinder, we run the five SSP scenarios (Riahi et al., 2017) for which climate and carbon cycle projections were reported by a large enough number of models in the AR6 (namely,
ssp119, ssp126, ssp245, ssp370, and ssp585). These simulations are run with prescribed CO[2] concentration and non-CO[2] ERF (the latter is taken from Smith et al., 2021). Time series of GMST and
cumulative land and ocean sinks are shown in Fig. 9. Table 3 shows a comparison of the projected changes in GMST to the CMIP6 estimates (Lee et al., 2021, Table 4.2) and of carbon pools to Liddicoat
et al. (2021) (since this was not directly reported in the AR6).
Be it on short-, mid- or long-term scales, Pathfinder's projections of GMST are very much in line with the one assessed by the IPCC in the AR6 based on multiple lines of evidence (Lee et al., 2021
, Table 4.5). The only significant difference is a smaller uncertainty range in our projections for the longer-term periods. Although this is the result of the efficiency of the Bayesian calibration,
one might wonder whether the climate module is over-constrained (or equivalently, too limited in its number of parameters).
The ocean carbon storage appears to be overestimated by 5% to 20% by Pathfinder across SSP scenarios. This is consistent with the upward adjustment of the ocean carbon sink stemming from our
Bayesian calibration. To compare the land carbon storage with CMIP6 models, because our land carbon module does not include land use change processes, we correct the value reported by complex models
by the cumulative land use change emissions of each scenario (Riahi et al., 2017; Gidden et al., 2019). While the land carbon storage of Pathfinder is well in line under ssp126 (a scenario consistent
with the 2^∘C target), it is underestimated in ssp119 (consistent with the 1.5^∘C target) and increasingly overestimated in higher warming scenarios. A likely explanation is that the climate–carbon
feedback on land is underestimated in Pathfinder, as suggested by the γ metric seen in Sect. 4.2. Alternatively (or concurrently), the absence of loss of sink capacity caused by land cover change (
Gasser and Ciais, 2013; Gasser et al., 2020) can explain the overestimation of the land sink under high CO[2]. The Pathfinder model's estimates of both sinks nonetheless remain well within the CMIP6
models' uncertainty ranges.
Our SLR emulator gives estimates (Table 4) that are always on the lower end of the range reported in the AR6 (Fox-Kemper et al., 2021, Table 9.9). This can be explained by the fact that our
individual SLR rate estimates are on the lower end of their respective constraints (see Sect. 3.5.3). This discrepancy also highlights potential structural limitations in the SLR module (e.g. too few
separate contributions) and the difficulty of calibrating the module given the short time period of data available, both from complex models (that cover the 21st century only) and observations,
compared to the long timescale of the SLR processes. Nevertheless, our estimates remain within the uncertainty range of the IPCC assessment, especially as we do not account for contribution from land
water storage that causes an additional 0.03 [0.01, 0.04]m of SLR in all scenarios in 2100 (Fox-Kemper et al., 2021).
In this paper, we have presented the Pathfinder model: a simple global carbon–climate model with selected impact variables, carefully designed to balance accuracy of representation and simplicity of
formulation and calibrated through Bayesian inference on the latest data from Earth system models and observations. Pathfinder has been shown to perform very well in comparison to complex models,
although there remains room for further improvement of the model and its calibration setup. We identify four main avenues to improve the model.
First, some parts of the model may well lean too much on the complexity side of the simplicity–accuracy balance we aimed to strike, owing to the creation process of Pathfinder that mostly compiled
existing formulations. Future development should therefore strive to reduce complexity wherever possible. The ocean carbon sub-pools and perhaps the land carbon pools are potential leads in this
Second, the ocean carbon module alone appears to be limited by its structure, which has been inherited from a 25-year-old (yet seminal) article (Joos et al., 1996). Although it is undoubtedly a
significant undertaking, developing an alternative formulation of the ocean carbon dynamic, calibrated on state-of-the-art ocean models and properly connected to ocean pH and the ocean of the climate
module, would benefit more than just the SCM community.
Third, integration of land use and land cover change in such a model appears warranted, despite the difficulty of doing so in a physically sensible yet simple manner. Given our expertise with the
OSCAR model and its bookkeeping module (Gasser et al., 2020), we are confident that this can be done, although it will demand extra care to keep the model compatible with the IAMs it is also meant to
be linked to.
Fourth, the Bayesian setup can be extended, notably by including more time periods for the existing constraints but also by introducing and constraining entirely new variables such as isotopic ratios
(Hellevang and Aagaard, 2015) or inter-hemispheric gradients (Ciais et al., 2019); however, a balance must be struck with respect to the calibration's computation time. Here, we caution against
including complex models' results as constraints in the Bayesian calibration, as was done for the IPCC AR6 (Smith et al., 2021; Nicholls et al., 2021), as it goes against the philosophy of Pathfinder
to use complex models' results as prior information only.
Fifth, although our model is restricted to CO[2] by design because of how IAMs like DICE (Nordhaus, 2017) are also limited to CO[2] emissions, we can imagine many reasons why one would want to add
non-CO[2] climate forcers into Pathfinder. We would suggest doing so by following the model's philosophy: that is, by taking existing reduced-complexity formulations such as something between FaIR (
Leach et al., 2020) and OSCAR (Gasser et al., 2017) and adding the new parameters into the Bayesian setup with the relevant observational constraints.
In spite of these few shortcomings and potential development leads, Pathfinder v1.0.1 is a powerful tool that fits the niche it has been created for perfectly. We will further demonstrate the
strengths and flexibility of Pathfinder in other publications. Meanwhile, we invite the community to seize this open-source model and use it in any study that could benefit from a simple, fast, and
accurate carbon–climate model aligned with the latest climate science.
Appendix A:Additional information about the model
A1Technical requirements
Pathfinder has been developed and run in Python (v3.7.6) (https://docs.python.org/3.7/, last access: 5 December 2022), preferentially using IPython (v7.19.0) (Pérez and Granger, 2007). Currently,
packages required to run it are NumPy (v1.19.2) (Harris et al., 2020), SciPy (v1.5.2) (Virtanen et al., 2020), and Xarray (v0.16.0) (Hoyer and Hamman, 2017), and it has hard-coded dependencies on
PyMC3 (v3.8) (Salvatier et al., 2016) and Theano (v1.0.4) (Theano Development Team, 2016) that are in fact used only for calibration. Other versions of Python or these packages were not tested.
The calibration procedure takes about 9h to run on a desktop computer (with a base speed of 3.4GHz). Simple use of the model is much faster: the idealized experiments and SSP scenarios for this
description paper, which represent 2984 simulated years, were run in about 20min for all 2000 configurations and on a single core. A single simulated year takes a few tenths of a second, although a
number of options in the model can drastically alter this performance. Note also that this scales sub-linearly with the amount of configurations or scenarios because of the internal workings of the
Xarray package, albeit at the cost of increased demand in random-access memory.
A2Known issues
Two relatively benign issues that have been identified during development remain unsolved. First, the model requires a high number of sub-time steps (i.e. high n[t]) to remain stable under high CO[2]
because of the ocean carbon cycle. Second, the version of the model that is driven by T and R[x] time series is extremely sensitive to its inputs because it mathematically requires the first two
derivatives of T and the first derivative of R[x].
Brief description of the successive versions of Pathfinder.
Exact same physical equations and numerical values as v1.0. Added best-guess parameters calculated as the average of the posterior distribution, and corresponding historical outputs, for
single-configuration runs. Improved README and MANUAL files.
First release. Exact model described in the preprint version of this paper (Bossy et al., 2022).
Appendix B:Additional figures and tables
Code and data availability
The source code of Pathfinder is openly available at https://github.com/tgasser/Pathfinder (last access: 5 December 2022). A frozen version of the code and data as developed in the paper can be found
on Zenodo at https://doi.org/10.5281/zenodo.7003848 (Gasser, 2022).
TG developed the model, with contributions from TB regarding sea level rise and PC regarding land carbon cycle. TG coded the model and its calibration. TB ran the diagnostic simulations and made the
final figures. TB and TG wrote the manuscript with input from PC.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Andrew H. MacDougall for sharing permafrost simulations made with UVic, and Côme Cheritel for the first version of Figs. 1–4.
Development of Pathfinder was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement no. 820829 (CONSTRAIN project) and by the Austrian Science Fund
(FWF) under grant agreement no. P31796-N29 (ERM project).
This paper was edited by Marko Scholze and reviewed by Ian Enting and one anonymous referee.
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What is Leverage in Trading? Unlocking Leveraged Trading - TradeXplore
Leveraged trading, also refers to as margin trading, enables traders to open larger positions with only a small margin. While traders often boast about their profit multiples, few truly understand
the risks associated with leverage.
Leverage is a double-edged sword, amplifying both potential profits and losses proportionally. In this article, we aim to provide a detailed explanation of leverage in trading and the associated
risks, helping new traders understand and master leveraged trading.
What is Leverage in Trading
Leveraged trading is familiar to the public, seen in activities like home loans, which are a form of leveraged investment. For instance, buying a $5 million property with a $1 million deposit and a
$4 million loan creates a trading leverage of 5x, allowing control of a $5 million asset with a $1 million investment.
In finance trading, leverage empowers investors to open sizable positions with minimal deposits. In the context of property purchase, if the aim is to speculate on housing prices, it essentially
mirrors leveraged trading in finance.
Following the property purchase, the profit and loss (PnL) of the investment is calculated based on the entire property value rather than the deposit. For instance, if the property price increases by
10%, the investment’s PnL would amount to $500K (5M * 10%). Conversely, if the property price decreases by 5%, the investment would incur a loss of $250K.
Leverage & Margin Connection
Margin, or margin requirement, essentially is equivalent to the deposit required to initiate a trade. It acts as a threshold for trading, as one must possess at least the deposit amount to enter into
a trade.
Leveraged trading often refers to margin trading, as leverage and margin rate are interchangeable concepts. For instance, a 5x leverage equates to a 20% margin rate. Leverage can be articulated in
various ways, which may lead to confusion. Below, we compare three different expressions:
Leverage is the most straightforward expression, we use 100× leverage to signify leverage of 100 times.
Similarly, 200× leverage indicates a leverage of 200 times.
Leverage ratio
Leverage ratio holds a distinct meaning in trading compared to finance, it’s essentially another term for leverage.
For example, 100× leverage equals to 1:100 leverage ratio.
Margin rate
Margin rate = Margin ÷ Position vavlue × 100%
Although leverage isn’t explicitly stated, margin rate serves as another indicator of leverage. For instance, 100× leverage is synonymous with a 1% margin rate, while a 0.5% margin rate represents a
leverage ratio of 1:200.
Leverage & Margin Calculation
In the property market, leverage depends on how much we can borrow from the bank and how much deposit we can afford. However, in online trading, leverage is determined by the exchange or broker.
There isn’t a formula for calculating leverage in online trading because it varies among brokers, markets, and instruments. For instance, leverage ratios in futures markets typically range from 1:10
to 1:20, while in margin forex trading, leverage can exceed 100×.
The margin calculation varies across different markets. In markets like futures trading, the margin requirement is set by the exchange and may fluctuate over time based on market volatility. However,
in certain other derivatives trading, leverage is a fixed trading condition and can be calculated using the formula below:
Margin requirement = Value of position ÷ leverage
The formula applies to stock, margin forex, and CFD trading. In fact, traders don’t need to manually perform the calculations themselves, the margin information is readily available on the trading
platform at the instrument level.
The content in the previous chapters has already covered the function of leverage, however, we will revisit it in this dedicated section. This is because many people, especially new traders, often
misinterpret leverage, believing that it can increase their account capital.
For example, let’s say we have $200 capital, and 100x leverage can supposedly increase it by 100 times to $20,000. This is a very common misunderstanding. Leverage does not augment the account
capital, instead, it amplifies the position size that traders can control.
Let’s illustrate this with an example of gold trading. Suppose we have $200 in capital and want to buy 1 ounce of gold with 100x leverage, when the gold is trading at $2000 per ounce.
The value of 1 ounce of gold is $2000, but we only have $200 in capital. How can we make the trading happen? It will be easier to understand if we convert the 100x leverage to a 1% margin. With a 1%
margin ($20), we can open a position valued 100 times higher ($2000).
Risk of Leveraged Trading
Leverage is a double-edged sword, amplifying both potential profits and potential losses proportionally. In comparison to traditional finance trading, leveraged trading presents three additional
1). Risk of losing rapidly
Let’s compare two examples of gold trading: one without leverage and another with a leverage ratio of 1:100. In both examples, we have a capital of $200.
Gold trading without leverage
Buy 0.1 ounces when gold is traded at 2000 per ournce
The PnL would be $20 for a 10% price fluctuation.
Gold trading with 100× leverage
Buy 1 oz with a margin of $20 when gold is traded at 2000
The PnL would be $200 for a 10% price fluctuation.
With a capital of $200, the maximum amount of gold we can purchase without leverage is 0.1 ounces. Consequently, the profit and loss (PnL) will be based on the movement of 0.1 ounces.
In contrast, with a leverage ratio of 1:100, we can buy 1 ounce of gold with a small margin of $20. The resulting PnL will be for the entire 1 ounce. With the same 10% price fluctuation, the PnL for
0.1 ounce is just $20, while it rises to $200 for 1 ounce.
In the leveraged trading example, we could incur losses rapidly if the market moves against us. In some cases, we might lose all of our capital in just a few minutes or even a few seconds.
2). Risk of Over-leveraged
Being over-leveraged doesn’t necessarily mean a trader is using an extremely high leverage ratio, it rather indicates that the position opened is too large in comparison to the account capital. Even
a small price fluctuation could result in significant losses and potential liquidation.
Let’s illustrate this with two gold trading examples again, assuming the account capital is $200 and the leverage is 100x:
Buy 1 ounce gold
Only requires $20 margin when gold is traded at 2000.
After opening the position, the free margin available to absorb floating PnL is $180
The recent daily gold price fluctuation is about $30, a free margin of $180 is sufficient to maintain the position safely for a week.
Buy 8 ounces gold
The margin is $160 when gold is traded at 2000/ounce.
The free margin available to absort floating PnL is only $40 after the position is opened.
A free margin of $40 can only accommodate a price movement of $5, which could occur in a minute or even a second on gold.
With the same $200 account capital, a position size of 1 ounce can survive for a week, indicating it is not over-leveraged. However, when the position size increases to 8 ounces, even a slight price
fluctuation within a minute or even a second can lead to forced liquidation. Therefore, this is absolutely over-leveraged.
Although there is no universal standard, but a common measure of over-leveraging is how long the position can sustain or how much unfavorable price fluctuation the account can withstand.
Over-leverage falls into the category of trading psychology. Many traders tend to open additional positions when facing losses in the hope of quickly recovering their losses, which can ultimately
result in even larger losses.
Margin call is a risk inherent in all leveraged trading. While the specific rules may vary, the underlying logic remains consistent. When a trader’s account equity falls below a certain level due to
running losses, the broker will initiate forced liquidation of the open position, thereby preventing the trader from losing more than the account capital.
In high leveraged trading, such as margin forex trading and CFDs trading, forced liquidation is a common occurrence for many traders. To learn more about margin calls, please read our other article
titled「How to Lower Margin Call Risk? Understanding Forex Margin Call」.
High Leverage vs Low Leverage
Ther is no definitive answer for the debate between high leverage and low leverage in trading. Low leverage is commonly perceived as less risky due to reduced margin call risk, but it also implies a
higher trading threshold.
However, the primary determinant of trading risk is not leverage, but rather position size. The larger the position size, the higher the trading risk. Surprisingly, with the same account capital and
position size, higher leverage can help traders withstand more market volatility, offering a better chance of closing with profit.
To illustrate this point, let’s consider two gold trading examples. In both cases, we have a $500 account capital and open a long position of 10 ounces of gold at $2000 per ounce, assuming the margin
call will be triggered when the account equity drops to the same level as the margin. One example uses a 1:100 leverage ratio, while the other uses a 1:200 ratio. The table below compares the key
factors for these two examples:
100× leverage 200× leverage
Margin $200 $100
Free margin $300 $400
Margin call price $1970 $1960
In the example of 100x leverage, the remaining free margin after opening the position can absorb a $30 price decline, indicating that the position will reach margin call when the gold price drops to
$1970. However, in the case of 200x leverage, the available free margin will be $10 higher than that of the 100x leverage example, delaying the margin call trigger until the gold price falls to
$1960. From this perspective, higher leverage appears more favorable to traders.
However, human nature often leads many traders to become emotional and unable to control themselves, resulting in the opening of larger positions with high leverage. As the position size increases,
so does the trading risk.
There are several factors influencing leverage, including capital amount, trading goals, trading strategy, risk preference, and even the characteristics of the traded instrument. For new traders, the
typical leverages such as 200x and 500x provided by high leverage brokers are generally sufficient.
High Leveraged Trading Markets
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Chiang And Wainwright Mathematical Economics Pdf
Mathematical Economics Chiang Wainwright Solutions
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Required Text: “Fundamental Methods of Mathematical Economics, 4th Edition” by Alpha Chiang and Kevin Wainwright (Custom edition available in UF Bookstore) FIRST WEEK 1) Read the Syllabus . READING
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The Course Links are three different courses of Mathematical Economics taught by the K. Wainwright OPMT 7701 - Calculus for Business (second or third year university) ECON 331 Introduction to
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Expand Your Academic Horizons with Our Comprehensive Linguix Dictionary
1. a system of logic whose formal properties resemble certain moral and epistemological concepts
2. the logical study of necessity and possibility
How To Use modal logic In A Sentence
• This can be remedied by hybridization, that is, hybridization of modal logics enables the formulation of uniform tableau, Gentzen, and natural deduction systems for wide classes of logics. Hybrid
• There is one context in which the language of possible worlds is undoubtedly useful and even illuminating, namely, in the study of formal axiomatic systems of modal logic.
• Modal logic has a more sophisticated truth definition in which formulas are not simply globally true or false; their truth depends on your point of view.
• His arguments regarding this are presented in which also examines more generally his views on modal logic.
• The three most important parts of this definition for quantified modal logic are the clauses for atomic, quantified, and modal formulas.
• By far, alethic logic has been the field of modal logic which has received the greatest attention.
• The “actually” operator in modal logic is supposed to mirror the behavior of the English adverb ˜actually™ and adjective ˜actual™ in the examples below: Names
• Doxastic logic is a modal logic that is concerned with reasoning about beliefs.
• (Philosophy/Logic) denoting the branch of modal logic that deals with the formalization of certain epistemological concepts, such as knowledge, certainty, and ignorance See also doxastic The
Volokh Conspiracy » Words I never expected to write
• Philosophy / Logic denoting the branch of modal logic that deals with the formalization of certain epistemological concepts, such as knowledge, certainty, and ignorance See also doxastic The
Volokh Conspiracy » Words I never expected to write
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Decimal point truncation - query
This is Maria. I have query on decimal values.
i have a decimal value a= 8.34512
when im rounding it or finding the abs … it is giving like 8.35. trunc is truncating all the decimals.
i would like to get 8.34, i dont want to round off … i want to truncate the remaining decimal. I meant i wanted to get the result as 8.34 and i dont want 0.00512
How should i get these ?
Please advise.
Truncate function in AX will remove all the decimal places from a real number. Why you don’t want to round off it?
Hi ,
Thanks for your reply.
For Example in some record CreditAmt is 500. Here, as per my requirment i’m splitting this value in to 3, based on rate values. and inserting 3 records(with split value- debit value) in to another
table. so at last when i’m checking the total of credit and debit amount, i’m getting some difference in points like 500.02 like that. I would like to avoid these. I believe this is happening because
of rounding the value in to 2 digits. so instead of rounding i want to truncate the remaining digits.
If you have any idea pls help me to resolve this.
You’re not getting to the root cause of this issue, which has to do with the precision of your numbers. Cutting off decimals only appears to make the error go away in that particular example, it
doesn’t really anything. Do the math with a credit limit of 100. You’d end up with three records of 33.33, which adds up to 99.99. If you cut off those decimals, you end up with a total of 99.
All that you’re doing is rounding intermediate amounts to too little precision, which causes rounding errors when you add them back together again. In your case, with a credit limit of 500 divided
into three records, with a precision of 2 decimals, you get three records with 166.67, which add up to 500.01. If you would round to 3 decimals, you get 166.667, and those would add up to 500.001,
which rounds perfectly to the original amount, and all you’d have to do is define the precision in the display controls. Personally I would even consider using a precision of 5 decimals.
Kranthi, this is one example of when precision makes a difference, and it is definitely not just for display purposes.
I do understand It[:)]
Maria i think these code will help you.
real val,val2;
str StrVal;
val = Actualvalue // value to be truncate
StrVal = num2str(val,8,3,1,0);
StrVal = strdel(StrVal,strlen(StrVal),1);
val2 = str2num(StrVal);
RAM [:)]
I can’t say anything about the code there, but I disagree that this should be fixed by code. With a proper design, there should not be a need for any code. This is a design issue, and if they would
just think about setting the proper precision, the values would add up perfectly.
Hi Maria,
a*100 = 834.512;
trunc(a) = 834 → Axapta function
a/100 = 8.34.
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