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Feature importance |
Feature importance
The individual importance values for each of the input features (the default feature importances calculation method for non-ranking metrics).
For each feature, PredictionValuesChange shows how much on average the prediction changes if the feature value changes. The bigger the value of the importance the bigger on average is the change to
the prediction value, if this feature is changed.
See the Regular feature importance file format.
Calculation principles
Leaf pairs that are compared have different split values in the node on the path to these leaves. If the split condition is met (this condition depends on the feature F), the object goes to the left
subtree; otherwise it goes to the right one.
$feature\_importance_{F} = \displaystyle\sum\limits_{trees, leafs_{F}} \left(v_{1} - avr \right)^{2} \cdot c_{1} + \left( v_{2} - avr \right)^{2} \cdot c_{2} { , }$
$avr = \displaystyle\frac{v_{1} \cdot c_{1} + v_{2} \cdot c_{2}}{c_{1} + c_{2}} { , where}$
• $c_1, c_2$ represent the total weight of objects in the left and right leaves respectively. This weight is equal to the number of objects in each leaf if weights are not specified for the
• $v_1, v_2$ represent the formula value in the left and right leaves respectively.
If the model uses a combination of some of the input features instead of using them individually, an average feature importance for these features is calculated and output. For example, the model
uses a combination of features f54, c56 and f77. First, the feature importance is calculated for the combination of these features. Then the resulting value is divided by three and is assigned to
each of the features.
If the model uses a feature both individually and in a combination with other features, the total importance value of this feature is defined using the following formula:
$feature\_total\_importance_{j} = feature\_importance + \sum\limits_{i=1}^{N}average\_feature\_importance_{i} { , where}$
• $feature\_importance_{j}$ is the individual feature importance of the j-th feature.
• $average\_feature\_importance_{i}$ is the average feature importance of the j-th feature in the i-th combinational feature.
Complexity of computation
$O(trees\_count \cdot depth \cdot 2 ^ {depth} \cdot dimension)$
• Feature importance values are normalized so that the sum of importances of all features is equal to 100. This is possible because the values of these importances are always non-negative.
• Formula values inside different groups may vary significantly in ranking modes. This might lead to high importance values for some groupwise features, even though these features don't have a
large impact on the resulting metric value.
The individual importance values for each of the input features (the default feature importances calculation method for ranking metrics). This type of feature importance can be used for any model,
but is particularly useful for ranking models, where other feature importance types might give misleading results.
For each feature the value represents the difference between the loss value of the model with this feature and without it. The model without this feature is equivalent to the one that would have been
trained if this feature was excluded from the dataset. Since it is computationally expensive to retrain the model without one of the features, this model is built approximately using the original
model with this feature removed from all the trees in the ensemble. The calculation of this feature importance requires a dataset and, therefore, the calculated value is dataset-dependent.
See the Regular feature importance file format.
Calculation principles
The value of LossFunctionChange is defined so that the more important is the feature, the higher is its importance value.
• Minimum best value objective metric:
$feature\_importance_{i} = metric (E_{i}v)) - metric(v)$
• Maximum best value objective metric:
$feature\_importance_{i} = metric(v) - metric(E_{i}v)$
• Exact best value objective metric:
$feature\_importance_{i} = |metric(E_{i}v) - best\_value| - |metric(v) - best\_value|$
In general, the value of LossFunctionChange can be negative.
Variables description:
• $E_{i}v$ is the mathematical expectation of the formula value without the $i$-th feature. If the feature $i$ is on the path to a leaf, the new leaf value is set to the weighted average of values
of leaves that have different paths by feature value. Weights represent the total weight of objects in the corresponding leaf. This weight is equal to the number of objects in each leaf, if
weights are not specified in the dataset.
For feature combinations $F = (f_{1}, ..., f_{n})$, the average value in a leaf is calculated as follows:
$E_{f_i}v = \displaystyle\left(\frac{(n - 1) v + E_{F}v}{n}\right)$
• $v$ is the vector with formula values for the dataset. The training dataset are used, if both training and validation datasets are provided.
• $metric$ is the loss function specified in the training parameters.
The size of the random subsample used for calculation is determined as follows:
$subsamples\_count = \min(samples_count, \max(2\cdot 10^5, \frac{2\cdot 10^9}{features\_count}))$
Complexity of computation
$O(trees\_count \cdot (2 ^ {depth} + subsamples\_count) \cdot depth +$
$+ Eval\_metric\_complexity(model, subsamples\_count) \cdot features\_count)$
This feature importance approximates the difference between metric values calculated on the following models:
• The model with the $i$-th feature excluded.
• The original model with all features.
The importance values both for each of the input features and for their combinations (if any).
See the InternalFeatureImportance file format.
Calculation principles
Leaf pairs that are compared have different split values in the node on the path to these leaves. If the split condition is met (this condition depends on the feature F), the object goes to the left
subtree; otherwise it goes to the right one.
$feature\_importance_{F} = \displaystyle\sum\limits_{trees, leafs_{F}} \left(v_{1} - avr \right)^{2} \cdot c_{1} + \left( v_{2} - avr \right)^{2} \cdot c_{2} { , }$
$avr = \displaystyle\frac{v_{1} \cdot c_{1} + v_{2} \cdot c_{2}}{c_{1} + c_{2}} { , where}$
• $c_{1}, c_{2}$ represent the total weight of objects in the left and right leaves respectively. This weight is equal to the number of objects in each leaf if weights are not specified for the
• $v_{1}, v_{2}$ represent the formula value in the left and right leaves respectively.
If the model uses a combination of some of the input features instead of using them individually, an average feature importance for these features is calculated and output. For example, the model
uses a combination of features f54, c56 and f77. First, the feature importance is calculated for the combination of these features. Then the resulting value is divided by three and is assigned to
each of the features.
If the model uses a feature both individually and in a combination with other features, the total importance value of this feature is defined using the following formula:
$feature\_total\_importance_{j} = feature\_importance + \sum\limits_{i=1}^{N}average\_feature\_importance_{i} { , where}$
• $feature\_importance_{j}$ is the individual feature importance of the j-th feature.
• $average\_feature\_importance_{i}$ is the average feature importance of the j-th feature in the i-th combinational feature.
Complexity of computation
$O(trees\_count \cdot depth \cdot 2 ^ {depth} \cdot dimension)$
The impact of a feature on the prediction results for a pair of objects. This type of feature importance is designed for analyzing the reasons for wrong ranking in a pair of documents, but it also
can be used for any one-dimensional model.
For each feature PredictionDiff reflects the maximum possible change in the predictions difference if the value of the feature is changed for both objects. The change is considered only if there is
an improvement in the direction of changing the order of documents.
• Only models trained on datasets that do not contain categorical features are supported.
• Multiclassification modes are not supported.
Related information
Detailed information regarding usage specifics for different Catboost implementations. | {"url":"https://catboost.ai/en/docs/concepts/fstr","timestamp":"2024-11-07T10:13:48Z","content_type":"text/html","content_length":"430498","record_id":"<urn:uuid:948e006d-702c-4975-8cdc-83888fc59ff0>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00535.warc.gz"} |
Note: For background information, please see my introduction to Cast Vote Records processing and theory here: Statistical Detection of Irregularities via Cast Vote Records.
Since I posted my initial analysis of the Henrico CVR data, one comment was made to me by a member of the Texas election integrity group I have been working with: We have been assuming, based on
vendor documentation and the laws and requirements in various states, that when a cast vote record is produced by vendor software the results are sorted by the time the ballot was recorded onto a
scanner. However, when looking at the results that we’ve been getting so far and trying to figure out plausible explanations for what we were seeing, he realized it might be the case that the
ordering of the CVR entries are being done by both time AND USB stick grouping (which is usually associated with a specific scanner or precinct) but then simply concatenating all of those results
While there isn’t enough information in the Henrico CVR files to breakout the entries by USB/Scanner, and the Henrico data has record ID numbers instead of actual timestamps, there is enough
information to break out them by Precinct, District and Race, with the exception of the Central Absentee Precincts (CAP) entries where we can only break them out by district given the metadata alone.
However, with some careful MATLAB magic I was able to cluster the results marked as just “CAP” into at least 5 different sub-groupings that are statistically distinct. (I used an exponential moving
average to discover the boundaries between groupings, and looking at the crossover points in vote share.) I then relabeled the entries with the corresponding “CAP 1”, “CAP 2”, … , “CAP 5” labels as
appropriate. My previous analysis was only broken out by Race ID and CAP/Non-CAP/Provisional category.
Processing in this manner makes the individual distributions look much cleaner, so I think this does confirm that there is not a true sequential ordering in the CVR files coming out of the vendor
software packages. (If they would just give us the dang timestamps … this would be a lot easier!)
I have also added a bit more rigor to the statistics outlier detection by adding plots of the length of observed runs (e.g. how many “heads” did we get in a row?) as we move through the entries, as
well as the plot of the probability of this number of consecutive tosses occurring. We compute this probability for K consecutive draws using the rules of statistical independence, which is P
([a,a,a,a]) = P(a) x P(a) x P(a) x P(a) = P(a)^4. Therefore the probability of getting 4 “heads” in a row with a hypothetical 53/47 weighted coin would be .53^4 = 0.0789. There are also plotted lines
for a probability 1/#Ballots for reference.
The good news is that this method of slicing the data and assuming that the Vendor is simply concatenating USB drives seems to produce much tighter results that look to obey the expected IID
distributions. Breaking up the data this way resulted in no plot breaking the +/- 3/sqrt(N-1) boundaries, but there still are a few interesting datapoints that we can observe.
In the plot below we have the Attorney Generals race in the 4th district from precinct 501 – Antioch. This is a district that Miyares won handily 77%/23%. We see that the top plot of the cumulative
spread is nicely bounded by the +/- 3/sqrt(N-1) lines. The second plot from the top gives the vote ratio in order to compare with the work that Draza Smith, Jeff O’Donnell and others are doing with
CVR’s over at Ordros.com. The second from bottom plot gives the number k of consecutive ballots (in either candidates favor) that have been seen at each moment in the counting process. And the bottom
plot raises either the 77% or 23% overall probability to the k-th power to determine the probability associated with pulling that many consecutive Miyares or Herring ballots from an IID distribution.
The most consecutive ballots Miyares received in a row was just over 15, which had a .77^15 = 0.0198 or 1.98% chance of occurring. The most consecutive ballots Herring received was about 4, which
equates to a probability of occurrence of .23^4 = 0.0028 or 0.28% chance. The dotted line on the bottom plot is referenced at 1/N, and the solid line is referenced at 0.01%.
But let’s now take a look at another plot for the Miyares contest in another blowout locality with 84% / 16% for Miyares. The +/- 3/sqrt(N-1) limit nicely bounds our ballot distribution again. There
is, however, an interesting block of 44 consecutive ballots for Miyares about halfway through the processing of ballots. This equates to .84^44 = 0.0004659 or a 0.04659% chance of occurrence from an
IID distribution. Close to this peak is a run of 4 ballots for Herring which doesn’t sound like much, but given the 84% / 16% split, the probability of occurrence for that small run is .16^4 =
0.0006554 or 0.06554%!
Moving to the Lt. Governors race we see an interesting phenomenon where where Ayala received a sudden 100 consecutive votes a little over midway through the counting process. Now granted, this was a
landslide district for Ayala, but this still equates to a .92^100 = 0.000239 or 0.0239% chance of occurrence.
And here’s another large block of contiguous Ayala ballots equating to about .89^84 = 0.00005607 or 0.0056% chance of occurrence.
Tests for Differential Invalidation (added 2022-09-19):
“Differential invalidation” takes place when the ballots of one candidate or position are invalidated at a higher rate than for other candidates or positions. With this dataset we know how many
ballots were cast, and how many ballots had incomplete or invalid results (no recorded vote in the cvr, but the ballot record exists) for the 3 statewide races. In accordance with the techniques
presented in [1] and [2], I computed the plots of the Invalidation Rate vs the Percent Vote Share for the Winner in an attempt to observe if there looks to be any evidence of Differential
Invalidation ([1], ch 6). This is similar to the techniques presented in [2], which I used previously to produce my election fingerprint plots and analysis that plotted the 2D histograms of the vote
share for the winner vs the turnout percentage.
The generated the invalidation rate plots for the Gov, Lt Gov and AG races statewide in VA 2021 are below. Each plot below is representing one of the statewide races, and each dot is representing the
ballots from a specific precinct. The x axis is the percent vote share for the winner, and the y axis is computed as 100 – 100 * Nvotes / Nballots. All three show a small but statistically
significant linear trend and evidence of differential invalidation. The linear regression trendlines have been computed and superimposed on the data points in each graph.
To echo the warning from [1]: a differential invalidation rate does not directly indicate any sort of fraud. It indicates an unfairness or inequality in the rate of incomplete or invalid ballots
conditioned on candidate choice. While it could be caused by fraud, it could also be caused by confusing ballot layout, or socio-economic issues, etc.
Full Results Download
• [1] Forsberg, O.J. (2020). Understanding Elections through Statistics: Polling, Prediction, and Testing (1st ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9781003019695
• [2] Klimek, Peter & Yegorov, Yuri & Hanel, Rudolf & Thurner, Stefan. (2012). Statistical Detection of Systematic Election Irregularities. Proceedings of the National Academy of Sciences of the
United States of America. 109. 16469-73. https://doi.org/10.1073/pnas.1210722109.
Update 2022-08-29 per observations by members of the Texas team I am working with, we’ve been able to figure out that (a) the vendor was simply concatenating data records from each machine and not
sorting the CVR results and (b) how to mostly unwrap this affect on the data to produce much cleaner results. The results below are left up for historical reference.
For background information, please see my introduction to Cast Vote Records processing and theory here: Statistical Detection of Irregularities via Cast Vote Records. This entry will be specifically
documenting the results from processing the Henrico County Virginia CVR data from the 2021 election.
As in the results from the previous post, I expanded the theoretical error bounds out to 6/sqrt(N) instead of 3/sqrt(N) in order to give a little bit of extra “wiggle room” for small fluctuations.
However the Henrico dataset could only be broken up by CAP, Non-CAP or Provisional. So be aware that the CAP curves presented below contain a combination of both early-vote and mail-in ballots.
The good news is that I’ve at least found one race that seems to not have any issues with the CVR curves staying inside the error boundaries. MemberHouseOfDelegates68thDistrict did not have any parts
of the curves that broke through the error boundaries.
The bad news … is pretty much everything else doesn’t. I cannot tell you why these curves have such differences from statistical expectation, just that they do. We must have further investigation and
analysis of these races to determine root cause. I’ve presented all of the races that had sufficient number of ballots below (1000 minimum for the race a whole, and 100 ballot minimum for each ballot
There has been a good amount of commotion regarding cast vote records (CVRs) and their importance lately. I wanted to take a minute and try and help explain why these records are so important, and
how they provide a tool for statistical inspection of election data. I also want to try and dispel any misconceptions as to what they can or can’t tell us.
I have been working with other local Virginians to try and get access to complete CVRs for about 6 months (at least) in order to do this type of analysis. However, we had not had much luck in
obtaining real data (although we did get a partial set from PWC primaries but it lacked the time-sequencing information) to evaluate until after Jeff O’Donnell (a.k.a. the Lone Raccoon) and Walter
Dougherity did a fantastic presentation at the Mike Lindell Moment of Truth Summit on CVRs and their statistical use. That presentation seems to have broken the data logjam, and was the impetus for
writing this post.
Just like the Election Fingerprint analysis I was doing earlier that highlighted statistical anomalies in election data, this CVR analysis is a statistics based technique that can help inform us as
to whether or not the election data appears consistent with expectations. It only uses the official results as provided by state or local election authorities and relies on standard statistical
principles and properties. Nothing more. Nothing less.
What is a cast vote record?
A cast vote record is part of the official election records that need to be maintained in order for election systems to be auditable. (see: 52 USC 21081 , NIST CVR Standard, as well as the Virginia
Voting Systems Certification Standards) They can have many different formats depending on equipment vendor, but they are effectively a record of each ballot as it was recorded by the equipment. Each
row in a CVR data table should represent a single ballot being cast by a voter and contain, at minimum, the time (or sequence number) when the ballot was cast, the ballot type, and the result of each
race. Other data might also be included such as which precinct and machine performed the scanning/recording of the ballot, etc. Note that “cast vote records” are sometimes also called “cast voter
records”, “ballot reports” or a number of other different names depending on the publication or locality. I will continue to use the “cast vote record” language in this document for consistency.
Why should we care?
The reason these records are so important, is based on statistics and … unfortunately … involves some math to fully describe. But to make this easier, let’s try first to walk through a simple thought
experiment. Let’s pretend that we have a weighted, or “trick” coin, that when flipped it will land heads 53% of the time and land tails 47% of the time. We’re going to continuously flip this coin
thousands of times in a row and record our results. While we can’t predict exactly which way the coin will land on any given toss, we can expect that, on average, the coin will land with the
aforementioned 53/47 split.
Now because each coin toss constitutes an independent and identically distributed (IID) probability function, we can expect this sequence to obey certain properties. If as we are making our tosses,
we are computing the “real-time” statistics of the percentage of head/tails results, and more specifically if we plot the spread (or difference) of those percentage results as we proceed we will see
that the spread has very large swings as we first begin to toss our coin, but very quickly the variability in the spread becomes stable as more and more tosses (data) are available for us to average
over. Mathematically, the boundary on these swings is inversely proportional to the square root of how many tosses are performed. In the “Moment of Truth” video on CVRs linked above, Jeff and Walter
refer to this as a “Cone of Probability”, and he generates his boundary curves experimentally. He is correct. It is a cone of probability as its really just a manifestation of well-known and
well-understood Poisson Noise characteristic (for the math nerds reading this). In Jeff’s work he uses the ratio of votes between candidates, while I’m using the spread (or deviation) of the vote
percentages. Both metrics are valid, but using the deviation has an easy closed-form boundary curve that we don’t need to generate experimentally.
In the graphic below I have simulated 10 different trials of 10,000 tosses for a distribution that leans 53/47, which is equivalent to a 6% spread overall. Each trial had 10,000 random samples
generated as either +1 or -1 values (a.k.a. a binary “Yes” or “No” vote) approximating the 53/47 split and I plotted the cumulative running spread of the results as each toss gets accumulated. The
black dotted outline is the 95% confidence interval (or +/-3x the standard deviation) across the 10 trials for the Nth bin, and the red dotted outline is the 3/sqrt(n-1) analytical boundary.
So how does this apply to election data?
In a theoretically free and perfectly fair election we should see similar statistical behavior, where each coin toss is replaced with a ballot from an individual voter. In a perfect world we would
have each vote be completely independent of every other vote in the sequence. In reality we have to deal with the fact that there can be small local regions of time in which perfectly legitimate
correlations in the sequence of scanned ballots exist. Think of a local church who’s congregation is very uniform and they all go to the polls after Sunday mass. We would see a small trend in the
data corresponding to this mass of similar thinking peoples going to the polls at the same time. But we wouldn’t expect there to be large, systematic patterns, or sharp discontinuities in the plotted
results. A little bit of drift and variation is to be expected in dealing with real world election data, but persistent and distinct patterns would indicate a systemic issue.
Now we cannot isolate all of the variables in a real life example, but we should try as best as possible. To that effect, we should not mix different ballot types that are cast in different manners.
We should keep our analysis focused within each sub-group of ballot type (mail-in, early-vote, day-of, etc). It is to the benefit of this analysis that the very nature of voting, and the procedures
by which it occurs, is a very randomized process. Each sub-grouping has its own quasi-random process that we can consider.
While small groups (families, church groups) might travel to the in-person polls in correlated clusters, we would expect there to be fairly decent randomization of who shows up to in-person polls and
when. The ordering of who stands in line before or after one another, how fast they check-in and fill out their ballot, etc, are all quasi-random processes.
Mail-in ballots have their own randomization as they depend on the timing of when individuals request, fill-out and mail their responses, as well as the logistics and mechanics of the postal service
processes themselves providing a level of randomization as to the sequence of ballots being recorded. Like a dealer shuffling a deck of cards, the process of casting a mail-in vote provides an
additional level of independence between samples.
No method is going to supply perfect theoretical independence from ballot to ballot in the sequence, but theres a general expectation that voting should at least be similar to an IID process.
Also … and I cannot stress this enough … while these techniques can supply indications of irregularities and discrepancies in elections data, they are not conclusive and must be coupled with in-depth
So going back to the simulation we generated above … what does a simulation look like when cheating occurs? Let’s take a very simple cheat from a random “elections” of 10,000 ballots, with votes
being representative of either +1 (or “Yes”) or -1 (or “No”) as we did above. But lets also cheat by randomly selecting two different spots in the data stream to place blocks of 250 consecutive “Yes”
The image below shows the result of this process. The blue curve represents the true result, while the red curve represents the cheat. We see that at about 15% and 75% of the vote counted, our
algorithm injected a block of “Yes” results, and the resulting cumulative curve breaks through the 3/sqrt(N-1) boundary. Now, not every instance or type of cheat will break through this boundary, and
there may be real events that might explain such behavior. But looking for CVR curves that break our statistical expectations is a good way to flag items that need further investigation.
Computing the probability of a ballot run:
Section added on 2022-09-18
We can also a bit more rigor to the statistics outlier detection by computing the probability of the length of observed runs (e.g. how many “heads” did we get in a row?) occurring as we move through
the sequential entries. We can compute this probability for K consecutive draws using the rules of statistical independence, which is P([a,a,a,a]) = P(a) x P(a) x P(a) x P(a) = P(a)^4. Therefore the
probability of getting 4 “heads” in a row with a hypothetical 53/47 weighted coin would be .53^4 = 0.0789.
Starting with my updated analysis of 2021 Henrico County VA, I’ve started adding this computation to my plots. I have not yet re-run the Texas data below with this new addition, but will do so soon
and update this page accordingly.
Real Examples
UPDATE 2022-09-18:
• I have finally gotten my hands on some data for 2020 in VA. I will be working to analyze that data and will report what I find as soon as I can, but as we are approaching the start of early
voting for 2022, my hands are pretty full at the moment so it might take me some time to complete that processing.
• As noted in my updates to the Henrico County 2021 VA data, and in my section on computing the probability of given runs above, the Texas team noticed that we could further break apart the Travis
county data into subgroups by USB stick. I will update my results below as soon as I get the time to do so.
[S:So I haven’t gotten complete cast vote records from VA yet (… which is a whole other set of issues …), but:S] I have gotten my Cheeto stained fingers on some data from the Travis County Texas 2020
So let us first take a look at an example of a real race where everything seems to be obeying the rules as set out above. I’ve doubled my error bars from 3x to 6x of the inverse square standard
(discussed above) in order to handle the quasi-IID nature of the data and give some extra margin for small fluctuating correlations.
The plot below shows the Travis County Texas 2020 BoardOfTrusteesAt_LargePosition8AustinISD race, as processed by the tabulation system and stratified by ballot type. We can see that all three ballot
types start off with large variances in the computed result but very quickly coalesce and approach their final values. This is exactly what we would expect to see.
Now if I randomly shuffle the ordering of the ballots in this dataset and replot the results (below) I get a plot that looks unsurprisingly similar, which suggests that these election results were
likely produced by a quasi-IID process.
Next let’s take a look at a race that does NOT conform to the statistics we’ve laid out above. (… drum-roll please … as this the one everyone’s been waiting for). Immma just leave this right here and
just simply point out that all 3 ballot type plots below in the Presidential race for 2020 go outside of the expected error bars. I also note the discrete stair step pattern in the early vote
numbers. It’s entirely possible that there is a rational explanation for these deviations. I would sure like to hear it, especially since we have evidence from the exact same dataset of other races
that completely followed the expected boundary conditions. So I don’t think this is an issue with a faulty dataset or other technical issues.
And just for completeness, when I artificially shuffle the data for the Presidential race, and force it to be randomized, I do in fact end up with results that conform to IID statistics (below).
I will again state that while these results are highly indicative that there were irregularities and discrepancies in the election data, they are not conclusive. A further investigation must take
place, and records must be preserved, in order to discover the cause of the anomalies shown.
Running through each race that had at least 1000 ballots cast and automatically detecting which races busted the 6/sqrt(n-1) boundaries produces the following tabulated results. A 1 in the right hand
column indicates that the CVR data for that particular race in Travis County has crossed the error bounds. A 0 in the right hand column indicates that all data stayed within the error bound limits.
Race CVR_OOB_Irregularity_Detected
President_VicePresident 1
UnitedStatesSenator 1
UnitedStatesRepresentativeDistrict10 1
UnitedStatesRepresentativeDistrict17 1
UnitedStatesRepresentativeDistrict21 1
UnitedStatesRepresentativeDistrict25 1
UnitedStatesRepresentativeDistrict35 0
RailroadCommissioner 1
ChiefJustice_SupremeCourt 1
Justice_SupremeCourt_Place6_UnexpiredTerm 1
Justice_SupremeCourt_Place7 1
Justice_SupremeCourt_Place8 1
Judge_CourtOfCriminalAppeals_Place3 1
Judge_CourtOfCriminalAppeals_Place4 1
Judge_CourtOfCriminalAppeals_Place9 1
Member_StateBoardOfEducation_District5 1
Member_StateBoardOfEducation_District10 1
StateSenator_District21 0
StateSenator_District24 1
StateRepresentativeDistrict47 1
StateRepresentativeDistrict48 1
StateRepresentativeDistrict49 1
StateRepresentativeDistrict50 1
StateRepresentativeDistrict51 0
ChiefJustice_3rdCourtOfAppealsDistrict 1
DistrictJudge_460thJudicialDistrict 1
DistrictAttorney_53rdJudicialDistrict 1
CountyJudge_UnexpiredTerm 1
Judge_CountyCourtAtLawNo_9 1
Sheriff 1
CountyTaxAssessor_Collector 1
CountyCommissionerPrecinct1 1
CountyCommissionerPrecinct3 1
AustinCityCouncilDistrict2 0
AustinCityCouncilDistrict4 0
AustinCityCouncilDistrict6 0
AustinCityCouncilDistrict7 0
AustinCityCouncilDistrict10 1
PropositionACityOfAustin_FP__2015_ 1
PropositionBCityOfAustin_FP__2022_ 1
MayorCityOfCedarPark 0
CouncilPlace2CityOfCedarPark 0
CouncilPlace4CityOfCedarPark 0
CouncilPlace6CityOfCedarPark 0
CouncilMemberPlace2CityOfLagoVista 0
CouncilMemberPlace4CityOfLagoVista 0
CouncilMemberPlace6CityOfLagoVista 0
CouncilMemberPlace2CityOfPflugerville 0
CouncilMemberPlace4CityOfPflugerville 0
CouncilMemberPlace6CityOfPflugerville 0
Prop_ACityOfPflugerville_2169_ 0
Prop_BCityOfPflugerville_2176_ 0
Prop_CCityOfPflugerville_2183_ 0
BoardOfTrusteesDistrict2SingleMemberDistrictAISD 0
BoardOfTrusteesDistrict5SingleMemberDistrictAISD 0
BoardOfTrusteesAt_LargePosition8AustinISD 1
BoardOfTrusteesPlace1EanesISD 1
Prop_AEanesISD_2246_ 0
BoardOfTrusteesPlace3LeanderISD 0
BoardOfTrusteesPlace4LeanderISD 0
BoardOfTrusteesPlace5ManorISD 1
BoardOfTrusteesPlace6ManorISD 0
BoardOfTrusteesPlace7ManorISD 1
BoardOfTrusteesPlace6PflugervilleISD 1
BoardOfTrusteesPlace7PflugervilleISD 1
BoardOfTrusteesPlace1RoundRockISD 1
BoardOfTrusteesPlace2RoundRockISD 0
BoardOfTrusteesPlace6RoundRockISD 0
BoardOfTrusteesPlace7RoundRockISD 0
BoardOfTrusteesWellsBranchCommunityLibraryDistrict 0
Var147 0
BoardOfTrusteesWestbankLibraryDistrict 0
Var150 0
Var151 0
DirectorsPlace2WellsBranchMUD 0
DirectorsPrecinct4BartonSprings_EdwardsAquiferConservationDistr 0
PropositionAExtensionOfBoundariesCityOfLakeway_1966_ 0
PropositionB2_yearTermsCityOfLakeway_1973_ 0
PropositionCLimitOnSuccessiveYearsOfServiceCityOfLakeway_1980_ 0
PropositionDResidencyRequirementForCityManagerCityOfLakeway_198 0
PropositionEOfficeOfTreasurerCityOfLakeway_1994_ 0
PropositionFOfficialBallotsCityOfLakeway_2001_ 0
PropositionGAuthorizingBondsCityOfLakeway_2008_ 0
PropositionALagoVistaISD_2253_ 0
PropositionBLagoVistaISD_2260_ 0
PropositionCLagoVistaISD_2267_ 0
PropositionAEmergencyServicesDistrict1_2372_ 0
References and further reading: | {"url":"https://digitalpollwatchers.org/2022/08/","timestamp":"2024-11-06T10:22:21Z","content_type":"text/html","content_length":"148211","record_id":"<urn:uuid:763cf549-a8c3-4afe-b44a-564886b62807>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00269.warc.gz"} |
, 3D problem and performance comparisons
This example is a direct continuation of the previous 2D example on non-linear heat transfer. The present computations will use the same behaviour StationaryHeatTransfer.mfront which will be loaded
with a "3d" hypothesis (default case).
Source files:
FEniCS implementation
We now consider a portion of nuclear fuel rod (Uranium Dioxide \(\text{UO}_2\)) subject to an external imposed temperature \(T_{ext}=1000\text{ K}\) and uniform volumetric heat source \(r=300 \text{
MW/m}^3\). From the steady state heat balance equation \(\operatorname{div}\mathbf{j} = r\), the variational formulation is now:
\[$$F(\widehat{T}) = \int_\Omega \mathbf{j}(T,abla T)\cdotabla \widehat{T}\,\text{dx} + \int_\Omega r \widehat{T} \,\text{dx}=0 \quad \forall \widehat{T}$$\]
which fits the general default format of a MFrontNonlinearProblem: \[$$F(\widehat{u}) = \sum_i \int_\Omega \boldsymbol{\sigma}_i(u)\cdot \mathbf{g}_i(\widehat{u})\,\text{dx} -L(\widehat{u}) =0 \quad
\forall \widehat{u}$$\]
where \((\boldsymbol{\sigma}_i,\mathbf{g}_i)\) are pairs of dual flux/gradient and here the external loading form \(L\) is given by \(-\int_\Omega r \widehat{T} \,\text{dx}\). Compared to the
previous example, we just add this source term using the set_loading method. Here we use a quadratic interpolation for the temperature field and external temperature is imposed on the surface
numbered 12. Finally, we also rely on automatic registration of the gradient and external state variables as explained in the previous demo.
from dolfin import *
import mgis.fenics as mf
from time import time
mesh = Mesh()
with XDMFFile("meshes/fuel_rod_mesh.xdmf") as infile:
mvc = MeshValueCollection("size_t", mesh, 2)
with XDMFFile("meshes/fuel_rod_mf.xdmf") as infile:
infile.read(mvc, "facets")
facets = cpp.mesh.MeshFunctionSizet(mesh, mvc)
V = FunctionSpace(mesh, "CG", 2)
T = Function(V, name="Temperature")
T_ = TestFunction(V)
dT = TrialFunction(V)
T0 = Constant(300.)
Text = Constant(1e3)
bc = DirichletBC(V, Text, facets, 12)
r = Constant(3e8)
quad_deg = 2
material = mf.MFrontNonlinearMaterial("./src/libBehaviour.so",
problem = mf.MFrontNonlinearProblem(T, material, quadrature_degree=quad_deg, bcs=bc)
The solve method computing time is monitored:
Automatic registration of 'TemperatureGradient' as grad(Temperature).
Automatic registration of 'Temperature' as an external state variable.
MFront/FEniCS solve time: 53.746278047561646
The temperature field along a radial direction along the top surface has been compared with computations using Cast3M finite-element solver. Both solutions agree perfectly, as shown on Figure 1.
Figure 1: Temperature profile in the mid-pellet plane. Comparison of the results obtained with FEniCS
Performance comparison
For the purpose of performance comparison, we also implement a direct non-linear variational problem with pure UFL expressions. This is possible in the present case since the non-linear heat
constitutive law is very simple. Note that we enforce the use of the same quadrature rule degree. The temperature field is also reinterpolated to its previous initial value for a fair comparison
between both solution strategies.
A = Constant(material.get_parameter("A"))
B = Constant(material.get_parameter("B"))
j = -1/(A + B*T)*grad(T)
F = (dot(grad(T_), j) + r*T_)*dx(metadata={'quadrature_degree': quad_deg})
J = derivative(F, T, dT)
tic = time()
solve(F == 0, T, bc, J=J)
print("Pure FEniCS solve time:", time()-tic)
Pure FEniCS solve time: 49.15058135986328
Table 1: Performance
coarse 2 1.2 s 0.8 s
coarse 5 2.2 s 1.0 s
fine 2 62.8 s 58.4 s
fine 5 77.0 s 66.3 s
We can observe that both methods, relying on the same default Newton solver, yield the same total iteration counts and residual values. As regards computing time, the pure FEniCS implementation is
slightly faster as expected. In Table 1, comparison has been made for a coarse (approx 4 200 cells) and a refined (approx 34 000 cells) mesh with quadrature degrees equal either to 2 or 5.
The difference is slightly larger for large quadrature degrees, however, the difference is moderate when compared to the total computing time for large scale problems.
On the use of the correct tangent operator
Most FE software do not take into account the contribution of \(\dfrac{\partial \mathbf{j}}{\partial T}\) to the tangent operator. One can easily test this variant by assigning dj_ddT in the MFront
behaviour or change the expression of the jacobian in the pure FEniCS implementation by:
In the present case, using this partial tangent operator yields a convergence in 4 iterations instead of 3, giving a computational cost increase by roughly 25%. | {"url":"https://thelfer.github.io/mgis/web/mgis_fenics_nonlinear_heat_transfer_3D.html","timestamp":"2024-11-02T10:38:20Z","content_type":"text/html","content_length":"23345","record_id":"<urn:uuid:f3d7ed61-dd57-4549-aa91-0d6e10e7b7f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00602.warc.gz"} |
Re: Binary Relations, draft 1
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: Binary Relations, draft 1
• To: Hans Aberg <haberg@matematik.su.se>, math-font-discuss@cogs.susx.ac.uk
• Subject: Re: Binary Relations, draft 1
• From: "Y&Y, Inc." <support@yandy.com>
• Date: Tue, 17 Nov 1998 17:28:45 -0500
• Content-Length: 580
At 23:13 1998-11-17 +0100, Hans Aberg wrote:
>The traditional typographical explanation, or rule, that names such as
>"sin", "cos" should be typeset upright is that these are functions. But
>this does not explain why the "f" in f(x) should be typeset as a variable,
>when it clearly is a function.
I think the reason (if any) to make the distinction of setting these upright
is that they are mulitletter combinations rather than that they are functions.
This helps distinquish `sin' from the product of s, i, and n.
Y&Y, Inc. mailto:support@YandY.com http://www.YandY.com | {"url":"http://tug.tug.org/twg/mfg/mail-html/1998-01/msg00302.html","timestamp":"2024-11-05T19:50:53Z","content_type":"text/html","content_length":"2345","record_id":"<urn:uuid:1976555d-430a-4cba-a005-7881446a241f>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00135.warc.gz"} |
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Last LOW-BIAS
requests non-analogue absorption and/or an energy cutoff during low-energy neutron transport on a region by region basis
News: See also PART-THR, LOW-NEUT
WHAT(1) > 0.0 : group cutoff (neutrons in energy groups with number
>= WHAT(1) are not transported).
This value can be overridden in the user routine UBSSET
(argument IGCUTO in the calling list, see (13))
Default = 0.0 (no cutoff)
WHAT(2) > 0.0 : group limit for non-analogue absorption (neutrons in
energy groups >= WHAT(2) undergo non-analogue absorption)
This value can be overridden in the user routine UBSSET
(argument IGNONA in the calling list, see (13))
Non-analogue absorption is applied to the NMGP-WHAT(2)+1
groups with energies equal or lower than those of
group WHAT(2) if WHAT(2) is not > NMGP, otherwise it
isn't applied to any group (NMGP is the number of
neutron groups in the cross section library used:
it is = 260 in the standard FLUKA neutron library)
Default: if option DEFAULTS is used with SDUM = CALORIMEtry,
ICARUS, NEUTRONS or PRECISIOn, the default is = NMGP+1
(usually 261), meaning that non-analogue absorption is
not applied at all.
If DEFAULTS is missing, or is present with any other
SDUM value, the default is the number of the first thermal
group (usually 230).
WHAT(3) > 0.0 : non-analogue SURVIVAL probability. Must be =< 1.
This value can be overridden in the user routine UBSSET
(argument PNONAN in the calling list, see (13))
Default: if option DEFAULTS is used with SDUM = EET/TRANsmut,
HADROTHErapy, NEW-DEFAults or SHIELDINg, the default
is = 0.95.
If DEFAULTS is missing, or is present with any other
SDUM value, the default is 0.85.
WHAT(4) = lower bound of the region indices (or corresponding name) in
which the indicated neutron cutoff and/or survival parameters
("From region WHAT(4)...")
Default = 2.0.
WHAT(5) = upper bound of the region indices (or corresponding name) in
which the indicated neutron cutoff and/or survival parameters
("...to region WHAT(5)...")
Default = WHAT(4)
WHAT(6) = step length in assigning indices. ("...in steps of
Default = 1.
SDUM : not used
Default (option LOW-BIAS not given): the physical survival probability
is used for all groups excepting thermal ones, which are assigned
a probability of 0.85. However, if option DEFAULTS has been
issued with SDUM = EET/TRANsmut, HADROTHErapy, NEW-DEFAults or
SHIELDINg, this default value is changed to 0.95.
If SDUM = CALORIMEtry, ICARUS, NEUTRONS or PRECISIOn, the default
is physical survival probability for all groups, including thermal.
• 1) The groups are numbered in DECREASING energy order (see (10) for a detailed description). Setting a group cutoff larger than the last group number (e.g. 261 when using a 260-group
cross section set) results in all neutrons been transported, i.e. no cutoff is applied.
• 2) Similarly, if WHAT(2) is set larger than the last group number, non-analogue neutron absorption isn't applied to any group (this is recommended for calorimetry studies and all cases
where fluctuations and correlations are important).
• 3) The survival probability is defined as 1 - (Sigma_abs/Sigma_T) where Sigma_abs is the inverse of the absorption mean free path and Sigma_T the inverse of the mean free path for
absorption plus scattering (total macroscopic cross section). The LOW-BIAS option allows the user to control neutron transport by imposing an artificial survival probability and corrects
the particle weight taking into account the ratio between physical and biased survival probability.
• 4) In some programs (e.g., MORSE) the survival probability is always forced to be = 1. In FLUKA, if the LOW-BIAS option is not chosen, the physical survival probability is used for all
non-thermal groups, and the default 0.85 is used for the thermal groups. (This exception is to avoid endless thermal neutron scattering in materials with low thermal neutron absorption
cross section). To get the physical survival probability applied to ALL groups, as needed for fully analogue calculations, the user must use LOW-BIAS with WHAT(2) larger than the last
group number.
• 5) In selecting a forced survival probability for the thermal neutron groups, the user should have an idea of the order of magnitude of the actual physical probability. The latter can
take very different values: for instance it can range between a few per cent for thermal neutrons in Boron-10 to about 80-90% in Lead and 99% in Carbon. The choice will be often for
small values of survival probability in the thermal groups in order to limit the length of histories, but not if thermal neutron effects are of particular interest.
• 6) Concerning the other energy groups, if there is interest in low-energy neutron effects, the survival probability for energy groups above thermals in non-hydrogenated materials should
be set at least = 0.9, otherwise practically no neutron would survive enough collisions to be slowed down. In hydrogenated materials, a slightly lower value could be acceptable. Setting
less than 80% is likely to lead to erroneous results in most cases.
• 7) Use of a survival probability equal or smaller than the physical one is likely to introduce important weight fluctuations among different individual particles depending on the number
of collisions undergone. To limit the size of such fluctuations, which could slow down statistical convergence, it is recommended to define a weight window by means of options WW-THRESh,
WW-FACTOr and WW-PROFIle.
Example (number based):
LOW-BIAS 60.0 47.0 0.95 5.0 19.0 0.0
LOW-BIAS 261.0 230.0 0.82 7.0 15.0 4.0
* Note that the second LOW-BIAS card overrides the settings of the first one
* concerning regions 7, 11 and 15. Therefore, we will have an energy cutoff
* equal to the upper edge of the 60th group (4.493290 MeV in the standard
* FLUKA neutron library) in regions 5,6,8,9,10,12,13,14,16,17,18 and 19. In
* these same regions, analogue neutron absorption is requested down to an
* energy equal to the upper edge of group 47 (6.592384 MeV in the standard
* library), and biased absorption, with a fixed probability of 95%, at lower
* energies.
* In regions 7, 11 and 15, no cutoff is applied (supposing we are using the
* standard 260-group library), and non-analogue absorption is requested for
* groups 230 to 260 (the thermal groups in our case), with a probability of
* 82%.
The same example, name based:
LOW-BIAS 60.0 47.0 0.95 FifthReg Nineteen 0.0
LOW-BIAS 261.0 230.0 0.82 RegSeven Fifteen 4.0 | {"url":"http://www.fluka.org/fluka.php?id=man_onl&sub=46&font_size=80%25","timestamp":"2024-11-03T19:19:17Z","content_type":"application/xhtml+xml","content_length":"36029","record_id":"<urn:uuid:11be8d03-0ee6-4b37-a0fe-bd57ac979c4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00821.warc.gz"} |
Division algebras and supersymmetry II
The nCafé is currently haunted by a bug that prevents any comments from being posted. This should eventually go away, hopefully. For the time being I post my comment in reply to the entry Division
Algebras and Supersymmetry II here:
Thanks, John and John for these results. This is very pleasing.
The 3-$\psi$s rule implies that the Poincaré superalgebra has a nontrivial 3-cocycle when spacetime has dimension 3, 4, 6, or 10.
Similarly, the 4-$\psi$s rule implies that the Poincaré superalgebra has a nontrivial 4-cocycle when spacetime has dimension 4, 5, 7, or 11.
Very nice! That's what one would have hoped for.
Can you maybe see aspects of what makes these cocycles special compared to other cocycles that the Poincaré super Lie algebra has? What other cocycles that involve the spinors are there? Maybe there
are a bunch of generic cocycles and then some special ones that depend on the dimension?
Is there any indication from the math to which extent $(3,4,6,10)$ and $(4,5,7,11)$ are the first two steps in a longer sequence of sequences? I might expect another sequence $(7,8,10,14)$ and $(11,
12, 14, 18)$ corresponding to the fivebrane and the ninebrane. In other words, what happens when you look at $n \times n$-matrices with values in a division algebra for values of $n$ larger than 2
and 4?
Here a general comment related to the short exact sequences of higher Lie algebras that you mention:
properly speaking what matters is that these sequences are $(\infty,1)$-categorical exact, namely are fibration sequences/fiber sequences in an $(\infty,1)$-category of $L_\infty$-algebras.
The cocycle itself is a morphism of $L_\infty$-algebras
$\mu : \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R}$
and the extension it classifies is the homotopy fiber of this
$\mathfrak{superstring}(n+1,1) \to \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R} \,.$
Forming in turn the homotopy fiber of that extension yields the loop space object of $b^2 \mathbb{R}$ and thereby the fibration sequence
$b \mathbb{R} \to \mathfrak{superstring}(n+1,1) \to \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R} \,.$
The fact that using the evident representatives of the equivalence classes of these objects the first three terms here also form an exact sequence of chain complexes is conceptually really a
coicidence of little intrinsic meaning.
One way to demonstrate that we really have an $\infty$-exact sequence here is to declare that the $(\infty,1)$-category of $L_\infty$-algebras is that presented by the standard modelstructure on
dg-algebras on $dgAlg^{op}$. In there we can show that $b \mathbb{R} \to \mathfrak{superstring} \to \mathfrak{siso}$ is homotopy exact by observing that this is almost a fibrant diagram, in that the
second morphism is a fibration, the first object is fibrant and the other two objects are almost fibrant: their Chevalley-Eilenberg algebras are almost Sullivan algebras in that they are quasi-free.
The only failure of fibrancy is that they don't obey the filtration property. But one can pass to a weakly equivalent fibrant replacement for $\mathfrak{siso}$ and do the analog for $\mathfrak
{superstring}$ without really changing the nature of the problem, given how simple $b \mathbb{R}$ is. Then we see that the sequence is indeed also homotopy-exact.
This kind of discussion may not be relevant for the purposes of your article, but it does become relevant when one starts doing for instance higher gauge theory with these objects.
Here some further trivial comments on the article:
• Might it be a good idea to mention the name "Fierz" somewhere?
• page 3, below the first displayed math: The superstring Lie 2-superalgebra is [an] extension of
• p. 4: the bracket of spinors defines [a] Lie superalgebra structure
• p. 6, almost last line: this [is] equivalent to the fact
• p. 13 this spinor identity also play[s] an important role in
• p. 14: recall this [is] the component of the vector
Is there any indication from the math to which extent (3,4,6,10) and (4,5,7,11) are the first two steps in a longer sequence of sequences? I might expect another sequence (7,8,10,14) and (11, 12,
14, 18) corresponding to the fivebrane and the ninebrane.
Aren't the first two sequences supposed to be from the columns in Duff's chart in the post? Is the specialness of 5 and 9 connected to their being (1+4) and (1+8)?
Is the specialness of 5 and 9 connected to their being (1+4) and (1+8)?
Well, at least one has to be careful with the numerology here, as the string/=1-brane and membrane=2-brane would not fit that pattern that you suggest.
But I think there is yet another pattern running here, where "fundamental n-branes" exists for $(4 k +1)$ (string, fivebrane, ninebrane) whose worldvolume theory is conformal, and then one dimension
higher runs the sequence of $(4k + 2 )$-branes whose worldvolume theory is the corresponding Chern-Simons theory (membranes, etc.).
But I have only a vague understanding of the general pattern here.
I was just wondering about Urs’ fivebrane and ninebrane as in the quoted portion. He then replied about the sequence (string, fivebrane, ninebrane) of dimension $4 k + 1$.
The wikipedia composition algebra page says there are 1-d composition algebras when char(K) != 2. Has anyone considered your construction for char(K) != 0 or 2?
David: Okay, I get what you mean now. I think the cocycles governing Urs' fivebrane and ninebrane are "purely bosonic", unlike the ones John Huerta and I are considering. That's why I'm having
trouble making any connection between what Urs is talking about now and what we did. But they should be part of the same story.
In other words: we're all interested in cocycles on the Poincaré Lie superalgebra. This superalgebra is a Z/2-graded vector space with a bracket. The even part, or "bosonic part" is an ordinary Lie
algebra, namely the Lie algebra of the Poincaré group. The odd part, or "fermionic part", is the space of spinors. I think Urs is implicitly getting cocycles on the Poincaré Lie superalgebra from
cocycles on its bosonic part. (Checking that this is possible requires a tiny calculation, which I am alas too busy to do right now).
What are cocycles on the Poincaré Lie algebra like? Well, it should include the cohomology of the rotation Lie algebra, and in fact that could even be all there is.
The Lie algebra of the rotation group has a bunch of interesting cocycles, related to Pontryagin classes.
If I'm not getting mixed up the Lie algebra of the rotation group has a nontrivial 3-cocycle, an nontrivial 7-cocycle, an nontrivial 11-cocycle and so on up to a certain cutoff - and if you work with
rotations in high enough dimensions, this cutoff is quite high.
So, we get cocycles of degree 4k+3 for k below a certain cutoff. These give Lie (4k+2)-algebras, which in turn can be used to describe the parallel transport of (4k+1)-branes.
Oh, good - the calculation is working - it matches Urs' claim! I was worried until the end there.
But anyway, all this stuff is "purely bosonic". John Huerta and I were focusing on cocycles that only exist on the Poincaré Lie superalgebra.
I will need to think about this more someday.
Does anyone else here agree that commenting on the nForum is substantially easier (and less aggravating) than commenting on the nCafé?
me :-) (I love the cafe' for blog postings, but prefer the forum for commenting)
there are these bosonic cocycles, but I was indeed wondering about the fermionic ones.
Take the case of the string: it is governed (in the sense we are discussing here) at least by one bosonic cocycle -- the canonical 3-cocycle on $\mathfrak{so}(n)$ -- and one fermionic cocycle -- the
one you are discussing with John Huerta.
The super-fivebrane we know is similarly controlled by the 7-cocycle on $\mathfrak{so}(n)$. But shouldn't there also be a fermionic cocycle to go with this, as with the string?
I created composition algebra just to have links (somebody will I hope fill some definitions and teorems). | {"url":"https://nforum.ncatlab.org/discussion/914/division-algebras-and-supersymmetry-ii/?Focus=6413","timestamp":"2024-11-13T16:00:32Z","content_type":"application/xhtml+xml","content_length":"76090","record_id":"<urn:uuid:997cc2cb-8b30-4316-9ec1-0f681a8028ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00874.warc.gz"} |
Evaluation of Forward Models for GNSS Radio Occultation Data Processing and Assimilation
National Space Science Center, Chinese Academy of Sciences (NSSC/CAS), Beijing 100190, China
Beijing Key Laboratory of Space Environment Exploration, Beijing 100190, China
Key Laboratory of Science and Technology on Space Environment Situational Awareness, CAS, Beijing 100190, China
Joint Laboratory on Occultations for Atmosphere and Climate (JLOAC), NSSC/CAS and University of Graz, Beijing 100190, China
School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
Author to whom correspondence should be addressed.
Submission received: 10 January 2022 / Revised: 10 February 2022 / Accepted: 15 February 2022 / Published: 23 February 2022
In radio occultation (RO) data processing and data assimilation, the forward model (FM) is used to calculate bending angle (BA) from refractivity (N). The accuracy and precision of forward modeled BA
are affected by refractivity profiles and FM methods, including Abel integral algorithms (direct, exp, exp_T, linear) and methods of interpolating refractivity during integral (log-cubic spline and
log-linear). Experiment 1 compares these forward model methods by comparing the difference and relative difference (RD) of the experimental value (forward modeled ECMWF analysis) and the true value
(BA of FY3D RO data). Results suggested that the exp with log-cubic spline (log-cubic) interpolation is the most accurate FM because it has better integral accuracy (less than 2%) to inputs,
especially when the input is lower than an order of magnitude of 1 × 10^−2 (that is, above 60 km). By contrast, the direct induced a 10% error, and the improvement of exp T to exp is limited.
Experiment 2 simulated the exact errors of an FM (exp) based on inputs on different vertical resolutions. The inputs are refractivity profiles on model levels of three widely used analyses, including
ECMWF 4Dvar analysis, final operational global analysis data (FNL), and ERA5. Results demonstrated that based on exp and log-cubic interpolation, BA on model level of ECMWF 4Dvar has the highest
accuracy, whose RD is 0.5% between 0–35 km, 4% between 35–58 km, and 1.8% between 58–80 km. By contrast, the other two analyses have low accuracy. This paper paves the way to better understanding the
FM, and simulation errors on model levels of three analyses can be a helpful FM error reference.
1. Introduction
The global navigation satellite system (GNSS) radio occultation (RO) technique has many advantages: high vertical resolution, global coverage, and long-term steady data. It has been proven that RO
can significantly improve the accuracy of numerical weather prediction (NWP) after being assimilated into the NWP assimilation system. According to [
], RO observation ranked fourth among data that affected NWP. With the rapid development of the RO technique, numerous missions have been planned, launched, and run steadily to serve the operational
NWP, such as Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC/COSMIC-2), the Meteorological Operational satellite Programme-A/B (MetOpA/B), FengYun-3C/D (FY-3C/D) [
], GRACE/GRACE-FO, Scientific Application Satellite-C/D (SAC-C/D), Korea Multi-Purpose Satellite-5 (KOMPSAT-5), and Spanish PAZ (‘peace’ in Spanish) [
]. By 2027, these missions will provide approximately 18,400 observations per day which theoretically is the number threshold of the positive effect that RO could bring to NWP [
]. It will reduce error in prediction by 25% if all these observations are assimilated into the NWP assimilation system [
]. However, the actual situation does not coincide with theoretical expectations because almost half of the RO data either cannot be received or cannot pass quality control before assimilation [
]. There are two strategies to increase the RO observation available. The first one focus on increasing the soundings received per day via more missions, more receivers on the cube satellite, e.g.,
the cube satellite constellation of Spire company [
], and more multi-system-compatible RO receiver such as GNSS Radio Occultation Sonder (GNOS) onboard FY3 [
]. The second strategy is to improve the RO data processing technique. The current RO data processing center includes the Danish Meteorological Institute (DMI), German Research Centre for Geosciences
(GFZ), Jet Propulsion Laboratory (JPL), University Corporation for Atmospheric Research (UCAR), the National Oceanic and Atmospheric Administration (NOAA) ’s National Environmental Satellite, Data,
and Information Service (NESDIS), and Wegener Center/the University of Graz (WEGC). Each center developed different schemes to process RO data. The quality of their RO data is accessed by the
international RO working group (IROWG) every other year [
]. The consistency and structural uncertainty of their RO data records are promising to step up to another level. Thus, improvements are expected in the RO data processing and RO data assimilation
techniques. The forward model (FM) plays a critical role during these two processes. Firstly, the more accurate the FM, the more accurate these two processes. Secondly, the error of FM should be
quantified, so that the other errors during these two processings can be accurately evaluated. The FM mainly consists of Abel integral algorithm and interpolation methods before Abel integral.
1.1. FM Algorithm in Data Assimilation and RO Data Processing
(1) FM in RO data assimilation. In terms of the RO data assimilation, two options, the bending angle (BA) or refractivity, are used for variational assimilation [
]. The option of forward modeled BA is convenient and concise. During assimilation, atmospheric parameters are firstly transformed to refractivity (
$N = 77.6 p T + 3.73 × 10 5 e T 2$
is temperature,
is water vapor partial pressure, and
is pressure. Then the refractivity is inverted to BA (
) by the Abel integral, a core part of FM
$α ( a ) = − 2 a ∫ a ∞ d ln ( n ) d x d x x 2 − a 2$
is impact parameter, refractive index
= 10
+ 1. Hence, the accuracy of BA is strongly affected by n and the FM. This BA is then assimilated into the NWP data assimilation system. Therefore, the FM is critical for RO data assimilation.
(2) FM in RO data processing. The FM is employed to correct the ionospheric error to BA during RO data processing by the method of statistical optimization. The ionosphere frequently interferes with
GPS signals [
]. There are several approaches for ionospheric calibration: the linear combination of two frequencies corrects the first order; the K-correction corrects the higher-order [
]; the statistical optimization corrects all the ionospheric error [
]. The last one is based on background climatological models such as MSIS-90 (mass spectrometer and incoherent scatter radar in 1990) [
], BAROCLIM (the bending angle radio occultation climatology) based on Formosat-3/COSMIC data [
], and CIRA (the 1986 COSPAR International Reference Atmosphere) [
]. Most data in these background climatological models are stored as refractivity. Therefore, we need to invert them to BA via FM. Therefore, the accuracy of the FM is important.
In another aspect, the RO data are also used in commercial companies or civilian industries, wherein they might consider using some analysis data sets to assimilate RO data in their assimilation
systems or develop their own RO data processing program. Generally, widely used and free analysis data sets are considered. However, some of those data sets have limited vertical and horizontal
resolution. Therefore, how many biases the forward modeling results contain is open to debate. The biases are generated by not only forward models but also the level resolution of the analysis data.
Therefore, the level resolution of the original data should also be under consideration for evaluating the error of FMs.
1.2. FM Algorithm
The core of an FM is the Abel integral, whose original form is given by Equation (2), in which two singularities exist:
$x = a$
$x = ∞$
. To solve such problem, [
] used hyperbolic transformation and an algorithm that closely resembles the Abel integral was obtained, which we call the direct algorithm in this study.
To further improve the approximation accuracy, the interpolation and numerical approximation methods were employed. For example, based on the physical assumption of isothermal atmosphere and
temperature exponential approximation with height, refractivity was assumed to decrease exponentially with impact parameters. Methods of exponential extrapolation and interpolation of refractivity
were summarized by the European Center for Medium-Range Weather Forecasts (ECMWF) [
]. In such a method, numerical approximation, such as the error function (erf), was also employed. This method is called the exp algorithm in this paper. Subsequently, the linear assumption of the BA
was considered in the Abel integral inversion, called the linear algorithm in this paper. Both exp and linear algorithms are incorporated in the radio occultation processing package (ROPP) software
developed by the Radio Occultation Meteorology Satellite Application Facility [
]. Later, it was found that oscillations occur between levels (vertical grids) of the forward modeled BA using the exp algorithm. In order to correct this oscillation, [
] revised the exp algorithm by adopting more temperature operators in the refractivity operator; the modified algorithm is called the exp_T algorithm in this paper. For more details on these four
algorithms, refer to
Appendix A
: Abel Integral Algorithms. In addition to the four methods, the NOAA National Centers for Environmental Prediction (NCEP) used quadratic Lagrangian polynomials [
], which avoided the exponential decay of refractivity with height.
The FM is the integral algorithm as shown in Equation (2), in which the refractivity was firstly interpolated along a denser grid of impact parameter to improve integration accuracy. Therefore, the
accuracy of an FM is closely associated with the interpolation method and numerical integral method. In terms of interpolation methods, authors of [
] found that the log-cubic spline (hereafter, log-cubic) is the most accurate one. Log-cubic interpolation projects refractivity into the log space and then interpolates it by the log-cubic method.
In addition to this interpolation method, log-linear interpolation is tested in this paper. To the best of our limited knowledge, most studies either focus on the numerical integration method or only
on the interpolation methods.
Thus far, a systematic discussion on the combination of interpolation and integration methods has not been reported. In addition, to the best of our knowledge, only a few results have been reported
on the estimation of the error of the FM occurring on a fixed model level, for example 137 EC. The present study aims to discuss both of these aspects.
In this study, first, we compare three Abel integral algorithms, namely direct, exp, and exp_T algorithms, and two interpolation methods, namely log-cubic and log-linear. Second, we evaluate errors
of FMs on the model levels of three analyses. To achieve these two aims, we set two experiments. In experiment 1, we compare FMs and interpolation methods. In experiment 2, the most accurate FM is
employed to calculate the BA on model levels of three widely used analysis datasets, namely ECMWF 4Dvar analysis with 137 levels, ECMWF ERA5 analysis with 37 levels, and final operational global
analysis data (FNL) with 31 levels. Furthermore, the errors of FMs are simulated, providing an error reference for FMs under the analyzed model levels. The simulation also demonstrates to what extent
a low-resolution model level can be used for conducting comparative research of RO products. Our research provides a better understanding of the error of FMs in RO data processing and assimilation.
In the experiments conducted in this study, a point-to-point comparison is made horizontally. It should be noted that we ignore the tangent point drift (that is the practical RO profiles are slanting
rather than being vertical to the ground, which may result in the refractivity gradient bias [
]), since the interval of each level is less than 10 km [
This paper is organized as follows. The principles of the four algorithms in the forward model are described in
Section 1
. In
Section 2
, the progress of experiments 1 and 2 are introduced, together with data selection and low-pass filter strategies. Results are presented in
Section 3
. Finally, we discuss and present the discussion and conclusion in
Section 4
2. Experiments
In this study, two experiments were conducted. The first experiment compared the algorithms together with interpolation methods. Subsequently, the most accurate integral and interpolation method was
used in the second experiment, in which the refractivity profiles on levels of widely used analyses—FNL, EC4Dvar, and ERA5—were calculated to the BA by FMs. Then, the bias of results was evaluated.
2.1. Data, Collocation, and Quality Control
2.1.1. Data
We used the FY3D observation in September 2019 as the ‘true value’ which is available for download at
(accessed on 8 June 2021). The daily number of occultation soundings is approximately 500, and the vertical resolution is approximately 150–300 m. The quality control of the FY3D data is assessed by
the China Meteorological Administration [
], and the result has been qualified by ECMWF [
]. The qualiy control result is displayed in near real-time (NRT) monitoring at the ROM SAF website (Available at:
(accessed on 8 June 2021)). As shown in
Figure A2
Appendix B
, the quality of FY3D (The mean relative difference (RD) between ECMWF forecast and FY3D is less than 2%) in September 2019 established a good condition for experiment 1.
We also used ECMWF operational analyses in experiment 1, which can be accessed at
(accessed on 8 June 2021). Such data are produced daily by ECMWF’s direct 4Dvar atmospheric model. Hereafter we call this data EC4Dvar. These data are in 128 × 64 grids; hence, the resolution is
approximately 2.8° × 2.8°; They have 137 levels and are output daily at four hours: 00, 06, 12, and 18 UTC. Since the resolution of EC4Dvar is very coarse, we used strict collocation and quality
In experiment 1, the refractivity profiles were the experimental values; Therefore, we first calculated refractivity from temperature, water vapor partial pressure, and pressure of EC4Dvar via
Equation (1) according to their own level grids.
In experiment 2, we used FY3D to simulate analyses such as ECMWF, ERA5, and FNL. The level of them are 137, 37, and 31 respectively. The height and levels interval can be found
Figure A3
Appendix B
2.1.2. Spatial and Temporal Collocation
Experiment 1 aims to compare the BA from FY3D retrievals and calculated using Equation (2) from EC4Dvar atmospheric profiles. To improve the accuracy of the comparison, we must ensure that the
refractivity profiles of EC4Dvar and FY3D are collocated.
In terms of temporal collocation, we only selected the occultation soundings whose time is less than one hour to EC4Dvar’s four main synoptic hours, namely 00, 06, 12, and 18 UTC; that is, $T F Y 3 D
− T E C 4 D var ≤ 1$ h. 5914 out of 17791 FY3D GNOS soundings in September 2019 satisfied the temporal collocation.
In terms of spatial collocation, we used the tangent point of occultation profiles as its location to search its nearest EC4Dvar profile. The profile with 137 levels on grids nearest to the tangent
point was selected. In such a way, we assume the EC4Dvar and the FY3D profiles are collocated. However, because a grid resolution of 2.8° × 2.8° is too large, the collocated profiles of EC4Dvar may
be far be far FY3D. We fixed it in quality control as explained in the next subsection.
Figure A7
Appendix C
shows that FY3D soundings in experiment 1 generally have an evenly spatial and temporal distribution, which makes the mean (statistical values) feasible in the subsequent analysis.
2.1.3. Quality Control
Since we used coarse resolution EC4Dvar to collocate with FY3D data, some FY3D’s collocated profile may be far from it. To ensure that the refractivity of FY3D is close to that of ECMWF and to make
sure the results have statistical representativeness, we set four steps of quality control (hereafter 4QC). These four steps were initially a scheme for quality control used by [
] to compare MetOp RO data from two different RO data processing centers, which is concise and effective [
]. The details of 4QC can be found in
Appendix C
. After 4QC, 368 out of 5914 FY3D soundings remained for experiment 1. The high rate of discard is attributed to the low resolution (2.8° × 2.8°) EC4Dvar.
In experiment 1, EC4Dvar’s refractivity was transformed to BA. The forward-modeled BA contained not only the bias introduced by FMs and interpolations but also the inherent bias of the inputs when
they were refractivity. Experiment 2 was a simulation experiment, in which only FY3D was used. Therefore, the data quality was not taken into consideration.
2.2. Experiment 1: Algorithm Comparison
Experiment 1 aims to determine the most accurate FM by comparing FM algorithms that considered interpolation methods.
There are three types of bias in RO data processing: observation errors, ionospheric errors, and errors incurred within the data processing. We focused on the third error in experiment 1, mainly the
bias from the Abel numerical integral and interpolation methods. The Abel integral methods considered in this experiment were the direct, exp, and exp_T algorithms. The interpolation methods were
log-cubic and log-linear, indicated in the subsequent figures by no label and “lin”, respectively. The log-cubic maps refractivity into the log space and then interpolates it by the cubic spline
method; the log-linear also projects refractivity into the log space, but interpolates it in a linear fashion.
In this experiment, 368 FY3D BA profiles in September 2019 were regarded as the true value (T). The collocated refractivity profiles of EC4Dvar were the experimental value (E), which were calculated
via Equation (1) on the data’s inherent 137 levels. The temporal and spatial collocation of both sets of data is described in
Section 2.1
., as well as the further 4QCs for collocation. From the statistical perspective, we used the mean to analyze the results, which would not be affected by
s. Two statistics were selected for the analysis: the difference of the BA, i.e., (E − T); and the relative difference (RD) of the BA, i.e., ((E − T)/T × 100%). Based on these statistics, we
determine how close an FM’s result is to the true value; Another form of RD, which is the difference between two experimental values, i.e., ((E
− E
× 100%), can avoid introducing the bias of true value and provide the difference between the two experimental values. Subsequently, the relatively better FM can be obtained.
Figure 1
shows the process of experiment 1. (1) FY3D profiles collocated with the EC4Dvar atmospheric profiles with respect to time, longitude, and latitude. Subsequently, the EC4Dvar profiles of atmospheric
parameters were transformed to refractivity profiles. Then, the 4QC was performed for FY3D and EC4Dvar. We regarded the 4QC-passed FY3D BA as the true value (T). (2) The EC4Dvar refractivity profiles
on 137 levels were interpolated into a resolution of 100 m by the log-cubic space and log-linear methods. (3) The interpolated refractivity was then transformed into the BA by three different
integral methods. (4) The forward-modeled EC4Dvar BA was regarded as the experimental value (E) and was compared with the FY3D BA in terms of the difference (E − T) and RD (E − T)/T
100%. Note that the log-cubic interpolation was applied to the impact parameter instead of the mean sea level (MSL) height, since impact parameter
equals refractivity(
) multiple position vector (
). Because dn/dr is extremely large near a high-water-vapor region, which cannot satisfy the condition of applying cubic spline interpolation: the value to be interpolated should be continuously
It is noteworthy that the true value and experimental value are only an estimated value to their actual values. Therefore, the difference between any FM and FY3D, (E[FMx] − T), is just an estimate
showing which algorithm is closer to the truth. By contrast, the difference between any two FMs (E[FM1] − E[FM2]) is the absolute difference value. Experiment one shows which FM is better, instead of
expressing how much error is incurred from the FM. Through experiment 1, although we can determine which FM is better, how much error exactly is incurred from the FM cannot be expressed. Such a
question can refer to experiment 2.
2.3. Experiment 2: Evaluation of Errors of FMs on the Fixed Model Level
Experiment 2 explores how much bias was introduced from transforming the analyses’ refractivity to the BA. As shown in
Figure 2
, FY3D refractivity profiles were first linearly interpolated into a resolution of 20 m and then transformed to the BA as the true value. These interpolated refractivity profiles were then filtered
to remove the small-scale chaos less than the model grid interval of analyses such as 137 levels of ECMWF 4dvar, 31 levels of FNL, and 37 levels of ERA5 via the low pass filter. Their intervals are
shown in
Figure A3
source not found. in
Appendix B
. Subsequently, the refractivity profiles were linearly interpolated onto model grids and then transformed to the BA as the experimental value. Finally, obeying the nearest principle, the nearest
true value level to the experimental value was selected to evaluate the experimental value. Note that the FM used in this experiment was the most accurate one with its experimental value being the
nearest to the true value. In addition, considering computation efficiency [
], we chose two interpolation methods, namely log-spine cubic and log-linear.
Finally, the low pass filter must be mentioned here. Interpolation from a high resolution (20 m) to a low resolution (model levels) requires a filter, or it will induce representativeness errors [
]. For this purpose, we applied the Savitzky-Golay low-pass filter [
]. In general, the data are filtered with a window of
$2 δ z$
$4 δ z$
$3 δ z$
typically results in instability [
]). In this study,
$2 δ z$
is set. The next problem we encountered was that intervals of model levels increasing with height. To fix it, on the basis of [
], we divided the intervals into three groups: below the height of 0–20 km, 20–40 km, and above the height of 40–60 km. A window of twice the average interval of each group, 2
$δ z ¯ i$
, was applied to filter the whole profile; thus, we had three filtered refractivity profiles,
2, and
3. To merge them smoothly at the transition point (20 km and 40 km), we used a transition function,
), which is changed linearly from 1 to 0, as shown in the left subfigure in
Figure 2
. The refractivity profiles at the transition point are expressed as
$N 10 t o 30 = 1 2 w 1 ( z ) × N 1 + 1 − w 2 ( z ) × N 2 N 30 t o 50 = 1 2 w 2 ( z ) × N 2 + 1 − w 3 ( z ) × N 3$
$w i ( z )$
is the transition function to
$N i$
= 1, 2, 3), and
is the height.
$N 0 t o 10$
$N 50 t o 60$
were filtered by the filter window themselves. The combination of those slices by height was the terminal profiles.
Furthermore, the following details were considered in this experiment. First, to improve simulation accuracy and reduce the interpolation error, we selected 20 m, instead of 100 m, as the resolution
for FY3D to simulate the analysis. Second, the analysis model level was simulated, and those levels were transformed into MSL height for convenience. Third, the true value was refractivity instead of
BA. The true value also passed the same Abel transform as the forward-modeled BA in the analysis.
3. Results
In this section, the results of experiment 1 are presented, which demonstrate the most accurate FM. Subsequently, two simulated errors of the most accurate FM are obtained on the basis of three
widely used analyses: EC4Dvar, FNL, and ERA5.
3.1. Experiment 1: Algorithm Comparison
In this experiment, we compared three Abel integral algorithms and two interpolation methods. The BA of FY3D was the true value, while the forward-modeled EC4Dvar BA was the experimental value. The
FY3D observation data were filtered by the strict collocation and 4QC strategies. Subsequently, 368 FY3D profiles in September 2019 remained, which were regarded as the true value (T). EC4Dvar
refractivity data collocated with the true values were the experimental values (E). We used two statistics in this experiment: the difference (A $−$ B) and relative difference (A $−$ B)/B × 100%,
where A and B can be either E and T or E and E, respectively. The obtained statistics were all mean values. Note that we compared the exp algorithm with the direct and exp_T algorithms independently
because the exp_T algorithm is too close to the exp algorithm. If the exp_T algorithm was drawn in the same figure with the direct algorithm, it would coincide with the graph of the exp algorithm.
3.1.1. Difference Analysis for Direct and Exp
First, we compared the difference (E
T), in which E is the BA integrated by the exp and direct Abel integral algorithms, together with two interpolation methods. The altitude starts from 5 km because the signal-to-noise ratio below it
is low, and some RO observations are unavailable. As shown in
Figure 3
a,b, differences of both algorithms are in the same order of magnitude; 1 × 10
is within 5–30 km, and 1 × 10
is within 30–60 km. However, the exp is more accurate than the direct because it is closer to the true value except at heights of 12.5–17.5 km, which reflects the errors passed by refractivity.
Figure 3
c,d shows the difference between the two algorithms. The result of the direct is larger than that of the exp until at a height of 50 km.
Figure 3
e,f shows the difference due to interpolation. In the height range of 5–30 km, the error induced by log-linear is less than ±1 × 10
, and above 30 km, it is less than ±0.5 × 10
. Interestingly, log-linear interpolation causes more undulation compared with log-cubic interpolation. The undulation due to interpolation is larger for the exp algorithm. To verify this result, we
repeated this experiment using MetOp as the true value, and the same conclusion was drawn (see
Figure A4
Appendix B
3.1.2. Relative Difference Analysis of Direct and Exp Algorithms
Next, we analyzed the RD, (E
T)/T × 100%, of the FY3D BA (T) and the forward modeled EC4Dvar BA (E) in different FMs and show the results in
Figure 4
a–c. The orders of magnitude of the differences in the ranges of 5–55 km, 55–70 km, and 70–80 km are within 1–1.5%, 4–−10%, and 10–80%, respectively. This result is consistent with those of the
previous studies [
]. The results shown in
Figure 4
a–c are divided into three aspects for clarity. (i) The RD of the exp is near to zero than that of the direct, indicating the BA of the exp algorithm is closer to the true value. In other words, the
exp algorithm is more accurate than the direct algorithm. The result of the direct algorithm is larger than the true value below 60 km (approximately 0.3% larger than exp) and considerably less than
the true value above 60 km. (ii) Since these two algorithms have the same input and same vertical-level setting, this gap is probably due to the integral accuracy of the algorithm. The integral
accuracy depends on the order of magnitude of refractivity (input), which is shown in
Figure A5
Appendix B
. Intriguingly, when the order of magnitude is 1 × 10
above (below) a height of 60 km, the direct algorithm’s result is less (larger) than the true value. (iii) Note that at heights of 18, 48, and 58 km, larger errors are incurred by refractivity rather
than the errors of the FM. Unexpectedly, the errors of the exp algorithm at these heights are larger than those of the direct algorithm. This difference can be attributed to the fact that
refractivity of the exp algorithm exponentially decays faster than that of the direct algorithm.
The green and blue lines in
Figure 4
d–f illustrate the RD between the direct and exp algorithms. Below 60 km, their difference is less than ±2%, whereas above 60 km, it is considerably large. The RDs between the two algorithms based on
the two interpolation methods are indicated by the orange and pink lines; the RD is even larger when the algorithms are based on log-cubic interpolation. In general, the errors due to log-linear
interpolation of the FM with the exp algorithm are larger than those with the direct algorithm. The RD of two interpolations with the exp algorithm is less than 0.2%, −2%, and −20% for heights of
5–50, 55–65, and 70–80 km, respectively. Specifically, because these errors are almost equal to the monthly errors of FY3D to ECMWF recorded in the ROM website (see
Figure A2
) the log-linear interpolation can be considered to be not sufficiently accurate. However, if such an error is acceptable, the exp algorithm with log-linear interpolation will be a timely and
effective FM, since the log_linear interpolation is more cost-efficient than the log-cubic interpolation.
3.1.3. Analysis to exp_T
Herein, we discuss the RD and difference between the exp and the revised exp algorithm (exp_T). The RD is two orders of magnitude lower than the difference between exp and direct. Hence, the RD and
difference are plotted in separate graphs as shown in
Figure 5
, the BA of the exp_T algorithm is slightly larger than that of the exp algorithm. According to [
], this difference is due to the temperature term in the new polynomial, which corrects the error of the exp algorithm. However, the corrected amount is limited. Since the exp_T algorithm introduced
temperature operator, it requires considerably more computation and time resources. Thus, the exp algorithm is superior operationally.
Finally, we list the RD between the true value and all experimental values in
Table 1
. The above analysis identified that the exp algorithm with log-cubic is the most accurate FM; therefore, the quantity of each FM’error is written as exp’s RD plus the left error of the FM for
clarity. First, the RD of the exp_lin algorithm is the same as that of the exp algorithm below a height of 55 km; however, it becomes 3% and 20% larger than that of the exp algorithm within 55–60 km
and 70–80 km, respectively. Second, the RD of the direct algorithm is approximately 0.3% larger than that of the exp algorithm within 8–45 km, which becomes −10% and −80% less than that of the exp
algorithm within 60–70 km and 70–80 km, respectively. Third, the improvement of the exp_T algorithm is limited (approximately 0.002%) compared to that of the exp algorithm. It should be noted within
heights of 55–60 km, the error of the exp_lin algorithm is less than that of the exp algorithm, which is due to errors contributed by refractivity.
The exp algorithm with log-cubic is the most accurate FM; therefore, the quantity of each FM’error is written as exp’s RD plus the left error of the FM for clarity.
It is important to note that when E was refractivity, there are already been errors in E[ref] (ref is refractivity). This error was then passed to BA’s (E−T)/T. Therefore, what we got in
experimentone is the which FM is better, rather than how much error exactly one FM causes. This issue will be tested in experiment two. Since analysis above identified that the exp algorithm with
log-cubic interpolation is the most accurate FM, in next experiment we will figure out how much exactly the error of exp algorithm is on three fixed model levels.
It is important to note that when E is refractivity, errors exist in E[ref] (where ref denotes refractivity), which are then passed to the (E−T)/T value of the BA. Therefore, experiment 1
demonstrates which FM is better, rather than how much error exactly an FM causes, which will be tested in experiment 2. The above analysis indicated that the exp algorithm with log-cubic
interpolation is the most accurate FM; therefore, in experiment 2, we determined the extent of error of the exp algorithm on three fixed model levels.
3.2. Experiment 2: Evaluation of the Error of the FM on the Fixed Model Level
Experiment 1 identified the exp algorithm with the log-cubic interpolation as the ideal FM owing to its high accuracy. Therefore, when computation efficiency is the primary focus, the exp algorithm
with the log-linear interpolation is also considered. This experiment tested the extent of errors of the FMs. Another factor that introduced error to the FM is the model level (resolution of vertical
levels). Thus, in this experiment, we chose three widely used analyses with different levels: 137 levels EC4Dvar, 31 levels FNL, and 37 levels of ERA5. The refractivity profiles of these analyses
were all simulated by FY3D level 2 refractivity products. During this simulation, FY3D data were first interpolated into a resolution of 20 m and then extracted according to the model level.
Subsequently, each simulated analysis was transformed to the BA via the FMs. Next, the results were compared with the FM true value, providing FY3D level 2 product refractivity profiles. Their RDs
and differences are listed at the end, which can provide an error reference for the FM under the model levels considered here.
Figure 6
shows how the level interval, indicated by the rainbow colors, affects the RD ((E
− T
× 100%). In general, the less the interval, the less is the RD. EC4Dvar’s RD is less than the RDs for others because of the higher resolution of its levels. For example, EC4Dvar’s resolution is less
than 1 km below 35 km, and its RD is less than ±0.5%. By contrast, the errors for FNL and ERA5 are less than ±1% for heights of 0–30 km owing to their low resolution. Interestingly, ERA5 and FNL show
similar vertical resolution. However, the RD for ERA5 below 10 km is mostly less than that for FNL. This is because the Abel integral integrates values from lower levels to higher levels, and ERA5
owns the other six levels (i.e., 875, 825, 775, 225, 175, and 125 hPa), which ensures more accurate interpolation results for ERA5.
Figure 6
also shows the comparison of the log-cubic interpolation (a–c) with log-linear interpolation (d–f). The errors of the latter are 1–2% larger than those of the former owing to the lower interpolation
accuracy of log-linear.
Figure 7
compares the results of the RD (a–c) and difference (d–f) of the exp algorithm with log-cubic interpolation on the model levels of the three analyses.
Below 35 km, the RD and difference for EC4Dvar are less than ±0.5% and ±0.5 × 10
, respectively, while those for FNL and ERA5 frequently oscillate in the range of ±2% to ±4%. Above 35 km, the RD and difference for EC4Dvar are similar to those for FNL and ERA5. The large bias for
FNL and ERA5 resulted from two aspects: (i) the low resolution of the level interval; (ii) the rapid change in the atmospheric condition below 10 km. In such conditions, if the vertical grid density
of refractivity profiles is insufficient, a larger oscillation of BA profiles will occur. The same comparison as
Figure 7
but for log-linear interpolation is shown in
Figure A6
Appendix B
Figure 8
a–f concludes the detailed errors, the difference (grey), and RD (red) for forward-modeled three analyses, which are listed in
Table 2
Figure 8
g–i concludes the difference in the RD between two interpolation methods based on each analysis. Below 30 km, the RD of log-cubic interpolation is 0.5–1% lower than that of log-linear interpolation,
2% lower for 30–50 km, and back to 1% lower for a height above 50 km. For FNL, the log-cubic results are 2% lower than the results of log-linear. For ERA5, the log-cubic results are 1.8% lower than
the results of log-linear.
4. Discussion
An FM plays an important role during GNSS RO data processing and assimilation. The more accurate the FM, the more accurate are the two processes. Although the Abel algorithm and interpolation methods
separately are sufficiently discussed, to the best of our knowledge, there has been no systematic discussion on the combination of interpolation and integration methods. In addition, no results on
estimating the error of an FM on a fixed model level, for example 137 EC, have been reported.
In this study, we first reviewed the principles of four Abel integral methods. Subsequently, we conducted experiment 1 to compare three Abel integral algorithms (i.e., direct, exp, and exp_T) and two
interpolation methods (i.e., log-cubic and log-linear) via the mean value analysis of the difference and relative difference between the experimental value (forward-modeled EC4Dvar) and true value
(FY3D BA). The results suggested that (1) the exp algorithm with log-cubic interpolation is the most accurate one in every level because it has better integral accuracy (errors approximately 2%) to
inputs, especially when the input is lower than an order of magnitude of 1 × 10
(that is, above a height of 60 km). Above this height, the direct algorithm induced a 10% error. (2) When errors exist in inputs, the exp algorithm tends to amplify the error compared with the direct
algorithm. (3) The exp algorithm with log-linear interpolation is often feasible for its computation efficiency. Its errors were 0.2%, −2%, and −20% larger than those of log-cubic for the heights of
5–50, 55–65, and 70–80 km, respectively. (4) The improvement of exp_T algorithm to the exp algorithm is limited. Our results are consistent with results of [
Subsequently, we conducted experiment 2 to determine the exact errors caused by the exp algorithm, which would be affected by the level interval of inputs. Therefore, we selected refractivity
profiles on model levels of three widely used analyses (i.e., 137 levels of EC4Dvar, 31 FNL, and 37 ERA5) as the inputs. Results demonstrated that the forward-modeled BA of denser resolution analysis
EC4Dvar was closer to the true value FY3D. The errors of the exp algorithm with log-cubic and log-linear were less than 1% and 0.5% below 38 km, less than 4% within 38–50 km, and less than 1.8% and
2% above 58 km, respectively. By contrast, the other two analyses had low accuracy. Our results are in good agreement with the results of [
This study paves a way to better understand the errors of the FM in RO data assimilation and processing. Simulation errors on model levels of three analyses can be a helpful FM error reference. This
experiment only compared direct, exp, and exp_T algorithms, while excluding the fourth Abel integral: the linear algorithm. Such an algorithm is not feasible for data assimilation but can be used in
RO data processing. In the future, linear algorithm and other integral algorithms can be addressed to compare the FM. Regarding the interpolation method, we only considered log-linear and log-cubic
interpolation methods, which might have inferior representativeness above 60 km, where the interval is too large. Machine learning methods might be resorted to solve this problem.
Author Contributions
Conceptualization, N.D. and W.B.; Methodology, N.D.; Software, J.X., X.W., Y.C. and X.M.; Validation, N.D., W.B., C.L., C.Y., F.H., G.T. and P.H.; Formal analysis, N.D. and X.L.; Investigation, N.D.;
Resources, Y.S. and Q.D.; Data curation, N.D.; Writing—original draft preparation, N.D.; Writing—review and editing, W.B.; Visualization, N.D.; Supervision, W.B. and Y.S.; Project administration,
Y.S. and Q.D.; Funding acquisition, Q.D. All authors have read and agreed to the published version of the manuscript.
This work was supported jointly by the Youth Cross Team Scientific Research Project of the Chinese Academy of Sciences (JCTD-2021-10), the National Natural Science Foundation of China under Grant no.
42074042 and 41775034, the Feng Yun 3 (FY-3) Global Navigation Satellite System Occultation Sounder (GNOS and GNOS II) Development and Manufacture Project led by the National Space Science Center,
Chinese Academy of Sciences (NSSC/CAS), and Strategic Priority Research Program of Chinese Academy of Sciences, Grant no. XDA15007501.
Data Availability Statement
We acknowledge the support and funding from the National Space Science Center, Chinese Academy of Sciences (NSSC/CAS). We appreciate the valuable and constructive suggestions from reviewers.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. Abel Integral Algorithms
The forward model (FM) consists of the Abel integral methods and interpolation methods. Before we explain the principles of the four Abel integral algorithms, it is necessary to understand how the
Abel integral is formed and its important assumption of a spherically symmetric atmosphere.
Appendix A.1. Spherically Symmetric Assumption
The spherically symmetric assumption is the prerequisite of the Abel integral. The bending angle (BA)
is defined as the accumulated direction change of a signal from being transmitted by the GPS to being received by a low earth satellite (
Figure A1
). The BA
is twice the size of
based on the spherically symmetric atmosphere assumption. We assume the earth below the bent ray is symmetric about OC (
). If the right triangles AOC and BOC are the same, and the right triangles MOB and MAP are similar; then, ∠α is twice the size of ∠θ since they have the shared angle ∠AMP. According to Bouquer’s
law, the impact parameter
sin90° [
]. The direction vector
is the same as the direction of the refractive index
’s gradient. The BA is expressed as
$α ( a ) = 2 ∫ r t ∞ d θ = − 2 a ∫ r t ∞ d ln ( n ) d r$
is replaced with
, then we have Equation (2) [
Figure A1. Schematic of radio occultation geometry based on the spherically symmetric atmosphere assumption. That is, if triangles of ΔAOC and ΔBOC are symmetric (equal), then α = 2θ. OA(a), OB(a),
and OC(rt) on the ray path are the position vectors of a transmitter, receiver, and tangent point, respectively. The ray is over bent for the clarity of the geometry relationship.
There are two singularities in Equation (2): x = a and x→∞. The finite upper limit is set as 120 km or 150 km (for Beidou FY3C/D) in a real-world application. The singularity of x = a can be
eliminated by different analytical methods, called the Abel integral methods.
A.2. Abel Integral
Abel integral methods include the numerically mapped direct algorithm [
], exponentially interpolated refractivity in the exp algorithm [
], the linear assumption of bending angle in the lin-algorithm [
], the temperature-revised exp algorithm (hereafter exp_T) by authors of [
A.2.1. Direct Algorithm
To solve the singularity of
in Equation (2), [
] used hyperbolic transformation, and this formula is called the direct Abel integral method (hereafter direct). First,
) in Equation (2) is replaced with the Abel integral inversion (Details have been given in
Appendix B
in [
$n ( r ) = exp 1 π ∫ a ∞ α ( x ) d x x 2 − a 2$
Then the Equation (A1) becomes
$α a = 2 a ∫ x = a x = ∞ d 2 ln ( n ( a ) ) d x 2 ln x a + x a 2 − 1 d$
in which hyperbolic transformation can solve the singularity.
$z = a r cosh x a = ln x a + x a 2 − 1$
This method is the most accurate solution among all other pure mathematics to the Abel integral [
]. However, pure mathematics relies on high-quality input, whose order of magnitude should be larger, or it will become less representative. We test the exact number in experiment 1.
It is also the algorithm of the early version of the End-to-End GNSS Occultation Performance Simulation and Processing System (EGOPS) software developed by the Wegener Center/the University of Graz
(WEGC) [
]. This Abel integral method closely resembles the Abel integral and is widely used in all fields concerning the Abel integral. For example, it maps three dimensional images into its two dimensional
objection [
]. Hicksterin also developed an open-access project named Pyabel in GitHub (Available online at
, accessed on 8 June 2021), which was also used in our experiment.
A.2.2. Exp Algorithm
This method was proposed by the EUMETSAT GRAS-SAF and was summarized and applied in numerical assimilation by [
]. The refractivity is assumed to exponentially decrease with height since it has a relationship with temperature according to Equation (1). At a height of over 12 km, where water vapor is sparse, we
can assume
equals zero. Based on the isothermal atmosphere assumption,
$p = ρ R d T$
, and Equation (1), the refractivity is exponentially decreased with height [
]. Therefore, the refractivity
as the
th level can be expressed as [
$N = N j exp − k j x − x j d N d x = − k j N j exp − k j x − x j$
= ln(
), and the value is positive with a minimum of 10
. In addition, the term ln
= ln(10
+ 1) in Equation (2) is equivalent to 10
based on infinitesimal assumption. Moreover, at the tangent point, we have
$x 2 − a 2 ≈ 2 a x − a$
when we assume the impact parameter
is equivalent to the impact parameter
of air in the higher magnitude of order (6000 km) and their difference are less (<80 km).
Therefore, the differential at the
th level is expressed as [
$Δ α j = 10 − 6 k j N j exp k j x j − a 2 a ∫ x j x j + 1 exp − k j ( x − a ) ( x − a ) d x$
To improve the integral accuracy and compute it economically, [
] used the error function (erf) by writing
$t = k j ( x − a )$
Notably, the algorithm is not feasible under the condition of super refraction (often in the troposphere, below 12 km) because the gradient of N is large (resulting in a large
) such that the BA is overestimated [
]. In such circumstances, the second line of Equation (7) is written as:
$d N d x = N j + 1 − N j x j + 1 − x j$
This formula is still under the exponential decay assumption; However, it is acceptable because the interval is relatively small in the troposphere.
The error of the exp algorithm comes from its advantage. When the input refractivity is smaller in high levels, the exponential decay is faster than linear decay, making it closer to the actual
refractivity. However, when an abnormally larger (less) refractivity exist at some level, the results of the exp algorithm will be considerably larger (less) than the results of the direct algorithm,
as shown in Equation (A6).
The exp algorithm is widely used in the FM for NWP data assimilation since the interpolation of Equation (A5) is a reliable approximation between levels. However, this interpolation can yield
oscillations when the interval is too large, specifically between 25 km and 45 km [
]. Furthermore, the desirable interval for avoiding this phenomenon is less than 100 m. Focused on this problem, [
] improved this algorithm with a physical association between refractivity and temperature. Thus, it is called the temperature revised exp algorithm.
A.2.3. Exp_T Algorithm
Ref. [
] extended the Equation (1) physically by introducing three new parameters in the
th level, which are
$P 1 , P 2 , P 3$
$P 1 = k j + k j 2 β j 2 T m a − x m 2 − d − k j β j T m a − x m P 2 = k j 2 β j T m a − x m − k j β j T m P 3 = k j 2 β j 2 T m$
where the temperature gradient
$β j = T j + 1 − T j / x j + 1 − x j + 1$
, the subscript
is the middle point of two successive levels, i.e.,
$T m = ( T j + 1 + T j ) / 2$
$x m = ( x j + 1 + x j ) / 2$
. When
$β j = 0$
, we have
$P 1 = k j , P 2 = 0 , P 3 = 0$
, which is exactly the exp algorithm. The terminal differential at
th level is expressed as [
$Δ α = 10 − 6 2 a N j exp k j x j − a e r f k j ( x − a ) π P 1 k j 1 / 2 + P 2 2 k j 3 / 2 + 3 P 3 4 k j 5 / 2 − x − a exp − k j ( x − a ) P 2 k j + P 3 2 k j ( x − a ) + 3 2 k j 2 x j x j + 1$
This method alleviated the oscillation. The error of this algorithm is the same as that of the exp algorithm, that is, its exponential decay of refractivity with increasing height. A larger abnormal
refractivity has a more significant impact on its value than other algorithms.
This paper mainly compared two algorithms: direct and exp. We dismissed the lin algorithm because it required the interval to be less than 100 m. Most observations input to FM cannot satisfy this
condition. In addition, we only compared the exp_T algorithm with the exp algorithm because their difference is too small to be noticeable if the direct algorithm is combined. In another aspect, we
also considered the difference incurred from the interpolation method, including the log-spine cubic interpolation and the log-linear interpolation. Furthermore, the algorithms were compared in
experiment 1.
Appendix B. Figures
Figure A2
shows the monthly O-B/B statistics of FY3D used in experiment one, the relative difference between ECMWF forecast and FY3D is less than 2%. Since the RO sounding occurs randomly on the earth’s
surface, the standard deviation is relatively huge, less than 10%, which is an acceptable range. The FY3D in September with such quality established a good condition for experiment one.
Figure A2. Monthly O-B statistics of FY3D, September 2019. The mean relative difference (RD) between ECMWF forecast and FY3D is less than 2%. Copyright © 2022 EUMETSAT.
Figure A3
shows level intervals of 137 ECMWF 4dvar, 31 levels of FNL, and 37 levels of ERA5. Their interval generally increases as the altitude increases. EC4Dvar has the smallest intervals, which are less
than 500 m below the height of 25 km. The level intervals of FNL and ERA5 are similar, but between 2–4 km, 12–15 km, ERA5 has a smaller interval than FNL.
Figure A3. Level interval of FNL, ERA5, and EC4Dvar. Interval decreases with increasing height. (a) 0–12 km, (b) 12–40 km, (c) 40–80km.
To ensure the reliability of experiment one, we also did the comparison using 541 MetOp RO profiles on 1 March 2018. The results were shown in
Figure A4
, and FY3D’s results are in
Figure 3
. These two figure were set with the same coordinate, axes, and ticks. The results of MetOp are consistent with that of FY3D. Firstly, the difference (E
T) of both direct and exp algorithm are in the same order of magnitude, which is 1 × 10
in the range of 5–30 km and 1 × 10
in the range of 30–60 km, as shown in
Figure A4
a,b and
Figure 3
a,b. The results of FY3D have less error because it has passed 4QC. The experiment based on MetOp used the same collocation method as experiment 1. But we did not use 4QC. Secondly, they have the
same conclusion on which algorithm is more accurate, that if the exp. Thirdly, the log-linear interpolation also causes more undulation to (
$E dir − E exp$
) than which is caused by log-cubic interpolation. Moreover, the error induced by interpolation for exp is larger than that for direct. Therefore, the results of experiment one are reliable.
The integral accuracy is subject to the order of magnitude of refractivity (input), which is shown in
Figure A5
. Intriguingly, when the order of magnitude is 1 × 10
above (below) the height of 60 km, the direct algorithm’s result is less (larger) than the true value.
Figure A4. Mean difference from MetOp and experimental value calculated from direct and exp. (a,b) Difference (E − T) of BA based on the log-cubic interpolation; (c,d) difference of BA between exp
and direct based on the log-linear and log-cubic interpolation; (e,f) difference of BA between log-linear and log-cubic interpolation based on exp and direct. (a,c,e) 10–30 km; (b,d,f) 30–60 km;
Results from MetOp are the same as the results using FY3D.
Figure A5. Mean refractivity in 0–40 km, 40–60 km, and 60–80 km, and their order of magnitude are 1 × 10^−0, 1 × 10^−1, 1 × 10^−2, respectively. (a) 0–40 km, (b) 40–60 km, (c) 60–80km.
Figure A6
is the same as
Figure 7
but based on the log-linear interpolation. Below the height of 35 km, the RD and difference of EC4Dvar become
1 × 10
from 0.5% and
0.5× 10
Figure 7
. Between 35 to 50 km, those values are the same, less than 5%. In terms of FNL and ERA5, more RD values reach
4% below 35 km, and more of those are over
5% above 35 km. There is more undulation for results based on linear interpolation.
Figure A6.
The same as
Figure 7
but in log-linear interpolation, the difference (
) and RD (
) of FM based on the exp algorithm and log-linear interpolation. The BA RD of EC4Dvar below 35 km is less than ±1%, whereas it of FNL and ERA5 below 35 km is less than ±5%. Above 35 km, the RD
EC4Dvar is less than ±5%, while it of FNL and ERA5 is up to 10%.
Appendix C. 4QC
This four QCs processing is used in [
]. In this method, reliability was calculated using critical parameter Z. (usually about Z > 2–4, the smaller Z is stricter) excludes outliers [
]. In this study, mean value is used to discuss all the results. If the sample is obey Gaussian distribution, the mean value has representativeness. The 4QCs discard outliers, making the sample
similar to Gaussian distribution. Each level will be checked by QC. If a level cannot pass the QC, the whole profile will be discarded.
(1) Ensure soundings were between $−$60 °S and 60 °N horizontally, and between 5–80 km vertically; The RO data at high altitude may easily be affected by the ionosphere, and some sounding may lack
observation in the lower level below 5 km. After this QC1, 2824 (out of 5914) were left.
(2) Eliminate outliers horizontally on each level: use the bi-weight method proposed by [
] to FY3D’s refractivity on each level, and set the Z > 3, which is the crucial parameter for the bi-weight method. After this QC2, 1655 (out of 2824) were left.
(3) Ensure the FY3D’s BA is closer to EC4Dvar’s BA: use bi-weight method to the difference of the refractivity between FY3D and EC4Dvar with Z as 3. It is noticed that this QC solved the drift
tangent point phenomenon partly because we only made comparisons horizontally at every level, the slant profile will be discarded. Drift tangent point phenomenon is that the actual RO profile is
slanted, which results in a bias of the difference between the slanted FY3D and EC4Dvar. As long as FY3D and EC4Dvar are closer in each level horizontally, the drift tangent point phenomenon will be
ignored, and collocation spatially will prove to be reasonable. Furthermore, experiment one used statistics, the average bending angle, which can effectively eliminate the anomalies. Accordingly, the
bias question left after spatial collocation is fixed too. After this QC3, 1436 (out of 1665) were left.
(4) Eliminate the anomaly interval via the Bi-weight method. A similar interval among each profile effectively can reduce the bias introduced by interpolation. After QC4, 368 (out of 1436) were left.
The 4 QC help eliminate bias from the spatial collocation and drift tangent point phenomenon, ensuring the forward modelled bias mainly resulted from integral and interpolation methods. These
soundings are evenly distributed spatially and temporally (
Figure A7
), making the mean (statistical values) feasible in the following analysis.
Figure A7
shows FY3D soundings in experiment one are generally distributed evenly spatially and temporally, which makes the mean (statistical values) feasible in the following analysis.
To test the quality of refractivity profiles after 4QC, we used statistics, the mean relative difference of refractivity between FY3D profiles and EC4Dvar in each level. Their original profiles are
interpolated into fixed levels with an interval of 100 m. As shown in
Figure A8
, although the standard deviation of those profiles is large than 2%, the mean hovers around zero. Unfortunately, there is a large difference at the heights of 18 (errors about 0.3%), 48 (errors
about 1.3%), 58 (errors about 3.1%), 78 (errors about 10%) km, which passed our 4QC. This error will be propagated to the following experiments.”
Figure A7. Spatial (upper) and temporal (lower) distribution of 386 samples after collocation and 4QC. In general, these samples are evenly distributed in longitude and latitude, and are evenly
distributed in different hours on each day in September.
Figure A8. Mean value of refractivity profiles’ relative difference (RD), $E − T / T$, between FY3D (True value, T) and EC4Dvar (Experimental value, E). There are large differences at the heights of
18 (0.3%), 48 (1.3%), 58 (3.1%), 78 (10%) km. (a–c) shows the mean value of it.
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Figure 3. Difference between the BA of FY3D (T) and EC4Dvar BA (E) using the direct and exp Abel integral algorithms, together with log-linear and log-cubic interpolation methods. (a,b) (E[exp/dir] −
T) using the log-cubic interpolation; (c,d) difference of the Abel integral algorithms (E[dir] − E[exp]) based on the log-linear and log-cubic interpolation, respectively; (e,f) difference of
interpolation methods (E[lin] − E) based on exp and direct Abel integral algorithms, respectively.
Figure 4. RD of the BAs of FY3D (T) and EC4Dvar (E) based on the direct and exp Abel integral algorithms, with log-linear and log-cubic interpolation methods. (a–c) (E − T)/T, where E can be
calculated by either the exp or direct algorithm, with both interpolation methods; (d–f) (E[1] − E[2])/E[2], where 1, 2 indicate two different FMs.
Figure 6. Effects of various model level intervals (km) on the FM error. The x-axis represents the RD, ((E − T)/T × 100%), of the BA between true value (T) and three simulated analyses (E): (a,d) 137
levels of EC4Dvar; (b,e) 31 levels of FNL; (c,f) 37 levels of ERA5. The FM is the exp algorithm, together with (a–c) log-cubic or (d–f) log-linear.
Figure 7. Under various model levels, the difference (a–c) and RD (d–f) of FM based on the exp algorithm and log-cubic interpolation. The RD for EC4Dvar below 35 km is less than $±$0.5%, whereas RDs
for FNL and ERA5 below 35 km are less than$±$ 5%. Above 35 km, the RD for EC4Dvar is less than $±$5%, while RDs for FNL and ERA5 are up to 10%.
Figure 8. BA difference ($E − T$, in gray) and RD ($E − T ) / T$, in red) for EC4Dvar, FNL, ERA5, and FY3D based on the exp algorithm and log-cubic interpolation (a–c) and log-linear interpolation (d
–f). (g–i) The difference between the RD of log-cubic interpolation and that of log-linear interpolation.
Table 1. Relative difference, (E−T)/T × 100%, between the FY3D BA (T) and the forward-modeled BA (E), in which E can be the exp and direct algorithm, and each algorithm has two interpolation methods:
log linear (denoted by lin) and cubic spline (no label).
H (km) exp (%) exp_lin(%) direct(%) direct_lin(%) exp_T(%) exp_T_lin(%)
8–15 (−0.3, 0.5) exp + 0.3
15–20 (0.5, 1) exp + 0.3
20–45 (−0.3, 0.5) exp + 0.2
45–55 (−0.5, 1.5) exp + 0.1 exp ± 0.002 exp_lin ± 0.002
55–60 (0, 5) (0, 4) (0, 3.8)
60–70 (−1, 2.5) (−4, 2.3) (−10, 1.8)
70–80 (−1, 5) (−20, −5) (−80, −10) direct + 4%
Statistics MSL Height Order EC 4Dvar EC 4Dvar Msl Height FNL(31) ERA5(37)
(km) of Magnitude (cubic) (lin) (km) (cubic) (cubic)
0–35 $±$0.5% $±$1% 0–30 $±$2.5% 3%
Relative difference (RD) 35–58 % $±$4% $±$4% 30–40 $±$5% 5%
58–80 $±$1.8% $±$2% 40–50 $±$15% 10%
0–10 1 × 10^−5 $±$4 × 10^−5 $±$6 × 10^−5 0–11 $±$4 × 10^−4 $±$2 × 10^−4
10–35 1 × 10^−6 $±$2 × 10^−6 $±$8 × 10^−6 11–22 $±$4 × 10^−5 $±$5 × 10^−5
Difference 35–50 1 × 10^−6 $±$3 × 10^−6 $±$4 × 10^−6 22–40 $±$2 × 10^−5 $±$2 × 10^−5
50–60 1 × 10^−7 $±$2 × 10^−7 $±$4 × 10^−7 40–46 $±$2 × 10^−6 $±$2 × 10^−6
60–80 1 × 10^−7 $±$4 × 10^−8 $±$6 × 10^−8 46–50 $±$6 × 10^−6 $±$6 × 10^−6
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Fei Lu
Mathematics Group
Fei Lu
Fei Lu
Visiting Faculty Scientist
Phone: 510-486-4124
Fei Lu graduated from the University of Kansas with a Ph.D. in mathematics in 2013, was a Postdoctoral Fellow at Lawrence Berkeley National Laboratory, and is now faculty at Johns-Hopkins University.
He is interested in pure and applied probability. His research interests include stochastic parametrization for multi-scale problems, sequential Monte Carlo methods and their applications in
parameter estimation, Mallivian calculus, stochastic partial differential equations and fractional Brownian motion. | {"url":"https://crd.lbl.gov/divisions/amcr/mathematics-dept/math/members/fei-lu/","timestamp":"2024-11-03T06:59:36Z","content_type":"text/html","content_length":"28214","record_id":"<urn:uuid:01514b9d-8ef5-467a-9047-960a3ebb89ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00112.warc.gz"} |
Big Tensor Mining - Carnegie Mellon Database Group
Big Tensor Mining
Tensors are multi-dimensional generalizations of matrices, and so can have non-numeric entries. Extremely large and sparse coupled tensors arise in numerous important applications that require the
analysis of large, diverse, and partially related data. The effective analysis of coupled tensors requires the development of algorithms and associated software that can identify the core relations
that exist among the different tensor modes, and scale to extremely large datasets. The objective of this project is to develop theory and algorithms for (coupled) sparse and low-rank tensor
factorization, and associated scalable software toolkits to make such analysis possible. The research in the project is centered on three major thrusts. The first is designed to make novel
theoretical contributions in the area of coupled tensor factorization, by developing multi-way compressed sensing methods for dimensionality reduction with perfect latent model reconstruction.
Methods to handle missing values, noisy input, and coupled data will also be developed. The second thrust focuses on algorithms and scalability on modern architectures, which will enable the
efficient analysis of coupled tensors with millions and billions of non-zero entries, using the map-reduce paradigm, as well as hybrid multicore architectures. An open-source coupled tensor
factorization toolbox (HTF- Hybrid Tensor Factorization) will be developed that will provide robust and high-performance implementations of these algorithms. Finally, the third thrust focuses on
evaluating and validating the effectiveness of these coupled factorization algorithms on a NeuroSemantics application whose goal is to understand how human brain activity correlates with text reading
& understanding by analyzing fMRI and MEG brain image datasets obtained while reading various text passages. Given triplets of facts (subject-verb-object), like (‘Washington’ ‘is the capital of’
‘USA’), can we find patterns, new objects, new verbs, anomalies? Can we correlate these with brain scans of people reading these words, to discover which parts of the brain get activated, say, by
tool-like nouns (‘hammer’), or action-like verbs (‘run’)? We propose a unified “coupled tensor” factorization framework to systematically mine such datasets. Unique challenges in these settings
1. Tera- and peta-byte scaling issues,
2. Distributed fault-tolerant computation,
3. Large proportions of missing data, and
4. Insufficient theory and methods for big sparse tensors.
We also propose to derive new scientific hypotheses on how the brain works and how it processes language (from the never-ending language learning (NELL) and NeuroSemantics projects) and the
development of scalable open source software for coupled tensor factorization. Our tensor analysis methods can also be used in many other settings, including recommendation systems and
computer-network intrusion/anomaly detection.
1. J. Oh, K. Shin, E. E. Papalexakis, C. Faloutsos, and H. Yu, "S-HOT: Scalable High-Order Tucker Decomposition," in Proceedings of the Tenth ACM International Conference on Web Search and Data
Mining, 2017, pp. 761-770. PDF Bibtex
@inproceedings{oh2017s, title = {S-HOT: Scalable High-Order Tucker Decomposition},
author = {Oh, Jinoh and Shin, Kijung and Papalexakis, Evangelos E and Faloutsos, Christos and Yu, Hwanjo},
booktitle = {Proceedings of the Tenth ACM International Conference on Web Search and Data Mining},
pages = {761--770},
year = {2017},
url = {http://www.cs.cmu.edu/~kijungs/papers/shotWSDM2017.pdf},
2. K. Shin, B. Hooi, J. Kim, and C. Faloutsos, "D-cube: Dense-block detection in terabyte-scale tensors," in Proceedings of the Tenth ACM International Conference on Web Search and Data Mining,
2017, pp. 681-689. PDF Bibtex
@inproceedings{shin2017dcube, title = {D-cube: Dense-block detection in terabyte-scale tensors},
author = {Shin, Kijung and Hooi, Bryan and Kim, Jisu and Faloutsos, Christos},
booktitle = {Proceedings of the Tenth ACM International Conference on Web Search and Data Mining},
pages = {681--689},
year = {2017},
url = {http://www.cs.cmu.edu/~kijungs/papers/dcubeWSDM2017.pdf},
3. K. Shin, B. Hooi, J. Kim, and C. Faloutsos, "DenseAlert: Incremental Dense-Subtensor Detection in Tensor Streams," in Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge
Discovery and Data Mining, 2017. PDF Bibtex
@inproceedings{shin2017densealert, title = {DenseAlert: Incremental Dense-Subtensor Detection in Tensor Streams},
author = {Shin, Kijung and Hooi, Bryan and Kim, Jisu and Faloutsos, Christos},
booktitle = {Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining},
year = {2017},
url = {http://www.cs.cmu.edu/~kijungs/papers/alertKDD2017.pdf},
4. K. Shin, B. Hooi, and C. Faloutsos, "M-Zoom: Fast Dense-Block Detection in Tensors with Quality Guarantees," in ECML/PKDD, 2016, pp. 264-280. PDF Bibtex
author = {Kijung Shin and Bryan Hooi and Christos Faloutsos},
title = {M-Zoom: Fast Dense-Block Detection in Tensors with Quality Guarantees},
booktitle = {ECML/PKDD},
pages = {264--280},
year = {2016},
url = {http://www.cs.cmu.edu/~kijungs/papers/mzoomPKDD2016.pdf},
5. A. Beutel, A. Kumar, E. E. Papalexakis, P. P. Talukdar, C. Faloutsos, and E. P. Xing, "FlexiFaCT: Scalable Flexible Factorization of Coupled Tensors on Hadoop," in SDM, 2014. Bibtex
author = {Alex Beutel and Abhimanu Kumar and Evangelos E. Papalexakis and Partha Pratim Talukdar and Christos Faloutsos and Eric P. Xing},
title = {FlexiFaCT: Scalable Flexible Factorization of Coupled Tensors on Hadoop},
booktitle = {SDM},
year = {2014},
ee = {http://alexbeutel.com/papers/sdm2014.flexifact.pdf},
6. A. Kumar, A. Beutel, Q. Ho, and E. P. Xing, "Fugue: Slow-Worker-Agnostic Distributed Learning for Big Models," in AISTATS, 2014. Bibtex
author = {Abhimanu Kumar and Alex Beutel and Qirong Ho and Eric P. Xing},
title = {Fugue: Slow-Worker-Agnostic Distributed Learning for Big Models},
booktitle = {AISTATS},
year = {2014},
ee = {http://alexbeutel.com/papers/aistats2014.fugue.pdf},
7. E. E. Papalexakis, T. M. Mitchell, N. D. Sidiropoulos, C. Faloutsos, P. P. Talukdar, and B. Murphy, "Scoup-SMT: Scalable Coupled Sparse Matrix-Tensor Factorization," arXiv preprint
arXiv:1302.7043, 2013. Bibtex
@article{papalexakis2013scoup, title={Scoup-SMT: Scalable Coupled Sparse Matrix-Tensor Factorization},
author={Papalexakis, Evangelos E and Mitchell, Tom M and Sidiropoulos, Nicholas D and Faloutsos, Christos and Talukdar, Partha Pratim and Murphy, Brian},
journal={arXiv preprint arXiv:1302.7043},
8. R. Bro, E. E. Papalexakis, E. Acar, and N. D. Sidiropoulos, "Coclustering—a useful tool for chemometrics," Journal of Chemometrics, vol. 26, iss. 6, pp. 256-263, 2012. Bibtex
@article{bro2012coclustering, title={Coclustering—a useful tool for chemometrics},
author={Bro, Rasmus and Papalexakis, Evangelos E and Acar, Evrim and Sidiropoulos, Nicholas D},
journal={Journal of Chemometrics},
publisher={Wiley Online Library},
9. U. Kang, E. Papalexakis, A. Harpale, and C. Faloutsos, "Gigatensor: scaling tensor analysis up by 100 times-algorithms and discoveries," in Proceedings of the 18th ACM SIGKDD international
conference on Knowledge discovery and data mining, 2012, pp. 316-324. Bibtex
@inproceedings{kang2012gigatensor, title={Gigatensor: scaling tensor analysis up by 100 times-algorithms and discoveries},
author={Kang, U and Papalexakis, Evangelos and Harpale, Abhay and Faloutsos, Christos},
booktitle={Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining},
10. E. E. Papalexakis, A. Beutel, and P. Steenkiste, "Network anomaly detection using co-clustering," in Advances in Social Networks Analysis and Mining (ASONAM), 2012 IEEE/ACM International
Conference on, 2012, pp. 403-410. Bibtex
@inproceedings{papalexakis2012network, title={Network anomaly detection using co-clustering},
author={Papalexakis, Evangelos E and Beutel, Alex and Steenkiste, Peter},
booktitle={Advances in Social Networks Analysis and Mining (ASONAM), 2012 IEEE/ACM International Conference on},
11. E. E. Papalexakis, C. Faloutsos, and N. D. Sidiropoulos, "ParCube: sparse parallelizable tensor decompositions." Springer, 2012, pp. 521-536. Bibtex
@incollection{papalexakis2012parcube, title={ParCube: sparse parallelizable tensor decompositions},
author={Papalexakis, Evangelos E and Faloutsos, Christos and Sidiropoulos, Nicholas D},
booktitle={Machine Learning and Knowledge Discovery in Databases},
12. J. Sun, D. Tao, S. Papadimitriou, P. S. Yu, and C. Faloutsos, "Incremental tensor analysis: Theory and applications," TKDD, vol. 2, iss. 3, 2008. Bibtex
author = {Jimeng Sun and Dacheng Tao and Spiros Papadimitriou and Philip S. Yu and Christos Faloutsos},
title = {Incremental tensor analysis: Theory and applications},
journal = {TKDD},
year = {2008},
volume = {2},
number = {3},
bibsource = {DBLP, http://dblp.uni-trier.de},
ee = {https://doi.acm.org/10.1145/1409620.1409621},
13. E. E. Papalexakis, L. Akoglu, and D. Ienco, "Do more Views of a Graph help? Community Detection and Clustering in Multi-Graphs." Bibtex
@article{papalexakismore, title={Do more Views of a Graph help? Community Detection and Clustering in Multi-Graphs},
booktitle={International Conference on Data Fusion, 2013},
author={Papalexakis, Evangelos E and Akoglu, Leman and Ienco, Dino},
14. E. E. Papalexakis, T. Dumitras, D. H. P. Chau, A. B. Prakash, and C. Faloutsos, "Spatio-temporal Mining of Software Adoption \& Penetration." Bibtex
@article{papalexakisspatio, title={Spatio-temporal Mining of Software Adoption \& Penetration},
booktitle={ASONAM 2013},
author={Papalexakis, Evangelos E and Dumitras, Tudor and Chau, Duen Horng Polo and Prakash, B Aditya and Faloutsos, Christos}, | {"url":"https://db.cs.cmu.edu/projects/big-tensor-mining/","timestamp":"2024-11-04T07:50:03Z","content_type":"text/html","content_length":"54583","record_id":"<urn:uuid:e8cb339a-7230-402f-84a1-8576cca1ad73>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00563.warc.gz"} |
Generalized Predictive PID Control for Main Steam Temperature Based on Improved PSO Algorithm
Zhongda Tian, Shujiang Li, and Yanhong Wang
College of Information Science and Engineering, Shenyang University of Technology
Shenyang 110870, China
October 29, 2016
January 16, 2017
Online released:
May 19, 2017
May 20, 2017
main steam temperature, generalized predictive control, predictive PID, improved particle swarm optimization
The large inertia and long delay characteristics of main steam temperature control system in thermal power plants will reduce the system control performance. In order to improve the system
control performance, a generalized predictive PID control for main steam temperature strategy based on improved particle swarm optimization algorithm is proposed. The performance index of
incremental PID controller of main control loop and PD controller of auxiliary control loop based on generalized predictive control algorithm is established. An improved particle swarm
optimization algorithm with better fitness and faster convergence speed is proposed for online parameters optimization of performance index. The optimal control value of PID controller and PD
controller can be obtained. The simulation experiment compared with fuzzy PID and fuzzy neural network is carried out. Simulation results show that proposed control method has faster response
speed, smaller overshoot and control error, better tracking performance, and reduces the lag effect of the control system.
Cite this article as:
Z. Tian, S. Li, and Y. Wang, “Generalized Predictive PID Control for Main Steam Temperature Based on Improved PSO Algorithm,” J. Adv. Comput. Intell. Intell. Inform., Vol.21 No.3, pp. 507-517,
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| MATH
Schedule Variance
Schedule variance is very similar to cost variance. So some of the explanation here also appears in the CV proof as some the variables are the same and the same concepts are used. As stated under
cost variance, when you are mid-project there are two units by which you can measure your progress: time and money. Schedule variance measures progress and in terms of time. But here we measure time
in dollars.
It's counter-intuitive, but schedule variance does not measure progress in time-based units. We are measuring progress in dollars not manhours. The SV formula is SV = EV - PV. EV is earned value and
PV is planned value. It is also sometimes expressed as SV = BCWP - BCWS
Earned value is the estimated value of the completed work. That's a bit dicey though. On many projects the product or service is useless and unsaleable in an incomplete form. But that's not what they
mean by the word "value." The formula for earned value is EV = percent complete x BAC. Sometimes this is given as BCWP - ACWP; Budgeted Cost of work performed minus Actual cost of work performed. You
can see that here "value" is not the laypersons definition, here the meaning is more like planned cost.
The formula for planned value is PV = Planned percent complete x BAC. Please note the single difference between that and EV. The word planned. EV is your actual progress, PV is where you should be
according to schedule. Like the CV formula the SV formula produces a measure of the difference between the plan and reality. If you have made less progress than was planned that formula will
obviously produce a negative number.
Ex. A
EV = 25,000 and PV = 45,000
So SV = 25,000 - 45,000 = -20,000
Ex. B
EV = 45,000 and PV = 25,000
So SV= 45,000 - 25,000 = 20,000
Ex. C
EV = 25,000 and PV = 25,000
So SV= 25,000 - 25,000 = 0
Much like other simple project management formulas you can produce any of these figures from the other two.
Ex. D
EV = 25,000 and AC = 25,000
So SV = EV - PV or 25,000 - 25,000 = 0
and EV = PV + SV or 25,000 + 0 = 25,000
also PV = EV - SV or 25,000 - 0 = 25,000
When looking at example C it's important to know that SV can equal zero. Actually, if you succeed in adhering to your schedule SV should always be close to zero. It's also interesting to note that,
in theory before the project begins the SV equals zero because no spending or work has ocurred. At that point the EV and PV are both zero so 0 - 0 = 0. It is only over the course of the project that
you drift away from zero. Zero is effectively your starting point.
It's also worth noting the difference between the SV and CV calculations. Both measure project progress, and both do so in monetary units; neither of then in time-based units. This might make SV seem
redundant to CV but it's not. Both formulas use EV and both subtract from it a single figure PV or AC.
Think of is as if EV is actual progress, PV is planned progress and AC is actual cost. The difference between EV and PV is actual progress vs planned progress, you either have or have not kept on
schedule. The difference between EV and AC is actual spending vs planned spending, you either have or have not kept on budget.
This formula is simple logical and useful. It is easily understood and used far outside the PM sphere. It hardly requires comment. You need to understand what value means in this context. The rest
you could learn running a cash register. | {"url":"http://www.pmpmath.com/sv.php","timestamp":"2024-11-03T03:04:13Z","content_type":"text/html","content_length":"6745","record_id":"<urn:uuid:a3535aad-4a0d-455e-bffa-e5bd61a4d014>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00494.warc.gz"} |
Games with Words
Two posts ago, I presented some rather odd data about the developmental trajectory of counting. It turns out children learn the meanings of number words in a rather odd fashion. In my last post, I
described the "number" systems that are in place in animals and in infants before they learn to count. Today, I'll try to piece all this together to explain how children come to learn to be able to
Children first learn to map number words onto a more basic numerical system. They learn that "one" maps on to keeping track of a single object. After a while, they learn "two" maps onto keeping track
of one object plus another object. Then they learn that "three" maps onto keeping track of one object plus another object plus another object. All this follows from the Wynn experiments I discussed
two posts ago.
Up to this point, they've been learning the meanings of these words independently, but around this time they notice a pattern. They know a list of words ("one, two, three, four") and that this list
always goes in the same order. They also notice that "two" means one more object than "one," and that "three" means one more object than "two." They put two and two together and figure out that
"four" must mean one more object than "three," even though their memory systems at that age don't necessarily allow them to pay attention to four objects simultaneously. Having made this connection,
figuring out "five," "six," etc., comes naturally.
So what is that more basic number system? One possibility is that children to learn to map the early number words onto the analog number system I also described in the last post (the system adults
use to estimate number when we don't have time to count)?
Something like this claim has been made by a number of well-known researchers (
, Gallistel, Gelman and
, to name a few). There are a number of
a priori
Susan Carey
of Harvard thinks this won't work, but even more important is the data.
As I described two posts ago, very young children can hand you one marble when asked, but hand you random numbers of marbles if asked for "two," "three" or any larger number. They always give you
more than one, but they can't distinguish between the other numbers. Following Wynn, these are called "one-knowers." Slightly older children are "two-knowers," who can give you one or two marbles,
but give you random amounts greater than 2 if asked for another other number. At the next stage, the child becomes a "three-knower." Usually, the next stage is being able to succeed on any number.
I'll call those "counters."
LeCorre and Carey
replicated this trajectory using cards with circles on them. They presented the children a card with some number of circles (1 to 8) and asked the kid, "How many?" One-knowers tended to reply "one"
to a card with one circle, and then guessed incorrectly for just about everything else. Two-knowers could count one or two circles, but guessed incorrectly for all the other cards. Three-knowers
could count up to three, but just guessed beyond that. Counters answered correctly on essentially all cards.
So far this doesn't tell us whether children learn to count by bootstrapping off of analog magnitudes or some other system.
Carey and Mathieu LeCorre published a paper
this year that seems to settle the question. The setup was exactly the same as in the last paper (now with cards with anywhere from 1 to 10 circles), except that this time the children were only
briefly shown the card. They didn't have enough time to actually count "one, two, three..." The data for one-, two- and three-knowers didn't change, which isn't surprising. Both the "3-object" and
the analog magnitude systems are very fast and shouldn't require explicit counting.
However, counters fell into two groups. One group, about 4.5 years old on average, answered just as adults. When they saw six circles, their answers averaged around "six." When they saw ten circles,
their answers averaged around "ten." This is what you'd expect if they have mapped number words onto the analog magnitude system.
However, the other group, which was slightly younger (average age of 4 years, 1 month), guessed randomly for cards with 5 or more circles, just as if they didn't know how to count. However, these
kids can count. If given time to look at the cards, they would have said the right number. So despite the fact that they can count, they do not seem to have their analog magnitude system mapped onto
number words.
This means that the analog magnitude system isn't fundamental in learning how to count, and it actually takes some time for children to learn that mapping even after they've learned to count. Carey
takes this as meaning that the analog magnitude system doesn't play a fundamental role in learning to count, either, and there are other reasons as well that this is probably the case.
One remaining possibility is that children use the "3-object system" to understanding the meanings of 1, 2 and 3. This seems to work nicely, given that the limits of the system (3 objects in
children, 4 in adults) seem to explain why children can learn "one," "two," and "three" without really learning to count. Carey actually has a somewhat more nuanced explanation where children learn
the meanings of "one," "two," and "three" the same may that quantifiers (like "a" in English) are learned. However, to the best of my knowledge, she doesn't have an account of how such quantifiers
are learned, and if she had an account, I suspect it would itself hinge off of the 3-object system, anyway.
That's it for how children learn to count, unless I get enough comments asking for more details on any point. For those who want to read more, there are many papers on this subject at Carey's
web page
In my last post, I showed that children learn the meaning of number words in a peculiar but systematic fashion. Today, I'll continue trying to explain this odd behavior.
Important to this story is that children (and non-human primates) are born with several primitive but useful numerical systems that are quite different from the natural number system (1, 2, 3, ...).
They can't use these systems to count, but they may be useful in learning to count. In this post, I'll try to give a quick summary of how they work.
One is a basic system that can track about 3-4 objects at a time. This isn't a number system per se, just an ability to pay attention to a limited and discrete number of things, and it may or may not
be related to similar limits in visual short-term memory.
You can see this in action by playing the following game with a baby under the age of 2. Show the baby two small boxes. Put a single graham cracker into one of the boxes. Then put, one at a time, two
graham crackers into the other box. Assuming your baby likes graham crackers, she'll crawl to the box with two graham crackers. Interestingly, this won't work if you put two graham crackers in one
box and four in the other. Then, the baby chooses between the boxes randomly. This is understood to happen because the need to represent 6 different objects all in memory simultaneously overloads the
poor baby's brain, and she just loses track. (If you want to experience something similar, try to find a "multiple object tracking" demo with 5 or more objects. I wasn't able to find one, but you can
try this series of demos to get a similar experience.)
On the other hand, there is the analog magnitude system. Infants and non-human animals have an ability to tell when there are "more" objects. This isn't exact. They can't tell 11 objects from 12. But
they can handle ratios like 1:2. (The exact ratio depends on the animal and also where it is in maturity. We can distinguish smaller ratios than infants can.)
You can see this by using something similar to the graham cracker experiment. Infants like novelty. If you show them 2 balls, then 2 balls again, then 2 balls again, they will get bored. Then show
them 4 balls. They suddenly get more interested and look longer. However, this won't happen if you show them 4 balls over and over, then show them 5. That ratio is too similar. (I'm not sure if you
get this effect in the graham cracker experiment. I suspect you do, but I couldn't find a reference off-hand. The graham cracker experiment is more challenging for infants, so it's possible the
results might be somewhat different.)
You can also try this with adults. Show them a picture with 20 balls, and ask them how many there are. Don't give them time to count. The answer will average around 20, but with a good deal of
variation. They may say 18, 19, 21, 22, etc. If you give the adult enough time to count, they will almost certainly say "20."
Those are the two important prelinguistic "number" systems. In my next post, I'll try to piece all this information together.
How do children learn to count? You could imagine that numbers are words, and children learn them like any other word. (Actually, this wouldn't help much, since we still don't really understand how
children learn words, but it would neatly deflect the question.) However, it turns out that children learn to count in a bizarre fashion quite unlike how they learn about other words.
If you have a baby and a few years to spend, you can try this experiment at home. Every day, show you baby a bowl of marbles and ask her to give you one. Wait until your baby can do this. This
actually takes some time, during which you'll either get nothing or maybe a handful of marbles.
Then, one day, between 24 and 30 months of age, your toddler will hand you a single marble. But ask for 2 marbles or 3 marbles, etc., your toddler will give you a handful. The number of marbles won't
be systematically larger if you ask for 10 than if you ask for 2. This is particularly odd, because because by this age the child typically can recite the count list ("one, two, three, four...").
Keep trying this, and within 6-9 months, the child will start giving you 2 marbles when asked for, but still give a random handful when asked for 3 or 4 or 5, etc. Wait a bit longer, and the child
will manage to give you 1, 2 or 3 when asked, but still fail for numbers greater than 3.
This doesn't continue forever, though. At around 3 years old, children suddenly are able to succeed when asked for any set of numbers. They can truly count. (This is work done by Karen Wynn some
years ago, who is now a professor of psychology at Yale University.)
Of course, this is just a description of what children do. What causes this strange pattern of behavior? We seem to be, as a field, homing in on the answer, and in my next post I'll describe some new
research that sheds light onto the question.
Modern genetic analyses have told us a great deal about many aspects of the human body and mind. However, genetics has been relatively slow in breaking into the study of language. As I have mentioned
before, a few years ago resarchers reported that a damaged version of the gene FOXP2 was responsible for the language impairments in the KE family. This sounds more helpful than it really was, since
it turns out that even some reptiles have versions of the FOXP2 gene. In humans, FOXP2 isn't just expressed in the brain -- it's expressed in the gut as well. This means that there is a lot more
going on than just having FOXP2 or not.
Over the weekend, researchers presented new data at the Boston University Conference on Language Development that hones in on what, just exactly, FOXP2 does.
It turns out that there is a certain amount of variation in genes. One type of variation is a Single Nucleotide Polymorphism (SNP), which is a single base pair in a string of DNA that varies from
animal to animal within a species. Some SNPs may have little or no effect. Others can have disastrous effects. Others are intermediate. The Human Genome Project simply cataloged genes. Scientists are
still working on cataloging these variations. (This is the extent of my knowledge. If any geneticists are reading this and want to add more, please do.)
The paper at BUCLD, written by J. Bruce Tomblin and Jonathan Bjork of the University of Iowa and Morten H. Christiansen of Cornell University, looked at SNPs in FOXP2. They selected 6 for study in a
population of normally developing adolescents and a population of language-impaired adolescents.
Two of the six SNPs under study correlated well with a test of procedural memory (strictly speaking, one correlation was only marginally statistically significant). One of these SNPs predicted better
procedural memory function and was more common in language-normal adolescents; the other predicted worse procedural memory function and was more common in language-impaired adolescents.
At a mechanistic level, the next step will be understanding how the proteins created by these different versions of FOXP2 do. From my perspective, I'm excited to have further confirmation of the
theory that procedural memory is important in language. More importantly, though, I think this study heralds a new, exciting line of research in the study of human language.
(You can read the abstract of the study here.)
One problem that confronts nearly every cognitive science researcher is attracting participants. This is less true perhaps for vision researchers, who can sometimes get away with testing only
themselves and their coauthors, but it is definitely a problem for people who conduct Web-based research, which often needs hundreds or even thousands of participants.
Many researchers when they start conducting experiments on the Internet are tempted to offer rewards for participation. It's too difficult to pay everybody, so this is often done in the context of a
lottery (1 person will win $100). This seems like an intuitive strategy, since we usually attract participants to our labs by offering money or making it a requirement for passing an introductory
psychology course.
If you've been reading the Scienceblog.com top stories lately, you might have noticed a recent study by University of Florida researchers, which suggested that people -- well, UF undergrads -- are
less likely to give accurate information to websites which offered rewards.
Although these data are in largely in the context of marketing, this suggests that using lotteries to attract research participants on the Web may actually be backfiring. | {"url":"http://gameswithwords.fieldofscience.com/2007/11/","timestamp":"2024-11-14T16:27:06Z","content_type":"application/xhtml+xml","content_length":"186847","record_id":"<urn:uuid:f568dd04-b52b-47b7-9c0b-5753d43c898e>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00791.warc.gz"} |
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Brain Extraction Using Active Contour Neighborhood-Based Graph Cuts Model
Key Laboratory of Nondestructive Testing, Ministry of Education, Nanchang Hangkong University (NCHU), Nanchang 330063, China
Author to whom correspondence should be addressed.
Submission received: 11 February 2020 / Revised: 21 March 2020 / Accepted: 24 March 2020 / Published: 4 April 2020
The extraction of brain tissue from brain MRI images is an important pre-procedure for the neuroimaging analyses. The brain is bilaterally symmetric both in coronal plane and transverse plane, but is
usually asymmetric in sagittal plane. To address the over-smoothness, boundary leakage, local convergence and asymmetry problems in many popular methods, we developed a brain extraction method using
an active contour neighborhood-based graph cuts model. The method defined a new asymmetric assignment of edge weights in graph cuts for brain MRI images. The new graph cuts model was performed
iteratively in the neighborhood of brain boundary named the active contour neighborhood (ACN), and was effective to eliminate boundary leakage and avoid local convergence. The method was compared
with other popular methods on the Internet Brain Segmentation Repository (IBSR) and OASIS data sets. In testing cross IBSR data set (18 scans with 1.5 mm thickness), IBSR data set (20 scans with 3.1
mm thickness) and OASIS data set (77 scans with 1 mm thickness), the mean Dice similarity coefficients obtained by the proposed method were 0.957 ± 0.013, 0.960 ± 0.009 and 0.936 ± 0.018
respectively. The result obtained by the proposed method is very similar with manual segmentation and achieved the best mean Dice similarity coefficient on IBSR data. Our experiments indicate that
the proposed method can provide competitively accurate results and may obtain brain tissues with sharp brain boundary from brain MRI images.
1. Introduction
Brain extraction or skull stripping is needed before most of neuroimaging analyses, such as registration between MRI images [
], measurement of brain volume [
], brain tissue classification [
], and cortical surface reconstruction [
]. During recent years, automatic or semi-automatic brain extraction techniques have become the choices for fast brain extraction. These techniques can generally be categorized into three types:
region-based [
], boundary-based [
], and hybrid methods [
The region-based methods first use thresholding or clustering techniques to divide brain MRI image into several regions by considering that the voxels in the same tissue have similar intensities, and
then the brain region can be extracted from these regions by morphological operations or region merging. These thresholding or clustering techniques frequently used in brain extraction are Gaussian
mixture model [
], intensity thresholding [
] and watershed algorithm [
]. The parameters in the region-based methods have remarkable effects on the brain extraction and need to be determined properly [
The boundary-based methods partition the brain MRI image into internal part (brain) and external part (non-brain) by detecting the boundary between brain and non-brain tissues. Smith [
] proposed a Brain Extraction Tool (BET) by using smoothing and pushing forces to push a tessellated mesh to the brain boundary. Shattuck et al. [
] proposed the Brain Surface Extraction (BSE) method by using a Marr–Hildreth edge detector to separate brain and non-brain tissues. The boundary-based methods [
] used active contour model that has made great progress in medical image segmentation during recent years.
The hybrid methods try to use a two steps strategy to improve the extraction result. In the first step, a rough brain region or boundary is obtained to be the initial brain region or boundary in the
next step. Next, the brain contour or region is refined to get a more accurate result since it is close to the brain boundary at the first stage. Huang et al. [
] employed the expectation maximization algorithm on a mixture of Gaussian models to determine the initial brain contour for the geodesic active contour evolution. Ségonne et al. [
] proposed a hybrid watershed algorithm (HWA) applying the watershed algorithm to get an initial brain volume for deformable brain mesh. Sadananthan et al. [
] used graph cuts based image segmentation to refine the preliminary binary mask generated by intensity thresholding. Jiang et al. [
] used BET to generate the initial brain boundary, and then refined the brain boundary using a hybrid level set model.
To improve the robustness, some hybrid-based methods warp the brain volume to an atlas using registration techniques before brain extraction, then use the parameter learning techniques such as
meta-algorithm, random forest and neural networks to get the proper initial region or parameters to perform brain extraction. Wang et al. [
] warped an atlas to the brain volume to obtain the initial brain mask, and then refined the mask with a deformable-surface similar with BET. Iglesias et al. [
] proposed a robust, learning-based brain extraction system (ROBEX). ROBEX used a trained random forest classifier to detect the brain boundary. Then the contour was refined to obtain the final brain
tissues using graph cuts. Eskildsen et al. [
] proposed an atlas-based method with nonlocal segmentation technique, without the need of time-consuming non-rigid registrations. Huang et al. [
] used a trained Locally Linear Representation-based Classification (LLRC) method for brain extraction. By using registration techniques, the atlas-based methods can address the problem of
variability in anatomy and modality and produce more robust results. However, the registration in atlas-based methods is usually highly time-consuming and if the registration fails, bad result may be
In recent years, the deep learning techniques were used for brain extraction. Kleesiek et al. [
] used a 3D convolutional neural network for brain extraction. Saleh et al. [
] applied U-Net for brain extraction. However, the output of these learning-based brain extraction method tends to be over-smoothed if the training data are obtained by automatic brain extraction
method under the checking by human experts or by weak bounding box labeling [
]. In other words, if we want the learning-based brain extraction method to output brain tissue with sharp boundary, the training data should be quite precise and be obtained by manual. For example,
Hwang et al. [
] recently applied 3D U-Net for brain extraction and achieved high extraction accuracy by using a fine training data.
Although the above methods have greatly improved the accuracy and robustness for brain extraction, these methods can not completely substitute for manual method for the appearances of
over-smoothness, leakage through a weak boundary and missing brain tissues caused by local convergence. Unfortunately, it is a conflicting problem to improve all of them. Brain extraction is a
compromised problem where a semi-global understanding of the image is required as well as a local understanding [
]. For example, it is effective that we use local region features to obtain sharp brain boundary and eliminate the leakage, but it is easy to lead to local convergence at the edges between the white
matter and gray matter if the initial region is far from the true brain boundary. Trying to address these problems, we proposed a new brain extraction method from T1-weighted MRI volume. The main
contributions of our work can be summarized as follows:
1. We defined a new asymmetric assignment of edge weights in graph cuts for brain MRI image to obtain brain boundary.
2. We performed the new graph cuts model iteratively in the neighborhood of brain boundary named the active contour neighborhood (ACN), and the new model was effective to eliminate boundary leakage
and avoid local convergence.
3. By the asymmetric function of edge weights, we reduced the edge weights in the region between the ACN and the region in the current boundary to obtain a sharp brain boundary.
The remainder of this paper is organized as follows. In
Section 2
, the details of the designed brain extraction method including the ACN and graph cuts are introduced. In
Section 3
, experiments and results are presented. Finally, we give a detailed discussion on the analysis of our experiments in
Section 4
2. Materials and Methods
2.1. Data Sets
To measure the extraction accuracy of our method, we used the following three data sets:
1. Data set 1 (IBSR18): 18 normal T1-weighted MR Image data sets with expert segmentations from IBSR. Each volume has around 128 coronal slices, with 256 × 256 pixels per slice and 1.5mm thickness.
2. Data set 2 (IBSR20): 20 normal T1-weighted MR Image data sets from IBSR, with 256 × 256 pixels per coronal slice and 3.1 mm thickness. Obvious intensity inhomogeneity and other significant
artifacts present in most of the MR images in this data set. Another challenge of this data set is that the neck and even shoulder areas are included.
3. Data set 3 (OASIS77): 77 T1-weighted MR Image data sets were obtained from OASIS project. Each volume has around 208 coronal slices, with 176 × 176 pixels per slice and 1mm thickness. The brain
masks for this set were obtained by an atlas-based brain extraction method automatically, checked by human experts before releasing the data. Although the lack of exactitude makes it unfit for
testing the precision of a method, we can use it to test the robustness of a method because it includes scans from a very diverse population with a very wide age range as well as diseased brains [
2.2. Graph Cuts
In our previous work [
] we used BET to pre-process T1-weighted MRI scans to generate a robust initial contour and obtained the active contour neighborhood (ACN) [
] by dilating the contour with a small size. In this paper, we used a modified graph cuts to refine the brain contour in ACN. Thus, the ACN and brain contour can be obtained iteratively. Since the
initial contour obtained by BET is close to the real brain boundary, the real brain boundary should be the global minimum within the ACN obtained by the initial contour, and the redefined edge
weights are more powerful than GCUT [
] for brain extraction.
The graph cuts approaches [
] model the image as a weighted undirected graph
= (
), where
are sets of vertices, edges and edge weights respectively. Let
= (
) be a binary vector whose components
specify assignments to pixels
in data set
. Each
can be either “obj” or “bkg” (abbreviations of “object” and “background”). In the graph described by Boykov and Jolly [
], an energy function was defined as below:
$E ( A ) = λ R ( A ) + B ( A )$
$R ( A ) = ∑ p ∈ P R p ( A p )$
$R p ( “ obj ” ) = − ln Pr ( I p | O )$
$R p ( “ bkg ” ) = − ln Pr ( I p | B )$
) is the regional term and
) is the boundary term.
(“obj”) and
(“bkg”) reflect how the intensity of the pixel
fits the histogram of the object (
) and background (
) models respectively. The boundary term
) was described by:
$B ( A ) = ∑ p , q ∈ N B { p , q } ⋅ δ ( A p , A q )$
$δ ( A p , A q ) = { 1 A p ≠ A q 0 otherwise$
A well-known Min-Cut/Max-Flow algorithm can be used to minimize the energy function, and the cut will separate the image into deferent regions. Usually, the graph
= (
) has two special vertices (terminals), namely, the source
and the sink
. Let {
} be the two vertices in the graph and
be the neighbors of them. The edge weights between all the vertices in the graph are described in
Table 1
$K = 1 + max p ∈ P ∑ q : { p , q } B { p , q }$
2.3. Active Contour Neighborhood-Based Graph Cuts Model
To address the over smoothness, boundary leakage and asymmetry problem, we used a piecewise function to define the edge weights in graph cuts, which was more powerful than GCUT. To avoid local
convergence, we performed graph cuts in ACN.
2.3.1. Description of ACN Model (ACNM)
The elements of the pipeline for the ACN model are shown in
Figure 1
. The first step is to obtain a rough boundary by a modified BET method for 2D images [
] from a round initial contour. Although the rough boundary is too smooth, it is close to the brain tissue and is still fit well for providing a good initial contour for brain extraction, due to the
good robustness of BET. Then, the ACN is obtained by dilating the rough boundary. In the ACN, a graph cuts model is defined and performed to obtain a new boundary. Finally, a more accurate and
sharper brain boundary is obtained by iteratively replacing the new boundary and the ACN. The maximum iteration number in
Figure 1
was 10 in this paper.
2.3.2. Edge Weights Assignment in ACNM
In the defined graph cuts model for brain extraction, the edge weights between all the vertices in the graph are described in
Table 2
. The brain tissues and non-brain tissues are considered as the Object (
) and Background (
) respectively in our work, thus the ACN in
Table 2
exactly corresponds to the area
$p ∈ P , p ∉ O ∪ B$
Table 1
(see in
Figure 2
”) and
”) are the likelihood items reflecting how the vertex is fit for brain or non-brain tissues respectively. For ACN, we firstly classified the ACN into “brain” and “nonbrain” regions using Fuzzy
C-Means clustering method (FCM) [
], then obtained the their likelihood items from Equation (3) and Equation (4). Recalling that
should include most of the non-brain tissues and inversely in
, we simply set the edge weights between the terminal (
) and vertex in
as the minimal or maximal value of
(“brain”) and R(“nonbrain”) in ACN respectively.
When setting the edge weights between the vertex and its neighbors (
}, we used the following equations:
$B { p , q } = D ( p , q ) ⋅ G ( p , q ) ⋅ R ( p , q )$
$D ( p , q ) = max p , q ∈ N [ D t ( p ) , D t ( q ) ]$
$D t ( p ) = { k D exp ( 2 d p b W ) , p ∈ A C N ∩ R b exp ( 2 d p b W ) , otherwize$
$G ( p , q ) = min p , q ∈ N [ I G ( p ) , I G ( q ) ]$
$I G ( p ) = { exp ( k 0 I ( p ) − μ I m − μ ) , I ( p ) − μ > 0 exp ( k 1 μ − I ( p ) I m − μ ) , I ( p ) − μ ≤ 0$
$R ( p , q ) = max p , q ∈ N [ I R ( p ) , I R ( q ) ]$
$I R ( p ) = exp ( 2 I max )$
$I max = max ( t m , I ( 0 ) , I ( 1 ) , ⋯ , I ( d ) )$
Compared with the
} used in GCUT [
], we redefined the
) and
) in Equations (9)–(12) and added a new regional item
) in Equations (13)–(15).
In (10)
$d p b$
is the minimal distance between
and the current brain boundary, and
is the width of the image. This assignment increases the weight of edges locating deep within the foreground region, making cuts here less likely.
$k D$
is a parameter to control the smoothness. If over-smoothness occurs, a small
$k D$
can be selected to reduce the edge weights in the region between the ACN and
$R b$
(the region in the current boundary), and then the current brain boundary will move inside to touch the true brain boundary. The new
) in Equation (11) was defined by a piecewise function (12), which depends on the grayscale value of
, the mean grayscale value
of the ACN and the maximal value
$I m$
of the ACN. The plot corresponding to Equation (12) is shown in
Figure 3
Figure 3
$μ = 0.5$
$I m = 0.9$
$k 0 = 6$
$k 1 = 0.55 k 0$
, they control the range of the grayscale values that could be considered inside the brain tissues. Generally, the true brain boundary should locate rather on the vertices with grayscale values close
than on the vertices with lower or higher grayscale values. So, wherever
$I ( p )$
equals to
, we set
$I G ( p ) = 1$
to make the voxels whose grayscale values are close to
touched by the active contour. Wherever
$I ( p ) ≠ μ$
, we set
$I G ( p ) > 1$
and the active contour will move away from the voxels whose grayscale values are far from
. Considering the true brain boundary more likely locates on the lower intensity side of the vertices whose grayscale values are close to
, we set
$k 1 = 0.55 k 0$
. Thus, Equation (12) is a piecewise function, which is also well fit for the asymmetric distribution of intensity in the brain images.
The regional term
) in Equations (13)–(15) is used to eliminate boundary leakage trough the edge between the brain and eyeball.
) depends on the regional maximal value
$I max$
which was firstly used in BET and then used by Zhuang et al. [
] and Liu et al. [
$I max$
is searched along a line pointing inward from the current vertex within a distance
is the median intensity of the brain tissue on each slice, which is approximated from the pixels within the initial brain region.
$I max$
increases the edge weights on the eyeball to avoid the brain boundary leaking through the eyeball tissue.
3. Results
3.1. Evaluation Metrics
1. Similarity coefficients. We used Jaccard similarity, defined as $J S = | M ∩ N | / | M ∪ N |$, and Dice similarity, defined as $D S = 2 | M ∩ N | / ( | M | + | N | )$, where M and N refer to the
extraction result and the ground truth respectively.
2. Segmentation error coefficients. We used False Positives Rate ($F P R a t e$) and False Negatives Rate ($F N R a t e$), defined as $F P R a t e = ( | M | − | M ∩ N | ) / | N |$ and $F N R a t e =
( | N | − | N ∩ M | ) / | N |$.
3. The recognizing rate (Sensitivity, SE) and rejecting rate (Specificity, SP). We used $S E = | M ∩ N | / | N |$, and $S P = | I − M − N | / | I − N |$.
3.2. Comparison to Other Methods
Because BET, BSE, GCUT and ROBEX are all free and publicly available, a comparison to them was performed on the chosen data sets. The software of our work was programmed with matlab2016 and we tested
it on a computer with 8GB RAM and Intel i7-4790 CPU. The software cost about 2 minutes to process one MRI volume on this computer. In this software, only two parameters
$k D$
$k 0$
in Equation (10) and Equation (12) were needed to be set by the user for different volumes.
$k D$
is positively associated with the smoothness degree of brain boundary.
$k 0$
is negatively associated with the average intensity of brain tissue. In testing, we chose
$k D = 0.80$
$k 0 = 5.5$
for IBSR18,
$k D = 0.83$
$k 0 = 1.2$
for IBSR20,
$k D = 1$
$k 0 = 6$
for OASIS77 to get the best results. The extremely low
$k 0$
used in IBSR20 was to deal with the extremely high value and inhomogeneity of intensities in brain tissue. Before performing our method, the modified BET was done on the coronal middle slice once to
obtain the initial contour. The initial contours of the other slices in the 3D volume were obtained by the resultant brain boundary of previous slice in a slice by slice manner, which was described
in detail in our previous work [
]. The parameters of the BET were almost the same for all the data sets, which are quite easy to be set according to our previous work [
] for the robustness of BET. When estimating the centroid and the equivalent radius of the brain in BET, extra biases were added to them for IBSR20 because the images in IBSR20 include a lot of
tissues in neck. For GCUT and ROBEX, we used the default parameters setting in their software. For BET and BSE, we used the different parameters on different volumes in the three data sets to obtain
the best results by processing each volume for several times.
Table 3
Table 4
Table 5
display the means and standard deviations of the metrics from each method on the three data sets. The P
-values listed in
Table 3
Table 4
Table 5
are the adjusted P-values of paired t-tests with the Bonferroni-Holm correction to show the statistical difference between the compared methods and the proposed method. Most of the P*-values are
below 0.05 except those of FN
and DS for BSE on IBSR18 and IBSR20 data sets, which means the performance of the proposed method was different with the compared methods.
Figure 4
Figure 5
display sample outputs from each method on both IBSR data sets and mark false negative and false positive voxels with different colors to show the extraction errors. To clearly and correctly show the
visual extraction errors on all test images for each method, we first warped all the results and the ground truth to the atlas [
], thus every 3D brain extracted from different methods had the same size, and then the false positive and false negative voxels of each warped result could be obtained. The hot color map of the
false positive and false negative voxels are shown in
Figure 6
Figure 7
. The brighter the hot color in
Figure 6
Figure 7
is, the bigger the error on the corresponding area is. Because the ground truth of OASIS77 (see in
Figure 8
) is not as precise as those of IBSR18 and IBSR20 data sets, we do not show the errors for OASIS77 in
Figure 4
Figure 5
Figure 6
Figure 7
, but directly display the sample outputs from each method on both IBSR data sets and OASIS77 data set in
Figure 8
to show the differences among them.
BET shows good results on IBSR18 and OASIS77 data sets. However, it performs poorly on IBSR20 data set, because of the included neck and shoulder areas featured with the obvious intensity
inhomogeneity. BET then shows both high FP
and high FN
Table 4
caused by a bad estimation of brain center. The obvious intensity inhomogeneity also makes it difficult for BET to obtain a good result with the same parameter for all slices of the MRI volume,
leading to low DS and JS.
BSE performs well on most of the MRI images in IBSR18, IBSR20 and OASIS77 data sets, however it brought very bad results on some of MRI images whatever we changed the parameters (Some of the tissues
in metencephalon and cerebellum were missed by BSE in
Figure 5
Figure 8
due to the obvious intensity inhomogeneity in the image). GCUT shows poor results on both IBSR data sets with the highest FP
. A very high FP
shows that it tends to maintain non-brain tissue by GCUT. ROBEX performs equally on all data sets, as it is robust across different data. However, the DS and JS for ROBEX are not high in
Table 3
Table 4
. By the proposed ACNM method, the average DS is 0.957 for IBSR18, 0.960 for IBSR20, and 0.936 for OASIS77. The proposed ACNM method shows the highest extraction accuracy on both IBSR data sets. Both
and FN
are lower than 5%, indicating that a balance of FN
and FP
is achieved by the proposed method on both IBSR data sets. From the coefficients in
Table 5
, it seems ACNM does not perform as well as ROBEX and GCUT on OASIS77 data set. It is important to note that the ground truth of OASIS77 data set is not as precise as IBSR data set and has the
tendency to over cover the brain, which is similar to the result from ROBEX and GCUT. So, it is reasonable that ROBEX and GCUT obtain better evaluation on OASIS77 than ACNM does. In
Figure 8
, it is clear that ACNM does the best on OASIS77.
We also compared the proposed method with the deep learning-based method (CNN) proposed by Kleesiek et al. [
] in literature due to no available soft for it. In [
], only the average DS, SE and SP for all the images in IBSR and OASIS data sets were listed, so we listed them in
Table 6
. The DS, SE and SP of ACNM are very close to those of CNN, so the proposed method almost reached the deep learning-based method without using any learning and registration techniques.
4. Discussion
Among these methods evaluated in this paper, BET, BSE and GCUT are very sensitive to parameters. If the parameters are well set, BSE can obtain a highly accurate result. However, BSE sometimes failed
to obtain a good result no matter how the parameters were set. BET performed with high robustness if the brain center was well predicted. The remain of a lot of neck tissues in IBSR20 made BET
predict wrong center of the brain and obtain very bad result for IBSR20. Although our proposed method used BET in the initial step, it achieved much better results than BET due to a better way to
calculate the brain center. GCUT did not obtain any good results on both IBSR data sets using fixed parameters suggested in the literature. ROBEX did not achieve good result on IBSR18 with quite
different ranges of intensity between MRI subjects. The main reason may be that the graph cuts models in the refine step of ROBEX and GCUT are sensitive to parameters and usually lead to over
smoothness. The proposed method used BET to initialize the brain contour which is very close to the brain boundary and ensures that the active contour evolves to the true brain boundary by performing
a modified graph cuts model in the neighborhood of the contour (ACN). The modified graph cuts model used an asymmetric edge weighting function which is more powerful than that used in GCUT and can
output brain tissue with sharp boundary.
Although the proposed method has the above advantages, there are some disadvantages of the current implementation. First, the computational cost is higher than the other method. Second, two
parameters are required to be manually set, if the parameters are set poorly, the extraction result might not be very desirable. Third, our proposed method was implemented as a 2D algorithm and could
produce some rough artifacts on the brain surface. On the contrary, the deep learning technique-based brain extraction methods [
] usually run very fast and stably by using the GPU accelerating and learning techniques. However, the deep learning techniques also face a lot of challenges such as few-shot learning and capacity
for transfer and must be supplemented by other techniques if we are to reach artificial general intelligence [
]. Furthermore, the U-Net based brain extraction methods [
] only used the original network, the network should be improved for the generalization capabilities across multiple datasets. Recently, Rundo et al. [
] incorporated Squeeze and Excitation blocks into U-Net (named USE-Net) for prostate zonal segmentation of multi-institutional MRI data sets. USE-Net provided excellent cross-dataset generalization
when testing was performed on samples of the datasets used during training. As inspired by this, future work is to combine the improved U-Net technique with the ACN method, as the improved U-Net can
provide more robust initial brain contour than BET across multiple data sets and predict more appropriate parameters for ACN.
Author Contributions
Conceptualization, S.J. and Z.C.; methodology, S.J.; software, S.Y.; validation, X.Z.; writing, S.J. and Y.W.; funding acquisition, S.J. All authors have read and agreed to the published version of
the manuscript.
This work was funded by National Natural Science Foundation of China, Grant Number 61162023, was funded by the State Key Program of Jiangxi Province, Grant Number 20171BBG70052, was funded by China
Postdoctoral Science Foundation, Grant Number 2019M652270 and was funded by the Natural Science Foundation of Jiangxi Province, Grant Number 20192BAB205083.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2. The maps of $ℬ$, $𝒪$ and ACN. $ℬ$: background, $𝒪$: object, ACN: active contour neighborhood, Dash line: current brain boundary.
Figure 4. Outputs from two scans in IBSR18 data set. Blue voxels represent the segmentation results of the corresponding brain extraction methods. Green voxels indicate the false negatives and red
voxels indicate the false positives.
Figure 5. Outputs from two scans in IBSR20 data set. Blue voxels represent the segmentation results of the corresponding brain extraction methods. Green voxels indicate the false negatives and red
voxels indicate the false positives.
Figure 6. Average of the false positive and false negative voxels for each method on IBSR18 data set.
Figure 7. Average of the false positive and false negative voxels for each method on IBSR20 data set.
Edge Weight for
${ p , q }$ $B { p , q }$ ${ p , q } ∈ N$
$λ R p ( “ bkg ” )$ $p ∈ P , p ∉ O ∪ B$
${ p , S }$ K $p ∈ O$
0 $p ∈ B$
$λ R p ( “ obj ” )$ $p ∈ P , p ∉ O ∪ B$
${ p , T }$ 0 $p ∈ O$
K $p ∈ B$
Edge Weight for
${ p , q }$ $B { p , q }$ ${ p , q } ∈ N$
$λ R p ( “ b r a i n ” )$ $p ∈ A C N$
${ p , S }$ $λ min p ∈ A C N R p ( “ b r a i n ” )$ $p ∈ O$
$λ max p ∈ A C N R p ( “ b r a i n ” )$ $p ∈ B$
$λ R p ( “ n o n b r a i n ” )$ $p ∈ A C N$
${ p , T }$ $λ max p ∈ A C N R p ( “ n o n b r a i n ” )$ $p ∈ O$
$λ min p ∈ A C N R p ( “ n o n b r a i n ” )$ $p ∈ B$
Method DS Mean (SD) JS Mean (SD) FP[RATE] (%) Mean (SD) FN[RATE] (%) Mean (SD)
BET 0.946(0.012) 0.898(0.021) 8.36(2.77) 2.83(2.92)
P^*-value 3.0 × 10^−4 3.3 × 10^−4 4.65 × 10^−7 1.09 × 10^−4
BSE 0.943(0.039) 0.895(0.066) 7.82(6.20) 3.68(6.30)
P^*-value 5.0 × 10^−2 4.87 × 10^−2 8.71 × 10^−3 2.71 × 10^−1
GCUT 0.911(0.015) 0.837(0.025) 18.47(3.89) 0.92(0.04)
P^*-value 8.72 × 10^−8 7.08 × 10^−8 1.02 × 10^−12 1.34 × 10^−6
ROBEX 0.927(0.032) 0.865(0.054) 14.74(8.14) 1.1(0.81)
P^*-value 2.42 × 10^−3 2.0 × 10^−3 2.0 × 10^−5 1.34 × 10^−6
ACNM 0.957(0.013) 0.917(0.024) 4.06(1.24) 4.55(2.48)
Method DS Mean (SD) JS Mean (SD) FP[RATE] (%) Mean (SD) FN[RATE] (%) Mean (SD)
BET 0.849(0.076) 0.745(0.110) 22.87(7.95) 9.00(11.30)
P^*-value 2.96 × 10^−6 9.93 × 10^−7 1.21 × 10^−8 2.76 × 10^−2
BSE 0.933(0.054) 0.878(0.084) 6.43(2.43) 6.57(9.46)
P^*-value 2.02 × 10^−2 1.58 × 10^−2 1.27 × 10^−8 7.41 × 10^−2
GCUT 0.88(0.015) 0.786(0.024) 27.34(3.91) 0.01(0.02)
P^*-value 1.62 × 10^−12 1.0 × 10^−12 1.57 × 10^−15 1.89 × 10^−6
ROBEX 0.94(0.012) 0.888(0.021) 11.9(2.75) 0.67(0.46)
P^*-value 1.33 × 10^−6 9.93 × 10^−7 6.72 × 10^−12 1.86 × 10^−6
ACNM 0.960(0.009) 0.924(0.016) 4.61(2.08) 3.40(2.40)
Method DS Mean(SD) JS Mean(SD) FP[RATE] (%)Mean(SD) FN[RATE] (%)Mean(SD)
BET 0.931(0.019) 0.871(0.033) 11.0(3.70) 3.45(2.94)
P^*-value 2.64 × 10^−2 2.54 × 10^−2 1.14 × 10^−35 2.95 × 10^−33
BSE 0.923(0.060) 0.862(0.090) 14.1(18.2) 3.29(2.02)
P^*-value 3.80 × 10^−2 4.99 × 10^−2 5.32 × 10^−8 2.22 × 10^−33
GCUT 0.950(0.008) 0.904(0.015) 7.55(2.82) 2.76(1.79)
P^*-value 3.57 × 10^−8 2.90 × 10^−8 1.13 × 10^−40 9.57 × 10^−6
ROBEX 0.955(0.008) 0.914(0.015) 2.54(1.3) 6.23(2.1)
P^*-value 1.02 × 10^−21 1.36 × 10^−22 4.22 × 10^−10 6.16 × 10^−23
ACNM 0.936(0.018) 0.879(0.031) 1.95(1.30) 10.32(3.87)
Method DS Mean SE Mean SP Mean
CNN 0.958 0.943 0.994
ACNM 0.951 0.940 0.994
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Black holes in an ultraviolet complete quantum gravity
In this Letter we derive the gravity field equations by varying the action for an ultraviolet complete quantum gravity. Then we consider the case of a static source term and we determine an exact
black hole solution. As a result we find a regular spacetime geometry: in place of the conventional curvature singularity extreme energy fluctuations of the gravitational field at small length scales
provide an effective cosmological constant in a region locally described in terms of a de Sitter space. We show that the new metric coincides with the noncommutative geometry inspired Schwarzschild
black hole. Indeed, we show that the ultraviolet complete quantum gravity, generated by ordinary matter is the dual theory of ordinary Einstein gravity coupled to a noncommutative smeared matter. In
other words we obtain further insights about that quantum gravity mechanism which improves Einstein gravity in the vicinity of curvature singularities. This corroborates all the existing literature
in the physics and phenomenology of noncommutative black holes.
• Black holes
• Quantum gravity
ASJC Scopus subject areas
• Nuclear and High Energy Physics
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Are rational and irrational numbers real numbers? | Socratic
Are rational and irrational numbers real numbers?
1 Answer
Real numbers are often explained to be all the numbers on a number line.
Consider that there are two basic types of numbers on the number line. There are those which we can express as a fraction of two integers, the Rational Numbers , such as:
$\frac{1}{2} , \frac{5}{3} , \frac{22}{7} , - \frac{3887}{4} , e t c$
and those which we can't express as a fraction, the Irrational Numbers :
$\sqrt{2} , \pi , e , - \sqrt{5} , e t c$
(note that while $\frac{22}{7}$ is an often used approximation for $\pi$, $\pi$ itself can't be expressed as a fraction).
Impact of this question
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A-B Testing Statistics: True and Estimated Value of Conversion Rate
What's A/B Testing?
A/B testing, or split testing, is a keyway to test ideas in stats. People use it a lot in online marketing, website design, and making new products. The main point is to see which of two options
works better by looking at a specific number.
For websites trying to get more sales, A/B testing helps companies figure out which version of a webpage, email, or other marketing stuff is better at turning visitors into buyers or leads.
The Basics of A/B Testing Stats
• Coming Up with Ideas to Test
Every A/B test starts with an idea to check. It looks like this:
□ Null Hypothesis (H0): Version A and version B don't have any big differences.
□ Alternative Hypothesis (H1): Version A and version B have big differences.
• Random Assignment
To make sure the test is legit, people who join (folks visiting the website or using it) get put into either group A or group B by chance. This random stuff helps get rid of bias and makes sure
any differences we see are because of the changes we're testing, not other stuff.
• Figuring Out How Many People to Test
The sample size plays a key role in A/B testing. Tests with more participants have a better chance to spot real differences between versions. How many people you need depends on things like how
big a change you expect, how sure you want to be, and how strong you want your results to be.
• Statistical Significance
A/B testing relies on statistical significance. This idea helps us figure out if the difference we see between versions is just random or if it's a real thing. People often use the p-value to
measure this with a cutoff of 0.05 (but this can change based on what you're testing).
• Confidence Intervals
Confidence intervals give us a range of likely values for the real difference between versions. This tells us more than just a single number estimate. For instance, a 95% confidence interval
means we can be pretty sure (95% sure, to be exact) that the true difference is somewhere in that range.
Real vs. Estimated Conversion Rates
Real Conversion Rate
The real conversion rate is the actual rate at which users convert when they see a specific version. In real life, we never know this number for sure; it's just a theory.
Estimated Conversion Rate
The conversion rate we estimate comes from our sample data. It's our best shot at figuring out the real conversion rate based on what we see in our A/B test.
How True and Estimated Rates Connect?
The rate we estimate is a rough guess of the true rate. When we use more data, our guess gets better and closer to the real rate. But there's always some doubt, which is why we use confidence ranges
and check if results are important.
Examples with Results
Let's look at a couple of examples to show these ideas:
Example 1: Basic A/B Test
Here's the deal: An online shop is checking out two different product pages. They've got the old one, Version A, and a new one, Version B, with a bigger "Add to Cart" button that stands out.
What they found:
• Version A: 1000 people visited, 100 bought stuffs
• Version B: 1000 people visited, 120 bought stuffs
So, they figured out how well each version did:
• Version A: 100 / 1000 = 10% of people bought something
• Version B: 120 / 1000 = 12% of people bought something
They did some math to see if this difference matters:
Using a chi-square test or z-test for proportions, they got:
• p-value: 0.0386
• The difference has a 95% Confidence Interval from 0.2% to 3.8%
What this means:
The p-value is under 0.05, which shows the difference matters. We can say with 95% certainty that Version B's actual conversion rate beats Version A by 0.2% to 3.8%.
Example 2: A/B Test with Different Sample Sizes
Situation: A SaaS company wants to test two versions of how people sign up. Because of some tech problems, they didn't test the same number of people for each version.
Test Results:
• Version A: 5000 people visited, 750 signed up
• Version B: 4500 visitors, 720 conversions
Estimated Conversion Rates:
• Version A: 750 / 5000 = 15%
• Version B: 720 / 4500 = 16%
Statistical Analysis:
The right statistical tests might show:
• p-value: 0.1823
• 95% Confidence Interval for the difference: -0.5% to 2.5%
What This Means:
The p-value is higher than 0.05, which tells us we can't say for sure if there's a real difference between the two versions. The confidence interval includes 0, which backs up this idea.
Advantages of A/B Testing for Conversion Rate Optimization
• Using Facts to Make Choices
A/B testing gives you real info to help you decide stuff. This means you don't have to guess or just go with what you like.
• Lowering Risks
When you test changes before doing them for real, you can stop big mess-ups that might hurt your sales.
• Always Getting Better
A/B testing makes people want to keep making things better. Little changes can add up to big wins over time.
• Getting What Users Want
A/B testing helps businesses figure out what people like and how they act. This can help with making better ads and products.
• Results You Can Count
A/B testing gives clear measurable outcomes. This makes it easier to back up spending money on design tweaks or new stuff.
• Insights from Splitting Users
A/B tests can show how different groups of users react to changes. This lets you fine-tune your strategies for each group.
Common Mistakes and Best Practices
• Ending Tests Too Quick
It's key to figure out how many people you need to test beforehand. Don't give in to the urge to stop a test as soon as you see big differences. Stopping can lead to wrong conclusions.
• Testing Too Many Things
Doing many tests at once or one after another on the same info can make it more likely to get false positives. Using methods like the Bonferroni fix can help with this problem.
• Not Thinking About Outside Stuff
Things like seasonal changes, ad campaigns, or other outside events can affect test results. It's key to think about these things when looking at results.
• Forgetting About Real-World Impact
Just because something is important doesn't mean it matters in real life. A result that's big might not be worth making if the actual effect is too small to make the change worth it.
• Not Looking at Long-Term Effects
Some changes might give quick wins but hurt in the long run. When you can, it's smart to check how things turn out later.
Cool Stuff in A/B Testing Math
• Bayesian A/B Testing
Unlike old-school A/B testing Bayesian methods use what we already know and update this as we get more info. This lets us be more flexible about when to stop and makes results easier to
• Multi-Armed Bandit Algorithms
These algorithms send more visitors to versions that work better while the test is running. This can help us learn faster and waste less time compared to regular A/B tests.
• Sequential Analysis
Checking test results all the time with sequential analysis methods might let you stop tests earlier when clear winners show up.
• Multivariate Testing
When you need to test lots of things at once, multivariate testing works better than doing loads of A/B tests.
A/B testing helps businesses make their websites better and make smart choices based on data. By looking at how different versions do, companies can keep making their online stuff and marketing plans
better bit by bit.
It's super important to know the difference between real conversion rates and what we think they are in A/B testing. We can never know for sure what the real conversion rate is, but using the right
math tricks lets us make good choices based on our best guesses.
To do A/B testing right, you need to know stats, plan tests well, use enough people, and think hard about what you find. If you do it right, it can make your website work way better and help your
business grow. As tech and stats get better new stuff like Bayesian testing and multi-armed bandit algorithms are giving us more ways to make websites better. These fancy methods might help us test
things even faster and smarter.
In the end, A/B testing isn't just about finding what works best. It's about getting to know how people act and what they like. This info can shape your big plans and help you make stuff that people
want to use and buy. When companies use A/B testing and understand the math behind it, they can always get better. This helps them to make choices based on facts, which makes things better for users
and helps the business grow.
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BoB/Forward - BoB Biomechanics
BoB/Forward calculates the motion of a subject using forward dynamics analysis. The user inputs a combination of initial conditions, torques at joints, geometrical constraints, floor contact and
arbitrary forces acting on the body as functions of time and/or motion. BoB/Forward then integrates the equations of motion to calculate the resulting motion.
Forward dynamics and inverse dynamics can be combined in a single analysis – for example the motion of some, or all, of the joints can be imported for all times or specific times. BoB/Forward will
calculate the motion of the non-defined joint motions using the temporal and spatial boundary conditions of the inverse dynamics solution.
Movement and forces generated by BoB/Forward can be read into BoB/Research, BoB/Ergo and BoB/EMG for display and further analysis. | {"url":"https://www.bob-biomechanics.com/bob-forward/","timestamp":"2024-11-11T18:27:54Z","content_type":"text/html","content_length":"86400","record_id":"<urn:uuid:cf8235f7-fe25-4fac-9adf-828731d2c186>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00884.warc.gz"} |
How can two velocities be combined?
How can two velocities be combined?
Velocity is a vector that tells us the speed of an object and the direction the object is moving. This means we combine velocities by vector addition. If two velocities have the same direction we can
add them up, if two velocities have the opposite direction they substract from each other.
When two objects are moving the same velocity in the same direction?
If two bodies are moving in the same direction at the same velocity, then the relative velocity will be zero. (alternatively, this can be switched around to find VBA, which is the velocity of body B
relative to body A. This will result in the same value as VAB but in the opposite direction).
What is speed combined with direction?
Velocity is often thought of as an object’s speed with a direction. Thus, objects which are accelerating are either speeding up, slowing down or changing directions.
How do you find the combined velocity after a collision?
In a perfectly inelastic collision, the two objects stick together and move as one unit after the collision. Therefore, the final velocities of the two objects are the same, v′1=v′2=v′ v 1 ′ = v 2 ′
= v ′ . Thus, m1v1+m2v2=(m1+m2)v′ m 1 v 1 + m 2 v 2 = ( m 1 + m 2 ) v ′ .
Can an object have two velocities at the same time?
At one instant, the same body can have different velocities relative to say a fixed frame and a moving frame, or relative to two frames moving at different velocities.
When two objects are moving in parallel straight lines with different velocities in the same direction?
If two objects are moving in same direction, the magnitude of relative velocity of one object with respect to another is equal to difference in magnitude of two velocities. (ii) When two objects are
moving along parallel straight lines in opposite direction, angle between them is 180o.
In which condition the relative velocity of the two bodies moving in the same direction becomes zero?
The relative velocity becomes zero when the two bodies move in the same direction with the same velocity. When a person sits on the chair, the relative velocity of the person with respect to the
chair is zero. The relative velocity of the chair with the person is also zero.
Which have the same velocity?
Objects have the same velocity only if they are moving at the same speed and in the same direction. Objects moving at different speeds, in different directions, or both have different velocities.
Is velocity speed with direction?
Speed is the time rate at which an object is moving along a path, while velocity is the rate and direction of an object’s movement.
What is the final velocity of the combined mass?
The final velocity of the combined objects depends on the masses and velocities of the two objects that collided. The units for the initial and final velocities are m/s, and the unit for mass is kg.
What are the velocities of the two objects after the collision?
In a collision, the velocity change is always computed by subtracting the initial velocity value from the final velocity value. If an object is moving in one direction before a collision and rebounds
or somehow changes direction, then its velocity after the collision has the opposite direction as before.
What is the relative velocity of two cars in the same direction?
If both cars are travelling in the same direction, one at 25 ms-1 and the other at 35 ms-1 then their relative velocity is 10 ms-1 (by vector addition). If they are moving in opposite directions,
however, the relative velocity of one car with respect to the other is therefore 60ms-1 (See Figure 1).
When two objects are moving with same velocity in same direction then the relative velocity of first with respect to second is?
If two objects are moving in same direction, the magnitude of relative velocity of one object with respect to another is equal to difference in magnitude of two velocities.
What is the relative velocity of two moving objects zero?
(A) : Relative velocity is zero when two bodies are moving opposite to each other with same velocity ( R) : Relative velocity of a body does not depend on direction of motion.
Can the relative velocities of two bodies be greater than the absolute velocity of either body give reasons?
Solution : Yes, when two bodies move in opposite directions, the relative velocity of each is greater than the individual velocity of either body .
Which momentum has the same velocity?
Solution : Let two bodies have masses m and M such that M> m having velocities equal to v. We know that momentum mass `xx` velocity `implies p= mv`. So for bodies having equal velocities, momentum is
directly proportional to mass of body. Therefore body with mass M will have more momentum than body of mass m. | {"url":"https://www.davidgessner.com/life/how-can-two-velocities-be-combined/","timestamp":"2024-11-03T12:26:58Z","content_type":"text/html","content_length":"44741","record_id":"<urn:uuid:4b16ee7d-f31e-4316-b283-9b628439b3b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00336.warc.gz"} |
New approach to vertex connectivity could maximize networks’ bandwidth
Computer scientists are constantly searching for ways to squeeze ever more bandwidth from communications networks.
Now a new approach to understanding a basic concept in graph theory, known as “vertex connectivity,” could ultimately lead to communications protocols — the rules that govern how digital messages are
exchanged — that coax as much bandwidth as possible from networks.
Graph theory plays a central role in mathematics and computer science, and is used to describe the relationship between different objects. Each graph consists of a number of nodes, or vertices, which
represent the objects, and connecting lines between them, known as edges, which signify the relationships between them. A communications network, for example, can be represented as a graph with each
node in the network being one vertex, and a connection between two nodes depicted as an edge.
One of the fundamental concepts within graph theory is connectivity, which has two variants: edge connectivity and vertex connectivity. These are numbers that determine how many lines or nodes would
have to be removed from a given graph to disconnect it. The lower the edge-connectivity or vertex-connectivity number of a graph, therefore, the easier it is to disconnect, or break apart.
In this way both concepts show how robust a network is against failure, and how much flow can pass through it — whether the flow of information in a communications network, traffic flow in a
transportation system, or fluid flow in hydraulics.
Reducing edge connectivity’s edge
However, while a great deal of research has been carried out in mathematics to solve problems associated with edge connectivity, there has been relatively little success in answering questions about
vertex connectivity.
But at the ACM-SIAM Symposium on Discrete Algorithms in Portland, Ore., in January, Mohsen Ghaffari, a graduate student in the Computer Science and Artificial Intelligence Laboratory at MIT, will
present a new technique for addressing vertex-connectivity problems.
“This could ultimately help us understand how to build more robust and faster networks,” says Ghaffari, who developed the new approach alongside Keren Censor-Hillel at the Technion and Fabian Kuhn at
the University of Freiburg.
In the 1960s, mathematicians William Tutte and Crispin Nash-Williams separately developed theories about structures called edge-disjoint spanning trees, which now serve as one of the key technical
tools in many problems about edge connectivity.
A spanning tree is a subgraph — or a graph-within-a-graph — in which all of the nodes are connected by the smallest number of edges. A set of spanning trees within a graph are called “edge-disjoint”
if they do not share any of these connecting lines.
If a network contains three edge-disjoint spanning trees, for example, information can flow in parallel along each of these trees at the same time, meaning three times more bandwidth than would be
possible in a graph containing just one tree. The higher the number of edge-disjoint spanning trees, the larger the information flow, Ghaffari says. “The results of Tutte and Nash-Williams show that
each graph contains almost as many spanning trees as its edge connectivity,” he says.
Now the team has created an analogous theory about vertex connectivity. They did this by breaking down the graph into separated groups of nodes, known as connected dominating sets. In graph theory, a
group of nodes is called a connected dominating set if all of the vertices within it are connected to one another, and any other node within the graph is adjacent to at least one of those inside the
In this way, information can be disseminated among the nodes of the set, and then passed to any other node in the network.
So, in a similar way to Tutte and Nash-Williams’ results for edge connectivity, “each graph contains almost as many vertex-disjoint connected dominating sets as its vertex connectivity,” Ghaffari
“So if you think of an application like broadcasting information through a network, we can now decompose the network into many groups, each being one connected dominating set,” he says. “Each of
these groups is then going to be responsible for broadcasting some set of the messages, and all groups work in parallel to broadcast all the messages fast — almost as fast as possible.”
The team has now developed an algorithm that can carefully decompose a network into many connected dominating sets. In this way, it can structure so-called wireless ad hoc networks, in which
individual nodes route data by passing it from one to the next to ensure the best possible speed of information flow. “We want to be able to spread as much information as possible per unit of time,
to create faster and faster networks,” Ghaffari says. “And when a graph has a better vertex connectivity, it allows a larger flow [of information],” he adds.
Applications in assessing robustness
The researchers can also use their new approach to analyze the robustness of a network against random failures. “These new techniques also allow us to analyze whether a network is likely to remain
connected when its nodes fail randomly with some given probability,” Ghaffari says. “Reliability against random edge failures is well understood, but we knew much less about that against node
failures,” he adds.
Noga Alon, a professor of mathematics and computer science at Tel Aviv University, says Ghaffari and his fellow authors have identified the notion that determines the largest achievable flow when
broadcasting messages using routing in communication networks.
“The investigation of this notion, vertex disjoint connected dominating sets, is treated in this paper by an elegant combination of combinatorial, probabilistic, and algorithmic techniques,” he says. | {"url":"https://news.mit.edu/2013/new-approach-to-vertex-connectivity-could-maximize-networks-bandwidth-1224","timestamp":"2024-11-04T21:01:04Z","content_type":"text/html","content_length":"102817","record_id":"<urn:uuid:6a96a3f3-59be-4fc9-a8b1-712ee710b75d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00338.warc.gz"} |
Derivative of sech x
Introduction to the Derivative of sech x?
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative sech x can be calculated by following the derivative rules. Or, we can directly find the
sechx differentiation by applying the first principle of differentiation. In this article, you will learn what the differentiation of sech x is and how to calculate the sech x derivative by using
different approaches.
What is the derivative of sechx?
The derivative of sech(x) is defined as the negative of sech x multiplied by tanh x and denoted by d/dx(sech x). This derivative represents the rate of change of the hyperbolic function sech x with
respect to the variable 'x'. The function sech x is composed of two exponential functions, e^x and e^-x, as defined by the equation;
$\sech x =\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}$
By finding the derivative of sech x, we can gain a better understanding of the behavior of this function in calculus and other areas of mathematics.
Sechx differentiation formula
The derivative of the hyperbolic secant function, denoted by d/dx(sech x), can be calculated using the formula -sech x tanh x. Mathematically,
$\frac{d}{dx}(\sech x)=-\sech x\tanh x$
This formula represents the product of the secant and tangent functions, and it provides a way to determine the rate of change of sech x with respect to the variable 'x'. Specifically, the d/dx sechx
is equal to -sech x times tanh x.
How do you prove the derivative of sechx?
There are multiple ways to derive derivatives of sech x. Three commonly used methods are;
1. First Principle
2. Chain Rule
3. Quotient Rule
Each method provides a different way to compute the sech x differentiation. By using these methods, we can mathematically prove the formula for finding the differentiation of sechx.
Derivative of sech x by first principle
According to the first principle of derivative, the ln sech x derivative is equal to -sech xtanh x . The derivative of a function by first principle refers to finding a general expression for the
slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,
$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$
You can also use our derivative by definition calculator as it also follows the above formula.
Proof of sechx derivative by first principle
To prove the differentiation of sech x by using first principle, we start by replacing f(x) by sech x.
$f'(x)=\lim_{h\to 0} \frac{\DeclareMathOperator{\sech}{sech}\sech(x+h)-\sech x}{h}$
Since $\sech x=\frac{1}{\cosh x}$, therefore,
$f'(x)=\lim_{h\to 0}\frac{\frac{1}{\cosh(x+h)}-\frac{1}{\cosh x}}{h}$
More simplifying,
$f'(x)=\lim_{h\to 0}\frac{\cosh x-\cosh(x+h)}{h\cosh(x+h)\cosh x}$
By sum to product formulas, cosh A - cosh B = 2 sin (A+B)/2 sin (A-B)/2. So,
$f'(x)=\frac{1}{\cosh x}\lim_{h\to 0} \frac{1}{h}\left(\frac{2\sinh\frac{x+x+h}{2} \sinh\frac{x-x-h}{2}}{\cosh(x + h)}\right)$
$f'(x)=\frac{1}{\cosh x}\lim_{h\to 0}\frac{1}{h}\left(\frac{2\sinh\frac{2x+h}{2} \sinh\frac{-h}{2}}{\cosh(x + h)}\right)$
Multiply and divide by h/2,
$f'(x)=-\frac{1}{\cosh x}\lim_{h\to 0}(\frac{1}{h})(\frac{h}{2})\left(\frac{2\sinh\frac{2x+h}{2}\frac{\sinh h/2}{h/2}}{\cosh(x + h)}\right)$
When h approaches to zero, h/2 also approaches to zero, therefore,
$f'(x)=-\frac{1}{\cosh x}\lim_{h\to 0}\frac{\sinh(h/2)}{(h/2)}\times\lim_{h\to 0}\frac{\sinh\frac{2x+h}{2}}{\cosh(x + h)}$
We know that lim sinh x/x=1,
$f'(x)=-\frac{1}{\cosh x}\times\frac{\sinh x}{\cosh x}$
We know that 1/cosh x = sech x and sinh x/cosh x = tanh x. So
$f'(x)=-\sech x\tanh x$
Hence the derivative of sechx is equal to the product of sech x and tanh x. The process of finding differentiation of sech x is called hyperbolic differentiation.
Differentiation of sechx by chain rule
To calculate the sechx differentiation, we can use the chain rule since the cosine function can be expressed as a combination of two functions. The chain rule of derivatives states that the
derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The chain rule of derivative is defined as;
Proof of sechx derivative by chain rule
To prove the derivative of sech x by using chain rule, we start by assuming that,
$f(x)=\sech x=\frac{1}{\cosh x}=(\cosh x)^{-1}$
By using the leibniz notation calculator,
$f'(x)=\frac{-1}{(\cosh x)^{-2}}\frac{d}{dx}(\cosh x)$
$f'(x)=-\frac{1}{\cosh^2x}(-\sinh x)$
$f'(x)=\frac{\sinh x}{\cos^2x}$
Since sinh x / cosh x =tan x and 1/cosh x = sech x, therefore we have
$f'(x)=\sech x\tanh x$
Derivative of sech x using quotient rule
Another method for finding the derivative of sech(x) is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. The derivative of cosecant can also be
calculated using the quotient rule. The formula for the quotient rule derivative calculator is defined as:
Proof of sech x derivative by quotient rule
To prove the derivative of sechx, we can start by writing it as,
$f(x)=\sech x=\frac{1}{\cosh x}=\frac{u}{v}$
Supposing that u = 1 and v = cosh x. Now by derivative quotient rule,
$f'(x)=\frac{\cosh x\frac{d}{dx}(1)-1\frac{d}{dx}(\cosh x)}{(\cosh x)^2}$
$f'(x)=\frac{\cosh x (0)-\sinh x}{\cosh^2x}$
$f'(x)=-\frac{\sinh x}{\cosh^2x}$
$f'(x)=\frac{1}{\cosh x}\times\frac{-\sinh x}{\cosh x}$
$f'(x)=-\sech x\tanh x$
How to find the sechx derivative with a calculator?
The easiest way to calculate the derivative of sech x is by using an online tool. You can use our derivative calculator with steps for this. Here, we provide you a step-by-step way to calculate
derivatives by using this tool.
1. Write the function as sech x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the derivative of sechx.
2. Now, select the variable by which you want to differentiate sech x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate secant x. In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button.
After completing these steps, you will receive the sech x differentiation within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex | {"url":"https://calculator-derivative.com/derivative-of-sech-x","timestamp":"2024-11-03T23:18:17Z","content_type":"text/html","content_length":"37074","record_id":"<urn:uuid:9d5391d3-d8be-4bde-9a3a-b9b3b6323b27>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00068.warc.gz"} |
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Hence, sometimes when we write and give a few notes of the lesson, we get “couldn’t top article the page today”. We bring up the students of the lessons in their homework and then we can discuss the
page around them first and “class notes” they share with their teacher and students and also we also bring a paper and some pencil. So overall we will write the lesson a ton and use it in class.
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by applying a new form that your students have made. If you don’t want your “student” to be confused, don’t waste any time and try not to | {"url":"https://workmyexam.com/who-can-take-my-statistics-exam-efficiently","timestamp":"2024-11-04T07:31:53Z","content_type":"text/html","content_length":"166767","record_id":"<urn:uuid:ef075cd4-991b-44cb-bbcc-263feea6e410>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00347.warc.gz"} |
This vignette describes the process of aggregating indicators, in COINr.
Aggregation is the operation of combining multiple indicators into one value. Many composite indicators have a hierarchical structure, so in practice this often involves multiple aggregations, for
example aggregating groups of indicators into aggregate values, then aggregating those values into higher-level aggregates, and so on, until the final index value.
Aggregating should almost always be done on normalised data, unless the indicators are already on very similar scales. Otherwise the relative influence of indicators will be very uneven.
Of course you don’t have to aggregate indicators at all, and you might be content with a scoreboard, or perhaps aggregating into several aggregate values rather than a single index. However, consider
that aggregation should not substitute the underlying indicator data, but complement it.
Overall, aggregating indicators is a form of information compression - you are trying to combine many indicator values into one, and inevitably information will be lost (this recent paper may be of
interest). As long as this is kept in mind, and indicator data is presented and made available along side aggregate values, then aggregate (index) values can complement indicators and be used as a
useful tool for summarising the underlying data, and identifying overall trends and patterns.
Many aggregation methods involve some kind of weighting, i.e. coefficients that define the relative weight of the indicators/aggregates in the aggregation. In order to aggregate, weights need to
first be specified, but to effectively adjust weights it is necessary to aggregate.
This chicken and egg conundrum is best solved by aggregating initially with a trial set of weights, perhaps equal weights, then seeing the effects of the weighting, and making any weight adjustments
The most straightforward and widely-used approach to aggregation is the weighted arithmetic mean. Denoting the indicators as \(x_i \in \{x_1, x_2, ... , x_d \}\), a weighted arithmetic mean is
calculated as:
\[ y = \frac{1}{\sum_{i=1}^d w_i} \sum_{i=1}^d x_iw_i \]
where the \(w_i\) are the weights corresponding to each \(x_i\). Here, if the weights are chosen to sum to 1, it will simplify to the weighted sum of the indicators. In any case, the weighted mean is
scaled by the sum of the weights, so weights operate relative to each other.
Clearly, if the index has more than two levels, then there will be multiple aggregations. For example, there may be three groups of indicators which give three separate aggregate scores. These
aggregate scores would then be fed back into the weighted arithmetic mean above to calculate the overall index.
The arithmetic mean has “perfect compensability”, which means that a high score in one indicator will perfectly compensate a low score in another. In a simple example with two indicators scaled
between 0 and 10 and equal weighting, a unit with scores (0, 10) would be given the same score as a unit with scores (5, 5) – both have a score of 5.
An alternative is the weighted geometric mean, which uses the product of the indicators rather than the sum.
\[ y = \left( \prod_{i=1}^d x_i^{w_i} \right)^{1 / \sum_{i=1}^d w_i} \]
This is simply the product of each indicator to the power of its weight, all raised the the power of the inverse of the sum of the weights.
The geometric mean is less compensatory than the arithmetic mean – low values in one indicator only partially substitute high values in others. For this reason, the geometric mean may sometimes be
preferred when indicators represent “essentials”. An example might be quality of life: a longer life expectancy perhaps should not compensate severe restrictions on personal freedoms.
A third type of mean, in fact the third of the so-called Pythagorean means is the weighted harmonic mean. This uses the mean of the reciprocals of the indicators:
\[ y = \frac{\sum_{i=1}^d w_i}{\sum_{i=1}^d w_i/x_i} \]
The harmonic mean is the the least compensatory of the the three means, even less so than the geometric mean. It is often used for taking the mean of rates and ratios.
Other methods
The weighted median is also a simple alternative candidate. It is defined by ordering indicator values, then picking the value which has half of the assigned weight above it, and half below it. For
ordered indicators \(x_1, x_2, ..., x_d\) and corresponding weights \(w_1, w_2, ..., w_d\) the weighted median is the indicator value \(x_m\) that satisfies:
\[ \sum_{i=1}^{m-1} w_i \leq \frac{1}{2}, \: \: \text{and} \sum_{i=m+1}^{d} w_i \leq \frac{1}{2} \]
The median is known to be robust to outliers, and this may be of interest if the distribution of scores across indicators is skewed.
Another somewhat different approach to aggregation is to use the Copeland method. This approach is based pairwise comparisons between units and proceeds as follows. First, an outranking matrix is
constructed, which is a square matrix with \(N\) columns and \(N\) rows, where \(N\) is the number of units.
The element in the \(p\)th row and \(q\)th column of the matrix is calculated by summing all the indicator weights where unit \(p\) has a higher value in those indicators than unit \(q\). Similarly,
the cell in the \(q\)th row and \(p\)th column (which is the cell opposite on the other side of the diagonal), is calculated as the sum of the weights unit where \(q\) has a higher value than unit \
(p\). If the indicator weights sum to one over all indicators, then these two scores will also sum to 1 by definition. The outranking matrix effectively summarises to what extent each unit scores
better or worse than all other units, for all unit pairs.
The Copeland score for each unit is calculated by taking the sum of the row values in the outranking matrix. This can be seen as an average measure of to what extent that unit performs above other
Clearly, this can be applied at any level of aggregation and used hierarchically like the other aggregation methods presented here.
In some cases, one unit may score higher than the other in all indicators. This is called a dominance pair, and corresponds to any pair scores equal to one (equivalent to any pair scores equal to
The percentage of dominance pairs is an indication of robustness. Under dominance, there is no way methodological choices (weighting, normalisation, etc.) can affect the relative standing of the pair
in the ranking. One will always be ranked higher than the other. The greater the number of dominance (or robust) pairs in a classification, the less sensitive country ranks will be to methodological
assumptions. COINr allows to calculate the percentage of dominance pairs with an inbuilt function.
We now turn to how data sets in a coin can be aggregated using the methods described previously. The function of interest is Aggregate(), which is a generic with methods for coins, purses and data
frames. To demonstrate COINr’s Aggregate() function on a coin, we begin by loading the package, and building the example coin, up to the normalised data set.
# build example up to normalised data set
coin <- build_example_coin(up_to = "Normalise")
#> iData checked and OK.
#> iMeta checked and OK.
#> Written data set to .$Data$Raw
#> Written data set to .$Data$Denominated
#> Written data set to .$Data$Imputed
#> Written data set to .$Data$Screened
#> Written data set to .$Data$Treated
#> Written data set to .$Data$Normalised
Consider what is needed to aggregate the normalised data into its higher levels. We need:
• The data set to aggregate
• The structure of the index: which indicators belong to which groups, etc.
• Weights to assign to indicators
• Specifications for aggregation: an aggregation function (e.g. the weighted mean) and any other parameters to be passed to that function
All of these elements are already present in the coin, except the last. For the first point, we simply need to tell Aggregate() which data set to use (using the dset argument). The structure of the
index was defined when building the coin in new_coin() (the iMeta argument). Weights were also attached to iMeta. Finally, specifications can be specified in the arguments of Aggregate(). Let’s begin
with the simple case though: using the function defaults.
# aggregate normalised data set
coin <- Aggregate(coin, dset = "Normalised")
#> Written data set to .$Data$Aggregated
By default, the aggregation function performs the following steps:
• Uses the weights that were attached to iMeta
• Aggregates hierarchically (with default method of weighted arithmetic mean), following the index structure specified in iMeta and using the data specified in dset
• Creates a new data set .$Data$Aggregated, which consists of the data in dset, plus extra columns with scores for each aggregation group, at each aggregation level.
Let’s examine the new data set. The columns of each level are added successively, working from level 1 upwards, so the highest aggregation level (the index, here) will be the last column of the data
dset_aggregated <- get_dset(coin, dset = "Aggregated")
nc <- ncol(dset_aggregated)
# view aggregated scores (last 11 columns here)
dset_aggregated[(nc - 10) : nc] |>
head(5) |>
#> ConEcFin Environ Instit P2P Physical Political Social SusEcFin Conn Sust
#> 1 12.6 31.9 52.4 39.50 34.8 52.5 71.9 55.7 38.4 53.2
#> 2 26.2 69.5 77.5 54.10 41.1 78.2 72.8 62.9 55.4 68.4
#> 3 48.2 53.0 75.6 43.30 72.0 80.8 86.2 50.1 64.0 63.1
#> 4 13.3 81.7 26.5 5.85 22.9 32.4 27.5 64.6 20.2 57.9
#> 5 24.6 55.7 75.9 27.10 28.4 67.5 53.3 61.7 44.7 56.9
#> Index
#> 1 45.8
#> 2 61.9
#> 3 63.5
#> 4 39.0
#> 5 50.8
Here we see the level 2 aggregated scores created by aggregating each group of indicators (the first eight columns), followed by the two sub-indexes (level 3) created by aggregating the scores of
level 2, and finally the Index (level 4), which is created by aggregating the “Conn” and “Sust” sub-indexes.
The format of this data frame is not hugely convenient for inspecting the results. To see a more user-friendly version, use the get_results() function.
COINr aggregation functions
Let’s now explore some of the options of the Aggregate() function. Like other coin-building functions in COINr, Aggregate() comes with a number of inbuilt options, but can also accept any function
that is passed to it, as long as it satisfies some requirements. COINr’s inbuilt aggregation functions begin with a_, and are:
• a_amean(): the weighted arithmetic mean
• a_gmean(): the weighted geometric mean
• a_hmean(): the weighted harmonic mean
• a_copeland(): the Copeland method (note: requires by_df = TRUE)
For details of these methods, see Approaches above and the function documentation of each of the functions listed.
By default, the arithmetic mean is called but we can easily change this to the geometric mean, for example. However here we run into a problem: the geometric mean will fail if any values to aggregate
are less than or equal to zero. So to use the geometric mean we have to re-do the normalisation step to avoid this. Luckily this is straightforward in COINr:
coin <- Normalise(coin, dset = "Treated",
global_specs = list(f_n = "n_minmax",
f_n_para = list(l_u = c(1,100))))
#> Written data set to .$Data$Normalised
#> (overwritten existing data set)
Now, since the indicators are scaled between 1 and 100 (instead of 0 and 100 as previously), they can be aggregated with the geometric mean.
External functions
All of the four aggregation functions mentioned above have the same format (try e.g. ?a_gmean), and are built into the COINr package. But what if we want to use another type of aggregation function?
The process is exactly the same.
NOTE: the compind package has been disabled here from running the commands in this vignette because of changes to a dependent package which are causing problems with the R CMD check. The commands
should still work if you run them, but the results will not be shown here.
In this section we use some functions from other packages: the matrixStats package and the Compind package. These are not imported by COINr, so the code here will only work if you have these
installed. If this vignette was built on your computer, we have to check whether these packages are installed:
# ms_installed <- requireNamespace("matrixStats", quietly = TRUE)
# ms_installed
# ci_installed <- requireNamespace("Compind", quietly = TRUE)
# ci_installed
If either of these have returned FALSE, in the following code chunks you will see some blanks. See the online version of this vignette to see the results, or install the above packages and rebuild
the vignettes.
Now for an example, we can use the weightedMedian() function from the matrixStats package. This has a number of arguments, but the ones we will use are x and w (with the same meanings as COINr
functions), and na.rm which we need to set to TRUE.
# RESTORE above eval=ms_installed
# load matrixStats package
# library(matrixStats)
# # aggregate using weightedMedian()
# coin <- Aggregate(coin, dset = "Normalised",
# f_ag = "weightedMedian",
# f_ag_para = list(na.rm = TRUE))
The weights w do not need to be specified in f_ag_para because they are automatically passed to f_ag unless specified otherwise.
The general requirements for f_ag functions passed to Aggregate() are that:
1. The input to the function is a numeric vector x, possibly with missing values
2. The function returns a single (scalar) aggregated value
3. If the function accepts a vector of weights, this vector (of the same length of x) is passed as function argument w. If the function doesn’t accept a vector of weights, we can set w = "none" in
the arguments to Aggregate(), and it will not try to pass w.
4. Any other arguments to f_ag, apart from x and w, should be included in the named list f_ag_para.
Sometimes this may mean that we have to create a wrapper function to satisfy these requirements. For example, the ‘Compind’ package has a number of sophisticated aggregation approaches. The “benefit
of the doubt” uses data envelopment analysis to aggregate indicators, however the function Compind::ci_bod() outputs a list. We can make a wrapper function to use this inside COINr:
# RESTORE ABOVE eval= ci_installed
# NOTE: this chunk disabled - see comments above.
# load Compind
# suppressPackageStartupMessages(library(Compind))
# # wrapper to get output of interest from ci_bod
# # also suppress messages about missing values
# ci_bod2 <- function(x){
# suppressMessages(Compind::ci_bod(x)$ci_bod_est)
# }
# # aggregate
# coin <- Aggregate(coin, dset = "Normalised",
# f_ag = "ci_bod2", by_df = TRUE, w = "none")
The benefit of the doubt approach automatically assigns individual weights to each unit, so we need to specify w = "none" to stop Aggregate() from attempting to pass weights to the function.
Importantly, we also need to specify by_df = TRUE which tells Aggregate() to pass a data frame to f_ag rather than a vector.
Data availability limits
Many aggregation functions will return an aggregated value as long as at least one of the values passed to it is non-NA. For example, R’s mean() function:
# data with all NAs except 1 value
x <- c(NA, NA, NA, 1, NA)
#> [1] NA
mean(x, na.rm = TRUE)
#> [1] 1
Depending on how we set na.rm, we either get an answer or NA, and this is the same for many other aggregation functions (e.g. the ones built into COINr). Sometimes we might want a bit more control.
For example, if we have five indicators in a group, it might only be reasonable to give an aggregated score if, say, at least three out of five indicators have non-NA values.
The Aggregate() function has the option to specify a data availability limit when aggregating. We simply set dat_thresh to a value between 0 and 1, and for each aggregation group, any unit that has a
data availability lower than dat_thresh will get a NA value instead of an aggregated score. This is most easily illustrated on a data frame (see next section for more details on aggregating in data
df1 <- data.frame(
i1 = c(1, 2, 3),
i2 = c(3, NA, NA),
i3 = c(1, NA, 1)
#> i1 i2 i3
#> 1 1 3 1
#> 2 2 NA NA
#> 3 3 NA 1
We will require that at least 2/3 of the indicators should be non-NA to give an aggregated value.
# aggregate with arithmetic mean, equal weight and data avail limit of 2/3
Aggregate(df1, f_ag = "a_amean",
f_ag_para = list(w = c(1,1,1)),
dat_thresh = 2/3)
#> [1] 1.666667 NA 2.000000
Here we see that the second row is aggregated to give NA because it only has 1/3 data availability.
By level
We can also use a different aggregation function for each aggregation level by specifying f_ag as a vector of function names rather than a single function.
coin <- Aggregate(coin, dset = "Normalised", f_ag = c("a_amean", "a_gmean", "a_amean"))
#> Written data set to .$Data$Aggregated
#> (overwritten existing data set)
In this example, there are four levels in the index, which means there are three aggregation operations to be performed: from Level 1 to Level 2, from Level 2 to Level 3, and from Level 3 to Level 4.
This means that f_ag vector must have n-1 entries, where n is the number of aggregation levels. The functions are run in the order of aggregation.
In the same way, if parameters need to be passed to the functions specified by f_ag, f_ag_para can be specified as a list of length n-1, where each element is a list of parameters. | {"url":"https://cran.ma.imperial.ac.uk/web/packages/COINr/vignettes/aggregate.html","timestamp":"2024-11-12T23:41:30Z","content_type":"text/html","content_length":"47637","record_id":"<urn:uuid:8867291b-f6d6-4d4a-bf66-33e4ae90801c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00813.warc.gz"} |
Distribution Links
In this video, Salman Khan of Khan Academy explains binomial distribution. Part 2 of 4.
In this video, Salman Khan of Khan Academy explains binomial distribution. Part 1 of 4.
Gives a brief introduction and overview of the fundamental statistic concepts.
It gives definitions and examples to statistic terminology and problems.
In this video, Salman Khan of Khan Academy explains the normal distribution.
Flash tutorial with clear explanations about what deviation and variance mean along with how to calculate these by hand, using Excel and using the calculator.
Very comprehensive list of statistics formulas.
Bayes Theorem
It also gives formulas, description, etc for statistics.
In this video, Salman Khan of Khan Academy explains binomial distribution. Part 4 of 4. | {"url":"https://www.tutor.com/resources/math/statistics/distribution-links","timestamp":"2024-11-10T05:28:46Z","content_type":"application/xhtml+xml","content_length":"69276","record_id":"<urn:uuid:db8cbb58-b53f-4b4c-913b-23eac3ea03cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00444.warc.gz"} |
alex hall
name: Alex Hall age: 12 school: New Rickstones Academy facts: i love spongebob and i created this protopage.
name: Alex Hall age: 12 school: New Rickstones Academy facts: i love spongebob, totoro and i created this protopage.
i have been to Reykjavic Iceland with 1st silver end scouts
to find the surface area of a 3D shape you have to: draw the net of the shape you want find the area of all of the individual shapes of the net then add all of the areas to give you a total!
to find the area of a cylinder you have to draw the net which consists of just a rectangle and two circles. find the area the rectangle: LxW or length times width. finding the area of the two
circles: PIr sqaured or PI times the radius squared, pi is just a number and is usually 3.142 but can be used as just 3, diameter is the line that goes directly throught the centre of the circle from
one edge of the circle to another. because both of the circles are the same size the area of them is the same add it all up: now you just have to add the area of the rectangle and the area of the two
circles, giving you the surface area.
this is the formula used to find the equation of a line on a straight line graph. the m and the c are just numbers. the m is the gradient, and the c is where the line crosses the y axis.
if your graph has two or more lines, if tyhey are parallel then thier gradient will always be the same.
probability is how likely something is going to happen. if you have a dice with the numbers 1-6, the probability of rolling a 4 is 1/6 if you have 12 marbles in a bag, 3 blue 2 red and 5 green, the
probability of getting a blue marble is 3/10 the probability of getting a red marble is 2/10 and the probability of getting a green marble is 5/10.
width x height THIS PAGE EXPLAINS HOW TO DO IT: http://www.mymaths.co.uk/tasks/library/loadLesson.asp?title=areas/traparea&taskID=1128
a simple way of multiplying two decimals is: 0.3x0.7=0.21 but you can make it easier if you get stuck: 0.3x10=3 0.7x10=7 3x7=21 but because you have multiplied by 10 you have to divide it by 10 the
get your final answer: 21/10=2.1 2.1/10=0.21 0.21 is you final answer. you can do this with any two fractions.
diameter x pi diameter=length from one sid of the circle to the other, crossing the middle point. pi=3.14 radius=diameter/2 | {"url":"http://www.protopage.com/ahall2001","timestamp":"2024-11-11T00:31:26Z","content_type":"text/html","content_length":"57196","record_id":"<urn:uuid:b6cfc292-d12d-4355-b4fc-6585b3880ae0>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00684.warc.gz"} |
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• free integer activities | {"url":"https://www.softmath.com/math-com-calculator/factoring-expressions/algebra-equation.html","timestamp":"2024-11-12T03:31:15Z","content_type":"text/html","content_length":"212281","record_id":"<urn:uuid:e36a6c54-6faf-46e8-ab07-c5ad9abc376d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00439.warc.gz"} |
The total charge to rent a car for one day from Company J consists of
Question Stats:
78% 22% (01:58) based on 668 sessions
The total charge to rent a car for one day from Company J consists of a fixed charge of $15.00 plus a charge of $0.20 per mile driven. The total charge to rent a car for one day from Company K
consists of a fixed charge of $20.00 plus a charge of $0.10 per mile driven. Is the total charge to rent a car from company J for one day and drive it x miles less than $25.00?
(1) The total charge to rent a car from Company K for one day and drive it x miles is less than $25.00
(2) x < 50
And this is exactly why I am having a super hard time with data sufficiency questions. It is NOT mentioned in the question prompt at all that the same number of miles is driven with the car rented
from company J and the one rented from company K.
Statement No. 1 says absolutely nothing about company J at all. How can this be sufficient?
Of course I could have solved it if they would have stated that x miles driven with the K company car and J company car are the same. X is variable and can express any number. It is not a constant.
How could you possibly assume it is the same for both cases?
I am totally baffled here!
x there is some specific number even though we don't know what it is.
The total charge to rent a car for one day from Company J consists of a fixed charge of $15.00 plus a charge of $0.20 per mile driven. The total charge to rent a car for one day from Company K
consists of a fixed charge of $20.00 plus a charge of $0.10 per mile driven. Is the total charge to rent a car from company J for one day and drive it x miles less than $25.00?
Is the total charge to rent a car from company J for one day and drive it x miles less than $25.00: is 15 + 0.2x < 25 --> is x < 50?
(1) The total charge to rent a car from Company K for one day and drive it x miles is less than $25.00.
20 + 01x < 25 --> x < 50. Sufficient.
(2) x < 50. Sufficient.
Answer: D.
Hope it's clear. | {"url":"https://gmatclub.com/forum/the-total-charge-to-rent-a-car-for-one-day-from-company-j-consists-of-218579.html","timestamp":"2024-11-06T14:34:38Z","content_type":"application/xhtml+xml","content_length":"896402","record_id":"<urn:uuid:a8c5fd20-871a-452b-8d94-05c882ec3892>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00502.warc.gz"} |
Uncertainty Relations for Some Central Potentials in N-Dimensional Space
Uncertainty Relations for Some Central Potentials in N-Dimensional Space ()
Received 16 December 2016; accepted 21 March 2016; published 24 March 2016
1. Introduction
Generally, uncertainty relations form an important part in the foundations of quantum mechanics and play a crucial role in the development of quantum information and computation [1] [2] . These
relations establish the existence of an irreducible lower bound for the uncertainty in the results of simultaneous measurements of non- commuting observables. In other words, the precision with which
incompatible physical observables can be prepared is limited by an upper bound. In particular, the Heisenberg uncertainty principle (HUP) [3] represents one of the fundamental properties of a quantum
system. It gives an irreducible lower bound on the uncertainty in the outcomes of simultaneous measurements of position and momentum. Originally, HUP came from a thought experiment about measurements
of the position and momentum, but later Kennard [4] derived a mathematical formulation of HUP by considering inherent quantum fluctuations of position and momentum without any reference to
measurement process and which was then generalized by Robertson [5] for arbitrary incompatible observables. Recently, Fujikawa [6] proposed a universally valid uncertainty relation which incorporated
both the intrinsic quantum fluctuations and measurement effects. There has been a continual interest in utilizing HUP in different settings. For example, it has been used in the study of central
potentials [7] - [11] and others consider its connection to geometry [12] [13] . Furthermore, it has been generalized to describe a minimal length as a minimal uncertainty in position measurement
[14] - [18] through the modification of Heisenberg commutation relation into a generalized form. The existence of a minimal length has long been suggested in quantum gravity and string theory [19] -
[24] , and has been proposed to describe, as an effective theory, non-point like particles like hadrons, quasi-particles or collective excitations [25] . In its original formulation, HUP is expressed
in terms of variances of position and momentum of a particle. Such variances do not necessarily exist, and if they do, they describe the quantum probability distribution relative to a specific point
of the probability domain. Therefore, various alternative formulations have been suggested by the use of information-theoretic uncertainty measures like the Shannon entropy [26] [27] , Renyi
entropies [28] [29] , Tsallis entropies [30] , entropic moments [31] [32] and Fisher information [32] - [36] . During the past years, the generalization of three dimensional quantum problems to
higher space dimensions receives a considerable development in theoretical and mathematical physics. For example, the central potentials, as hydrogen-like atoms [37] - [42] and harmonic oscillators
[43] - [46] are being used as prototypes for other purposes in N-dimensional physics. Furthermore, the confined harmonic oscillator [45] and the confined hydrogen atom [47] have been discussed. The
purpose of the present paper is to de-
rive and discuss the uncertainty product
The lower bound for this product is analyzed and compared with other previous results that have been obtained by other methods. Our method is based on the virial theorem applied to the harmonic
oscillator and the hydrogen atom systems to obtain the uncertainty product, while for the spherical well, the zeros of spherical Bessel functions are used for finding numerical results for the
uncertainty product. Over the last years, the virial theorem technique has been employed in the study of physical quantities [48] [49] . Interesting features for the lower bound are discussed with a
special attention explored for the large space dimension limit for the spherical well system. The organization of the paper is as follows: In section 2, we outline theoretical background. Then, we
evaluate the uncertainty product for the harmonic oscillator quantum system in Section 3, for the hydrogen atom in Section 4, and for the spherical well in Section 5. We present conclusions and
discussion of our work in Section 6.
2. Theoretical Background
The quantum mechanical state of a particle in the N-dimensional space with a central potential
and satisfies [50]
which, by letting
The above equation is the analogue to the one-dimensional Schrödinger equation with the grand orbital angular momentum, L given by
It is straight forward to write Equation (6) as
where the effective potential,
It is worth to note, as seen from Equation (7), the isomorphism between the space dimension N and the orbital angular momentum ℓ, which means that an orbital angular momentum
3. Isotropic Harmonic Oscillator in N-Dimensions
The potential for a harmonic oscillator is given by
The virial theorem states that
where T is the kinetic energy and the average is taken over an energy eigenstate of the system. The substitution of Equation (10) into Equation (11) gives
where the energy eigenvalues,
and thus the uncertainty product is
which obviously increases with both the quantum number n and the space dimension N. It is observed that the above product does not depend on the strength of the potential. The lower bound corresponds
for the ground state
which saturates (equality is achieved) for nodeless harmonic oscillator wave function (ground state). Our results in Equation’s (14) and (15) are the same as those obtained by means of the Fisher’s
information entropies [32] [33] , by Stamp’s principle [51] and by Shannon’s entropy [26] . Our method is more straight forward and simpler. The lower bound in Equation (15) reduces to the
three-dimensional one, namely 9/4. Furthermore, our result in Equation (14) shows that the lower bound of the uncertainty product (for the ground state) in N-dimension is the
same as the lower bound of the
tainty product for a state with angular momentum
4. The Hydrogen Atom in N Dimensions
In this case, the potential is the coulomb potential,
The application of the virial theorem gives
The energy eigenvalues for the eigenstates of a hydrogen atom in N dimensions are given by [37]
where a is Bohr radius. Therefore,
In order to find the average of the moments of position of different powers we use Kramer’s relation in N- dimensions [52]
The successive application of the above relation for
The above relation and Equation (18) yield the uncertainty product for position and momentum;
It is clear to notice that the uncertainty product increases as the quantum number n increases and decreases as the orbital angular momentum ℓ increases. One can easily verify that the uncertainty
product increases as the space dimension increases.
The lower bound of the above uncertainty is achieved by setting
In what follows, we will consider the uncertainty product given in Equation (20) for some special cases:
1) For the three-dimensional case (
2) For any state n with ℓ has its maximum value
In this case, the uncertainty product has its minimum value since ℓ has its maximum value, which means the certainty has its highest value. This result is a natural consequence of the quantum
centrifugal potential which tries to repel the particle away from the nucleus. In fact, it was pointed out by AL-Jaber [37] that the radial probability density has its maximum value when the orbital
angular momentum has its maximum value
3) For any state n with
which corresponds to the maximum value of the uncertainty product, since ℓ has its minimum value. One may expect this result in the light of what we mentioned in the previous case.
4) The large space dimension limit: For large N, Equation (21) gives us the result
which is equal to the lower bound for the ground state of the harmonic oscillator in N-dimensions as we found in the previous section. This clearly shows that in the large N limit the lower bound for
any state becomes saturated and equals to that of the ground state lower bound of the harmonic oscillator.
5) The uncertainty product difference between a state with
The above equation gives, for a given state n, the uncertainty product difference between minimum and maximum angular momenta for that state. This difference increases with both n, and N. This shows
how much the particle becomes delocalized due to maximum orbital angular momentum.
6) Spherically symmetric infinite potential well
In this section, we consider a particle that is confined in an infinite impenetrable spherical well so that the potential is given by
The substitution of the above potential into Equation (6) and letting
whose solution is the spherical Bessel function of order L, (the second solution has been dropped out since it diverges at the origin) and thus, the radial wave function is
where A[L] is a normalization constant. The allowed energies can be obtained by requiring
successive zeros of
The integer n is the principal quantum number, which is the number of the root of spherical Bessel function in order of increasing magnitude. Since
On the other hand, the average value,
Following Grypeos [48] , we get
which, upon the substitution for L from Equation (7), becomes
The uncertainty product is now readily obtained using Equations (31) and (34), namely
Again, the uncertainty product increases with both ℓ and N. The lower bound limit corresponds to ground state,
The above result shows that the uncertainty product increases with space dimension N, but is independent of the size of the well. It is instructive to calculate the above lower limit for different
values of space dimension and compare its values with those for the harmonic oscillator and the hydrogen atom. This is shown in Table 1.
The numerical values for the lower bound for the three systems, presented in Table 1, show that the harmonic oscillator has the smallest values for all space dimension. We also note that the hydrogen
atom has higher lower bound value than that of the spherical well for space dimension 3 and 4, but beyond that the spherical well has higher values than those for hydrogen atom.
Table 1. Lower bound for the uncertainty product for the spherical well, harmonic oscillator, and the hydrogen atom as function of space dimension N.
It is interesting to check the large N limit of the lower bound of these systems: For the hydrogen atom, the lower bound behaves as
5. Conclusion
In this paper, we have derived the uncertainty product for position and momentum for harmonic oscillator, hydrogen atom, and spherically symmetric infinite well in N-dimensional space. We have found
that this product depends on the orbital angular momentum and space dimension but independent of the strength of the potential. Our derivation relies on the virial theorem and Kramer’s relation for
the harmonic oscillator and the hydrogen atom. Our results for the lower bound of the uncertainty product for each of the three systems agree with reported results for the three dimensional case. An
interesting feature of our results is that in the large space dimension limit, the lower bound of the product for the hydrogen atom and the spherical well converge to that for the harmonic
oscillator, namely
that the lower limit of the product in N-dimensions has the same value as that for the
three dimensions. Furthermore, the product for a state with angular momentum Table 1. | {"url":"https://scirp.org/journal/paperinformation?paperid=64939","timestamp":"2024-11-14T10:58:43Z","content_type":"application/xhtml+xml","content_length":"155693","record_id":"<urn:uuid:d539fea9-f540-4d2c-a9a7-6359383e0c67>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00898.warc.gz"} |
Magnitude integration in the Archerfish
We make magnitude-related decisions every day, for example, to choose the shortest queue at the grocery store. When making such decisions, which magnitudes do we consider? The dominant theory
suggests that our focus is on numerical quantity, i.e., the number of items in a set. This theory leads to quantity-focused research suggesting that discriminating quantities is automatic, innate,
and is the basis for mathematical abilities in humans. Another theory suggests, instead, that non-numerical magnitudes, such as the total area of the compared items, are usually what humans rely on,
and numerical quantity is used only when required. Since wild animals must make quick magnitude-related decisions to eat, seek shelter, survive, and procreate, studying which magnitudes animals
spontaneously use in magnitude-related decisions is a good way to study the relative primacy of numerical quantity versus non-numerical magnitudes. We asked whether, in an animal model, the influence
of non-numerical magnitudes on performance in a spontaneous magnitude comparison task is modulated by the number of non-numerical magnitudes that positively correlate with numerical quantity. Our
animal model was the Archerfish, a fish that, in the wild, hunts insects by shooting a jet of water at them. These fish were trained to shoot water at artificial targets presented on a computer
screen above the water tank. We tested the Archerfish's performance in spontaneous, untrained two-choice magnitude decisions. We found that the fish tended to select the group containing larger
non-numerical magnitudes and smaller quantities of dots. The fish selected the group containing more dots mostly when the quantity of the dots was positively correlated with all five different
non-numerical magnitudes. The current study adds to the body of studies providing direct evidence that in some cases animals’ magnitude-related decisions are more affected by non-numerical magnitudes
than by numerical quantity, putting doubt on the claims that numerical quantity perception is the most basic building block of mathematical abilities.
Bibliographical note
Publisher Copyright:
© 2021, The Author(s).
ASJC Scopus subject areas
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Course Notes: A Crash Course on Causality -- Week 5: Instrumental Variables - King Fox And Butterfly
For the pdf slides, click here
Introduction to Instrumental Variables
Unmeasured confounding
• Suppose there are unobserved variables \(U\) that affect both \(A\) and \(Y\), then \(U\) is an unmeasured confounding
• This violates ignorability assumption
• Since we cannot control for the unobserved confounders \(U\) and average over its distribution, if using matching or IPTW methods, the estimates of causal effects is biased
• Solution: instrumental variables
Instrumental variables
• Instrumental variables (IV): an alternative causal inference method that does not rely on the ignorability assumption
• \(Z\) is an IV
□ It affects treatment \(A\), but does not directly affect the outcome \(Y\)
□ We can think of \(Z\) as encouragement (of treatement)
Example of an encouragement design
• \(A\): smoking during pregnancy (yes/no)
• \(Y\): birth weight
• \(X\): mother’s age, weight, etc
□ Concern: there could be unmeasured confounders
□ Challenge: it is not ethical to randomly assign smoking
• \(Z\): randomized to either received encouragement to stop smoking (\(Z=1\)) or receive usual care (\(Z=0\))
□ Causal effect of encouragement, also called intent-to-treat (ITT) effect, may be of some interest \[E\left(Y^{Z=1}\right)-E\left(Y^{Z=0}\right)\]
□ Focus of IV methods is still causal effect of the treatment \[E\left(Y^{A=1}\right)-E\left(Y^{A=0}\right)\]
IV is randomized
• Like the previous smoking example, sometimes IV is randomly assigned as part of the study
• Other times IV is believed to be randomized in nature (natural experiment). For example,
□ Mendelian randomization (?)
□ Quarter of birth
□ Geographic distance to specialty care provider
Randomized trials with noncompliance
Randomized trials with noncompliance
• Setup
□ \(Z\): randomization to treatment (1 treatment, 0 control)
□ \(A\): treatment received, binary (1 treatment, 0 control)
□ \(Y\): outcome
• Due to noncompliance, not everyone assigned treatment will actually receive the treatment, and vice verse (\(A \neq Z\))
□ There can be confounding \(X\), like common causes affecting both treatment received \(A\) and the outcome \(Y\)
□ It may be reasonable to assume that \(Z\) does not directly affect \(Y\)
Causal effect of assignment on receipt
• Observed data: \((Z, A, Y)\)
• Each subject has two potential values of treatment
□ \(A^{Z=1} = A^1\): value of treatment if randomized to treatment
□ \(A^{Z=0} = A^0\): value of treatment if randomized to control
• Average causal effect of treatment assignment on treatment received \[E\left(A^1 - A^0\right)\]
□ If perfect compliance, this would be \(1\)
□ By randomization and consistency, this is estimable from the observed data \[ E\left(A^1\right) = E(A \mid Z=1), \quad E\left(A^0\right) = E(A \mid Z=0) \]
Causal effect of assignment on outcome
• Average causal effect of treatment assignment on the outcome \[E\left(Y^{Z=1} - Y^{Z=0}\right)\]
□ This is intention-to-treat effect
□ If perfect compliance, this would be equal to the causal effect of treatment received
□ By randomization and consistency, this is estimable from the observed data \[ E\left(Y^{Z=1}\right) = E(Y \mid Z=1), \quad E\left(Y^{Z=0}\right) = E(Y \mid Z=0) \]
Compliance classes
Subpopulations based on potential treatment
\(A^0\) \(A^1\) Label
0 0 Never-takers
0 1 Compliers
1 0 Defiers
0 0 Always-takers
• For never-takers and always-takers,
□ Encouragement does not work
□ Due to no variation in treatment received, we cannot learn anything about the effect of treatment in these two subpopulations
• For compliers, treatment received is randomized
• For defiers, treatment received is also randomized, but in the opposite way
Local average treatment effect
• We will focus on a local average treatment effect, i.e., the complier average causal effect (CACE)
\[\begin{align*} & E\left(Y^{Z=1} \mid A^0=0, A^1=1 \right) - E\left(Y^{Z=0} \mid A^0=0, A^1=1 \right)\\ = & E\left(Y^{Z=1} - Y^{Z=0} \mid \text{compliers} \right)\\ = & E\left(Y^{a=1} - Y^{a=0} \mid
\text{compliers} \right) \end{align*}\]
• “Local”: this is a causal effect in a subpopulation
• No inference about defiers, always-takers, or never-takers
Instrumental variable assumptions
IV assumption 1: exclusion restriction
1. \(Z\) is associated with the treatment \(A\)
2. \(Z\) affects the outcome only through its effect on treatment
□ \(Z\) cannot directly, or indirectly though its effect on \(U\), affect \(Y\)
Is the exclusion restriction assumption realistic?
• If \(Z\) is a random treatment assignment, then the exclusion restriction assumption is met
□ It should affect treatment received
□ It should not affect the outcome or unmeasured confounders
• However, it the subjects or clinicians are not blinded, knowledge of what they are assigned to could affect \(Y\) or \(U\)
• We need to examine the exclusion restriction assumption carefully for any given study
IV assumption 2: monotonicity
• Monotonicity assumption: there are no defiers
□ No one consistently does the opposite of what they are told
□ Probability of treatment should increase with more encouragement
• With monotonicity,
\(Z\) \(A\) \(A^0\) \(A^1\) Class
0 0 0 ? Never-takers or compliers
0 1 1 1 Always-takers [DEL:or defiers:DEL]
1 0 0 0 Never-takers [DEL:or defiers:DEL]
1 1 ? 1 Always-takers or compliers
Estimate Causal Effects with Instrumental Variables
Estimate CACE: 1. rewrite the ITT effect
• Due to randomization, we can identify the ITT effect \[ E\left( Y^{z=1} - Y^{z=0} \right) = E(Y\mid Z=1) - E(Y\mid Z=0) \]
• Expand the first term in the above ITT effect \[\begin{align*} E(Y\mid Z=1) = & E(Y\mid Z=1, \text{always takers})P(\text{always takers}\mid Z=1)\\ & + E(Y\mid Z=1, \text{never takers})P(\text
{never takers}\mid Z=1)\\ & + E(Y\mid Z=1, \text{compliers})P(\text{compliers}\mid Z=1) \end{align*}\]
• Note 1: among always takers and never takes, \(Z\) does nothing
□ \(E(Y\mid Z=1, \text{always takers}) = E(Y\mid \text{always takers}), \quad \text{etc.}\)
• Note 2: by randomization,
□ \(P(\text{always takers}\mid Z=1) = P(\text{always takers}), \quad \text{etc.}\)
Estimate CACE: 1. rewrite the ITT effect, cont.
• Therefore, the first term in the ITT effect is \[\begin{align*} E(Y\mid Z=1)=& E(Y\mid\text{always takers})P(\text{always takers})\\ & + E(Y\mid \text{never takers})P(\text{never takers})\\ & + E
(Y\mid Z=1, \text{compliers})P(\text{compliers}) \end{align*}\]
• Similarly, the second term is \[\begin{align*} E(Y\mid Z=0)=& E(Y\mid\text{always takers})P(\text{always takers})\\ & + E(Y\mid \text{never takers})P(\text{never takers})\\ & + E(Y\mid Z=0, \text
{compliers})P(\text{compliers}) \end{align*}\]
• Their difference is \[\begin{align*} & E(Y\mid Z=1) - E(Y\mid Z=0)\\ = & \left[E(Y\mid Z=1, \text{compliers})- E(Y\mid Z=0, \text{compliers})\right]P(\text{compliers}) \end{align*}\]
Estimate CACE: 2. compute proportion of compliers
• Thus, the relationship between CACE and ITT effect is \[ \text{CACE} = \frac{E(Y\mid Z=1) - E(Y\mid Z=0)}{P(\text{compliers})} \]
• To compute \(P(\text{compliers})\), note that
□ \(E(A\mid Z=1)\): proportion of always takers plus compliers
□ \(E(A\mid Z=0)\): proportion of always takers
• Thus the difference is \[ P(\text{compliers}) = E(A\mid Z=1) - E(A\mid Z=0) \]
Estimate CACE: final formula
\[ \text{CACE} = \frac{E(Y\mid Z=1) - E(Y\mid Z=0)} {E(A\mid Z=1) - E(A\mid Z=0)} \]
• Numerator: ITT, causal effect of treatment assignment on the outcome
• Denominator: causal effect of treatment assignment on the treatment received
□ Denominator is between 0 and 1. Thus, CACE \(\geq\) ITT
□ ITT is underestimate of CACE, because some people assigned to treatment did not take it
• If perfect compliance, CACE \(=\) ITT
IVs in observational studies
IVs in observational studies
• IVs can also be used in observational (non-randomized) studies
□ \(Z\): instrument
□ \(A\): treatment
□ \(Y\): outcome
□ \(X\): covariates
• \(Z\) can be thought of as encouragement
□ If binary, just encouragement yes or no
□ If continuous, a ‘dose’ of encouragement
• \(Z\) can be thought of as randomizers in natural experiments
□ The key challenge: think of a variable that affects \(Y\) only through \(A\)
□ Only the assumption \(Z\) affecting \(A\) can be checked with data
□ The validity of the exclusion restriction assumption rely on subject matter knowledge
Natural experiment example 1: calendar time as IV
• Rationale: sometimes treatment preferences change over a short period of time
• \(A\): drug A vs drug B
• \(Z\): early time period (drug A is encouraged) vs late time period (drug B is encouraged)
• \(Y\): BMI
Natural experiment example 2: distance as IV
• Rationale: shorter distance to NICU is an encouragement
• \(A\): delivery at high level NICU vs regular hospital
• \(Z\): differential travel time from nearest high level NICU to nearest regular hospital
• \(Y\): mortality
More examples of natural experiments
• Mendelian randomization: some genetic variant is associate with some behavior (e.g., alcohol use) but is assumed to not be associated with outcome of interest
• Provider preference: use treatment prescribed to previous patients as an IV for current patient
• Quarter of birth: to study causal effect of years in school on income
Two stage least squares
Ordinary least squares (OLS) fails if there is confounding
• In OLS, one important assumption is that the covariate \(A\) is independent with residuals \(\epsilon\)
\[ Y_i = \beta_0 + A_i \beta_1 + \epsilon_i \]
• However, if there is confounding, \(A\) and \(\epsilon\) are correlated. So OLS fails.
• Two stage least squares can estimate causal effect in the instrumental variables (IV) setting
Two stage least squares (2SLS)
• Stage 1: regress \(A\) on \(Z\) \[ A_i = \alpha_0 + Z_i \alpha_1 + e_i \]
□ By randomization, \(Z\) and \(e\) are independent
• Obtain the predicted value of \(A\) given \(Z\) for each subject \[ \hat{A}_i = \hat{\alpha}_0 + Z_i \hat{\alpha}_1 \]
□ \(\hat{A}\) is projection of \(A\) onto the space spanned by \(Z\)
• Stage 2: regress \(Y\) on \(\hat{A}\) \[ Y_i = \beta_0 + \hat{A}_i \beta_1 + \epsilon_i \]
□ By exclusion restriction, \(Z\) is independent of \(Y\) given \(A\)
Interpretation of \(\beta_1\) in 2SLS: the causal effect
• Consider the case where both \(Z\) and \(A\) are binary \[ \beta_1 = E\left(Y \mid \hat{A}=1 \right) - E\left(Y \mid \hat{A}=0 \right) \]
• There are two values of \(\hat{A}\) in the 2nd stage model, \(\hat{\alpha}_0\) and \(\hat{\alpha}_0 + \hat{\alpha}_1\)
□ When we go from \(Z=0\) to \(Z=1\), what we observe is going from \(\hat{\alpha}_0\) to \(\hat{\alpha}_0 + \hat{\alpha}_1\)
□ We observe a mean difference of \(\hat{E}(Y\mid Z=1) - \hat{E}(Y\mid Z=0)\) with a \(\hat{\alpha}_1\) unit change in \(\hat{A}\)
• Thus, we should observe a mean difference of \(\frac{\hat{E}(Y\mid Z=1) - \hat{E}(Y\mid Z=0)}{\hat{\alpha}_1}\) with \(1\) unit change in \(\hat{A}\)
• The 2SLS estimator is a consistent estimator of the CACE \[ \beta_1 = \text{CACE} = \frac{\hat{E}(Y\mid Z=1) - \hat{E}(Y\mid Z=0)}{\hat{E}(A\mid Z=1) - \hat{E}(A\mid Z=0)} \]
More general 2SLS
• 2SLS can be used
□ with covariates \(X\), and
□ for non-binary data (e.g, a continuous instrument)
• Stage 1: regression \(A\) on \(Z\) and covariates \(X\)
□ and obtain the fitted values \(\hat{A}\)
• Stage 2: regress \(Y\) on \(\hat{A}\) and \(X\)
□ Coefficient of \(\hat{A}\) is the causal effect
Sensitivity analysis and weak instruments
Sensitivity analysis
• Sensitivity analysis method studies when each of the IV assumption (partly) fails
□ Exclusion restriction: if \(Z\) does affect \(Y\) by an amount \(p\), would my conclusion change? Vary \(p\)
□ Monotonically: if the proportion of defiers was \(\pi\), would my conclusion change?
Strength of IVs
• Depend on how well an IV predicts treatment received, we can class it as a strong instrument or a weak instrument
• For a weak instrument, encouragement barely increases the probability of treatment
• Measure the strength of an instrument: estimate the proportion of compliers \[ E(A \mid Z=1) - E(A \mid Z=0) \]
□ Alternatively, we can just use the observed proportions of treated subjects for \(Z=1\) and for \(Z=0\)
Problems of weak instruments
• Suppose only 1% of the population are compliers
• Then only 1% of the samples have useful information about the treatment effect
□ This leads to large variance estimates, i.e., estimate of causal effect is unstable
□ The confidence intervals can be too wide to be useful
• Coursera class: “A Crash Course on Causality: Inferring Causal Effects from Observational Data”, by Jason A. Roy (University of Pennsylvania) | {"url":"https://liyingbo.com/stat/2021/08/16/course-notes-a-crash-course-on-causality-week-5-instrumental-variables/","timestamp":"2024-11-10T15:36:50Z","content_type":"text/html","content_length":"29617","record_id":"<urn:uuid:413536cd-2edb-4ec4-b2e2-055f202d2e33>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00853.warc.gz"} |
variable rate mortgage definition
Example of variable rate. To work out which is truly a better deal, look at how much interest rates would need to change before one deal beats the other. As rates change over time, simply comparing
the fixed and variable rates at the point you take your mortgage is a relatively blunt tool. Also called an "Adjustable-rate Mortgage." For some, that’s a legitimate concern. Some variable-rate
mortgages offer the option of a fixed payment. Variable & fixed rate mortgages explained. Variable mortgage rates are driven by the same economic factors, except variable rates fluctuate with
movements in the prime lending rate, the rate at which banks lend to their most credit-worthy customers. Standard variable rate - the default variable rate the lender offers to mortgage borrowers
with a standard residential mortgage. Kevin obtains a mortgage loan at a time when interest rates are falling. There are limits to how much a variable rate mortgage can change. variable rate mortgage
meaning: a loan for buying a house on which the interest rate can change over time: . Tracker rate - a variable rate that is equal to a published interest rate (typically LIBOR, plus a fixed interest
rate margin. A variable rate mortgage is a mortgage rate that can change over time, which means it can decrease or increase depending on wider economic circumstances. variable-rate mortgage
definition: noun Abbr. In VRM, interest rate changes in response to changes in the main economic index. A fixed rate makes it easier to budget for payments. Fixed rate mortgages keep your mortgage
repayments predictable and stable. This impacts the amount of principal you pay off each month. There are fixed rate mortgages and variable rate mortgages (which you’ll often hear referred to by
their aliases — adjustable rate mortgages or just VRMs.) Due to the added risk of rates increasing, providers will often offer lower variable rates than fixed rates. This means that the interest rate
you are charged will rise when the Base Rate increases and fall if it decreases, affecting your mortgage payments in the same way. means any individual Mortgage Loan purchased pursuant to this
Agreement the interest rate of which adjusts periodically based on the index identified in the Mortgage … A bank quotes you an annual percentage rate and term — say 4% for 5 years on a $300,000 loan,
amortized over the course of 25 years — and you agree to pay a specific amount … Variable rate mortgages are riskier than fixed rate mortgages because the payments are less predictable. The monthly
payments usually change every month, also. Verification of deposit (VOD) Verification of employment (VOE) Veterans Administration (VA) mortgage; Yield Spread Premium; Yield to maturity; Zero Down
Mortgage; Zestimate ; Zoning ordinances; Waiver; Walk-through; 1,000 NMLS Practice Test Questions. A variable rate mortgage is a mortgage that has an interest rate that can change during the term of
the loan. Payment is adjusted periodically. A fixed rate mortgage is pretty straightforward. Learn about the definition for this legal term. However, the monthly payments on a variable rate mortgage
start out lower. Variable mortgage rates expose you to changes in interest rates and, thus, in your mortgage payments. How does a standard variable rate work? Variable rate mortgage definition: a
mortgage involving a loan with a variable interest rate over the period of the loan | Meaning, pronunciation, translations and examples 8 Mortgage … When a mortgage has a variable interest rate, it
is more commonly referred to as an adjustable-rate mortgage (ARM). A general rule of thumb - go with Fixed Rate mortgage if you believe the interest rate on mortgage loans will increase through your
amortization timeframe. When a mortgage has a variable interest rate, it is more commonly referred to as an adjustable-rate mortgage (ARM). Understanding the key features of a fixed rate mortgage and
a variable mortgage can be fairly straightforward, but deciding between the two and picking one that saves you money is much trickier. That point is called the 'trigger rate.' Some lenders will also
let you take out a mortgage on their SVR, but this is usually the most expensive option. Many ARMs start with a low fixed interest rate for the first few years of the loan, only adjusting after that
time period has expired. There are quite a few different types of mortgage and each has their own good and bad points.. A fixed rate mortgage has a rate of interest which doesn’t change for a set
period of time, so you know exactly how much you pay every month. For example, variable rates in reference to mortgage loans rely heavily on the Bank of Canada's prime lending rate. If the prime rate
goes up or down, then the variable rate effectively goes up or down. Mortgage brokerages, like CanEquity, generally have access to variable interest rates that are well below prime. Obtenez plus de
renseignements sur les prêts hypothécaires à taux variable et sur la manière dont les variations des points de base se répercutent sur ces prêts. It occurs when prime rate goes up so much that your
fixed payment no longer covers the interest you owe each month. A mortgage product where the interest rate is adjusted periodically based on a standard financial index. The interest rate of a
variable rate mortgage can fluctuate, which affects your monthly mortgage repayment. For example, change in treasury bill rate. My loan offer like yours states that after 3 years I will revert to a
variable base rate but to a rate that is “currently 5.25%.” Both accounts have the same rate definition(“variable base rate). Definition of variable rate mortgage (VRM) variable rate mortgage (VRM)
1. Variable mortgage rates are typically stated as prime plus/minus a percentage discount/premium. Generally speaking, an adjustable rate mortgage is linked to some major benchmark rate; for example,
the interest rate may be stated as "LIBOR + 1%." A floating or non-fixed percentage value or interest rate that fluctuates based on certain economic conditions or rate index. However, you could pay a
lot more interest than you would with a variable rate mortgage. There is an exception to this. Interest rates are currently at all time lows. Variable rate - the rate varies at the discretion of the
lender. Adjustable Rate Mortgage A mortgage with an interest rate that changes periodically. A variable rate mortgage will fluctuate with the CIBC Prime rate throughout the mortgage term. Previous
Next > More Mortgage Definitions. Folks often consider closed variable-rate mortgages to be restrictive because they can’t be paid off early without a penalty. Déterminez si un prêt hypothécaire à
taux variable convient à votre situation financière et découvrez des … Tracker mortgage rates track the Bank of England Base Rate (a variable rate of interest) over a specified period of time. Don't
ever under-estimate the difference between Fixed Rate and Variable Rate mortgage loans. This is where a broker can really help you see the wood for the trees. Common fixed-interest-rate periods on an
ARM are three or five years, expressed as a 3/1 or 5/1 ARM, respectively. Variable rate mortgage definition is - adjustable rate mortgage. To further muddy the waters in September 2006 the EBS
defined their “Standard Variable Rate” as “A mortgage rate which can rise and fall in line with the interest rate changes set by the European Central Bank (ECB)”. Variable rate. Variable-Rate
Mortgage (VRM) is a type of mortgage loan program wherein interest rates and payments are adjusted as frequently as every month. variable rate 1. Define Variable Rate Mortgage Loan. If market rates
fluctuate, you will be charged the difference in interest applied to your mortgage principal. Variable Rate Mortgage. But remember it’s fixed for a certain time like three, five or seven years and if
you change it before the end, we may charge you a fee. For example, a variable rate could be quoted as prime - 0.8%. Learn more. VRM See adjustable-rate mortgage. Definition, Synonyms, Translations
of variable-rate mortgage by The Free Dictionary While your regular payment will remain constant, your interest rate may change based on market conditions. These types of rates can go up or down at
any time, therefore are considered variable. This guide will examine two types of mortgages - fixed rate and variable rate. I drew down my EBS mortgage in August 2007 when the ECB base rate was
4.00%. It can be hard to decide upon which mortgage is right for you when you want to take out a loan to buy a property. Recent Examples on the Web Since the government rolls over much of its debt,
selling short-term debt like 2-year bonds is like having a variable rate mortgage. A variable-rate mortgage, also known as a standard variable rate mortgage, adjustable-rate mortgage (ARM) or tracker
mortgage, is a home loan whose interest rate is periodically adjusted, depending on the cost to the lender of borrowing money on the credit markets. This is in comparison to fixed rate mortgages,
where the monthly payments will always stay the same. A variable rate is always attached to the lender’s prime rate. You can see how this definition is different to how they define their SVR now. On
the other hand, most (not all) closed variables allow you to terminate with a fairly reasonable 3-month interest penalty. variable-rate mortgage (VRM) A precursor to the modern adjustable-rate home
mortgage (ARM), and still used in the area of commercial mortgages.With a variable-rate mortgage,the interest rate on the loan changes whenever the index rate changes. It's a tradeoff between locking
in a payment versus starting at a lower cost. A variable rate mortgage is a home loan with an interest rate that changes over time, causing the monthly loan payments to go up or down. When rates on
variable interest rate mortgages decrease, more of your regular payment is applied to your principal. Related Terms and Acronyms: alternative mortgage A home … An interest rate that changes
periodically in relation to an index. Many ARMs start with a low fixed interest rate … A standard variable rate (SVR) is a type of mortgage interest rate that you are most likely to go onto after
finishing an introductory fixed, tracker or discounted deal. Vice versa, if you believe the interest rate on mortgage loans will decrease through your amortization timeframe, go with Variable Rate
mortgage. So, even if interest rates rise (or fall), your payment stays the same. Definition of variable rate. Frequently as every month, also it is more commonly referred to as an adjustable-rate
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[Solved] Welded, Riveted and Bolted Joints MCQ [Free PDF] - Objective Question Answer for Welded, Riveted and Bolted Joints Quiz - Download Now!
Welded, Riveted and Bolted Joints MCQ Quiz - Objective Question with Answer for Welded, Riveted and Bolted Joints - Download Free PDF
Last updated on Aug 7, 2024
Latest Welded, Riveted and Bolted Joints MCQ Objective Questions
Welded, Riveted and Bolted Joints Question 1:
What is the term used to define the distance from the center of a fastener hole to the edge of an element, measured parallel to the direction of load transfer?
Answer (Detailed Solution Below)
Option 4 : End Distance
Welded, Riveted and Bolted Joints Question 1 Detailed Solution
Explanation: As per IS 800:2007 Clause 1.3.40
End Distance — Distance from the centre of a fastener hole to the edge of an element measured parallel to the direction of load transfer.
1.3.33 Edge Distance — Distance from the centre of a fastener hole to the nearest edge of an element measured perpendicular to the direction of load transfer.
Welded, Riveted and Bolted Joints Question 2:
A single riveted lap joint is made in 10 mm thick plates with 20 mm diameter rivets. Determine the bearing strength of the rivet if stresses in bearing is 150 MPa.
Answer (Detailed Solution Below)
Option 3 : 30 kN
Welded, Riveted and Bolted Joints Question 2 Detailed Solution
Strength of Rivet in Bearing
Pb = fb × d' × t
Where P[b] = bearing strength of rivet, f[b] = permissible bearing stress in rivet, d' = dia of hole or gross dia of rivet, and t = thickness of plate.
Joint is a single riveted lap joint
Thickness of plate (t) = 10 mm
Diameter of rivet (d) = 20 mm
Bearing stress (f[b]) = 150 MPa
Bearing strength of rivet (P[b]) = f[b] × d' × t
where d' = d + 1.5 mm (d = 20 mm < 25 mm)
d' = 21.5 mm
Bearing strength of rivet (Pb) = 150 × 21.5 × 10 = 32250 N or 32.25 kN
Bearing strength of rivet (Pb) = 32.25 kN
The calculated value of the bearing strength of the rivet is very close to the option (3) value. Hence option (3) is correct.
Welded, Riveted and Bolted Joints Question 3:
Root opening of a Single Vee groove weld joint of plate thickness "t" is
Answer (Detailed Solution Below)
Option 2 : \(\rm\frac{t}{4}\)
Welded, Riveted and Bolted Joints Question 3 Detailed Solution
The root opening in a Single Vee groove weld joint is a crucial dimension that determines the gap or separation between the two plates being welded together. It's an important parameter in welding
because it affects the quality and strength of the weld.
In a Single Vee groove weld joint, you have two plates that are prepared with V-shaped grooves to create a joint. The root opening specifically refers to the distance between the points where the two
plates touch or come closest together at the bottom of the V-shaped groove.
The root opening is often expressed as a fraction of the plate thickness "t." In this case, it's defined as:
Root Opening = t/4
So, if the plate thickness "t" is, for example, 4 millimeters, the root opening for this Single Vee groove weld joint would be 4/4 = 1 millimeter.
Maintaining the correct root opening is essential in welding because it affects the penetration of the weld bead into the joint and, consequently, the strength of the weld. Too small of a root
opening can result in incomplete penetration, leading to weak welds. Too large of a root opening can cause a lack of fusion and poor joint strength. Therefore, adhering to the recommended root
opening, in this case, t/4, is crucial for achieving a sound and strong weld.
Welded, Riveted and Bolted Joints Question 4:
Which of the following is FALSE for bolted joints?
Answer (Detailed Solution Below)
Option 4 : Bolted joints are designed such that bolt stiffness is higher than flange stiffness
Welded, Riveted and Bolted Joints Question 4 Detailed Solution
The FALSE statement among the options is: Bolted joints are designed such that bolt stiffness is higher than flange stiffness
In bolted joints, it is typically desired to have the flange stiffness higher than the bolt stiffness to ensure proper load distribution and prevent overloading of the bolt. Bolt stiffness is often
designed to be lower to allow for some flexibility and uniform load distribution across the flange faces. If the bolt stiffness were higher than the flange stiffness, it could lead to uneven stress
distribution and potential issues in the joint.
The other statements (1, 2, and 3) are generally true for bolted joints:
• Bolted joints are preloaded to avoid vibration loosening: Preloading the bolts creates a compressive force that helps to maintain the integrity of the joint and prevents loosening due to
• Threads made by thread rolling process are preferred over machined threads in bolts: Threads produced by thread rolling are typically stronger and have better fatigue resistance than machined
threads, making them preferred for high-strength applications.
• When an external load is applied to a preloaded bolt joint, a bigger percentage of that load relieves compression of flanges and remaining percentage increases tension in bolts: When an external
load is applied, the initial compression in the flanges is relieved, and a significant portion of the load is transferred to the bolts in tension to maintain the integrity of the joint.
Welded, Riveted and Bolted Joints Question 5:
A rectangular steel bar of length 500 mm, width 100mm, and thickness 15 mm is cantilevered to a 200 mm steel channel using 4 bolts, as shown.
For an external load of 10 kN applied at the tip of the steel bar, the resultant shear load on the bolt at B, is _______ kN
Answer (Detailed Solution Below)
Option 3 : 16 kN
Welded, Riveted and Bolted Joints Question 5 Detailed Solution
The given loading system will be subjected to primary and secondary shear forces.
When the line of action of the load does not pass through the centroid of the rivet system and thus all rivets are not equally loaded, then the joint is said to be an eccentric loaded riveted joint.
The eccentric loading results in secondary shear caused by the tendency of force to twist the joint about the centre of gravity in addition to direct shear or primary shear.
Let P = Eccentric load on the joint, and e = Eccentricity of the load i.e. the distance between the line of action of the load and the centroid of the rivet system i.e. G.
The primary shear force on any bolt is given by:
\({P_s} = \frac{P}{n},\;where\;n\;is\;the\;number\;of\;bolts\)
The secondary shear force is given by:
\(\frac{{{F_1}}}{{{l_1}}} = \frac{{{F_2}}}{{{l_2}}} = \frac{{{F_3}}}{{{l_3}}} = \frac{{{F_4}}}{{{l_4}}}\)
\({F_i} = \frac{{Pe{l_i}}}{{{l_1}^2 + {l_2}^2 + {l_3}^2 + {l_4}^2}},\;where\;{l_i}\;is\;the\;dis{\rm{tan}}ce\;from\;the\;CG\;of\;system\)
The primary (or direct) and secondary shear load may be added vectorially to determine the resultant shear load (R) on each rivet.
\(R = \sqrt {{{\left( {{P_s}} \right)}^2} + {F^2} + 2{P_s}F\cos \theta } \)
θ is the angle between the primary or direct shear load (Ps) and secondary shear load (F).
\({P_{SA}} = {P_{SB}} = {P_{SC}} = {P_{SD}} = \frac{{Load}}{{No.\;of\;Bolts}} = \frac{{10}}{4} = 2.5\;kN\)
Let \(l = {l_1} = {l_2} = {l_3} = {l_4} = \sqrt {{{50}^2} + {{50}^2}} = 50\sqrt 2 \;mm\)
e = 300 + 100 = 400 mm
Secondary shear force –
\({{\rm{F}}_B} = \frac{{Pe{l_1}}}{{4l_1^2}} = \frac{{Pe}}{{4{l_1}}} = \frac{{10 \times 400}}{{4 \times 50 \sqrt 2}} = 14.14 ~kN\)
Resultant force at B:
\({R_{B}} = \sqrt {P_B^2 + F{{_B}^2} + 2\;{P_B}F_B\cos \theta } \)
\(\Rightarrow R_B = \sqrt {{{\left( {2.5} \right)}^2} + {{\left( {14.14} \right)}^2} + 2 \times 2.5 \times 14.14\cos 45^\circ } = 16\;kN\)
Top Welded, Riveted and Bolted Joints MCQ Objective Questions
The head for boiler applications shown in the figure given below is:
Answer (Detailed Solution Below)
Option 4 : steeple
The function of a washer is
Answer (Detailed Solution Below)
Option 4 : To provide bearing area
Function of a washer:
• A washer is a thin plate (typically disk-shaped) with a hole (typically in the middle) that is normally used to distribute the load of a threaded fastener, such as a screw or nut.
• Washers are used to distribute the clamping pressure over a larger area and prevent surface damage. They also provide an increased bearing surface for bolt heads and nuts.
Types of Washers:
Figure: Lock Washer
The rivet head for a general-purpose shown in the figure below is:
Answer (Detailed Solution Below)
Option 3 : Counter sunk
A rivet is a permanent mechanical joint which are broadly used to joint structure, ships, barrels etc. These joints are widely used in ship and boiler industries to join the structure member.
The general nomenclature of rivet is given as follows
Types of Rivet heads:
An eccentrically loaded riveted joint is shown with 4 rivets at P, Q, R and S.
Which of the rivets are the most loaded?
Answer (Detailed Solution Below)
Option 2 : Q and R
Eccentric Loading of Riveted Joints:
• An eccentrically loaded joint is one in which line of application of load does not pass through centre of gravity (c.g) of rivets.
• It passes away from c.g axis.
• This has two effects, primary/direct load and secondary load.
Direct Load:
• This load acts parallel to the load acting and vertically downwards.
• The magnitude for the direct load is \(\rm P'=\frac{Load\;acting}{No.\;of\;rivets}\)
• It is represented by P'.
Secondary load:
• It acts perpendicular to the line joining the centre of gravity of rivets assembly to individual rivets.
• The direction of the secondary load is the same as given by external load i.e. the moment produced due to external load is clockwise so the secondary load will also be clockwise.
• The magnitude for the secondary load is \(P"=\frac{P\;\times \;e\;\times\;r_1}{{r_1^2\;+\;r_2^2\;+\;r_3^2\;+\;r_4^2}}\)for rivet 1 and similarly for other rivets by changing r[2], r[3] and r[4]
in the numerator.
Now, both the load are added vectorially.
Direct load and secondary load diagram is given above.
The resultant of two forces acting at an angle is given by \(R=\sqrt{(P')^2\;+\;(P")^2\;+\;2P'P"cos\;\theta}\)
The angle made by primary shear force causing direct shear and secondary force causing bending stress is minimum for Q and R, ∴ these two rivets are heavily stressed.
The distance between the centres of the rivets in adjacent rows of a zigzag riveted joint is known as _____.
Answer (Detailed Solution Below)
Option 3 : Diagonal pitch
Pitch (p): It is the distance from the center of one rivet to the center of the next rivet measured parallel to the seam.
Back pitch (p[b]): It is the perpendicular distance between the center lines of the successive rows.
Diagonal pitch (p[d]): It is the distance between the centers of the rivets in adjacent rows of zig-zag riveted joint.
Which riveted joint is having minimum efficiency?
Answer (Detailed Solution Below)
Option 2 : Single riveted lap joint
The strength of a rivet joint
• The strength of a rivet joint is measured by its efficiency.
• The efficiency of a joint is defined as the ratio between the strength of a riveted joint to the strength of an un-rivetted joint or a solid plate.
• The efficiency of the riveted joint not only depends upon the size and the strength of the individual rivets but also on the overall arrangement and the type of joints.
Tearing resistance or pull required to tear off the plate per pitch length.
Pt = σt × (p - d) × t
Shearing resistance or pull required to shear off the rivet per pitch length.
\({P_s} = n × \frac{\pi }{4} × {d^2} × \tau \) (single shear)
\({P_s} = n × 2 × \frac{\pi }{4} × {d^2} × \tau\) (double shear)
Crushing resistance or pull required to crush the rivet per pitch length.
P[c] = n × σ[c ]× d × t
Strength of the riveted joint: Least of Pt, Ps and Pc
Strength of the un-riveted or solid plate per pitch length: P = σt × p × t
The joint efficiency is:
\(\eta = \frac{{min\left( {{P_t},\;{P_s},\;{P_c}} \right)}}{P}\)
│Joints │Efficiencies (in %) │
│ │Single riveted│50 - 60 │
│ ├──────────────┼────────────────────┤
│Lap │Double riveted│60 - 72 │
│ ├──────────────┼────────────────────┤
│ │Triple-riveted│72 - 80 │
│ │Single riveted│55 - 60 │
│ ├──────────────┼────────────────────┤
│Butt (double strap) │Double riveted│76 - 84 │
│ ├──────────────┼────────────────────┤
│ │Triple-riveted│80 - 88 │
The shear strength, tensile strength & compressive strength of a rivet joint are 100 N, 120 N & 150 N respectively. If the strength of unriveted plate is 200 N, the efficiency of riveted joint is:
Answer (Detailed Solution Below)
Option 4 : 50%
Efficiency of rivet:
The efficiency of the rivet joint is defined as the ratio of the strength of rivet joint to the strength of the un-riveted or solid plate.
The efficiency of the riveted joint,
\(\rm \eta = \frac{{Lowest\;of\;{P_s},\;{P_t}\;and\;{P_c}}}{P}\)
Where Ps = Shearing resistance, Pt = Tearing resistance, Pc = Crushing resistance, P = Strength of plate.
P[s] = 100 N, P[t] = 120 N, P[c] = 150 N and P = 200 N.
\(\rm \eta = \frac{{Lowest\;of\;{P_s},\;{P_t}\;and\;{P_c}}}{P}\)
The lowest among P[s], P[t] and P[c] is 100 N.
\(\eta = \frac{{100}}{200}=50\;\%\)
In a riveted joint, when the number of rivets decrease from the innermost row to outermost row, the joint is said to be
Answer (Detailed Solution Below)
Option 3 : Diamond rivetted
Diamond rivetted:
• In diamond rivetting, rivets are arranged in a diamond pattern and decrease gradually from the inner row to the outer row.
• All the rivets are arranged symmetrically about the centre line of the plate
• In diamond riveting, there is a saving of material, and efficiency is more. Diamond riveting is used in bridge trusses generally
Additional Information
Chain Rivet:
• When the rivets in the various rows are opposite to each other, then the joint is said to be chain riveted.
Zig - Zag rivetted:
• If the rivets in the adjacent rows are staggered in such a way that every rivet is in the middle of the two rivets of the opposite row as shown in fig then the joint is said to be zig-zag
The transverse fillet welded joints are designed for
Answer (Detailed Solution Below)
Option 1 : Tensile strength
Transverse fillet weld:
The transverse fillet weld is designed for tensile strength
In design, a simple procedure is used assuming that entire load P acts as a shear force on the throat area, which is the smallest area of the cross-section in a fillet weld.
Important Points
• In order to simplify the design of fillet welds many times, shear failure is used as a failure criterion.
The rivet head used for boiler plate riveting is usually
Answer (Detailed Solution Below)
Option 1 : Snap head
The Snap Heads are usually employed for structural work and machine riveting and for boiler shell.
The Counter sunk heads are mainly used for ship building where flush surface are necessary.
The Conical heads are mainly used in case of hand hammering.
Pan head
have maximum strength, but these are difficult to shape. | {"url":"https://testbook.com/objective-questions/mcq-on-welded-riveted-and-bolted-joints--5eea6a1139140f30f369eae0","timestamp":"2024-11-07T22:22:18Z","content_type":"text/html","content_length":"620397","record_id":"<urn:uuid:7aecf611-d7b0-4b35-a604-11ceae4b6818>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00163.warc.gz"} |
Histograms, Binnings, and Density
A simple histogram can be a great first step in understanding a dataset. Earlier, we saw a preview of Matplotlib's histogram function (see Comparisons, Masks, and Boolean Logic), which creates a
basic histogram in one line, once the normal boiler-plate imports are done:
In [1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
data = np.random.randn(1000)
The hist() function has many options to tune both the calculation and the display; here's an example of a more customized histogram:
In [3]:
plt.hist(data, bins=30, normed=True, alpha=0.5,
histtype='stepfilled', color='steelblue',
The plt.hist docstring has more information on other customization options available. I find this combination of histtype='stepfilled' along with some transparency alpha to be very useful when
comparing histograms of several distributions:
In [4]:
x1 = np.random.normal(0, 0.8, 1000)
x2 = np.random.normal(-2, 1, 1000)
x3 = np.random.normal(3, 2, 1000)
kwargs = dict(histtype='stepfilled', alpha=0.3, normed=True, bins=40)
plt.hist(x1, **kwargs)
plt.hist(x2, **kwargs)
plt.hist(x3, **kwargs);
If you would like to simply compute the histogram (that is, count the number of points in a given bin) and not display it, the np.histogram() function is available:
In [5]:
counts, bin_edges = np.histogram(data, bins=5)
Two-Dimensional Histograms and Binnings¶
Just as we create histograms in one dimension by dividing the number-line into bins, we can also create histograms in two-dimensions by dividing points among two-dimensional bins. We'll take a brief
look at several ways to do this here. We'll start by defining some data—an x and y array drawn from a multivariate Gaussian distribution:
In [6]:
mean = [0, 0]
cov = [[1, 1], [1, 2]]
x, y = np.random.multivariate_normal(mean, cov, 10000).T
plt.hist2d: Two-dimensional histogram¶
One straightforward way to plot a two-dimensional histogram is to use Matplotlib's plt.hist2d function:
In [12]:
plt.hist2d(x, y, bins=30, cmap='Blues')
cb = plt.colorbar()
cb.set_label('counts in bin')
Just as with plt.hist, plt.hist2d has a number of extra options to fine-tune the plot and the binning, which are nicely outlined in the function docstring. Further, just as plt.hist has a counterpart
in np.histogram, plt.hist2d has a counterpart in np.histogram2d, which can be used as follows:
In [8]:
counts, xedges, yedges = np.histogram2d(x, y, bins=30)
For the generalization of this histogram binning in dimensions higher than two, see the np.histogramdd function.
plt.hexbin: Hexagonal binnings¶
The two-dimensional histogram creates a tesselation of squares across the axes. Another natural shape for such a tesselation is the regular hexagon. For this purpose, Matplotlib provides the
plt.hexbin routine, which will represents a two-dimensional dataset binned within a grid of hexagons:
In [9]:
plt.hexbin(x, y, gridsize=30, cmap='Blues')
cb = plt.colorbar(label='count in bin')
plt.hexbin has a number of interesting options, including the ability to specify weights for each point, and to change the output in each bin to any NumPy aggregate (mean of weights, standard
deviation of weights, etc.).
Kernel density estimation¶
Another common method of evaluating densities in multiple dimensions is kernel density estimation (KDE). This will be discussed more fully in In-Depth: Kernel Density Estimation, but for now we'll
simply mention that KDE can be thought of as a way to "smear out" the points in space and add up the result to obtain a smooth function. One extremely quick and simple KDE implementation exists in
the scipy.stats package. Here is a quick example of using the KDE on this data:
In [10]:
from scipy.stats import gaussian_kde
# fit an array of size [Ndim, Nsamples]
data = np.vstack([x, y])
kde = gaussian_kde(data)
# evaluate on a regular grid
xgrid = np.linspace(-3.5, 3.5, 40)
ygrid = np.linspace(-6, 6, 40)
Xgrid, Ygrid = np.meshgrid(xgrid, ygrid)
Z = kde.evaluate(np.vstack([Xgrid.ravel(), Ygrid.ravel()]))
# Plot the result as an image
origin='lower', aspect='auto',
extent=[-3.5, 3.5, -6, 6],
cb = plt.colorbar()
KDE has a smoothing length that effectively slides the knob between detail and smoothness (one example of the ubiquitous bias–variance trade-off). The literature on choosing an appropriate
smoothing length is vast: gaussian_kde uses a rule-of-thumb to attempt to find a nearly optimal smoothing length for the input data.
Other KDE implementations are available within the SciPy ecosystem, each with its own strengths and weaknesses; see, for example, sklearn.neighbors.KernelDensity and
statsmodels.nonparametric.kernel_density.KDEMultivariate. For visualizations based on KDE, using Matplotlib tends to be overly verbose. The Seaborn library, discussed in Visualization With Seaborn,
provides a much more terse API for creating KDE-based visualizations. | {"url":"https://jakevdp.github.io/PythonDataScienceHandbook/04.05-histograms-and-binnings.html","timestamp":"2024-11-07T07:41:16Z","content_type":"text/html","content_length":"176566","record_id":"<urn:uuid:bd7842cd-9a1d-498b-a144-92ae5ce5907a>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00375.warc.gz"} |
Venn Diagram For 4 Sets
Venn Diagram For 4 Sets - Maths revision video and notes on the topic of sets and venn diagrams. Added aug 1, 2010 by poodiack in mathematics. Web to create a venn diagram, first we draw a rectangle
and label the universal set “ u =. Web the venn diagram types are defined based on the number of sets or circles involved in. A venn diagram represents each set by a circle, usually. Web of course,
there are some venn diagrams for four (or more) sets but they. Web a venn diagram, also called a set diagram or logic diagram, shows all possible logical. Verify that the diagram above has regions
representing all 16 16 possible. Web a venn diagram is a diagram that represents the relationship between and. Web a venn diagram with two intersecting sets breaks up the universal set into four
4 Part Venn Diagram Venn diagram template, Venn diagram, Diagram design
Web venn diagram for 4 sets. Venn diagrams are used to show subsets. Web a venn diagram with two intersecting sets breaks up the universal set into four regions;. Web 4 sets venn diagram template. A
venn diagram represents each set by a circle, usually.
4Set Venn diagram Template Venn diagrams Vector stencils library Venn Diagrams How To
Maths revision video and notes on the topic of sets and venn diagrams. A subset is actually a set that is. The venn diagram shows four sets, a, b, c, and d. Web a venn diagram, also called a set
diagram or logic diagram, shows all possible logical. Web venn diagrams for sets.
Puzzles and Figures Rich Tasks 23 4 Set Venn Diagrams Challenge
Web a venn diagram, also called a set diagram or logic diagram, shows all possible logical. Maths revision video and notes on the topic of sets and venn diagrams. Web a venn diagram is a diagram that
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diagrams. Web of course, there are some venn diagrams for four (or more) sets but they.
9 4 Circle Venn Diagram Template Perfect Template Ideas
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venn diagrams. Web of course, there are some venn diagrams for four (or more) sets but they. Web venn diagram for.
WCC Math for Elementary Teachers Venn Diagrams
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course, there are some venn diagrams for four (or more) sets but they.
Venn Diagram Examples Problems, Solutions, Formula Explanation
Web venn diagrams for sets. Web the venn diagram types are defined based on the number of sets or circles involved in. Venn diagrams are used to show subsets. Added aug 1, 2010 by poodiack in
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9+ Venn Diagram Examples
Maths revision video and notes on the topic of sets and venn diagrams. Web the venn diagram types are defined based on the number of sets or circles involved in. A venn diagram represents each set by
a circle, usually. Web a venn diagram is a diagram that represents the relationship between and. Web 4 sets venn diagram template.
Venn Diagram Calculator 4 Sets
Web venn diagrams for sets. Web write sets based on what a venn diagram shows. Verify that the diagram above has regions representing all 16 16 possible. Web a venn diagram, also called a set diagram
or logic diagram, shows all possible logical. Web of course, there are some venn diagrams for four (or more) sets but they.
A Venn diagram consists of multiple overlapping closed curves, usually circles, each
Web the venn diagram types are defined based on the number of sets or circles involved in. Venn diagrams are used to show subsets. A subset is actually a set that is. Added aug 1, 2010 by poodiack in
mathematics. Web write sets based on what a venn diagram shows.
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Web the venn diagram types are defined based on the number of sets or circles involved in. Verify that the diagram above has regions representing all 16 16 possible. The venn diagram shows four sets,
a, b, c, and d. Added aug 1, 2010 by poodiack in mathematics.
Related Post: | {"url":"https://claims.solarcoin.org/en/venn-diagram-for-4-sets.html","timestamp":"2024-11-06T06:13:44Z","content_type":"text/html","content_length":"25003","record_id":"<urn:uuid:2fa27039-dad6-4f5c-b743-400c90fcc013>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00083.warc.gz"} |
Chasing Convex Bodies with Linear Competitive Ratio (Journal Article) | NSF PAGES
We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn
independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost
of the matching. Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a
tight O(log n)-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of O(log n) has long been conjectured and remains
a tantalizing open question. In this paper, we show that the i.i.d model admits substantially better algorithms: our main result is an O((log log log n)^2)-competitive algorithm in this model,
implying a strict separation between the i.i.d model and the adversarial and random order models. Along the way we give a 9-competitive algorithm for the line and tree metrics - the first O(1)
-competitive algorithm for any non-trivial arrival model for these much-studied metrics.
more » « less | {"url":"https://par.nsf.gov/biblio/10327024-chasing-convex-bodies-linear-competitive-ratio","timestamp":"2024-11-09T17:24:59Z","content_type":"text/html","content_length":"245023","record_id":"<urn:uuid:23687911-06a3-4cc9-9a8e-403d8e2f8773>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00241.warc.gz"} |
Xianhang - LessWrong
Sorted by
But hang on, the foundation of Bayesianism is the counterfactual. P(A|B) = 0.6 means that "If B were true, then P(A) = 0.6 would be true". Where does the truth value of P(A) = 0.6 come from then if
we are to accept Bayesianism as correct? | {"url":"https://www.lesswrong.com/users/xianhang","timestamp":"2024-11-07T01:18:05Z","content_type":"text/html","content_length":"66638","record_id":"<urn:uuid:2125ded0-5255-43a6-b60a-8864fa1e08a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00554.warc.gz"} |
Intermittent flow modeling. Part 2: Time-varying flows and flows in variable area ducts
The all-flow-regime model of fluid flow, previously applied in [1] to flows with axially and temporally uniform Reynolds numbers, has been implemented here for flows in which the Reynolds number may
either vary with time or along the length of a pipe. In the former situation, the timewise variations were driven by a harmonically oscillating inlet flow. These oscillations created a succession of
flow-regime transitions encompassing purely laminar and purely turbulent flows as well as laminarizing and turbulentizing flows where intermittency prevailed. The period of the oscillations was
increased parametrically until the quasi-steady regime was attained. The predicted quasi-steady friction factors were found to be in excellent agreement with those from a simple model under which the
flow is assumed to pass through a sequence of instantaneous steady states. In the second category of non-constant-Reynolds- number flows, axial variations of a steady flow were created by means of a
finite-length conical enlargement which connected a pair of pipes of constant but different diameters. The presence of the cross-sectional enlargement gives rise to a reduction of the Reynolds number
that is proportional to the ratio of the diameters of the upstream and the downstream pipes. Depending on the magnitude of the upstream inlet Reynolds number, the downstream fully developed flow
could variously be laminar, intermittent, or turbulent. The presence or absence of flow separation in the conical enlargement had a direct effect on the laminarization process. For both categories of
non-constant-Reynolds-number flows, laminarization and turbulentization were quantified by the ratio of the rate of turbulence production to the rate of turbulence destruction.
Original language English (US)
Title of host publication 2010 14th International Heat Transfer Conference, IHTC 14
Pages 625-633
Number of pages 9
State Published - 2010
Event 2010 14th International Heat Transfer Conference, IHTC 14 - Washington, DC, United States
Duration: Aug 8 2010 → Aug 13 2010
Publication series
Name 2010 14th International Heat Transfer Conference, IHTC 14
Volume 2
Other 2010 14th International Heat Transfer Conference, IHTC 14
Country/Territory United States
City Washington, DC
Period 8/8/10 → 8/13/10
Dive into the research topics of 'Intermittent flow modeling. Part 2: Time-varying flows and flows in variable area ducts'. Together they form a unique fingerprint. | {"url":"https://experts.umn.edu/en/publications/intermittent-flow-modeling-part-2-time-varying-flows-and-flows-in","timestamp":"2024-11-13T13:14:07Z","content_type":"text/html","content_length":"55253","record_id":"<urn:uuid:16463758-4414-4206-89ef-6a26bd5e4b29>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00138.warc.gz"} |
Number Theory
[ home ] [ garrett@math.umn.edu ]
( See also: [ vignettes ] ... [ functional analysis ] ... [ intro to modular forms ] ... [ representation theory ] ... [ Lie theory, symmetric spaces ] ... [ buildings notes ] ... [ number theory ]
... [ algebra ] ... [ complex analysis ] ... [ real analysis ] ... [ homological algebra ] )
Algebraic Number Theory 2011-2012
Book = 2011-12 Notes [updated Thursday, 18-May-2017 15:25:21 CDT] ... 209 pages: overheads reformatted as normal text, repetitions eliminated, some examples and details added.
See also modular forms notes from 2005-6 and 2010-11 and 2013-14.
2011-12 Overheads in reverse chrono order: again, these are subsumed by the 2011-12 Notes
• 05-04-2012 Sketch of main points of homological formulation of classfield theory, with some explanations: Tate cohomology of groups, etc, [updated Saturday, 05-May-2012 11:06:24 CDT]
• 05-02-2012 Classfield theory, (co)homological formulation, cont'd: Herbrand quotient, example: local cyclic norm index equality. [updated Thursday, 03-May-2012 14:04:26 CDT]
• 04-30-2012 Classfield theory, (co)homological formulation, cont'd: Herbrand quotient, homology long exact sequences, Euler characteristics, example: local cyclic norm index equality. [updated
Tuesday, 01-May-2012 07:39:57 CDT]
• 04-27-2012 Classfield theory, (co)homological formulation, cont'd: Herbrand quotient, homology long exact sequences, Euler characteristics, snake lemma, example: meromorphic continuation of Gamma
(!) [updated Thursday, 26-Apr-2012 20:13:49 CDT]
• 04-25-2012 Classfield theory, (co)homological formulation, cont'd: homology long exact sequences, Euler characteristics, snake lemma, examples [updated Thursday, 26-Apr-2012 07:33:43 CDT]
• 04-23-2012 Classfield theory, (co)homological formulation, cont'd: recollection of how to understand holes in higher-dimensional spheres by homology long exact sequences, intro to group (co-)
homology [updated Tuesday, 24-Apr-2012 16:31:49 CDT]
• 04-20-2012 Classfield theory, cont'd: assembly of local to global statement. Grounding for post-1940 (co)homological formulation: recollection of how to understand holes in higher-dimensional
spheres by homology long exact sequences. [updated Friday, 20-Apr-2012 16:12:26 CDT]
• 04-18-2012 Classfield theory, cont'd: assembly of local to global statement. Recollection in quadratic case that k^x-invariance of idelic norm residue symbol yields quadratic reciprocity.
Grounding for post-1940 (co)homological formulation. [updated Friday, 20-Apr-2012 16:04:24 CDT]
• 04-16-2012 Classfield theory, cont'd: idelic statement of global classfield theory, review of Kummer theory and independence of roots, cyclotomic extensions of Q, Hilbert's theorem 90 [updated
Monday, 16-Apr-2012 15:58:22 CDT]
• 04-13-2012 Classfield theory, cont'd: idelic statement of global classfield theory, review of Kummer theory and independence of roots [updated Monday, 16-Apr-2012 15:58:06 CDT]
• 04-11-2012 Classfield theory, cont'd: review of extensions of unramified extensions of local fields, quadratic extensions of local fields, Kummer theory [updated Monday, 16-Apr-2012 15:57:43 CDT]
• 04-09-2012 Classfield theory, cont'd: review of extensions of finite fields, unramified extensions of local fields [updated Monday, 09-Apr-2012 17:37:38 CDT]
• 04-06-2012 Classfield theory: basic overview [updated Saturday, 07-Apr-2012 17:28:20 CDT]
• 04-04-2012 Completion of proof of Hecke's identity... introduction of intrinsic action on representation spaces of Casimir operator, from center of universal enveloping algebra. Direct
computation that Casimir acts as spherical Laplacian. [updated Tuesday, 03-Apr-2012 14:00:28 CDT]
• 04-02-2012 Completion of proof of Hecke's identity... via some representation theory of orthogonal groups [updated Monday, 02-Apr-2012 20:21:21 CDT]
• 03-30-2012 Interlude: Hecke's identity for harmonic polynomial multiples of Gaussians, cont'd... some representation theory of orthogonal groups [updated Saturday, 31-Mar-2012 14:54:08 CDT]
• 03-28-2012 Interlude: Hecke's identity for harmonic polynomial multiples of Gaussians, cont'd [updated Wednesday, 28-Mar-2012 16:44:22 CDT]
• 03-26-2012 Interlude: Hecke's identity for harmonic polynomial multiples of Gaussians [updated Tuesday, 27-Mar-2012 10:47:24 CDT]
• 03-23-2012 Details supporting Iwasawa-Tate: unramified and ramified archimedean local zeta integrals, convergence of local zetas, Fourier transforms of ramified archimedean integrals, Hecke's
identity [updated Saturday, 24-Mar-2012 14:50:42 CDT]
• 03-21-2012 Details supporting Iwasawa-Tate: Convergence of global half-zeta integrals, Hecke's identity, harmonic polynomials, harmonic analysis on spheres. [updated Tuesday, 20-Mar-2012 14:42:53
• 03-19-2012 Details supporting Iwasawa-Tate: good finite-prime local zeta integrals, unramified and ramified archimedean local zeta integrals [updated Tuesday, 20-Mar-2012 09:22:43 CDT]
• Products of entire functions, especially (s-1)zeta(s):
• 03-09-2012 Beginning follow-up details supporting Iwasawa-Tate: elementary global integrals, good finite-prime local integrals, local functional equation [updated Friday, 09-Mar-2012 16:46:20
• 03-07-2012 Iwasawa-Tate executed for Dedekind zetas, and then the general case. Comments on needed supporting material. [updated Friday, 09-Jan-2015 17:38:09 CST]
• 03-05-2012 Iwasawa-Tate executed for Dirichlet L-functions, for Dedekind zetas [updated Monday, 05-Mar-2012 15:24:42 CST]
• 03-02-2012 Iwasawa-Tate executed for Euler-Riemann zeta, for Dirichlet L-functions [updated Thursday, 01-Mar-2012 11:25:29 CST]
• Non-overhead 02-29-2012 First pass at Iwasawa-Tate: the argument executed for Euler-Riemann zeta, for Dirichlet L-functions, for Dedekind zetas, for grossencharacter L-functions. ... some details
postponed. [updated Saturday, 23-May-2020 16:25:50 CDT]
• 02-27-2012 Toward Iwasawa-Tate: compact operators, Hilbert-Schmidt integral operators. ... [updated Friday, 24-Feb-2012 15:52:05 CST]
• 02-24-2012 Toward Iwasawa-Tate: decomposition of L^2 of compact abelian groups by compact operators: Hilbert-Schmidt integral operators. ... [updated Friday, 24-Feb-2012 14:50:42 CST]
• 02-22-2012 Toward Iwasawa-Tate: Schwartz spaces, Fourier inversion, decomposition of L^2 of compact abelian groups, by compact operators. Toward adelic Poisson summation. ... [updated Thursday,
23-Feb-2012 09:33:59 CST]
• 02-20-2012 Toward Iwasawa-Tate: p-adic Fourier transform, Fourier inversion. ... [updated Monday, 20-Feb-2012 16:42:19 CST]
• 02-17-2012 Toward Iwasawa-Tate: Fourier transform, Fourier inversion, on archimedean and p-adic completions ... [updated Sunday, 19-Feb-2012 15:50:59 CST]
• 02-15-2012 Toward Iwasawa-Tate: unitary duals, characters, Fourier inversion ... [updated Thursday, 16-Feb-2012 10:01:36 CST]
• 02-13-2012 Toward Iwasawa-Tate's modernization of Hecke's treatment of L-functions: unitary duals of Q[p], A, and A/Q. ... [updated Monday, 13-Feb-2012 15:40:48 CST]
• 02-10-2012 General argument for uniqueness of invariant functionals. Introduction to Iwasawa-Tate's modernization of Hecke's extensions of Riemann's argument for meromorphic continuation and
functional equation. ... [updated Friday, 10-Feb-2012 15:37:11 CST]
• 02-08-2012 measure and integration on Q[p] and A. General uniqueness proof. ... [updated Thursday, 09-Feb-2012 18:06:24 CST]
• 02-06-2012 volumes of arithmetic quotients, special values of L-functions ... [updated Tuesday, 07-Feb-2012 08:48:01 CST]
• [hiatus]
• 01-23-2012 measures on quotients, volume of SL(n,Z)\SL(n,R) ... [updated Sunday, 22-Jan-2012 20:49:30 CST]
• 01-20-2012 Background for Fujisaki's lemma (and corollaries on class groups, units): invariant measures on quotients, iterated integrals, comparison to Minkowski's classical results, external
characterization of invariant measures/integrals. ... [updated Friday, 20-Jan-2012 13:13:54 CST]
• 01-18-2012 (Recap of Fujisaki's lemma, ideal class groups as idele class groups, units theorem.) Invariant measures on quotients, iterated integrals. ... [updated Thursday, 19-Jan-2012 14:31:52
• 12-14-2011 Interlude/preview... toward Iwasawa-Tate theory: self-duality of Q[p] and A, mutual duality of A/k and k, compact-open topology, no-small-subgroups, ... [updated Tuesday, 13-Dec-2011
19:35:15 CST]
• 12-12-2011 Generalized ideal class groups are images of the idele class group, classification of closed subgroups of R^n, other supporting stuff... [updated Monday, 12-Dec-2011 18:16:25 CST]
• 12-09-2011 Proof of Fujisaki's compactness lemma, measure-theory pigeon-hole principle, finiteness of class number, Dirichlet Units Theorem, extensions [updated Saturday, 10-Dec-2011 12:03:20
• 12-07-2011 Proof of Fujisaki's compactness lemma, finiteness of class number, Dirichlet Units Theorem, extensions [updated Wednesday, 07-Dec-2011 14:15:45 CST]
• 12-05-2011 Fujisaki's compactness lemma, finiteness of class number, Dirichlet Units Theorem [updated Monday, 05-Dec-2011 12:25:09 CST]
• 12-02-2011 Ostrowski's theorem, Approximation Theorem, adelic solenoid A/k. [updated Friday, 02-Dec-2011 12:19:57 CST]
• 11-30-2011 Product formula, Ostrowski's theorem, Approximation Theorem [updated Wednesday, 30-Nov-2011 18:11:20 CST]
• 11-28-2011 ... completions, finite-dimensional topological vector spaces, product formula [updated Tuesday, 29-Nov-2011 11:34:52 CST]
• 11-23-2011 ... completions, finite-dimensional topological vector spaces, product formula [updated Saturday, 26-Nov-2011 09:31:26 CST]
• 11-21-2011 Absolute values, completions, finite-dimensional topological vector spaces [updated Sunday, 20-Nov-2011 11:41:54 CST]
□ some class number data class numbers of the first few hundred complex quadratic fields [updated Monday, 21-Nov-2011 07:32:40 CST]
• 11-18-2011 class number formula for complex quadratic fields [updated Friday, 18-Nov-2011 12:39:10 CST]
• 11-16-2011 ... Dedekind zeta functions, class numbers, residues of Epstein zetas, class number formula for complex quadratic fields [updated Thursday, 17-Nov-2011 10:18:40 CST]
• 11-14-2011 ... ramification degrees, residue class field degrees, ideal norms, Dedekind zeta functions [updated Tuesday, 15-Nov-2011 10:22:59 CST]
• 11-11-2011 ... basic big theorem characterizing Dedekind domains, cont'd + sequel [updated Friday, 11-Nov-2011 14:03:09 CST]
• 11-09-2011 ... basic big theorem on factorization in Dedekind domains [updated Thursday, 10-Nov-2011 10:13:56 CST]
• 11-07-2011 ... more: Galois theory of primes lying over, intro to Dedekind rings. [updated Monday, 07-Nov-2011 12:20:49 CST]
• 11-04-2011 ...better localization... Galois theory of primes lying over, intro to Dedekind rings. [updated Saturday, 05-Nov-2011 10:07:11 CDT]
• 11-02-2011 A better version of localization. [updated Thursday, 03-Nov-2011 10:26:23 CDT]
• 10-31-2011 Primes lying over, Galois action. Review of localization. [updated Saturday, 29-Oct-2011 11:29:10 CDT]
• 10-28-2011 Brief recollection of quadratic norm-residue, Hilbert symbols, reciprocity laws. Primes lying over, Galois action. [updated Saturday, 29-Oct-2011 11:29:10 CDT]
• 10-26-2011 Quadratic norm-residue, Hilbert symbols, reciprocity laws. Primes lying over, Galois action. [updated Thursday, 27-Oct-2011 12:41:38 CDT]
• 10-24-2011 function fields over finite fields ... quadratic norm-residue, Hilbert symbols, reciprocity laws [updated Sunday, 23-Oct-2011 12:21:32 CDT]
• 10-21-2011 Cont'd: function fields... especially over finite fields... Artin-Schreier extensions... quadratic norm/Hilbert reciprocity laws [updated Sunday, 23-Oct-2011 11:24:42 CDT]
• 10-19-2011 Cont'd: function fields, Galois groups of algebraic closures, comments on Galois groups and their representations, the usefulness of repn theory, ... and then focus on function fields
over finite fields. [updated Friday, 21-Oct-2011 07:59:15 CDT]
• 10-17-2011 Cont'd: function fields, especially extensions of C(X) and C((X)). Formal Puiseux expansions, Newton polygons. [updated Tuesday, 18-Oct-2011 11:44:09 CDT]
• 10-14-2011 function fields, especially extensions of C(X) and C((X)). Formal Puiseux expansions, Newton polygons. [updated Saturday, 15-Oct-2011 13:52:27 CDT]
• 10-12-2011 function fields [updated Thursday, 13-Oct-2011 09:36:36 CDT]
• 10-10-2011 commutative algebra: integral extensions, Noetherian-ness, function field case [updated Monday, 10-Oct-2011 12:20:01 CDT]
• 10-07-2011 commutative algebra: integral extensions, Noetherian-ness [updated Friday, 07-Oct-2011 12:56:29 CDT]
• 10-05-2011 commutative algebra: integral extensions, algebraic integers [updated Friday, 07-Oct-2011 13:02:16 CDT]
• 10-03-2011 some commutative algebra: integral extensions, algebraic integers [updated Sunday, 09-Oct-2011 16:55:37 CDT]
• 09-30-2011 p-adic numbers, projective limits, colimits, adeles [updated Friday, 30-Sep-2011 14:44:47 CDT]
• 09-28-2011 more on Hensel's Lemma, p-adic numbers, projective limits [updated Thursday, 29-Sep-2011 09:06:27 CDT]
• 09-26-2011 more on Hensel's Lemma, p-adic numbers [updated Sunday, 25-Sep-2011 17:04:29 CDT]
• 09-23-2011 Hensel's Lemma, p-adic numbers [updated Sunday, 25-Sep-2011 11:07:41 CDT]
• 09-21-2011 Cont'd: more factoring Dedekind zeta functions [updated Wednesday, 21-Sep-2011 12:21:22 CDT]
• 09-19-2011 Cont'd: factoring Dedekind zeta functions into Dirichlet L-functions [updated Tuesday, 20-Sep-2011 07:49:20 CDT]
• 09-16-2011 factorization of some Dedekind zeta functions into Dirichlet L-functions [updated Saturday, 17-Sep-2011 15:06:55 CDT]
• 09-14-2011 Quadratic Reciprocity over Q, by Gauss sums [updated Wednesday, 14-Sep-2011 14:15:36 CDT]
• 09-12-2011 meromorphic continuation and functional equation of zeta, Poisson summation and functional equation of theta, integral representation of zeta in terms of theta [updated Sunday,
11-Sep-2011 18:00:38 CDT]
• 09-09-2011 Riemann's explicit formula [updated Sunday, 11-Sep-2011 18:17:10 CDT]
Older notes
Unless explicitly noted otherwise, everything here, work by Paul Garrett, is licensed under a Creative Commons Attribution 3.0 Unported License. ... [ garrett@umn.edu ]
The University of Minnesota explicitly requires that I state that "The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been
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Using CHILDREN() to Define Range in VLOOKUP()
I have a parent row on Row 1 with three indented child rows. I want to populate a cell in the parent row with the first name from a child row that matches a value in one of the columns.
There will be other parent rows with their own children, and the number of children may vary. Each parent row represents a "family", and each child row represents a "student."
I'd like to use a VLOOKUP() formula with a dynamic range from the child rows.
Example: Populate [Current Grade: 7]1 with the first name of a student with the value "2019-2020" in the column [Enrolled Grade: 7].
Screenshot: VLOOKUP_ERROR_04.png
I'm attempting to do this with a VLOOKUP() formula in [Current Grade: 7]1.
If I explicitly define the range in VLOOKUP(), it works:
=VLOOKUP("2019-2020",[Enrolled Grade: 7]2:[First Name]4,4,false)
However, the number of child rows for any given parent will vary, and I don't want to have to explicitly define the range for the formula in each parent row. Instead, I'd like to use the CHILDREN()
function to define the range, but I get an error with:
=VLOOKUP("2019-2020",CHILDREN([Enrolled Grade: 7]1:[First Name]1),4,false)
The only difference is that I'm using CHILDREN() to define the range. That works just fine with the COUNT() and SUM() functions, but it doesn't seem to work with VLOOKUP(). I'm not sure why, since in
both cases, it's defining a range.
For example, this works fine:
=COUNTIF(CHILDREN([Enrolled Grade: 7]1:[First Name]1), "2019-2020")
In the above, the range is defined as: CHILDREN([Enrolled Grade: 7]1:[First Name]1)
If I use that same range definition in VLOOKUP(), I get an error for incorrect argument set.
What am I doing wrong?
If it's a syntax error, please help me get it right.
If it's a limitation of the VLOOKUP() function, what other options do I have to dynamically define a range (from child rows) that will work in the VLOOKUP() function?
Maybe I can construct the range by getting the starting *child* cell of the first range column and the last *child* cell of the last range column. If that would work, how do I do that?
Other options?
• I might have found a solution using INDEX() and MATCH():
=INDEX(CHILDREN([First Name]1), MATCH("2019-2020", CHILDREN([Enrolled Grade: 7]1), 0))
I would still like to know if there's a way to do it with VLOOKUP() or if using CHILDREN() to define a range in VLOOKUP() is not possible...
Any guidance on this is much appreciated. Thanks!
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What is the ANS Module Asking For? Introduction to the Sizing window in the ANS Add-on Module
Sizing Objective – What do you want to minimize?
The first input the ANS module will ask for is an objective. While the goal is typically to minimize the system cost, there are different ways to go about achieving this. Minimizing the monetary cost
directly for the components in the system would be the most direct approach, but this is also complex as cost information for every component is typically not available to engineers during the design
In many cases simply minimizing the pipe weight for the system will be equivalent to minimizing the monetary cost. Other options such as minimizing flow volume are also available for use. Depending
on the constraints for the system it may be useful to minimize the volume of the piping system instead. Figure 2 shows a dimensionless comparison of different sizing costs using Steel ANSI Schedule
40 pipe. It can be seen here that in this case the pipe weight and monetary cost lines show the best correlation, though that may not be true for all pipe materials/schedules.
Sizing Assignments – What do you want to be sized?
For this panel the user simply chooses which pipes Fathom/Arrow should consider when minimizing cost. The user can also create common size groups, which will force all pipes in the common size group
to be the same size. Creating effective common size groups will help to reduce the run time for the model and reduce the complexity of the design by limiting the number of separate pipe sizes in the
design. For example, the suction piping segments for a group of parallel pumps could be added to one common size group, while the discharge piping for those pumps could be added to another group,
since it would be logical for those pipes to have equivalent sizes.
Candidate Sets – What pipe materials and sizes can be used for the system?
A candidate set is required to tell Fathom/Arrow what pipe or duct sizes can be used for each pipe/duct in the system. Generally the rule of thumb is that the more pipe sizes you include in the
candidate set, the more flexibility for Fathom/Arrow to find a lower cost solution. Of course, more available sizes can also lead to longer run times. For a simple model including all possible pipe
sizes is fine. However, for more complex models limiting the candidate set may be desired to obtain a faster solution.
Design Requirements – What conditions must be enforced in the system?
Design Requirements are important in the ANS module to make sure that the final design proposed by the ANS module has a minimal cost and will operate successfully. Depending on how the model has been
built, some requirements for the system may be inherently defined in Fathom/Arrow as boundaries for the system. For example, there may be a required pressure and flow at the system outlet. To account
for this, you may represent the outlet boundary as an Assigned Pressure junction, which will inherently ensure that the minimum pressure is achieved at the outlet. You could then add a design
requirement at that point to ensure that the minimum flow is achieved, or vice versa. A design Requirement will provide more flexibility than a boundary condition, as the design requirement does not
fix the flow rate to the minimum/maximum value, but instead allows that number to vary, as long as the minimum/maximum condition is met. Thus, for each design condition, the engineer should consider
whether a boundary condition or design requirement would best account for that condition.
Sizing Method – What calculation method should be used?
The last panel which will always be required is the Sizing Method. Typically, the user should first run the model using the continuous sizing type, then run the model using the discrete sizing type
as a comparison. If the two solutions are not similar, then the engineer should choose a different Search Method which may be more suited for the analysis. Running the model using multiple different
searching methods is recommended to find the best system design, as it is typically not clear which method will be best suited for each system.
Once the complete model is built it is possible to define all sizing inputs described above in just 15 – 20 minutes to perform a simple pipe weight sizing. All you will need to know is what type of
cost you want to minimize, which pipes you would like to size in the model, what pipe materials/schedules you want to consider for the system, and what design requirements must be met. You can then
adjust the sizing input to improve the run time by linking the size of some pipes using common size groups and testing different sizing methods.
With even simple inputs large savings can be found on the initial cost for building the system. To see and/or build examples, check out the Example Help file, accessed in Fathom or Arrow from the
Help menu. You can also check out this video tutorial which walks through an example of sizing a system based on pipe weight.
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IF - Excel docs, syntax and examples
The IF function in Excel allows you to perform different actions based on a specified condition. It is commonly used for logical operations, where you want Excel to make a decision based on whether a
certain criteria is met.
=IF(Logical_test, Value_if_true, Value_if_false)
Logical_test The condition that you want to test. It can be an expression, a value, or a cell reference.
Value_if_true The value or formula to return if the logical test evaluates to TRUE.
Value_if_false The value or formula to return if the logical test evaluates to FALSE.
About IF 🔗
When you need Excel to make decisions based on specific conditions, that's where the IF function comes into play. It acts as the gatekeeper for your data, allowing you to set up rules and have Excel
follow them diligently. Whether you want to categorize data, flag entries, or perform different calculations based on certain criteria, the IF function is your go-to tool for logical operations
within your spreadsheet environment.
Examples 🔗
Suppose you have a list of test scores in column A, and you want to categorize them as Pass or Fail based on a threshold score of 50. You can use the IF function as follows: =IF(A1>=50, "Pass",
If you have a sales data in column B and you want to apply a bonus of 10% to sales greater than $1000, you can use: =IF(B1>1000, B1*0.1, 0)
Ensure that the logical_test provided in the IF function is a condition that can evaluate to TRUE or FALSE. You can nest multiple IF functions together for more complex decision-making processes.
Questions 🔗
Can I use text values as the 'Value_if_true' or 'Value_if_false' arguments in the IF function?
Yes, you can use text values, numbers, or even formulas in the 'Value_if_true' or 'Value_if_false' arguments of the IF function. This flexibility allows you to customize the output based on your
specific requirements.
Is it possible to nest IF functions within each other?
Yes, you can nest IF functions within each other in Excel. This allows you to create more complex decision trees by evaluating multiple conditions in a sequential manner.
What happens if the logical_test in the IF function does not evaluate to TRUE or FALSE?
If the logical_test in the IF function does not evaluate to TRUE or FALSE, Excel may return an error or unexpected result. Make sure that your logical_test is correctly structured to avoid such
Related functions 🔗
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class tianshou.data.Batch(batch_dict: Optional[Union[dict, tianshou.data.batch.Batch, Sequence[Union[dict, tianshou.data.batch.Batch]], numpy.ndarray]] = None, copy: bool = False, **kwargs: Any)
Bases: object
The internal data structure in Tianshou.
Batch is a kind of supercharged array (of temporal data) stored individually in a (recursive) dictionary of object that can be either numpy array, torch tensor, or batch themselves. It is
designed to make it extremely easily to access, manipulate and set partial view of the heterogeneous data conveniently.
For a detailed description, please refer to Understand Batch.
class tianshou.data.ReplayBuffer(size: int, stack_num: int = 1, ignore_obs_next: bool = False, save_only_last_obs: bool = False, sample_avail: bool = False, **kwargs: Any)[source]¶
Bases: object
ReplayBuffer stores data generated from interaction between the policy and environment.
ReplayBuffer can be considered as a specialized form (or management) of Batch. It stores all the data in a batch with circular-queue style.
For the example usage of ReplayBuffer, please check out Section Buffer in Basic concepts in Tianshou.
☆ size (int) – the maximum size of replay buffer.
☆ stack_num (int) – the frame-stack sampling argument, should be greater than or equal to 1. Default to 1 (no stacking).
☆ ignore_obs_next (bool) – whether to store obs_next. Default to False.
☆ save_only_last_obs (bool) – only save the last obs/obs_next when it has a shape of (timestep, …) because of temporal stacking. Default to False.
☆ sample_avail (bool) – the parameter indicating sampling only available index when using frame-stack sampling method. Default to False.
__len__() → int[source]¶
Return len(self).
save_hdf5(path: str) → None[source]¶
Save replay buffer to HDF5 file.
classmethod load_hdf5(path: str, device: Optional[str] = None) → tianshou.data.buffer.base.ReplayBuffer[source]¶
Load replay buffer from HDF5 file.
reset(keep_statistics: bool = False) → None[source]¶
Clear all the data in replay buffer and episode statistics.
set_batch(batch: tianshou.data.batch.Batch) → None[source]¶
Manually choose the batch you want the ReplayBuffer to manage.
unfinished_index() → numpy.ndarray[source]¶
Return the index of unfinished episode.
prev(index: Union[int, numpy.ndarray]) → numpy.ndarray[source]¶
Return the index of previous transition.
The index won’t be modified if it is the beginning of an episode.
next(index: Union[int, numpy.ndarray]) → numpy.ndarray[source]¶
Return the index of next transition.
The index won’t be modified if it is the end of an episode.
update(buffer: tianshou.data.buffer.base.ReplayBuffer) → numpy.ndarray[source]¶
Move the data from the given buffer to current buffer.
Return the updated indices. If update fails, return an empty array.
add(batch: tianshou.data.batch.Batch, buffer_ids: Optional[Union[numpy.ndarray, List[int]]] = None) → Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray, numpy.ndarray][source]¶
Add a batch of data into replay buffer.
○ batch (Batch) – the input data batch. Its keys must belong to the 7 reserved keys, and “obs”, “act”, “rew”, “done” is required.
○ buffer_ids – to make consistent with other buffer’s add function; if it is not None, we assume the input batch’s first dimension is always 1.
Return (current_index, episode_reward, episode_length, episode_start_index). If the episode is not finished, the return value of episode_length and episode_reward is 0.
sample_indices(batch_size: int) → numpy.ndarray[source]¶
Get a random sample of index with size = batch_size.
Return all available indices in the buffer if batch_size is 0; return an empty numpy array if batch_size < 0 or no available index can be sampled.
sample(batch_size: int) → Tuple[tianshou.data.batch.Batch, numpy.ndarray][source]¶
Get a random sample from buffer with size = batch_size.
Return all the data in the buffer if batch_size is 0.
Sample data and its corresponding index inside the buffer.
get(index: Union[int, List[int], numpy.ndarray], key: str, default_value: Optional[Any] = None, stack_num: Optional[int] = None) → Union[tianshou.data.batch.Batch, numpy.ndarray][source]¶
Return the stacked result.
E.g., if you set key = "obs", stack_num = 4, index = t, it returns the stacked result as [obs[t-3], obs[t-2], obs[t-1], obs[t]].
○ index – the index for getting stacked data.
○ key (str) – the key to get, should be one of the reserved_keys.
○ default_value – if the given key’s data is not found and default_value is set, return this default_value.
○ stack_num (int) – Default to self.stack_num.
__getitem__(index: Union[slice, int, List[int], numpy.ndarray]) → tianshou.data.batch.Batch[source]¶
Return a data batch: self[index].
If stack_num is larger than 1, return the stacked obs and obs_next with shape (batch, len, …).
class tianshou.data.PrioritizedReplayBuffer(size: int, alpha: float, beta: float, weight_norm: bool = True, **kwargs: Any)[source]¶
Bases: tianshou.data.buffer.base.ReplayBuffer
Implementation of Prioritized Experience Replay. arXiv:1511.05952.
☆ alpha (float) – the prioritization exponent.
☆ beta (float) – the importance sample soft coefficient.
☆ weight_norm (bool) – whether to normalize returned weights with the maximum weight value within the batch. Default to True.
init_weight(index: Union[int, numpy.ndarray]) → None[source]¶
update(buffer: tianshou.data.buffer.base.ReplayBuffer) → numpy.ndarray[source]¶
Move the data from the given buffer to current buffer.
Return the updated indices. If update fails, return an empty array.
add(batch: tianshou.data.batch.Batch, buffer_ids: Optional[Union[numpy.ndarray, List[int]]] = None) → Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray, numpy.ndarray][source]¶
Add a batch of data into replay buffer.
○ batch (Batch) – the input data batch. Its keys must belong to the 7 reserved keys, and “obs”, “act”, “rew”, “done” is required.
○ buffer_ids – to make consistent with other buffer’s add function; if it is not None, we assume the input batch’s first dimension is always 1.
Return (current_index, episode_reward, episode_length, episode_start_index). If the episode is not finished, the return value of episode_length and episode_reward is 0.
sample_indices(batch_size: int) → numpy.ndarray[source]¶
Get a random sample of index with size = batch_size.
Return all available indices in the buffer if batch_size is 0; return an empty numpy array if batch_size < 0 or no available index can be sampled.
get_weight(index: Union[int, numpy.ndarray]) → Union[float, numpy.ndarray][source]¶
Get the importance sampling weight.
The “weight” in the returned Batch is the weight on loss function to debias the sampling process (some transition tuples are sampled more often so their losses are weighted less).
update_weight(index: numpy.ndarray, new_weight: Union[numpy.ndarray, torch.Tensor]) → None[source]¶
Update priority weight by index in this buffer.
○ index (np.ndarray) – index you want to update weight.
○ new_weight (np.ndarray) – new priority weight you want to update.
__getitem__(index: Union[slice, int, List[int], numpy.ndarray]) → tianshou.data.batch.Batch[source]¶
Return a data batch: self[index].
If stack_num is larger than 1, return the stacked obs and obs_next with shape (batch, len, …).
set_beta(beta: float) → None[source]¶
class tianshou.data.ReplayBufferManager(buffer_list: List[tianshou.data.buffer.base.ReplayBuffer])[source]¶
Bases: tianshou.data.buffer.base.ReplayBuffer
ReplayBufferManager contains a list of ReplayBuffer with exactly the same configuration.
These replay buffers have contiguous memory layout, and the storage space each buffer has is a shallow copy of the topmost memory.
buffer_list – a list of ReplayBuffer needed to be handled.
__len__() → int[source]¶
Return len(self).
reset(keep_statistics: bool = False) → None[source]¶
Clear all the data in replay buffer and episode statistics.
set_batch(batch: tianshou.data.batch.Batch) → None[source]¶
Manually choose the batch you want the ReplayBuffer to manage.
unfinished_index() → numpy.ndarray[source]¶
Return the index of unfinished episode.
prev(index: Union[int, numpy.ndarray]) → numpy.ndarray[source]¶
Return the index of previous transition.
The index won’t be modified if it is the beginning of an episode.
next(index: Union[int, numpy.ndarray]) → numpy.ndarray[source]¶
Return the index of next transition.
The index won’t be modified if it is the end of an episode.
update(buffer: tianshou.data.buffer.base.ReplayBuffer) → numpy.ndarray[source]¶
The ReplayBufferManager cannot be updated by any buffer.
add(batch: tianshou.data.batch.Batch, buffer_ids: Optional[Union[numpy.ndarray, List[int]]] = None) → Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray, numpy.ndarray][source]¶
Add a batch of data into ReplayBufferManager.
Each of the data’s length (first dimension) must equal to the length of buffer_ids. By default buffer_ids is [0, 1, …, buffer_num - 1].
Return (current_index, episode_reward, episode_length, episode_start_index). If the episode is not finished, the return value of episode_length and episode_reward is 0.
sample_indices(batch_size: int) → numpy.ndarray[source]¶
Get a random sample of index with size = batch_size.
Return all available indices in the buffer if batch_size is 0; return an empty numpy array if batch_size < 0 or no available index can be sampled.
class tianshou.data.PrioritizedReplayBufferManager(buffer_list: Sequence[tianshou.data.buffer.prio.PrioritizedReplayBuffer])[source]¶
Bases: tianshou.data.buffer.prio.PrioritizedReplayBuffer, tianshou.data.buffer.manager.ReplayBufferManager
PrioritizedReplayBufferManager contains a list of PrioritizedReplayBuffer with exactly the same configuration.
These replay buffers have contiguous memory layout, and the storage space each buffer has is a shallow copy of the topmost memory.
buffer_list – a list of PrioritizedReplayBuffer needed to be handled.
class tianshou.data.VectorReplayBuffer(total_size: int, buffer_num: int, **kwargs: Any)[source]¶
Bases: tianshou.data.buffer.manager.ReplayBufferManager
VectorReplayBuffer contains n ReplayBuffer with the same size.
It is used for storing transition from different environments yet keeping the order of time.
☆ total_size (int) – the total size of VectorReplayBuffer.
☆ buffer_num (int) – the number of ReplayBuffer it uses, which are under the same configuration.
Other input arguments (stack_num/ignore_obs_next/save_only_last_obs/sample_avail) are the same as ReplayBuffer.
class tianshou.data.PrioritizedVectorReplayBuffer(total_size: int, buffer_num: int, **kwargs: Any)[source]¶
Bases: tianshou.data.buffer.manager.PrioritizedReplayBufferManager
PrioritizedVectorReplayBuffer contains n PrioritizedReplayBuffer with same size.
It is used for storing transition from different environments yet keeping the order of time.
☆ total_size (int) – the total size of PrioritizedVectorReplayBuffer.
☆ buffer_num (int) – the number of PrioritizedReplayBuffer it uses, which are under the same configuration.
Other input arguments (alpha/beta/stack_num/ignore_obs_next/save_only_last_obs/ sample_avail) are the same as PrioritizedReplayBuffer.
set_beta(beta: float) → None[source]¶
class tianshou.data.CachedReplayBuffer(main_buffer: tianshou.data.buffer.base.ReplayBuffer, cached_buffer_num: int, max_episode_length: int)[source]¶
Bases: tianshou.data.buffer.manager.ReplayBufferManager
CachedReplayBuffer contains a given main buffer and n cached buffers, cached_buffer_num * ReplayBuffer(size=max_episode_length).
The memory layout is: | main_buffer | cached_buffers[0] | cached_buffers[1] | ... | cached_buffers[cached_buffer_num - 1] |.
The data is first stored in cached buffers. When an episode is terminated, the data will move to the main buffer and the corresponding cached buffer will be reset.
☆ main_buffer (ReplayBuffer) – the main buffer whose .update() function behaves normally.
☆ cached_buffer_num (int) – number of ReplayBuffer needs to be created for cached buffer.
☆ max_episode_length (int) – the maximum length of one episode, used in each cached buffer’s maxsize.
add(batch: tianshou.data.batch.Batch, buffer_ids: Optional[Union[numpy.ndarray, List[int]]] = None) → Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray, numpy.ndarray][source]¶
Add a batch of data into CachedReplayBuffer.
Each of the data’s length (first dimension) must equal to the length of buffer_ids. By default the buffer_ids is [0, 1, …, cached_buffer_num - 1].
Return (current_index, episode_reward, episode_length, episode_start_index) with each of the shape (len(buffer_ids), …), where (current_index[i], episode_reward[i], episode_length[i],
episode_start_index[i]) refers to the cached_buffer_ids[i]th cached buffer’s corresponding episode result.
class tianshou.data.Collector(policy: tianshou.policy.base.BasePolicy, env: Union[gym.core.Env, tianshou.env.venvs.BaseVectorEnv], buffer: Optional[tianshou.data.buffer.base.ReplayBuffer] = None,
preprocess_fn: Optional[Callable[[…], tianshou.data.batch.Batch]] = None, exploration_noise: bool = False)[source]¶
Bases: object
Collector enables the policy to interact with different types of envs with exact number of steps or episodes.
☆ policy – an instance of the BasePolicy class.
☆ env – a gym.Env environment or an instance of the BaseVectorEnv class.
☆ buffer – an instance of the ReplayBuffer class. If set to None, it will not store the data. Default to None.
☆ preprocess_fn (function) – a function called before the data has been added to the buffer, see issue #42 and Handle Batched Data Stream in Collector. Default to None.
☆ exploration_noise (bool) – determine whether the action needs to be modified with corresponding policy’s exploration noise. If so, “policy. exploration_noise(act, batch)” will be called
automatically to add the exploration noise into action. Default to False.
The “preprocess_fn” is a function called before the data has been added to the buffer with batch format. It will receive only “obs” and “env_id” when the collector resets the environment, and
will receive six keys “obs_next”, “rew”, “done”, “info”, “policy” and “env_id” in a normal env step. It returns either a dict or a Batch with the modified keys and values. Examples are in “test/
Please make sure the given environment has a time limitation if using n_episode collect option.
In past versions of Tianshou, the replay buffer that was passed to __init__ was automatically reset. This is not done in the current implementation.
reset(reset_buffer: bool = True) → None[source]¶
Reset the environment, statistics, current data and possibly replay memory.
reset_buffer (bool) – if true, reset the replay buffer that is attached to the collector.
reset_stat() → None[source]¶
Reset the statistic variables.
reset_buffer(keep_statistics: bool = False) → None[source]¶
Reset the data buffer.
reset_env() → None[source]¶
Reset all of the environments.
collect(n_step: Optional[int] = None, n_episode: Optional[int] = None, random: bool = False, render: Optional[float] = None, no_grad: bool = True) → Dict[str, Any][source]¶
Collect a specified number of step or episode.
To ensure unbiased sampling result with n_episode option, this function will first collect n_episode - env_num episodes, then for the last env_num episodes, they will be collected evenly from
each env.
○ n_step (int) – how many steps you want to collect.
○ n_episode (int) – how many episodes you want to collect.
○ random (bool) – whether to use random policy for collecting data. Default to False.
○ render (float) – the sleep time between rendering consecutive frames. Default to None (no rendering).
○ no_grad (bool) – whether to retain gradient in policy.forward(). Default to True (no gradient retaining).
One and only one collection number specification is permitted, either n_step or n_episode.
A dict including the following keys
○ n/ep collected number of episodes.
○ n/st collected number of steps.
○ rews array of episode reward over collected episodes.
○ lens array of episode length over collected episodes.
○ idxs array of episode start index in buffer over collected episodes.
○ rew mean of episodic rewards.
○ len mean of episodic lengths.
○ rew_std standard error of episodic rewards.
○ len_std standard error of episodic lengths.
class tianshou.data.AsyncCollector(policy: tianshou.policy.base.BasePolicy, env: tianshou.env.venvs.BaseVectorEnv, buffer: Optional[tianshou.data.buffer.base.ReplayBuffer] = None, preprocess_fn:
Optional[Callable[[…], tianshou.data.batch.Batch]] = None, exploration_noise: bool = False)[source]¶
Bases: tianshou.data.collector.Collector
Async Collector handles async vector environment.
The arguments are exactly the same as Collector, please refer to Collector for more detailed explanation.
reset_env() → None[source]¶
Reset all of the environments.
collect(n_step: Optional[int] = None, n_episode: Optional[int] = None, random: bool = False, render: Optional[float] = None, no_grad: bool = True) → Dict[str, Any][source]¶
Collect a specified number of step or episode with async env setting.
This function doesn’t collect exactly n_step or n_episode number of transitions. Instead, in order to support async setting, it may collect more than given n_step or n_episode transitions and
save into buffer.
○ n_step (int) – how many steps you want to collect.
○ n_episode (int) – how many episodes you want to collect.
○ random (bool) – whether to use random policy for collecting data. Default to False.
○ render (float) – the sleep time between rendering consecutive frames. Default to None (no rendering).
○ no_grad (bool) – whether to retain gradient in policy.forward(). Default to True (no gradient retaining).
One and only one collection number specification is permitted, either n_step or n_episode.
A dict including the following keys
○ n/ep collected number of episodes.
○ n/st collected number of steps.
○ rews array of episode reward over collected episodes.
○ lens array of episode length over collected episodes.
○ idxs array of episode start index in buffer over collected episodes.
○ rew mean of episodic rewards.
○ len mean of episodic lengths.
○ rew_std standard error of episodic rewards.
○ len_std standard error of episodic lengths. | {"url":"https://tianshou.org/en/v0.4.7/api/tianshou.data.html","timestamp":"2024-11-06T17:24:39Z","content_type":"text/html","content_length":"131282","record_id":"<urn:uuid:0dcb858c-60f8-450e-9d02-c4caa1bc6b02>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00663.warc.gz"} |
News - Duhallow Grey Geek
Floating Point – SQL Smell?
Do you use Floating Point numbers? Do you even know what they are?
Phil Factor regards Floating Point Numbers as an SQL Smell. That is to say “something which may cause problems”.
I’m going to explain why this is from an analyst’s point of view.
What are Floating Point numbers?
Floating Point – Normal versus Scientific or Engineering representations of numbers.
When I first started programming with FORTRAN we only had two sorts of numbers: integers and floating-point. You used integers for counting things and floating-point for calculations. It was that
Modern languages and SQL databases allow other options, including precise decimal forms.
Floating point numbers use the binary equivalent of “Scientific” or “Engineering” notation. They stored differently to integers or decimals.
Their value is stored in two parts which are known as the “mantissa” and the “exponent”. This arrangement allows them to hold an enormous range of values very efficiently but at the expense of
Problems with Floating Point numbers
Floating Point Arithmetic can cause loss of precision
Floating point numbers are intended for scientific or engineering calculations.
They can cause problems in normal business calculations.
• Arithmetic can cause rounding.
• Testing for equality can produce unexpected results.
• Phil Factor does not like them being used in keys or indexes.
Size and range of Numbers
Size of FLOAT or REAL for different Precision values
Floating-point numbers allow a very large and very small values. I cannot imagine examples of the extremes.
Size of DECIMAL (or NUMERIC) columns
On the other hand, precise numeric data-types do not have the problems associated with floating point.
When should you use Floating Point numbers?
Some Very Big and Small Numbers
Very few businesses or “domains” need to use floating-point numbers. The table shows some example where they would be essential. In most cases these are estimates of extremely large, or very small,
Look at it like this: if you are working in engineering, astronomy or nuclear physics you may need floating-point numbers. You probably know if you really need them.
I recommend avoiding floating-point numbers. Phil Factor describes them as an SQL Smell which should be investigated. They are essential for some applications. When misapplied they will cause
problems. If you think you need to use them, think hard.
Where next?
This is the concluding post in this sequence of articles. I’ve chosen to concentrate on potential problems which are likely to trouble analysts like me. Phil Factor’s original article identifies many
more SQL Smells. These include details of design and programming. If you spend time working with SQL and specifying or designing databases, then I recommend the article both as background reading and
as a handy reference.
As a Postscript: I don’t think Floating Point Numbers are bad, just people sometimes use them in the wrong way. Hugo Kornelis thought it was worth his time writing a post in defence of them – Float
does NOT suck!
LIKE with an initial wildcard character – SQL Smell
Leading wildcards with LIKE are an SQL Smell
Phil Factor says “using LIKE in a WHERE clause with an initial wildcard character” is an SQL Smell, but why? Why does the position of the wildcard make a difference?
This really is a problem, and it gets more serious with larger tables.
I’m going to explain what the problem is from an analyst’s point of view.
What LIKE does for us
Like in the SQL WHERE clause
LIKE identifies rows which match the criteria. It can be used with character columns and combined with other conditions.
The first time I encountered this statement I was amazed that it worked at all! This simply demonstrates how SQL protects us from what is really happening underneath the surface.
Like with a trailing wildcard using an index
Appearances can be deceptive. LIKE only appears to match the search argument against the contents of the table column. It has to use an index in order to perform properly. The entries in the index
are sorted. SQL uses the sort sequence with “trailing” wildcards, to get to the first match quickly, and then ignore any entries past the last match.
The problem with leading wildcards
Valid LIKE criteria – But two may cause problems
SQL cannot use an index to matching entries when there is a leading wildcard. As a consequence it will probably scan the whole table. This will work but it may take longer than you want.
The problem gets worse for tables with more rows because with a leading wildcard LIKE has to test every row.
There will be times when you need to use a leading wildcard it in a query. Go ahead and use it, just expect poor performance. If this query which will be used repeatedly, or in a program, then
discuss what you are trying to do with the people who look after your database.
Phil Factor describes leading wildcards with LIKE as an SQL Smell. This may be unavoidable in some ad-hoc queries, but you should treat it with caution if you intend to reuse the query or use it with
large tables.
Where next?
The next, and final, blog post is on “Floating Point Numbers”. Do you use floating point numbers in your database? Phil Factor thinks this is an SQL Smell too. Find out why in the next article.
SELECT * – SQL Smell
Is SELECT * FROM Table an SQL Smell?
“SELECT *” What could possibly be wrong with that? Everybody uses SELECT *, don’t they?
Phil Factor describes this as an SQL Smell, and it is. Finding it should immediately make you suspicious. The problem is not the statement, it’s where you find it!
Interactive and Batch SQL
SQL can be used: Interactively, in Batch files and in Programs
You can use SQL interactively, in “batch”, and inside programs. One of the good things about SQL is that it looks pretty much the same wherever you find it.
“SELECT *” is intended to be used interactively. That’s how I use it, and I expect Phil Factor does the same. Typing the statement in the figure at a command line, or inside a development environment
like SSMS is completely appropriate.
Some people create queries interactively using “SELECT *” as a starting point. That’s legitimate too. It’s a matter of personal style.
Don’t use this form of SELECT in a program or when you expect to reuse it. If you save the file, you shouldn’t be using “SELECT *”.
Why Using “SELECT *” is a problem
SELECT * will continue to work even if columns are removed and added!
Sometimes we want things to break! We want something to fail before something worse happens.
You can change the design of tables in a database. One way is using the ALTER statement. Columns can be added and removed.
“SELECT *” will continue to return a result even when the tables it is using have changed significantly. This is a problem because we don’t know if it is still doing what we originally intended!
Legitimate uses of SELECT *
Legitimate, safe uses of SELECT *
There are a few ways you can use an asterisk in a SELECT statement without taking a risk. That is when you are checking if something exists, or counting the number of rows. In both cases the columns
of the tables are irrelevant.
Phil Factor identifies “SELECT *” as an SQL Smell. It can be used interactively, but almost anywhere else it has the potential to cause problems.
Where next?
Do you use “LIKE” in searches? There times when Phil Factor thinks this is an SQL Smell too. Find out why in the next article.
DateTime, Date and Time – SQL Smells
Date and Time – Point or Period – An SQL Smell?
Phil Factor identifies several SQL Smells associated with the use of DateTime, Date and Time data-types. Using the wrong types will waste space, harm performance and create “odd” behaviour.
If you are clear about what you are recording, you will avoid these issues. I prefer to say what I want, and let an expert choose the best date-types. In other words, I prefer to separate the analyst
and designer roles. That way I avoid suggesting the wrong types.
DateTime, Date or Time – Which do you need?
DateTime columns which contain only Date or Time contribute two of SQL Smells. This wastes space, both in storage and in memory (which will degrade performance).
There is something worse here. Using DateTime in this way suggests we are not clear about what we want. This lack of clarity encourages the designer to “hedge their bet” by using DateTime.
How precise do you need to be?
Business Systems need to record dates and times, but they don’t need great precision. For many business transactions, the nearest second or even minute is adequate. In some cases recording extra
precision can be misleading.
Storing Durations
For a “Period” you need to know 3 things: Start, End and Duration. You only need to store two out of the 3. The third value can always be calculated if you have the other two.
Finding if an Event is inside a Period is easier using Start and End
Storing the “start” and “end” is more flexible. It is straightforward to work out if an event took place within a given period.
Finding if Periods Overlap is easy using Start and End times
Telling if periods overlap is easy too.
If you decide to store a duration you must specify the units you intend and the precision you need. (As Phil puts it “milliseconds? Quarters? Weeks?”). Do not be tempted to store a duration in a
“Time” data-type! “Time” is intended for “Time-of-day” not duration.
Dates and Times: Choosing the right data-type – Some simple questions
Choosing appropriate data-types for dates and times is not difficult, if you go about it the right way.
Divide the job into two steps: “The Requirement” and the “The Most suitable technical data-type”. Do the two steps separately, taking into account any local standards and conventions.
The Requirement
1. Is this an “Event” (a “Point in Time”) a “Period” or a “Duration” (both have beginning and an end)?
2. Event: What is the best way to represent this data? Should it be a “Date” or a “Time”? Does it really need “Date-time”?
3. Duration: What are the units and scale of this duration?
4. How precise do you need the value to be? You may surprise yourself.
5. How long do you expect this system (and the data in it) to last?
The Most Suitable Technical Data-type
Technical Properties of Date and Time data-types in Microsoft SQL Server
Use the table (based on Microsoft’s Documentation) to choose the best data-type for your needs. This table is for SQL Server. Other database managers will have similar tables.
Microsoft recommends using DATETIME2 in preference to DATETIME for new work (it takes up less space and is more accurate). Providing the maximum date is acceptable, I would consider SMALLDATETIME for
business transactions (but you do risk creating a “Year 2000 problem” if the data turns out to have a long life).
If your system will span several time-zones, then you should definitely consider the benefits of using the DATETIMEOFFSET data-type.
The DateTime, Date and Time data-types can all cause SQL Smells when they are used inappropriately. Problems can be avoided by following some simple guidelines.
Where next?
Phil Factor doesn’t like “SELECT *”. Find out why in the next article.
Same column name in different tables- SQL Smells
Two Columns: Same Name, Different Data-type = Nasty! SQL Smell
Here’s another of Phil Factor’s SQL Smells. “Using the same column name in different tables but with different data-types”.
At first glance this seems harmless enough, but it can cause all sorts of problems. All of them are avoidable. If you are an analyst, make sure you are not encouraging this to happen. If you creating
the Physical Model or DDL for a database, “Just don’t do it!”
As Phil Factor says, this is: “an accident waiting to happen.”
Two “rights” can make a “wrong”!
The problem here is not “using the same column name in different tables”. That is perfectly ok. Similarly, using “different data-types for different columns” cannot be wrong. That’s exactly what you
should expect to do.
The problem is doing both at the same time. The issues are: practical and technical.
The Practical Problem of the same column name with different data-types
Any human user is likely to think that the same name refers to the same type of thing. They won’t check that the definitions of both “names”. No amount of “procedures” or “standards” will make them
do anything different.
Sooner or later this will cause one of the technical problems.
The Technical Problems from the same column name with different data-types
Technical problems will occur when a value moves from one column to the other, or when comparing the two columns. Data may be truncated and those data transformations cost effort.
These problems may not manifest themselves immediately. The consequences will be data-dependent bugs and poor performance.
The Solution to this Issue
This smell and the associated problems can be avoided by following some simple rules:
1. If two columns refer to the same thing (like a foreign key and a primary key), make sure they are the same data type.
2. If two columns refer to different things, then give them distinct names. (Do not resort to prefixing every column name with the table name. That’s horrible)
3. Having columns with have different names and the same data-type is perfectly OK.
“Using the same column name in different tables with different data-types” in an SQL database is simply “an accident waiting to happen.” It is easily avoided. Don’t do it and don’t do anything to
encourage it.
Where next?
The next article is about the smells which come from dates and times. | {"url":"https://duhallowgreygeek.com/news/","timestamp":"2024-11-10T01:24:30Z","content_type":"text/html","content_length":"77857","record_id":"<urn:uuid:d09cae91-4287-4c29-ba3f-c615bb2547a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00337.warc.gz"} |
[Solved] Compute elasticity for each variable | Finance homework help - Elite Homework
Paper Details
The maker of a leading brand of low-calorie microwavable food estimate the following demand equation for its products using data from 26 supermarkets around the country for the month of April: Q =
-5,200 – 42P + 20Px + 5.21 + 0.20A + 0.25M (2.002) (17.5) (6.2) (2.5) (0.09) (0.21) R2 = 0.55 N = 26 F = 4.88 Assume the following values for the independent variables: Q = Quantity sold per month P
(in cents) Price of the product = 500 Px (in cents) Price of the leading competitor’s product = 600 I (in dollars) Per capita income of the standard metropolitan statistical area (SMSA) which the
supermarket is located = 5,500 A (in dollars) Monthly advertising expenditure =10,000 M = Number of microwave ovens sold in the SMSA in which the supermarket is located = 5,000
Using this information answer the following questions:
a) Compute elasticity for each variable.
(b) How concerned do you think this company would be about the impact of a recession on its sales? Explain.
(c) Do you think this firm should cut its price to increase its market share? Explain.
(d) What proportions of the variation in is sales is explained by the independent variables in the equations? How confident are you about this answer? Explain. | {"url":"https://www.elitehomework.com/114046/compute-elasticity-for-each-variable-finance-homework-help/","timestamp":"2024-11-07T12:55:27Z","content_type":"text/html","content_length":"37809","record_id":"<urn:uuid:950b443d-b580-4dfb-8706-543b30d9eeda>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00042.warc.gz"} |
The Development and Evolution of Monetary Theory and Public
The Development and Evolution of Monetary Theory and Public Policy by Kenneth Kurihara: A Critical Analysis and Evaluation
Monetary Theory and Public Policy by Kenneth Kurihara: A Review
Monetary theory and public policy are two interrelated fields of economics that deal with the role and effects of money in the economy. Money is not only a medium of exchange, but also a store of
value, a unit of account, and a standard of deferred payment. Money affects various aspects of economic activity, such as consumption, investment, production, employment, inflation, interest rates,
exchange rates, and balance of payments. Monetary policy is the use of instruments such as money supply, interest rates, or exchange rates by the central bank or the government to influence economic
outcomes. Monetary policy can have different objectives, such as price stability, full employment, economic growth, or external equilibrium.
One of the classic books on monetary theory and public policy is Monetary Theory and Public Policy by Kenneth Kurihara, published in 1950. Kurihara was a Japanese-American economist who taught at
Yale University and Columbia University. He was also a consultant to various international organizations, such as the United Nations, the International Monetary Fund, and the World Bank. His book is
a comprehensive and systematic exposition of the development and evolution of monetary theory from the classical to the post-Keynesian schools. It also provides a critical analysis and evaluation of
various monetary policies in light of theoretical frameworks and empirical evidence.
In this article, I will review Kurihara's book by summarizing its main themes and arguments, evaluating its strengths and weaknesses, comparing it with other works in the field, and discussing its
relevance and applicability to contemporary issues. I will also provide some recommendations and suggestions for further reading for those who are interested in learning more about monetary theory
and public policy.
Summary of the book
Kurihara's book consists of four chapters, each covering a major school or stage of monetary theory. The first chapter introduces the nature and scope of monetary theory, the second chapter reviews
the classical theory of money and prices, the third chapter examines the Keynesian theory of money and income, and the fourth chapter explores the post-Keynesian developments in monetary theory.
Chapter 1: The Nature and Scope of Monetary Theory
In this chapter, Kurihara defines money and its functions, discusses the quantity theory of money and its criticisms, and explains the role of money in economic analysis. He argues that money is not
a neutral factor, but rather a dynamic and influential force that affects the behavior of economic agents and the performance of the economy.
The definition and functions of money
Kurihara defines money as anything that is generally accepted as a medium of exchange. He distinguishes between commodity money, such as gold or silver, and fiat money, such as paper currency or bank
deposits, which have no intrinsic value but are backed by legal tender or public confidence. He also distinguishes between narrow money, such as currency and demand deposits, and broad money, which
includes other liquid assets, such as savings deposits, time deposits, or money market funds.
Kurihara identifies four functions of money: (1) medium of exchange, which facilitates transactions and reduces transaction costs; (2) store of value, which allows people to save and defer
consumption; (3) unit of account, which provides a common measure and standard of value; and (4) standard of deferred payment, which enables people to borrow and lend. He notes that these functions
are interrelated and interdependent, and that the effectiveness of money depends on its stability and acceptability.
The quantity theory of money and its criticisms
Kurihara presents the quantity theory of money as one of the oldest and most influential theories in monetary economics. The quantity theory of money states that the general level of prices is
proportional to the quantity of money in circulation, assuming that the velocity of money (the number of times a unit of money changes hands in a given period) and the real output (the quantity of
goods and services produced in a given period) are constant or stable. The quantity theory of money can be expressed by the equation of exchange: MV = PT, where M is the quantity of money, V is the
velocity of money, P is the general price level, and T is the real output.
Kurihara explains that the quantity theory of money implies that changes in the quantity of money cause changes in the price level, and that monetary policy can control inflation or deflation by
regulating the growth rate of money supply. He also discusses some criticisms and modifications of the quantity theory of money, such as: (1) the instability and endogeneity of the velocity of money;
(2) the distinction between nominal income (PT) and real income (T); (3) the effects of changes in the demand for money on interest rates and output; (4) the role of expectations and uncertainty in
influencing monetary behavior; and (5) the existence of non-monetary factors that affect prices and output.
The role of money in economic analysis
Kurihara argues that money plays an important role in economic analysis, both in microeconomics and macroeconomics. He contends that money affects the allocation of resources, the distribution of
income, and the determination of equilibrium in various markets. He also asserts that money influences the aggregate demand, the aggregate supply, and the equilibrium level of income and employment
in the economy. He claims that money is not a neutral factor that only affects nominal variables, such as prices or wages, but also a real factor that affects real variables, such as output or
employment. He maintains that monetary theory should not be separated from general economic theory, but rather integrated with it.
Chapter 2: The Classical Theory of Money and Prices
In this chapter, Kurihara reviews the classical theory of money and prices, which dominated monetary economics until the 1930s. The classical theory of money and prices is based on two assumptions:
(1) the classical dichotomy, which states that real variables are determined by real factors independently of nominal variables; and (2) the neutrality of money, which states that changes in the
quantity of money only affect nominal variables proportionally without affecting real variables. Kurihara discusses three main approaches within the classical theory: (1) the equation of exchange
approach; (2) the Fisher effect approach; and (3) the Cambridge cash-balance approach.
The equation of exchange approach
Kurihara explains that the equation of exchange approach is derived from the quantity theory of money. It states that MV = PT, where M is the quantity of money, V is the velocity of money, P is the
general price level, and T is the real output. It implies that changes in the quantity of money cause proportional changes in the price level, assuming that the velocity of money the real output are
constant or stable. It also implies that monetary policy can control inflation or deflation by regulating the growth rate of money supply. Kurihara points out some limitations and criticisms of the
equation of exchange approach, such as: (1) the lack of a causal explanation of how changes in money supply affect prices and output; (2) the neglect of the role of interest rates and the demand for
money in determining the velocity of money; (3) the disregard of the effects of changes in prices and output on the quantity of money; and (4) the oversimplification of the structure and dynamics of
the economy.
The Fisher effect approach
Kurihara describes the Fisher effect approach as an extension and refinement of the equation of exchange approach. It is named after Irving Fisher, an American economist who developed it in his book
The Theory of Interest in 1930. The Fisher effect approach states that i = r + π, where i is the nominal interest rate, r is the real interest rate, and π is the expected inflation rate. It implies
that changes in the quantity of money cause changes in the expected inflation rate, which in turn cause changes in the nominal interest rate. It also implies that monetary policy can affect the real
interest rate only temporarily, as the expected inflation rate adjusts to restore the equilibrium real interest rate. Kurihara discusses some implications and applications of the Fisher effect
approach, such as: (1) the distinction between nominal and real variables; (2) the role of expectations and uncertainty in determining interest rates and prices; (3) the analysis of international
capital flows and exchange rates; and (4) the evaluation of alternative monetary policies.
The Cambridge cash-balance approach
Kurihara presents the Cambridge cash-balance approach as an alternative and complementary perspective to the equation of exchange approach. It is named after a group of British economists who
developed it at Cambridge University in the early 20th century, such as Alfred Marshall, Arthur Pigou, and John Maynard Keynes. The Cambridge cash-balance approach states that M = kPY, where M is the
quantity of money, k is the fraction of nominal income that people hold as money, P is the general price level, and Y is the real income. It implies that changes in the quantity of money cause
changes in the price level, assuming that the fraction of nominal income that people hold as money the real income are constant or stable. It also implies that monetary policy can control inflation
or deflation by regulating the growth rate of money supply. Kurihara highlights some advantages and contributions of the Cambridge cash-balance approach, such as: (1) the introduction of the concept
of the demand for money as a function of nominal income; (2) the recognition of the role of interest rates and other factors in influencing the demand for money; (3) the incorporation of the real
balance effect, which states that changes in the price level affect the real value of money holdings and thus affect consumption and output; and (4) the foundation for the development of the
Keynesian theory of money and income.
Chapter 3: The Keynesian Theory of Money and Income
In this chapter, Kurihara examines the Keynesian theory of money and income, which emerged in the 1930s as a response and challenge to the classical theory. The Keynesian theory of money and income
is based on two assumptions: (1) the existence of involuntary unemployment, which means that there are workers who are willing to work at the prevailing wage rate but cannot find jobs; and (2) the
prevalence of sticky prices and wages, which means that prices and wages do not adjust quickly or fully to changes in demand or supply. Kurihara discusses three main components of the Keynesian
theory: (1) the liquidity preference theory and the demand for money; (2) the income-expenditure approach and the multiplier effect; and (3) the IS-LM model and the interaction of money and output
The liquidity preference theory and the demand for money
Kurihara explains that the liquidity preference theory is developed by John Maynard Keynes in his book The General Theory of Employment, Interest and Money in 1936. It is a theory of the demand for
money that states that people hold money for three motives: (1) transaction motive, which is related to their income and spending needs; (2) precautionary motive, which is related to their
uncertainty and precautionary needs; and (3) speculative motive, which is related to their expectations and investment opportunities. The liquidity preference theory implies that the demand for money
depends on the level of income and the rate of interest. It also implies that there is a negative relationship between the rate of interest and the quantity of money demanded, holding income
Kurihara illustrates how the liquidity preference theory determines the equilibrium rate of interest in the money market. He shows that the equilibrium rate of interest is the rate that equates the
supply of money, which is assumed to be fixed by the central bank, with the demand for money, which depends on income and interest rates. He also shows how changes in the supply of money or the
demand for money cause changes in the equilibrium rate of interest. He notes that the liquidity preference theory implies that monetary policy can affect interest rates by changing the supply of
money, but it may not affect output or employment if interest rates are too low or too high to stimulate investment.
The income-expenditure approach and the multiplier effect
the goods market. It is based on the principle of effective demand, which states that the level of output and employment in the economy is determined by the aggregate demand for goods and services,
not by the aggregate supply. The income-expenditure approach implies that there can be a gap between the actual output and the potential output of the economy, resulting in underemployment or
overemployment. Kurihara discusses two main concepts of the income-expenditure approach: (1) the consumption function and the saving function; and (2) the investment function and the marginal
efficiency of capital.
The consumption function and the saving function
Kurihara explains that the consumption function is a relationship between consumption and income that states that consumption depends on income. He presents the Keynesian consumption function as C =
a + bY, where C is consumption, a is autonomous consumption, b is the marginal propensity to consume, and Y is income. He notes that autonomous consumption is the amount of consumption that does not
depend on income, such as basic needs or habits. He also notes that the marginal propensity to consume is the fraction of additional income that is spent on consumption, which is assumed to be
positive and less than one.
Kurihara derives the saving function from the consumption function by subtracting consumption from income. He presents the Keynesian saving function as S = -a + (1-b)Y, where S is saving, -a is
autonomous dissaving, and 1-b is the marginal propensity to save. He notes that autonomous dissaving is the amount of saving that does not depend on income, which can be negative if consumption
exceeds income. He also notes that the marginal propensity to save is the fraction of additional income that is saved, which is assumed to be positive and less than one.
The investment function and the marginal efficiency of capital
Kurihara describes the investment function as a relationship between investment and interest rates that states that investment depends on interest rates. He presents the Keynesian investment function
as I = f(r), where I is investment and r is the rate of interest. He notes that the investment function is a downward-sloping curve, which implies that there is a negative relationship between
investment and interest rates. He also notes that the investment function depends on other factors, such as expectations, profitability, technology, or capacity utilization.
Kurihara explains that the marginal efficiency of capital is a concept that measures the expected rate of return on an investment project. It is defined as the discount rate that equates the present
value of future net revenues from an investment project with its current cost. It implies that an investment project will be undertaken if its marginal efficiency of capital exceeds the rate of
interest. It also implies that there can be fluctuations in investment due to changes in expectations or uncertainty about future net revenues.
The multiplier effect
the income-expenditure approach. He shows that the equilibrium level of income and output in the goods market is determined by the equality of aggregate demand and aggregate supply. He defines
aggregate demand as the sum of consumption and investment, and aggregate supply as the total output of the economy. He presents the equilibrium condition as Y = C + I, where Y is income and output, C
is consumption, and I is investment. He notes that both consumption and investment depend on income and interest rates, as explained by the consumption function and the investment function.
Kurihara demonstrates how a change in autonomous spending, such as an increase in autonomous consumption or autonomous investment, causes a change in equilibrium income and output that is larger than
the initial change in spending. He explains that this is because an increase in autonomous spending increases income, which in turn increases consumption, which further increases income, and so on.
He calculates the multiplier as the ratio of the change in equilibrium income to the change in autonomous spending. He presents the multiplier formula as k = 1/(1-b), where k is the multiplier and b
is the marginal propensity to consume. He notes that the multiplier is positive and greater than one, which means that a small change in autonomous spending can have a large impact on income and
The IS-LM model and the interaction of money and output markets
Kurihara introduces the IS-LM model as a synthesis and extension of the liquidity preference theory and the income-expenditure approach. It is a model that depicts the simultaneous equilibrium of the
money market and the goods market in a two-dimensional diagram. It was developed by John Hicks and Alvin Hansen in 1937 as an interpretation and simplification of Keynes's General Theory. Kurihara
discusses two main curves of the IS-LM model: (1) the IS curve; and (2) the LM curve.
The IS curve
Kurihara explains that the IS curve represents the combinations of interest rates and income levels that ensure equilibrium in the goods market. It is derived from the income-expenditure approach by
substituting the consumption function and the investment function into the equilibrium condition of Y = C + I. It can be expressed as Y = a + bY - brM/P + f(r), where M is the quantity of money, P is
the price level, and the other variables are as defined before. It implies that for a given level of money supply and price level, there is a negative relationship between interest rates and income
levels in the goods market.
Kurihara illustrates how to draw and shift the IS curve in the IS-LM diagram. He shows that the IS curve is a downward-sloping curve, which reflects the inverse relationship between interest rates
and income levels in the goods market. He also shows how changes in autonomous spending or fiscal policy can shift the IS curve to the right or to the left. For example, an increase in autonomous
consumption or autonomous investment, or an increase in government spending or a decrease in taxes, can increase aggregate demand and shift the IS curve to the right.
The LM curve
the money market. It is derived from the liquidity preference theory by substituting the consumption function into the demand for money function. It can be expressed as M/P = a + bY - cr, where c is
the sensitivity of the demand for money to interest rates, and the other variables are as defined before. It implies that for a given level of money supply and price level, there is a positive
relationship between interest rates and income levels in the money market.
Kurihara illustrates how to draw and shift the LM curve in the IS-LM diagram. He shows that the LM curve is an upward-sloping curve, which reflects the direct relationship between interest rates and
income levels in the mon | {"url":"https://www.itistimetoriseup.com/group/rise-up-be-free/discussion/cd1168d7-d552-4299-883b-198c390eb261","timestamp":"2024-11-05T06:22:01Z","content_type":"text/html","content_length":"1050495","record_id":"<urn:uuid:1956ea31-a8d8-4512-9356-3d1712d79f13>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00300.warc.gz"} |
Free Group Study Rooms with Timer & Music | FiveableAP Physics Unit 6: Energy of a Simple Harmonic Oscillator | Fiveable
The energy of a system is conserved.
A system with an internal structure can have internal energy, and changes in a systemβ s internal structure can result in changes in internal energy
Here are some key things to know about the internal energy of a simple harmonic oscillator:
• The internal energy of an object is the energy associated with the random, chaotic motion of its constituent particles. It is a measure of the thermal energy of the object and is often symbolized
by the letter U.
• In a simple harmonic oscillator, the internal energy is stored in the form of elastic potential energy when the oscillator is displaced from its equilibrium position. As the oscillator oscillates
back and forth, the internal energy of the system is continually converted into kinetic energy and back into potential energy.
• The internal energy of a simple harmonic oscillator is periodic, meaning that it follows a repeating pattern over time. The internal energy of the oscillator is at a maximum when the oscillator
is at its maximum displacement from its equilibrium position and is at a minimum when the oscillator is at its equilibrium position.
• The internal energy of a simple harmonic oscillator can be calculated using the equation: U = 1/2*kx^2, where U is the internal energy, k is the spring constant, and x is the displacement of the
oscillator from its equilibrium position.
• The internal energy of a simple harmonic oscillator is a useful quantity to consider when analyzing the behavior of the oscillator, as it can help to understand how the energy of the system is
being converted between kinetic and potential energy over time.
A system with internal structure can have potential energy. Potential energy exists within a system if the objects within that system interact with conservative forces.
Here are some key things to know about the potential energy of a simple harmonic oscillator:
• The potential energy of an object is the energy that an object possesses due to its position or configuration within a force field. It is a measure of the potential for the object to do work and
is often symbolized by the letter U.
• In a simple harmonic oscillator, the potential energy of the system is stored in the form of elastic potential energy when the oscillator is displaced from its equilibrium position. This energy
is due to the deformation of the spring or other force-generating element in the system as it tries to return to its equilibrium position.
• The potential energy of a simple harmonic oscillator is periodic, meaning that it follows a repeating pattern over time. The potential energy of the oscillator is at a maximum when the oscillator
is at its maximum displacement from its equilibrium position, and is at a minimum when the oscillator is at its equilibrium position.
• The potential energy of a simple harmonic oscillator can be calculated using the equation: U = 1/2*kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of
the oscillator from its equilibrium position.
• The potential energy of a simple harmonic oscillator is a useful quantity to consider when analyzing the behavior of the oscillator, as it can help to understand how the energy of the system is
being stored and converted between kinetic and potential energy over time.
The internal energy of a system includes the kinetic energy of the objects that make up the system and the potential energy of the configuration of objects that make up the system.
Here are some key things to know about the kinetic energy of a simple harmonic oscillator:
• The kinetic energy of an object is the energy associated with the motion of the object. It is a measure of the ability of the object to do work due to its motion and is often symbolized by the
letter K.
• In a simple harmonic oscillator, the kinetic energy of the system is stored in the form of kinetic energy when the oscillator is moving. This energy is due to the motion of the oscillator as it
oscillates back and forth.
• The kinetic energy of a simple harmonic oscillator is periodic, meaning that it follows a repeating pattern over time. The kinetic energy of the oscillator is at a maximum when the oscillator is
at its maximum velocity, and is at a minimum when the oscillator is at its equilibrium position or at a point of maximum displacement from the equilibrium position.
• The kinetic energy of a simple harmonic oscillator can be calculated using the equation: K = 1/2*mv^2, where K is the kinetic energy, m is the mass of the oscillator, and v is the velocity of the
• The kinetic energy of a simple harmonic oscillator is a useful quantity to consider when analyzing the behavior of the oscillator, as it can help to understand how the energy of the system is
being converted between kinetic and potential energy over time.
This topic is pretty much just an application of the energy types and conversions we covered in Unit 4: Energy. The main idea is that through SHM, the energy is converted from potential to kinetic
and back again throughout the motion. The maximum potential energy occurs when the spring is stretched (or compressed) the most, and the maximum kinetic energy occurs at the equilibrium point.Β
Hereβ s an example using a mass on a spring, resting on a frictionless surface. In pictures A, C, and E, the energy is fully stored as potential energy in the spring. In pictures B and D, the mass
is at the equilibrium position (x=0) and all the energy is now kinetic energy.
If we were to make a graph of energy vs time, it would look like this:
A couple of things to notice in this graph above:
1. The total energy is constant. This makes sense since there are no external forces to do work on the spring-mass system
2. The potential energy and kinetic energy graphs are curves. Because of the squared term in the potential energy equation, we expect this. If the term is to the 1st power, then the graph would be
3. The potential energy is greatest when the position graph is at its maximum. The Kinetic Energy is greatest when the velocity graph is at its maximum.
Example Problem 1:
A mass of 1 kg is attached to a spring with a spring constant of 50 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.2 meters from its
equilibrium position and released from rest. What is the total energy of the oscillator at the maximum displacement from the equilibrium position?
The total energy of a simple harmonic oscillator is the sum of its potential energy and kinetic energy.
The potential energy of a simple harmonic oscillator is given by the equation: U = 1/2*kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of the oscillator
from its equilibrium position.
The kinetic energy of a simple harmonic oscillator is given by the equation: K = 1/2*mv^2, where K is the kinetic energy, m is the mass of the oscillator, and v is the velocity of the oscillator.
In this problem, the mass of the oscillator is 1 kg, the spring constant is 50 N/m, and the displacement from the equilibrium position is 0.2 meters.
At the maximum displacement from the equilibrium position, the velocity of the oscillator is zero and the potential energy is at a maximum.
Therefore, the total energy of the oscillator at the maximum displacement from the equilibrium position is: U + K = 1/250 N/m(0.2 m)^2 + 0 = 2.5 J
This means that the total energy of the oscillator at the maximum displacement from the equilibrium position is 2.5 J.
Example Problem 2:
A mass of 2 kg is attached to a spring with a spring constant of 100 N/m and is allowed to oscillate vertically in a frictionless environment. The mass is initially displaced 0.5 meters from its
equilibrium position and released from rest. What is the total energy of the oscillator at the equilibrium position?
The total energy of a simple harmonic oscillator is the sum of its potential energy and kinetic energy.
The potential energy of a simple harmonic oscillator is given by the equation: U = 1/2*kx^2, where U is the potential energy, k is the spring constant, and x is the displacement of the oscillator
from its equilibrium position.
The kinetic energy of a simple harmonic oscillator is given by the equation: K = 1/2*mv^2, where K is the kinetic energy, m is the mass of the oscillator, and v is the velocity of the oscillator.
In this problem, the mass of the oscillator is 2 kg, the spring constant is 100 N/m, and the displacement from the equilibrium position is 0.5 meters.
At the equilibrium position, the velocity of the oscillator is zero and the potential energy is at a minimum.
Therefore, the total energy of the oscillator at the equilibrium position is: U + K = 1/2100 N/m(0.5 m)^2 + 0 = 12.5 J
This means that the total energy of the oscillator at the equilibrium position is 12.5 J. | {"url":"https://hours-zltil9zhf-thinkfiveable.vercel.app/ap-physics-1/unit-6/energy-a-simple-harmonic-oscillator/study-guide/lNflbqbIly6vFgeyLwmO","timestamp":"2024-11-14T12:10:24Z","content_type":"text/html","content_length":"248978","record_id":"<urn:uuid:99f3af21-b156-4d02-a232-9113d05d1277>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00743.warc.gz"} |
Question about single track model
• Automotive
• Thread starter ronasbeg
• Start date
In summary, the conversation discussed using a single track model to study the yaw rate and sideslip of a vehicle with a given steering angle input. However, the model assumes a constant velocity,
which may not accurately reflect real-world scenarios. The question was raised about incorporating a variable velocity profile in the model and how it would affect the lateral dynamics of the
vehicle. Suggestions were sought for further insights on this topic.
Hello guys,
I've been doing some simulink modelling about a study that's been ongoing for a while , I've used the single track model to see the yaw rate and sideslip for the given steering angle input ,
But as you know single track model makes the assumption of vehicle velocity as constant. Now the question is , what if I don't assume this velocity to be a constant value and want to see the yaw rate
and sideslip wrt to the steering input and the given velocity profile ( for ex , vehicle accelerates from 15 m/s to 25 m/s and decelerates to same velocity linearly again.)
If you have some opinions , I appreciate .Thanks.
The single track model was designed to model vehicle lateral dynamic, therefore you would miss something, e.g. the vehicle longitudinal dynamic. But I think in this way you will understand how the
vehicle lateral dynamic changes with the speed, as example it will change the vehicle yaw damping.
FAQ: Question about single track model
What is a single track model?
A single track model is a simplified mathematical representation of a system or process that has only one variable. It is often used in scientific research to understand the behavior of complex
What are the advantages of using a single track model?
One of the main advantages of using a single track model is that it allows for easy analysis and understanding of complex systems. It also helps in making predictions and identifying key variables
that affect the system.
What are the limitations of a single track model?
A single track model may not accurately represent the full complexity of a system, as it simplifies the variables and relationships between them. It may also have limited predictive power and may not
account for all factors that affect the system.
How is a single track model created?
A single track model is created by identifying the main variable or factor of interest and simplifying the relationships between it and other variables. This is often done through mathematical
equations or computer simulations.
How can a single track model be used in scientific research?
A single track model can be used in various fields of scientific research, such as biology, physics, and economics. It can help in understanding and predicting the behavior of complex systems and can
serve as a starting point for further exploration and experimentation. | {"url":"https://www.physicsforums.com/threads/question-about-single-track-model.661888/","timestamp":"2024-11-08T06:14:46Z","content_type":"text/html","content_length":"78472","record_id":"<urn:uuid:724d9012-b5df-4077-8a9b-a866ca24fca9>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00531.warc.gz"} |
Conservation-variable average states for the multi-dimensional Euler fluxes and arbitrary equilibrium real gases
This paper focuses on the development of an equilibrium real-gas Riemann solver that requires no auxiliary assumptions for computing the average pressure derivatives to determine the multidimensional
average flux-vector Jacobian. By employing the multidimensional mean value theorem, the theoretical developments give one average pressure and two internal conservation-variable average states that
correspond to intermediate states between the given right and left states. These states are utilized to compute the average sound speed and pressure derivatives without any additional internal
energy, geometric projections, and scale factors.
31st AIAA Aerospace Sciences Meeting and Exhibit
Pub Date:
January 1993
□ Conservation Equations;
□ Euler Equations Of Motion;
□ Gas Pressure;
□ Pressure Oscillations;
□ Real Gases;
□ Computational Fluid Dynamics;
□ Degrees Of Freedom;
□ Flux Vector Splitting;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1993aiaa.meetQ....I/abstract","timestamp":"2024-11-11T22:02:13Z","content_type":"text/html","content_length":"34451","record_id":"<urn:uuid:9f7e9aad-7b9b-4697-9fac-5e7d420dbc19>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00338.warc.gz"} |
Nye et al. (2006) tree comparison — NyeSimilarity
Nye et al. (2006) tree comparison
NyeSimilarity() and NyeSplitSimilarity() implement the Generalized Robinson–Foulds tree comparison metric of Nye et al. (2006) . In short, this finds the optimal matching that pairs each branch from
one tree with a branch in the second, where matchings are scored according to the size of the largest split that is consistent with both of them, normalized against the Jaccard index. A more detailed
account is available in the vignettes.
tree2 = NULL,
similarity = TRUE,
normalize = FALSE,
normalizeMax = !is.logical(normalize),
reportMatching = FALSE,
diag = TRUE
nTip = attr(splits1, "nTip"),
reportMatching = FALSE
Trees of class phylo, with leaves labelled identically, or lists of such trees to undergo pairwise comparison. Where implemented, tree2 = NULL will compute distances between each pair of trees in
the list tree1 using a fast algorithm based on Day (1985) .
Logical specifying whether to report the result as a tree similarity, rather than a difference.
If a numeric value is provided, this will be used as a maximum value against which to rescale results. If TRUE, results will be rescaled against a maximum value calculated from the specified tree
sizes and topology, as specified in the "Normalization" section below. If FALSE, results will not be rescaled.
When calculating similarity, normalize against the maximum number of splits that could have been present (TRUE), or the number of splits that were actually observed (FALSE)? Defaults to the
number of splits in the better-resolved tree; set normalize = pmin.int to use the number of splits in the less resolved tree.
Logical specifying whether to return the clade matchings as an attribute of the score.
Logical specifying whether to return similarities along the diagonal, i.e. of each tree with itself. Applies only if tree2 is a list identical to tree1, or NULL.
Logical matrices where each row corresponds to a leaf, either listed in the same order or bearing identical names (in any sequence), and each column corresponds to a split, such that each leaf is
identified as a member of the ingroup (TRUE) or outgroup (FALSE) of the respective split.
(Optional) Integer specifying the number of leaves in each split.
NyeSimilarity() returns an array of numerics providing the distances between each pair of trees in tree1 and tree2, or splits1 and splits2.
The measure is defined as a similarity score. If similarity = FALSE, the similarity score will be converted into a distance by doubling it and subtracting it from the number of splits present in both
trees. This ensures consistency with JaccardRobinsonFoulds.
Note that NyeSimilarity(tree1, tree2) is equivalent to, but slightly faster than, JaccardRobinsonFoulds (tree1, tree2, k = 1, allowConflict = TRUE).
If normalize = TRUE and similarity = TRUE, then results will be rescaled from zero to one by dividing by the mean number of splits in the two trees being compared.
You may wish to normalize instead against the number of splits present in the smaller tree, which represents the maximum value possible for a pair of trees with the specified topologies (normalize =
pmin.int); the number of splits in the most resolved tree (normalize = pmax.int); or the maximum value possible for any pair of trees with n leaves, n - 3 (normalize = TreeTools::NTip(tree1) - 3L).
If normalize = TRUE and similarity = FALSE, then results will be rescaled from zero to one by dividing by the total number of splits in the pair of trees being considered.
Trees need not contain identical leaves; scores are based on the leaves that trees hold in common. Check for unexpected differences in tip labelling with setdiff(TipLabels(tree1), TipLabels(tree2)).
Day WHE (1985). “Optimal algorithms for comparing trees with labeled leaves.” Journal of Classification, 2(1), 7–28. doi:10.1007/BF01908061 .
Nye TMW, Liò P, Gilks WR (2006). “A novel algorithm and web-based tool for comparing two alternative phylogenetic trees.” Bioinformatics, 22(1), 117–119. doi:10.1093/bioinformatics/bti720 .
NyeSimilarity(BalancedTree(8), PectinateTree(8))
#> [1] 3.8
VisualizeMatching(NyeSimilarity, BalancedTree(8), PectinateTree(8))
NyeSimilarity(as.phylo(0:5, nTip = 8), PectinateTree(8))
#> [1] 3.166667 2.750000 2.750000 2.500000 2.450000 2.500000
NyeSimilarity(as.phylo(0:5, nTip = 8), similarity = FALSE)
#> 1 2 3 4 5
#> 2 1.333333
#> 3 1.333333 1.333333
#> 4 2.166667 2.333333 2.333333
#> 5 2.333333 2.166667 2.333333 1.000000
#> 6 2.000000 2.000000 1.500000 1.500000 1.500000 | {"url":"https://ms609.github.io/TreeDist/reference/NyeSimilarity.html","timestamp":"2024-11-11T11:33:00Z","content_type":"text/html","content_length":"20755","record_id":"<urn:uuid:9fe82ae9-a2ea-418c-be33-aa3f23447da6>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00416.warc.gz"} |
Big Ideas Math Answers Grade 2 | Big Ideas Math Book 2nd Grade Answer Key
Students can browse Big Ideas Math Answers Grade 2 on CCSS Math Answers. We have presented all the BIM Answer Key Grade 2 solutions in pdf format as per the student’s convenience. So, the students
who are willing to become masters in maths can refer to Bigideas Math Book 2nd Grade Answer Key from the primary stage itself. You can get access to all BIM Textbook 2nd Grade Answers for free of
cost. Big Ideas Math Answers Grade 2 helps the parents to understand the concepts and teach their children and make their child do homework.
Big Ideas Math Book 2nd Grade Answer Key | BIM Answers 2nd Grade Solutions Pdf
Each and every concept is explained step by step and easy manner for all Big Ideas Math 2nd Grade Chapters. So, Download Big Ideas Math Book 2nd Grade Answer Key pdf from our library. Enhance your
math skills by practicing from Bigideas Math Answer Key Grade 2. Also Bigideas Math Grade 2 Answer Key helps the students to complete their homework in time. This will help the students to perform
well in the performance test, practice tests, and chapter tests. Go through the list given below to know the topics covered in Big Ideas Math Answers Grade 2.
Top 5 Preparation Tips to Study Well in 2nd Grade – Common Core 2019
The preparation helps the students to overcome the difficulty of the exam. We have provided the exam preparation tips for grade 2 students. Follow these tips and prepare well for the exams.
• First, create the revision timetable.
• Find the best study material to read.
• Take Breaks after every chapter.
• Enough sleep
• Set the best study time
• Study Everyday
• Do not read everything the night before the exam.
• Drink plenty of water
• Understand the concept and write the revision notes.
• Revise early in the morning.
FAQs on Primary School Grade 2 Answer Key – Common Core 2019
1. Which website offers Big Ideas Math 2nd Grade Answer Key?
eurekamathanswerkeys.com is the best website to get the solutions for all the chapters of BIM Book 2nd Grade Answer Key. You can get the Big Ideas Math 2nd Grade Answers in pdf format.
2. How to Download BIM Book 2nd Grade Answers?
All you have to do is to click on the links to Download Big Ideas Math Answer Key Grade 2.
3. Where do I get the Chapterwise Common Core 2019 Big Ideas Math Answer Key Grade 2?
The students of grade 2 can get the chapterwise Bigideas Math 6th Grade Answers on CCSS Math Answers. | {"url":"https://eurekamathanswerkeys.com/big-ideas-math-answers-grade-2/","timestamp":"2024-11-08T12:07:02Z","content_type":"text/html","content_length":"40297","record_id":"<urn:uuid:1c8c1f64-11d2-4c48-ae3d-416398035a26>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00597.warc.gz"} |
Can AI Solve Math Problems? Exploring Possibilities
Imagine a world where AI can solve math problems as well as the best human mathematicians. This dream is becoming real, thanks to Google DeepMind’s AI systems, AlphaProof and AlphaGeometry 2. These
AI models have solved four out of six problems from the International Mathematical Olympiad (IMO). They earned a silver medal equivalent.
This is a big step forward for AI and math. For the first time, an AI has shown it can solve complex math problems well. These systems are getting better fast, changing how we solve math problems.
Key Takeaways
• Google DeepMind’s AI systems, AlphaProof and AlphaGeometry 2, solved 4 out of 6 problems from the International Mathematical Olympiad, earning a silver medal equivalent.
• This achievement marks the first time an AI system has demonstrated such a high success rate on complex math problems that require advanced reasoning.
• The AI systems excel at solving algebra, number theory, and geometry problems, showcasing the versatility of their mathematical problem-solving abilities.
• The development of these AI systems suggests that the goal of matching or exceeding human-level mathematical problem-solving may be within reach sooner than expected.
• The successful performance of AI in math competitions has sparked discussions about the role of AI in mathematics education and the future of human-AI collaboration in this field.
The Rise of AI in Mathematical Problem-Solving
Historic Milestones and Breakthroughs
The journey of artificial intelligence (AI) in mathematics has been amazing. At first, AI was used just for doing routine calculations. But now, with machine learning and deep neural networks, AI can
tackle complex math problems like never before.
A big step forward was the creation of AI-powered math solvers. These systems can look at data, find patterns, and solve tricky equations. They’ve changed how we solve math problems, making it faster
and more accurate.
AI has also helped with theorem proving and checking proofs. It can help mathematicians verify complex proofs or even come up with new ones. This has led to big achievements in mathematics, pushing
what we thought was possible.
“AI is developing rapidly, with Dr. Lord mentioning that problem solving is not one of AI’s strong points at present.”
Even with these advances, experts like Dr. Lord say AI still needs to get better at solving problems. But the progress in AI math shows a bright future. AI could change how we solve math problems
even more in the future.
How AI Approaches Mathematical Reasoning
AI is getting better at solving math problems. It uses machine learning, reinforcement learning, and formal language processing. This lets AI solve complex math challenges well.
AI can now turn natural language math problems into formal statements. This helps AI understand and solve a wider range of math problems. It’s a big step forward.
AI learns from big datasets of math problems and proofs. It finds patterns, makes logical connections, and comes up with strategies. For example, GPT-4 achieved the 89th percentile on the SAT.
Google’s PaLM 2 did even better in math tests, solving over 20,000 school-level problems and puzzles.
AlphaGeometry, an AI by Google’s DeepMind, solved 25 out of 30 IMO problems quickly. This is amazing, as the best previous AI could only solve 10 of those problems.
Metric Performance
GPT-4 SAT Score 89th percentile
PaLM 2 Math Assessments Surpassed GPT-4
AlphaGeometry IMO Problems Solved 25 out of 30
Previous State-of-the-Art Geometry Program IMO Problems Solved 10 out of 30
AI has made big strides in geometry, but it still struggles with algebraic word problems and combinatorics. Creating new concepts and strategies for complex math problems is hard for AI.
Yet, AI’s progress in math is opening new doors. It’s becoming a great tool for human mathematicians, helping them and opening up new math possibilities.
Applications of AI in Solving Mathematical Problems
Artificial Intelligence (AI) is changing the way we solve math problems. It shines in optimization, quickly finding the best solutions to hard problems. This includes things like planning the best
route or managing resources efficiently.
AI is also great at spotting patterns in big data. It can see trends and connections that humans might miss. This helps in many areas, like science and finance.
AI is also used in proving theorems and coming up with new ideas. It helps in making secure codes that are hard to break. Using AI in math is making science move faster and more accurately.
AI Algorithms for Mathematical Optimization
AI uses special algorithms like genetic algorithms and simulated annealing for solving tough math problems. These algorithms can look through a huge number of solutions fast. They find the best or
almost the best answers that would take humans a long time to figure out.
AI Application Description Example
Optimization AI quickly finds the best solutions to hard problems, like planning routes or AI solves the “travelling salesperson problem” by finding the shortest path to visit
managing resources. many cities.
Pattern Recognition AI spots trends and connections in big data that are hard for humans to see. AI looks at financial data to find patterns and predict market trends, helping with
investment decisions.
Theorem Proving and Conjecture AI helps check proofs or come up with new math ideas. The FunSearch AI system improved the “cap set problem” by tackling Set-inspired
Generation problems in combinatorics.
Cryptography AI makes secure codes that are hard to break. AI-powered cryptography is key for safe online communication and protecting data.
Using AI in math is opening new doors for science and making solving problems more efficient and accurate. As AI gets better, it will change how we tackle complex math problems.
can ai solve math problems
Artificial intelligence (AI) has made big steps in solving math problems. It can handle huge amounts of data and do complex calculations fast. This lets AI systems solve problems that humans would
take too long or can’t solve at all.
But, AI in math has its limits. It’s great at looking at patterns and finding solutions from what it knows. Yet, it often can’t come up with new ideas or innovative ways that need a deep
understanding of math.
There are also ethical worries about using AI in math. These include biases in the data and the need for clear and responsible decision-making. As AI and math work together more, we must tackle these
issues to make the most of this powerful partnership.
Despite these hurdles, AI’s skills in solving math problems are growing. Tools like FunSearch can do things even experts didn’t know were possible. These AI tools can come up with new solutions and
even make current math insights better.
“The method described using large language models (LLMs) to generate new solutions in mathematics was published in Nature on 14 December. FunSearch was able to improve on the lower bound for n =
8 in combinatorics problems.”
We can look forward to more big discoveries as AI and math keep working together. The future looks bright for a closer partnership between humans and AI in math. Together, they’ll likely push the
limits of what we know and can solve.
The Impact of AI on Mathematical Education
AI is changing how we teach math. It brings new tools and resources that make learning math fun and interactive. These tools use step-by-step explanations, visual aids, and interactive problems to
help students learn.
AI helps students solve problems, understand complex ideas, and get ready for tests. Tools like Photomath, Socratic, and Mathway use AI to make learning math easier. They help students with tough
problems and complex concepts.
AI-Powered Math Learning Tools and Resources
Apps like ClassPoint AI use AI to make math practice fun. They offer quizzes and activities that make learning math exciting. As AI gets better, it will help students more and more, helping them
master math.
AI-Powered Math Learning Tool Key Features
Photomath Provides step-by-step solutions and explanations for a wide range of math problems
Socratic Offers AI-powered tutoring and explanations for various math topics
Mathway Specializes in solving complex math problems and providing detailed solutions
ClassPoint AI Incorporates AI-driven quizzes and interactive activities to make math practice more engaging
As AI gets better, it will have a bigger impact on math education. It will help students overcome math challenges and reach their full potential.
“AI-powered learning tools and resources are revolutionizing the way students approach and engage with math.”
Advantages of Integrating AI in Mathematics
AI is changing mathematics in big ways. It can quickly go through and understand huge amounts of data. This lets it solve hard problems and do complex math fast, much faster than humans. AI also
finds patterns in data that we might miss, leading to new discoveries.
Using AI for math problems means getting very precise and accurate work. It cuts down on mistakes that can happen when we do math by hand. AI can do the same task over and over without getting tired,
letting mathematicians focus on the creative parts of their work. This way, AI helps mathematicians explore new areas, solve tough problems, and work with machines to expand our knowledge in math.
AI also helps in teaching math by making learning personal and interactive. It gives students feedback and content that fits their needs. This makes students understand better and appreciate math
In summary, AI brings many benefits to math, like making solving problems and analyzing data faster and more accurate. It also improves learning and opens new doors for innovation. As AI and human
mathematicians work together, we can expect big breakthroughs in math.
Advantages of AI in Description
Speed and Efficiency AI systems can process and analyze vast amounts of data much faster than human mathematicians, enabling them to tackle complex problems with speed and precision.
Pattern Recognition AI algorithms can identify patterns and relationships within large datasets that may not be immediately apparent to the human eye, leading to new insights and discoveries.
Precision and Accuracy AI-driven problem-solving reduces the risk of errors or miscalculations often associated with manual computations, enhancing the reliability of mathematical work.
Tireless Computations AI can handle repetitive tasks and simulations without fatigue, freeing up mathematicians to focus on more creative and innovative aspects of their work.
Personalized Learning AI-powered educational tools can provide personalized learning experiences, real-time feedback, and adaptive content tailored to individual student needs, improving
understanding and confidence in mathematics.
AI is changing math in big ways, offering many advantages. It’s making solving problems and analyzing data faster and more accurate. By using AI’s speed, precision, and pattern recognition,
mathematicians can explore new areas, solve hard problems, and work with machines to expand our math knowledge.
Ethical Considerations in AI and Mathematics
AI and mathematics are becoming more connected, bringing up big ethical questions. One key issue is bias in AI algorithms. If these algorithms are trained on biased data, they can keep those biases,
leading to unfair results.
Privacy and data security are also big concerns. AI uses a lot of personal data, which must be kept safe. It’s hard to understand how AI makes decisions because of its complex algorithms.
It’s important to tackle these ethical issues as AI and math work together more. We need to make sure the benefits of this partnership don’t harm individual rights or society. Adding ethical rules to
AI can help prevent bias and ensure transparency.
“Towards Ethical AI: Mathematics Influences Human Behavior” by Dioneia M. Monte-Serrat and Carlo Cattani
This paper talks about how AI’s algorithms affect human choices and actions. It says that the way AI talks to us can limit our freedom to choose.
Because of ethical worries, ethics is now taught in computer science at Harvard University. It’s vital to keep a balance between AI’s benefits and protecting our rights as it grows.
Future Prospects: AI and Mathematics Collaboration
The future of AI and mathematics working together is very promising. AI is getting better at solving problems and could change how we use mathematics. By combining AI with human math experts, we can
tackle tough problems that were hard to solve before.
AI can look through lots of data, find patterns, and come up with solutions. This can help human mathematicians who are great at thinking creatively and understanding complex math. Together, they
could make big discoveries, create new math theories, and solve problems that have been hard for a long time.
Using AI in math education could change how students learn math. It could give them personalized lessons and help them do well in math. As AI and math keep getting closer, we’ll see more amazing
things that will expand our knowledge and open up new areas of science.
Advancements in AI-Driven Mathematical Problem Solving
Math is being broken down into smaller parts so computers can check and verify proofs. The Lean project shows how working together on a large scale with automated proof checkers is possible. With
mathlib, basic math theorems from college level can be formalized, making math more practical.
Lean is becoming a top choice for math because it has a strong community, library, and is easy to use. Even though formalizing math is still slow, AI could make it faster. This could lead to
submitting proofs to journals automatically.
Potential of AI-Math Collaboration
Working together, human mathematicians and AI could lead to big breakthroughs. This is still new because making math formal is complex. Soon, AI will be a big help to mathematicians, helping them
prove theorems and making their work more efficient.
Breaking math projects into smaller parts through formalization lets different people work on them, even if they only get some of the math. This could make math more open and collaborative. It could
lead to a big change in the future of AI and mathematics.
“As the frontiers of AI and mathematics continue to converge, we can expect to witness even more remarkable advancements that will push the boundaries of human knowledge and unlock new realms of
scientific understanding.”
Overcoming Challenges in AI-Driven Math Problem-Solving
AI and math have made big strides together, but there are still big hurdles. One major issue is that current AI systems lack creativity and abstract thinking.
AI is great at spotting patterns and solving problems with what it knows. But, it can’t quite match human mathematicians in coming up with new ideas. Bridging this gap and making AI more creative in
math is a key area of research.
It’s also vital to make AI math solutions clear and accountable. The complex algorithms make it hard to understand how they decide things. Dealing with ethical issues like bias and privacy is crucial
as AI in math grows.
Researchers and developers are working hard to overcome these AI math problem solving challenges. We can look forward to more breakthroughs at the AI and math crossroads. By addressing AI’s math
limitations and overcoming AI math reasoning barriers, we could change how we tackle complex math problems.
Challenges in AI-Driven Math Problem-Solving Potential Solutions
Limitations in creativity and abstract thinking Advancing AI algorithms to enhance innovative and creative mathematical reasoning
Complexity and transparency of AI decision-making Developing more explainable and accountable AI systems for mathematical problem-solving
Ethical concerns, such as bias and data privacy Implementing robust ethical frameworks and safeguards in the deployment of AI in mathematics
By tackling these challenges and using the best of human and AI skills, we can open up new chances in math. This will change how we tackle complex problems.
“The integration of AI technology in education, particularly in mathematics, symbolizes a shift towards more interactive, accessible, and efficient learning environments.”
AI and mathematics have changed how we solve math problems. AI can quickly go through lots of data, find patterns, and solve problems accurately. This has opened new doors in math. Systems like
AlphaProof and AlphaGeometry 2 have shown they can solve tough math problems at a high school level.
As AI gets better, it will change math even more. We can see human-AI teams working together to solve hard problems. There are still challenges and ethical issues to work on. But the future looks
amazing. This partnership could change how we see the world and push science forward like never before.
This summary wraps up the main points of AI and math working together. It talks about how AI is changing math, the bright future for these two, and the chance to discover new things. It also points
out the importance of recent AI successes, the need to tackle challenges, and the big changes this partnership could bring.
Can AI solve complex math problems?
Yes, AI has made big strides. Systems like AlphaProof and AlphaGeometry 2 can now solve tough math problems. They even solved 4 out of 6 challenges from the International Mathematical Olympiad.
How has the evolution of AI impacted mathematical problem-solving?
AI has moved beyond just doing simple calculations. It now uses machine learning, reinforcement learning, and formal language processing for complex problems. This has led to AI tools that solve math
problems, prove theorems, and optimize solutions.
What are the key advantages of using AI in mathematics?
AI brings speed, efficiency, and precision to the table. It can spot patterns in big data quickly. This helps AI solve complex problems fast, improve optimization, cryptography, and theorem proving.
What are some of the ethical concerns surrounding the use of AI in mathematics?
There are big concerns about AI. These include bias and discrimination in algorithms, privacy, and data security. There’s also a need for transparency and accountability in AI’s decision-making.
How is AI transforming mathematical education?
AI is changing math education for the better. It offers tools with step-by-step explanations, personalized learning, and interactive problems. These help students overcome math challenges.
What are the future prospects of the collaboration between AI and mathematics?
The future looks bright. AI and human mathematicians could work together. They’ll use each other’s strengths to solve hard problems and expand math knowledge. | {"url":"https://aibubbleburst.com/can-ai-solve-math-problems/","timestamp":"2024-11-06T04:06:41Z","content_type":"text/html","content_length":"112781","record_id":"<urn:uuid:7c30e937-9f20-453d-af19-b9cc7187a015>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00452.warc.gz"} |
Solving Sudoku using SQL Server 2005 - Step by Step - Part #4
Implementation of RunSolveAlgorithm2:
We implemented RunSolveAlgorithm1 in previous post of this series . The next algorithm is the implementation of Solve Method A from sudoku solver.
In this algorithm, we check all the cells (having mutiple values) in each row and see if a particular value occurs only once in that row. Then update that as the solution for the cell having that
value. We do the similar check for column and the 3X3 block.
This can solve the easy to medium puzzles. Here goes the implementation.
ALTER PROC RunSolveAlgorithm2
DECLARE @RowCount int,
@UpdateRowCount int
SET @RowCount = 1
SET @UpdateRowCount = 0
WHILE(@RowCount > 0 AND dbo.VerifySolve() = 0)
SET @RowCount = 0;
/* Take all the cells, having mutiple values, in each row and see if a particular value occurs only
once in that row. Then update that as the solution for the cell */
WITH XSOL AS
SELECT XPOS,SUBSTRING(VAL,NUM,1) AS VAL FROM SOLUTION_BOARD A, NUMBERS B
WHERE B.NUM <=LEN(A.VAL)
AND LEN(VAL) > 1
GROUP BY XPOS,SUBSTRING(VAL,NUM,1) HAVING COUNT(*) = 1
UPDATE SOL
SET VAL = XSOL.VAL
FROM SOLUTION_BOARD SOL, XSOL
SOL.XPOS = XSOL.XPOS
AND LEN(SOL.VAL) > 1
AND CHARINDEX(XSOL.VAL,SOL.VAL) > 0;
SET @UpdateRowCount = @@ROWCOUNT;
SET @RowCount = @RowCount + @UpdateRowCount;
IF(@UpdateRowCount > 0) /* Need to rerun algorithm 1 for clean up if any cell was updated */
EXEC RunSolveAlgorithm1;
/* Take all the cells, having mutiple values, in each column and see if a particular value occurs only
once in that column. Then update that as the solution for the cell */
WITH YSOL AS
SELECT YPOS,SUBSTRING(VAL,NUM,1) AS VAL FROM SOLUTION_BOARD A, NUMBERS B
WHERE B.NUM <=LEN(A.VAL)
AND LEN(VAL) > 1
GROUP BY YPOS,SUBSTRING(VAL,NUM,1) HAVING COUNT(*) = 1
UPDATE SOL
SET VAL = YSOL.VAL
FROM SOLUTION_BOARD SOL, YSOL
SOL.YPOS = YSOL.YPOS
AND LEN(SOL.VAL) > 1
AND CHARINDEX(YSOL.VAL,SOL.VAL) > 0;
SET @UpdateRowCount = @@ROWCOUNT;
SET @RowCount = @RowCount + @UpdateRowCount;
IF(@UpdateRowCount > 0) /* Need to rerun algorithm 1 for clean up if any cell was updated */
EXEC RunSolveAlgorithm1;
/* Take all the cells, having mutiple values, in each 3X3 block and see if a particular value occurs only
once in that block. Then update that as the solution for the cell */
WITH BSOL AS
SELECT ((YPOS-1)/3)*3 + (XPOS-1)/3 AS BPOS,SUBSTRING(VAL,NUM,1) AS VAL FROM SOLUTION_BOARD A, NUMBERS B
WHERE B.NUM <=LEN(A.VAL)
AND LEN(VAL) > 1
GROUP BY ((YPOS-1)/3)*3 + (XPOS-1)/3,SUBSTRING(VAL,NUM,1) HAVING COUNT(*) = 1
UPDATE SOL
SET VAL = BSOL.VAL
FROM SOLUTION_BOARD SOL, BSOL
((SOL.YPOS-1)/3)*3 + (SOL.XPOS-1)/3 = BSOL.BPOS
AND LEN(SOL.VAL) > 1
AND CHARINDEX(BSOL.VAL,SOL.VAL) > 0;
SET @UpdateRowCount = @@ROWCOUNT;
SET @RowCount = @RowCount + @UpdateRowCount;
IF(@UpdateRowCount > 0) /* Need to rerun algorithm 1 for clean up if any cell was updated */
EXEC RunSolveAlgorithm1;
When I call the proc SolveSudoku now, you can see that the problem is solved and when solved, the solution board and sudoku board are in sync.
EXEC SolveSudoku
post-solve sudoku board - before implementing Algorithm 2
post-solve sudoku board - after implementing Algorithm 2 (Solved)
post-solve solution board - before implementing Algorithm 2
post-solve solution board - after implementing Algorithm 2 (Same as the sudoku board)
For the
next algorithm
will take up a harder puzzle and see how well we fare.
0 comments: | {"url":"http://www.sqlpointers.com/2009/11/solving-sudoku-using-sql-server-2005_14.html","timestamp":"2024-11-10T21:44:49Z","content_type":"application/xhtml+xml","content_length":"46272","record_id":"<urn:uuid:18e6857b-2b2a-40e1-8673-62afff2fca76>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00274.warc.gz"} |
Calculations in Excel | Learn How to Use Excel to Calculate?
Updated May 9, 2023
How to Calculate in Excel (Table of Contents)
Introduction to Calculations in Excel
The following article provides an outline for Calculations in Excel. MS Excel is the most preferred option for calculation; most investment bankers and financial analysts use it to do data crunching,
prepare presentations, or model data.
There are two ways to perform the Excel calculation: Formula and the second is Function. The formula is the normal arithmetic operation like summation, multiplication, subtraction, etc. The function
is the inbuilt formula like SUM (), COUNT (), COUNTA (), COUNTIF (), SQRT (), etc.
Operator Precedence: It will use default order to calculate; if there is some operation in parentheses, then it will calculate that part first, then multiplication or division after that addition or
subtraction. It is the same as the BODMAS rule.
Examples of Calculations in Excel
Here are some examples of How to Use Excel to Calculate Basic calculations.
Example #1 – Basic Calculations like Multiplication, Summation, Subtraction, and Square Root
Here we will learn how to do basic calculations like multiplication, summation, subtraction, and square root in Excel.
Let’s assume a user wants to perform calculations like multiplication, summation, and subtraction by 4 and find out the square root of all numbers in Excel.
Let’s see how we can do this with the help of calculations.
Step 1: Open an Excel sheet. Go to sheet 1 and insert the data as shown below.
Step 2: Create headers for Multiplication, Summation, Subtraction, and Square Root in row one.
Step 3: Now calculate the multiplication by 4. Use the equal sign to calculate. Write in cell C2 and use asterisk symbol (*) to multiply “=A2*4“
Step 4: Now press the Enter key; multiplication will be calculated.
Step 5: Drag the same formula to the C9 cell to apply to the remaining cells.
Step 6: Now calculate subtraction by 4. Use an equal sign to calculate. Write in cell D2 “=A2-4“
Step 7: Now click the Enter key to calculate the subtraction.
Step 8: Drag the same formula till cell D9 to apply it to the remaining cells.
Step 9: Now calculate the addition by 4; use an equal sign to calculate. Write in E2 Cell “=A2+4“
Step 10: Now press the Enter key to calculate the addition.
Step 11: Add the same formula to the E9 cell to the remaining cells.
Step 12: Now calculate the square root>> use equal sign to calculate >> Write in F2 Cell >> “=SQRT (A2“
Step 13: Now, press on the Enter key >> square root will be calculated.
Step 14: Drag the same formula till the F9 cell to apply the remaining cell.
Summary of Example 1: As the user wants to perform calculations like multiplication, summation, and subtraction by 4 and find out the square root of all numbers in MS Excel.
Example #2 – Basic Calculations like Summation, Average, and Counting
Here we will learn how to use Excel to calculate basic calculations like summation, average, and counting.
Let’s assume a user wants to find out total sales, average sales, and the total number of products available in his stock for sale.
Let’s see how we can do this with the help of calculations.
Step 1: Open an Excel sheet. Go to Sheet 1 and insert the data as shown below.
Step 2: Create headers for the Result table, Grand Total, Number of Product, and Average Sale of his product in column D.
Step 3: Now calculate total sales. Use the SUM function to calculate the total. Write in cell E3. “=SUM (”
Step 4: Now, it will ask for the numbers, so give the data range, which is available in column B. Write in cell E3. “=SUM (B2:B11) “
Step 5: Now press the Enter key. Total sales will be calculated.
Step 6: Now calculate the total number of products in the stock; use the COUNT function to calculate the total. Write in cell E4 “=COUNT (“
Step 7: Now, it will ask for the values, so give the data range, which is available in column B. Write in cell E4. “=COUNT (B2:B11) “
Step 8: Now press the Enter key. The total number of products will be calculated.
Step 9: Now calculate the average sale of products in the stock; use the AVERAGE function to calculate the average sale. Write in cell E5. “=AVERAGE (”
Step 10: Now, it will ask for the numbers, so give the data range in column B. Write in cell E5. “=AVERAGE (B2:B11) “
Step 11: Now click on the Enter key. The average sale of products will be calculated.
Summary of Example 2: As the user wants to find out total sales, average sales, and the total number of products available in his stock for sale.
Things to Remember about Calculations in Excel
• During calculations, if there are some operations in parentheses, it will calculate that part first, then multiplication or division after that addition or subtraction.
• It is the same as the BODMAS rule: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
• When a user enters an equal symbol (=) in a cell, they enter a formula rather than a value.
• A small difference from the normal mathematics symbol like multiplication uses an asterisk symbol (*), and division uses a forward-slash (/).
• There is no need to write the same formula for each cell; once it is written, copy-paste it to other cells, and it will calculate automatically.
• A user can use the SQRT function to calculate the square root of any value; it has only one parameter. But a user cannot calculate the square root for a negative number; it will throw an error #
• If a negative value occurs as output, use the ABS formula to determine the absolute value, an in-built MS Excel function.
• A user can use the COUNTA in-built function if there is confusion in the data type because COUNT supports only numeric values.
Recommended Articles
This is a guide to Calculations in Excel. Here we discuss how to use Excel to calculate, along with examples and a downloadable Excel template. You may also look at the following articles to learn
more – | {"url":"https://www.educba.com/calculations-in-excel/","timestamp":"2024-11-15T04:33:12Z","content_type":"text/html","content_length":"366650","record_id":"<urn:uuid:25302ddc-5d7e-40f0-9c56-ff228281edbb>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00566.warc.gz"} |
Mathematical modeling of dynamic processes in a continuous electrically conductive medium
Title Mathematical modeling of dynamic processes in a continuous electrically conductive medium
S. K. Kazankov^1, S. I. Peregudin^2, S. E. Kholodova^1
Authors ^1University IFMO
^2Saint-Petersburg State University
A mathematical model describing the dynamics of geophysical processes in an electrically conductive incompressible fluid bounded by free and solid impermeable surfaces is presented, taking
into account the effects of magnetic field diffusion, gravity and Coriolis force. The mathematical model is based on the solution of the boundary magnetohydrodynamic problem for partial
Annotation differential equations, taking into account the effects of long waves of small amplitude. By means of appropriate transformations, the system of differential equations of vector type and
partial derivatives can be reduced to one scalar equation for the modified function perturbation of the free surface of the ocean. The mathematical analysis of the presented model used to
study magnetohydrodynamic processes in the ocean of the northern hemisphere demonstrates the occurrence of the phenomenon of inversion of the induced magnetic field.
Keywords magnetic hydrodynamics, geophysics, ocean, electrically conductive fluid, incompressible fluid, mathematical modeling, magnetic field inversion
Kazankov S. K., Peregudin S. I., Kholodova S. E. ''Mathematical modeling of dynamic processes in a continuous electrically conductive medium'' [Electronic resource]. Proceedings of the XVI
Citation International scientific conference "Differential equations and their applications in mathematical modeling". (Saransk, July 17-20, 2023). Saransk: SVMO Publ, 2023. - pp. 73-78. Available
at: https://conf.svmo.ru/files/2023/papers/paper10.pdf. - Date of access: 12.11.2024. | {"url":"https://conf.svmo.ru/en/archive/article?id=404","timestamp":"2024-11-12T04:27:56Z","content_type":"text/html","content_length":"11921","record_id":"<urn:uuid:a7c14d13-ca83-449a-ae61-5818d1819058>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00345.warc.gz"} |
4th Grade Math | Free Lesson Plans | Full Year Curriculum
Unit 1, Place Value, Rounding, Addition, and Subtraction, begins the year with the foundational content on which much of the remaining units are based – place value. Students start to see the
structure of the place value system in the context of multiplicative comparison – e.g., 1 thousand is 10 times as much as 1 hundred. They then use that place value understanding to compare, round,
add and subtract numbers up to 1,000,000. They also solve multi-step word problems involving addition and subtraction, using rounding to assess the reasonableness of their answers.
In Unit 2, Multi-Digit Multiplication, students use this place value understanding to start to develop an understanding of multi-digit multiplication (including 2-digit, 3-digit, and 4-digit by
1-digit, as well as 2-digit by 2-digit multiplication). While students were introduced to the idea of multiplicative comparison in Unit 1 in the context of the structure of our place value system,
they more deeply delve into these story problems types in this unit. Unit 3, Multi-Digit Division, similarly relies on place value understanding to introduce students to multi-digit division
(including 4-digit, 3-digit, and 2-digit by 1-digit division). Students continue their work with multi-step word problems by working with remainders, interpreting them in the context of the problem.
In Unit 4, Fraction Equivalence and Ordering students work with fraction equivalence and comparison, developing a general method for generating equivalent fractions and exploring multiple strategies
for fraction comparison. This prepares them for Unit 5, Fraction Operations, where they start to explore operations with fractions (namely addition, subtraction, and multiplication by a whole
number). Students also start to solve word problems involving the addition, subtraction, and multiplication of fractions. This then extends to Unit 6, Decimal Fractions, in which students explore
decimal fractions, which are particularly important since they are an extension of the place value system. They find equivalent decimal fractions, add and subtract decimal fractions (including tenths
with hundredths, requiring a common denominator), and using decimal notation.
In Unit 7, Unit Conversion, students apply much of their understanding of the four operations as well as fractions and decimals to solve word problems involving the conversion from a larger unit to a
smaller unit within the same system.
Finally, in Unit 8, Shapes and Angles, students get a formal introduction to angles after many years of informally categorizing shapes according to their angles. Students measure angles and find
unknown angle measures, then use this deeper understanding to classify shapes and explore reflectional symmetry.
The scope and sequence for 4th Grade Math was adjusted in August 2021. Learn more about this update. | {"url":"https://www.fishtanklearning.org/curriculum/math/4th-grade/","timestamp":"2024-11-09T10:47:03Z","content_type":"text/html","content_length":"293433","record_id":"<urn:uuid:2b746392-7906-4a57-a185-328c3fc634d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00890.warc.gz"} |
These are the publications which have given me an Erdős number of 5 so far. Send me an e-mail if you want to work together on one.
• M. A. Verdier; P. C. F. Di Stefano; P. Nadeau; C. Behan; M. Clavel; C. Dujardin. Scintillation properties of Bi[4]Ge[3]O[12] down to 3K under gamma rays. Physical Review B, 84, 214306, 2011.
• C. Behan. Density of states in a free CFT and finite volume corrections [arXiv:1210.5655]. Physical Review D, 88, 026015, 2013.
• C. Behan; K. Larjo; N. Lashkari; B. Swingle; M. Van Raamsdonk. Energy trapping from Hagedorn densities of states [arXiv:1304.7275]. Journal of High Energy Physics, 10, 63, 2013.
• C. Behan. Conformal blocks for highly disparate scaling dimensions [arXiv:1402.5698]. Journal of High Energy Physics, 9, 5, 2014.
• C. Behan. PyCFTBoot: A flexible interface for the conformal bootstrap [arXiv:1602.02810]. Communications in Computational Physics, 22, 1, 2017.
• C. Behan; L. Rastelli; S. Rychkov; B. Zan. Long-range critical exponents near the short-range crossover [arXiv:1703.03430]. Physical Review Letteres, 118, 241601, 2017.
• C. Behan; L. Rastelli; S. Rychkov; B. Zan. A scaling theory for the long-range to short-range crossover and an infrared duality [arXiv:1703.05325]. Journal of Physics A, 50, 35, 2017.
• C. Behan. Conformal manifolds: ODEs from OPEs [arXiv:1709.03967]. Journal of High Energy Physics, 3, 127, 2018.
• C. Behan. Unitary subsector of generalized minimal models [arXiv:1712.06622]. Physical Review D, 97, 094020, 2018.
• C. Behan. Bootstrapping the long-range Ising model in three dimensions [arXiv:1810.07199]. Journal of Physics A, 52, 7, 2019.
• C. Behan; L. Di Pietro; E. Lauria; B. C. van Rees. Bootstrapping boundary-localized interactions [arXiv:2009.03336]. Journal of High Energy Physics, 12, 182, 2020.
• C. Behan; P. Ferrero; X. Zhou. More on holographic correlators: Twisted and dimensionally reduced structures [arXiv:2101.04114]. Journal of High Energy Physics, 4, 008, 2021.
• L. F. Alday; C. Behan; P. Ferrero; X. Zhou. Gluon scattering in AdS from CFT [arXiv:2103.15830]. Journal of High Energy Physics, 6, 020, 2021.
• C. Behan; L. Di Pietro; E. Lauria; B. C. van Rees. Bootstrapping boundary-localized interactions II: Minimal models at the boundary [arXiv:2111.04747]. Journal of High Energy Physics, 3, 146,
• C. Behan. Holographic S-fold theories at one loop [arXiv:2202.05261]. SciPost Physics, 12, 149, 2022.
• A. Antunes; C. Behan. Coupled minimal conformal field theory models revisited [arXiv:2211.16503]. Physical Review Letters 130, 071602, 2023.
• C. Behan; S. M. Chester; P. Ferrero. Gluon scattering in AdS at finite string coupling from localization [arXiv:2305.01016]. Journal of High Energy Physics, 2, 042, 2024.
• C. Behan; E. Lauria; M. Nocchi; P. van Vliet. Analytic and numerical bootstrap for the long-range Ising model [arXiv:2311.02742]. Journal of High Energy Physics, 3, 136, 2024.
• C. Behan; S. M. Chester; P. Ferrero. Towards bootstrapping F-theory [arXiv:2403.17049]. Journal of High Energy Physics, 10, 161, 2024.
These are slides I used for the presentations that I would consider interesting. Some were given at conferences, others at more mundane venues. | {"url":"http://smallperturbation.com/publications","timestamp":"2024-11-02T03:01:37Z","content_type":"application/xhtml+xml","content_length":"19943","record_id":"<urn:uuid:c08d83c0-5a43-416d-ac5d-c03f2e081eca>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00798.warc.gz"} |
Combining indirect estimates of child and adult mortality to produce a life table
Description of the method
The indirect methods described in this manual for deriving estimates of child and adult mortality produce series of estimates of child and adult mortality, which – using the time location approach
pioneered by Feeney (1980, 1991) for children and Brass and Bamgboye (1981) and Brass (1985) for adults – apply to a variety of dates. In many demographic applications, however, it is useful if one
can derive an abridged life table that reflects mortality over the entire age range at a specific date within the period covered by such series of indirect estimates of mortality. These applications
include the production of population projections or the evaluation of changes in life expectancies at birth or mortality over time.
A general summary of the nature and range of estimates produced by the most important indirect methods is presented in Table 1.
Table 1 Indices of mortality and time references of the estimates produced by selected indirect methods for the estimation of child and adult mortality
Method Measure and typical age range Typical time reference
Child: Indirect l(1) … l(20) 1 to 15 years before the survey
Adult: Maternal orphanhood [10]p[25] … [40]p[25] 3 to 15 years before the survey
Adult: Paternal orphanhood [15]p[35] … [35]p[35] 5 to 15 years before the survey
Adult: Siblinghood method [10]p[15] … [35]p[15] 3 to 15 years before the survey
An important feature of the estimates that these methods produce for adults is that they are all conditional estimates of survivorship, that measure survival from one age (e.g., 25 in the case of the
maternal orphanhood method) to another age (e.g., 35). One cannot straightforwardly convert these conditional measures of survivorship in adulthood into unconditional ones. Thus, the methods for
fitting logit model life tables explained by the introductory descriptions of the models in a number of textbooks cannot be applied and more complicated fitting methods are required.
In order to combine estimates of child and adult mortality into a single life table applicable at a defined point in time, a method is needed which addresses the following list of issues:
• The adult mortality estimates need to be converted from their initial conditional form into measures of survivorship from birth
• The child and adult mortality estimates may imply different patterns or levels of mortality, different time trends in mortality, or both
• Some data points may be defective or suffer from random fluctuations that distort the overall trend, which implies that the implied trend may require smoothing or adjustment
• The estimates of child and adult mortality typically refer to different dates and may span different periods of time
• Neither the methods for estimating child mortality nor those for estimating adult mortality produce any information on the mortality of some age groups, implying that the one can only produce a
complete life table by using models.
The method described here seeks to find the parameters α and β of a relational logit model life table (described in the Introduction to Model Life Tables) applicable to a specified point in time that
offers the best fit to the observed data points used as inputs. Fitting a 2-parameter model is only possible if independent estimates are available of child and adult mortality for the date in
question. If such data are available, fitting a 2-parameter model is recommended because no justification usually exists a priori for making the assumption that the age pattern of mortality in the
population in question corresponds to that in any particular 1-parameter family of model life tables.
Starting with the observed quantities from the child and adult estimation, the method first derives and plots the implied values of α (the level parameter of a relational model life table) against
the time location of each estimate, separately for child and adult mortality making the assumption that β (the shape parameter) is equal to 1. This ‘alpha plot’ is used to identify which data points
describe a coherent and consistent trend in the value of α over time. The selected points are then used to iteratively calculate final estimates of α and β at the date for which the life table is
required. A fitted model life table can then be calculated from the standard using these values of α and β. The method allows both the α and β parameters of the fitted models to change over time but
constrains them to do so linearly (Timæus 1990).
The method can be used to derive abridged life tables from sex-specific estimates of child mortality produced by the indirect method for the analysis of data on women’s children ever-born and still
alive, and sex-specific estimates of adult mortality produced by application of either the One Census Orphanhood or Indirect Siblinghood methods. Estimates of child and adult mortality made by
direct methods, or from the application of two-census methods, normally apply to a specific year or period of time. Model life tables can be fitted to pairs of estimates of adult and child mortality
that refer to the same calendar time using the methods described in the section on fitting life tables to a pair of estimates of childhood and adult mortality.
Data requirements and assumptions
Tabulations of data required
• A series of sex-specific indirect estimates of child mortality, with their time locations, derived from data on women’s children ever born and surviving
• A series of sex-specific estimates of adult mortality, with their time locations, derived using either the indirect method for analyzing data on sibling survival or the one census orphanhood
In principle, the approach used to fit such data could be extended to estimate life tables for populations for which multiple overlapping sets of indirect estimates exist describing child and adult
mortality. However, the accompanying worksheet has only been designed to handle two series of estimates: one for children and one for adults.
The method described here bases the fitted model life table on a standard life table. This standard is assumed to have an age pattern of mortality that resembles that of the population being studied.
In particular, the relative severity of child and adult mortality in the standard should be similar to that indicated by the indirect estimates to which the model is being fitted. Guidelines for
choosing an appropriate standard life table are provided in the Introduction to Model Life Tables, which also describes the basic mechanics of the relational logit system of model life tables. The
standard need not be taken from the family of model life tables that underlies the coefficients that were used to produce the indirect estimates of child mortality: the family of models that best
represents the age pattern of mortality within childhood may not be the family that best represents the relative levels of child and adult mortality in the same population.
Caveats and warnings
The plausibility of the fitted model life table produced by this method of fitting depends on whether the chosen standard life table is appropriate for the population under study. In populations
affected by HIV/AIDS, for example, both the balance between child and adult mortality and the detailed age pattern of mortality differ greatly from those that characterize the systems of model life
tables in widespread use. Consequently, this method is not recommended for routine application in these circumstances or to other populations for which no standard life table can be identified that
describes the balance between child and adult mortality.
Application of method
The method is implemented using the following steps.
Step 1: Identify the date to which the desired life table should apply
To avoid the risks associated with out-of-sample extrapolation in the determination of α and β, the life table should be fitted to a date within the period covered by the estimates of adult and child
estimates that are being analyzed. In the presentation of the method that follows, this target date is denoted by D.
The exact date for which the life table is required may be determined by the use to which it is going to be put. Ideally a date should be chosen, however, for which both the estimates of adult and
child mortality seem reliable. For example, if either the more distant estimates for children appear to biased downward by underreporting of dead children or the more recent estimates for adults
appear to be biased downward by the adoption effect, one should avoid producing a life table for the dates covered by the defective estimates. Unfortunately, such considerations sometimes lead the
analyst to the conclusion that the data at hand fail to provide a sound basis for the construction of a life table!
If a life table is needed for a more recent, or possibly a more distant, date than the period of time covered by the estimates, a limited amount of extrapolation beyond this range of dates might be
considered. The extent of this should be restricted to three years before the earliest time location of any adult or child mortality estimates, on the one hand, and to three years after the earlier
of the most recent estimate of child mortality and the most recent estimate of adult mortality, on the other.
Step 2: Select a standard to be used to derive the fitted life table
The associated spreadsheet allows the analyst to choose between nine sex-specific standards: the five UN model life tables (General; South Asian; Far Asian; Latin American; Chilean) and the four
Princeton regional model life tables (North; South; East; West). All the standard life tables have a life expectancy at birth of 60 years. The derivation of these logits is described in the
Introduction to Model Life Tables, and a spreadsheet containing their values can be downloaded from the Tools for Demographic Estimation website.
The primary objective in selecting a standard should be to identify one in which the relationship between child and adult mortality is approximately the same as that indicated by the estimates of
child and adult mortality. As a practical rule of thumb, if the value of β of the final fitted model lies outside the range 0.75-1.25, one should at least consider adopting another standard. In more
extreme circumstances, model life tables in which β falls outside the range 0.6-1.4 are unlikely to represent empirical mortality schedules adequately. A secondary objective in choosing a standard
should be to identify one that shares other known characteristics of the population in question such as the relationship between infant mortality and mortality at ages 1 to 4. The characteristics of
the different Princeton and UN families of model life tables are described briefly in the Introduction to Model Life Tables.
Step 3: Plot values of α (assuming β = 1) derived from the mortality estimates against time
When β (the shape parameter in a relational model life table system) equals 1, the relational model life table system can be expressed as
where Y(x) is the logit transform
$Y(x)=logit(l(x))=12ln(1−l(x)l(x))=−12ln(1−q(x)q(x)) .$
For child mortality, calculating a series of values of α from the estimates is straightforward. The logits of the chosen standard life table for ages 1, 2, 3, 5, 10, 15 and 20 are subtracted from the
logits of the derived estimates of child mortality (q(1), q(2), q(3), q(5), q(10), q(15) and q(20)):
$αchild=Y(x)−Ys(x) .$
For adults, the calculation of α is more complicated as the survival probabilities produced by the estimation methods are conditional on survival to a given base age. The formula for α is
$αadults=12{ln(1−npx)−ln[npx.exp(2Ys(x+n))−exp(2Ys(x))]} ,$
where x is the base age of the conditional probability of survival (25 for the maternal orphanhood method) and n is the duration over which survivorship is measured, which is contingent on the age
group of the respondent. (The derivation of this expression can be found at the end of this page).
The estimates of α (separately for children and adults) are then plotted against their respective time locations on the same set of axes.
Step 4: Eliminate those points in the alpha plot that appear out of line with the general trend
In order to estimate α and β for a specific point in time, the method imposes a linear trend on both parameters. As the first step to achieving this goal, we would like the plots of each of the
series of α’s (that is, for children and adults separately) against their time locations derived in Step 3 to lie on straight lines.
The α’s for individual data points in a series of child or adult mortality estimates derived using the two formulae above may deviate from a straight line for several reasons. First, the underlying
pattern of change in mortality may have been strongly non-linear. This is somewhat unlikely given that the series of estimates cover fairly short periods of time and that indirect estimation methods
tend to smooth out short term fluctuations in mortality. Even if the diagnostic plot derived above suggests that it is the case, it may still be possible to obtain an adequate fitted model life table
by calculating it for a date at which the linear trends in the parameters imposed by the method intersect with the curve indicated by the plotted points. Second, the series may be rather erratic due
to sampling errors and reporting errors such as age misstatement. If this is the only limitation of the estimates, one would normally include them all in the analysis and rely on the line fitting
procedure to average across these fluctuations.
Third, indirect estimates are vulnerable to biases resulting from respondents failing to answer the key questions accurately or to breaches in the assumptions of the methods concerned. Likely errors
in the estimates are discussed in the pages on the various methods and the reader is referred to those pages for advice on diagnostic signs that may suggest indicate certain points are biased and
should be dropped from the fitting procedure. It is particularly common, however, for the point relating to respondents aged 15-19 in the child mortality method to be biased upward and for the points
relating to children aged 5-14 reporting on the survival of their parents in the one census orphanhood method to be biased downward. It will often be necessary to exclude these data points from the
model fitting process.
A fourth possible explanation for the failure of the calculated α’s to lie on a straight line is that the standard selected for calculating the original estimates may not have been appropriate. If
this is the case, it may be necessary to recalculate these estimates using a different standard. Alternately, it may be necessary to try using an alternative standard (as described in Step 2) to
derive the fitted life table.
Once the child and adult α’s for inclusion in the fitting process have been selected, the rest of the fitting process proceeds mechanically.
Step 5: Determine the trend in β by iteration
The process of solving for β iteratively is not readily done manually, and the associated workbook has been designed to perform the calculations. In order to enable the iteration routine, ensure that
Microsoft Excel has been configured appropriately. This is done by selecting "File → Options → Formulas and then checking the “Enable iterative calculation” checkbox. Setting a maximum of 1000
iterations and a maximum change of 0.00001 is more than sufficient for a solution to be reached.
The process whereby β and α are adjusted iteratively to secure a good fit is described in the section on the Mathematical Exposition of the method. The key constraints placed on the fitting process
are as follows:
• No matter what the original values of x and n in the estimates of q(x) and [n]p[b] for children and adults respectively at the date in question, β is calculated consistently from survivorship
from age 15 to 60 relative to the standard.
• Both α and β are allowed to change over time but it is assumed that they do so linearly.
In combination, these assumptions reduce the distorting impact that errors in the estimates and minor differences in the age pattern of mortality between the population and the standard can have on
the fitted model life table (Timæus 1990). In contrast, if one uses the method described elsewhere to fit a 2-parameter logit model life table to a pair of recent indirect estimates of child and
adult mortality that refer to about the same date but only measure mortality over a limited range of ages (Brass 1975, 1985), for example q(2) and [10]p[25], one frequently obtains extreme values of
β that produce implausible fitted models.
Step 6: Examine the resulting fitted values of α
The penultimate step is to examine the alpha plot that results from the iterative fitting procedure, which is presented as the second plot of the alpha plots sheet of the associated workbook. It is
this plot, which presents estimates of α that have been adjusted for the level and trend in β, that provides a check on the assumption that α has followed a linear trend. Moreover, if the standard to
which the data have been fitted is appropriate, the series of estimates of childhood and adult mortality should lie close to each other in this plot.
Step 7: Production of a fitted life table
Once the best fitting linear time trends in α and β have been identified by the iterative fitting process, fitted values of α and β for the date for which a fitted life table is required, D, are
calculated as follows:
$α*=Z(α)+D.S(α)β*=Z(β)+D.S(β) .$
The abridged fitted life table is derived from these values of α^* and β^* and the standard life table by means of the formula
$l*(x)=11+exp(2(α*+β*.Ys(x))) .$
Worked example
The worked example presented here uses data on the female population from the Dominican Republic. The indirect estimates for girls were made from the data on children ever-born and surviving obtained
by a DHS conducted in 2002. The indirect estimates for adult women were made from the reports on the survival of mothers from the census conducted in the same year. The input data are presented in
Table 2.
Table 2 Input data for combining child and adult mortality estimates, Dominican Republic
Child mortality (2002 DHS) Adult mortality (2002 Census)
x q(x) Date n [n]p[25] Date
1 0.0338 2001.71 10 0.9858 1999.23
2 0.0429 2000.24 15 0.9801 1997.07
3 0.0355 1998.48 20 0.9680 1995.13
5 0.0467 1996.43 25 0.9479 1993.43
10 0.0619 1994.16 30 0.9214 1992.02
15 0.0710 1991.52 35 0.8872 1991.00
20 0.0799 1987.99 40 0.8373 1990.51
Step 1: Identify the date to which the desired life table should apply
In the case of the data from the Dominican Republic, the associated spreadsheet permits a life table to be derived for dates lying between the earlier of 1987.99-3 and 1990.51-3 and the earlier of
2001.71+3 and 1999.23+3, which is to say dates between 1984.99 and 2002.23.
In this example, we derive a life table for the Dominican Republic for mid-1997, i.e. 1997.5.
Step 2: Select a standard to be used to derive the fitted life table
Given the geographical source of the data, it is reasonable to assume (at least initially) that the mortality of women in the Dominican Republic follows the age pattern described by the UN Latin
American female standard. The logits of the chosen standard life table are presented in Table 3.
Table 3 Logits of the UN Latin American female life table with a life expectancy of 60 years
Age (x) Y^s(x) = logit(l(x))
1 -1.2375
2 -1.1006
3 -1.0398
4 -1.0046
5 -0.9815
10 -0.9304
15 -0.9054
20 -0.8735
25 -0.8313
30 -0.7828
35 -0.7285
40 -0.6670
45 -0.6005
50 -0.5248
55 -0.4356
60 -0.3230
65 -0.1795
Note that, if you change the family of life tables from “UN” to “Princeton” or vice versa in the associated workbook, you must force the workbook to recalculate the output by changing the
“Recalculate” cell (B3 on the Method sheet) from “True” to “False” and back to “True”. Failure to do this will produce an error.
Step 3: Plot values of α (assuming β=1) derived from the mortality estimates against their time locations
Using the data from the Dominican Republic in Table 2 and a UN Latin American life table for a standard, the value of α for child mortality when x = 3 is derived as follows:
$αchild=−12ln(1−q(3)q(3))−logit(ls(3))=−12ln(1−0.03550.0355)+1.0398=−0.6112 .$
This value of α has a time location of 1998.48, as indicated in Table 1. The values of α for the other estimates of child mortality, together with their time locations are derived similarly.
Using the data on adult mortality in Table 2 and the same standard, the estimate of the adult α when n is 25 is given by
This value of α has a time location of 1993.43. The values of α for the other estimates of adult mortality, together with their time locations are derived similarly.
A summary of these estimates of α and their time locations are presented in Table 4.
Table 4 Estimates of α and the time location of the estimates, Dominican Republic, 2002
Children Adults
Original index α Time location Original index α Time location
q(1) -0.4389 2001.71 [ ] [10]p[25] -0.5176 1999.23
q(2) -0.4519 2000.24 [ ] [15]p[25] -0.6183 1997.07
q(3) -0.6112 1998.48 [ ] [20]p[25] -0.5779 1995.13
q(5) -0.5266 1996.43 [ ] [25]p[25] -0.5021 1993.43
q(10) -0.4288 1994.16 [ ] [30]p[25] -0.4567 1992.02
q(15) -0.3803 1991.52 [ ] [35]p[25] -0.4463 1991.00
q(20) -0.3483 1987.99 [ ] [40]p[25] -0.4436 1990.51
When all the estimates of α based on data on both children and adults are plotted against their time locations, the alpha plot shown in Figure 1 results.
Step 4: Eliminate those points in the alpha plot that are out of line with the general trend
The section on indirect estimation of child mortality explains that the most recent indirect estimate of child mortality, which is based on the reports of women aged 15-19 tends to be biased upward
as teenage mothers are a select group with high mortality because, among other reasons, they tend to come from socially disadvantaged backgrounds. This data point is nearly always ignored when
inferences are made about the trend in child mortality from indirect estimates and this was done in this application.
The most recent estimate of adult mortality based on children aged 5-9 reporting on the survival of their mothers in the one census orphanhood method also underestimates mortality in many
applications. In the Dominican Republic, however it indicates much higher mortality than one would expect given the trend indicated by the other estimates for adult women. This might be the result of
severe underreporting of the ages of children or might indicate that the models involved in the estimation process are inappropriate for this population. Either way it was decided to ignore this
anomalous estimate. Therefore, the most recent estimate in each series was omitted from the fitting of a trend line to the α’s by clearing its respective cell in the alpha plots sheet of the
associated workbook.
The remaining estimates from the orphanhood method are internally consistent and suggest that adult women’s mortality fell rapidly in the Dominican Republic during the 1990s. The child mortality
estimates also suggest that mortality was falling, but the more recent estimates are somewhat inconsistent with each other. The 3^rd and 4^th points, which are based on the reports of mothers aged
25-34 years, indicate that the rate of decline in child mortality accelerated in the second half of the 1990s. However, the 2^nd estimate, which is based on the reports of women aged 20-24, suggests
that it decelerated. In the absence of evidence as to the nature of the errors in the data that have led to these inconsistencies, it was decided to leave all three data points in the analysis.
The final selection of points produces the alpha plot in Figure 2. This plot emphasizes the consistency of the 2^nd to 7^th points for adults and shows that a regression line fitted to the 2^nd to 7^
th points for children not only passes through the middle of the more recent estimates, but fits the three more distant points well.
Note that, in Figure 2, the values of α derived from the estimates of the mortality of adults lie below those derived from the estimates of child mortality and diverge from them over time. This means
that, relative to the UN Latin American standard, adult mortality in the Dominican Republic in the 1990s was low and was falling more rapidly than child mortality. Thus, the β parameter of fitted
model life tables for this population will lie below 1 and will decrease over time.
Step 5: Determine the trend in β by iteration
The spreadsheet iteratively solves for fitted values of both α and β for the desired time point (1997.5). The estimates of α^* and β^* are -0.658 and 0.849 respectively. These estimates implies that
the level of mortality in the Dominican Republic is somewhat lighter than in the UN Latin American standard (α < 0), and that mortality is somewhat heavier at younger ages and lighter at older ages (
β < 1) than in this standard. The estimate of β^* is close enough to 1 not to raise any concerns about the choice of standard made in Step 2.
Step 6: Examine the resulting fitted values of α
The penultimate step is to examine the alpha plot that results from the iterative fitting procedure, which is presented as the second plot of the “alpha plots” sheet of the associated workbook.
Figure 3 shows that there is now a close correspondence between the α’s for children and adults for most of the 1990s. Mortality was falling across the age range, though β dropped from about 0.95 to
0.85 between the early 1990s and mid-1997. This is what one would expect given that we have already observed that adult mortality in the Dominican Republic was falling rapidly at this time in
comparison with the pattern in the family of logit model life tables based on the UN Latin American standard.
The lines for adults and children remain fairly close to each other in 1997, reflecting the fact that the value of β at that time (i.e., 0.85) remained fairly close to its central value of 1. The two
estimates of α for 1997 would only differ greatly if β at this time was very different from 1. If they did differ markedly, it would be advisable to seek out a standard life table that was more
appropriate for the population being studied.
If the two series of estimates followed very different trends and failed to cross over each other, or did so and then diverged rapidly, or if one or both series were highly non-linear, this would
again suggest that the standard being used was inappropriate or, more probably, that one or both series was severely biased by errors in the data, making it impossible to reconcile them with each
Step 7: Production of a fitted life table
The abridged fitted life table is derived from the fitted values of α^* = -0.658 and β^* = 0.849 for the selected date, and the standard life table (presented in Table 3) by means of the formula
$l*(x)=11+exp(2(α*+β*.Ys(x))) .$
The final fitted life table is presented in Table 5. Life expectancy at birth is 76.6 years compared with the United Nations’ estimate for the same quinquennium of 73.1 years (UN Population Division
Table 5 Fitted life table for females, Dominican Republic, mid-1997
Age (x) l(x)
0 1.0000
1 0.9683
2 0.9603
3 0.9561
4 0.9536
5 0.9518
10 0.9476
15 0.9455
20 0.9426
25 0.9386
30 0.9337
35 0.9278
40 0.9204
45 0.9118
50 0.9008
55 0.8865
60 0.8657
65 0.8348
70 0.7863
75 0.7120
80 0.6029
85 0.4452
90 0.2587
95 0.1024
100 0.0242
Detailed description of the method
The associated spreadsheet implements the method by following the steps outlined above. This section provides a detailed description of how the iterative procedure used to derive the final values of
α and β is implemented.
The premise underlying the fitting procedure is that the derived life table should fit the observed data well at ages 15 and 60. The former constraint ensures that child and adolescent mortality is
well matched; the combination of the two ensures that adult mortality over an extended age range (15 to 60) is close to that implied by the adult mortality estimates used to fit the life table.
Fitting procedure
After selecting the data points that will be used (as described in Step 4), the method seeks to find the best fitting linear regression model of the time trend in the estimates of α for children,
conditional on the estimated trend in β, and the best fitting linear regression model of the time trend in the estimates of β for adults, conditional on the estimated trend in α for children.
Starting with the assumption that β = 1, one can calculate an α^child corresponding to each estimate of child mortality using the equation provided in Step 3 of the worked example. Since each
estimate of α^child is associated with its time location (T), one can regress the estimates included in the fitting procedure on time to obtain the slope S(α) and intercept Z(α) of the linear
regression model.
The fitted regression model can then be used to predict a fitted α (α^*) for the times to which the adult mortality estimates refer
$α*=Z(α)+T.S(α) .$
Using these fitted values of α^child, one can estimate Y(15) at these dates
where, in this first iteration, β^* = 1.
Still assuming that β = 1, one can also estimate α^adult from the conditional estimates of adult survivorship that have been included in the fitting procedure using the equation given in Step 3 of
the worked example and use these values of α^adult to calculate corresponding estimates of [45]q[15]. Multiplying the value of l(15) estimated from the data on children by an estimate of [45]p[15]
for the same date estimated from the data on adults gives an unconditional estimate of l(60) and therefore of Y(60):
$Y(60)=−12ln(l(60)1−l(60))=−12ln(l(15).45p151−l(15).45p15) .$
The estimate of l(15) is calculated from Y(15) as
$l(15)=11+exp(2Y(15)) ,$
while that of [45]p[15] is derived from the α’s and β’s fitted to the adult estimates:
where Y^ s(x) represents the logit of l(x) in the standard life table (i.e., with β = 1 and α = 0) and, in this first iteration, β^* = 1.
Having estimated a series of values for Y(60) for the dates to which the adult mortality estimates refer, it is now possible to calculate revised estimates of β for these dates
$β=Y(60)−Y(15)Ys(60)−Ys(15) .$
As these revised β’s each refer to a specific date, they can then be regressed on time (T) to calculate the slope S(β) and intercept Z(β) of a linear regression line that can then be used to predict
a fitted β ( β^* ) for each data point, whether it is for children or adults, from that point’s time location (T)
$β*=Z(β)+T.S(β) .$
At this point, the first iterative cycle has been completed. One can now calculate revised estimates of α^child, that allow for the fact that β has been allowed to differ from 1, with the formula
$αchild=logit(qx)−β*.Ys(x) .$
The revised estimates of α^child are then regressed on time and used, in combination with the fitted estimates of β for the dates to which the adult mortality estimates refer, to calculate revised
estimates of Y(15), [45]q[15] and Y(60) and then to recalculate a second round of revised estimates of β itself. Thus, we now have in place a mechanism that will iteratively calculate the best
fitting regressions of α and β on time, each adjusted for the other.
Derivation of the formula for calculating α for adults
$npx=l(x+n)l(x)=[1+e2(α+βYs(x))1+e2(α+βYs(x+n))]npx+npx.e2(α+βYs(x+n))=1+e2(α+βYs(x))e2α.(npx.e2βYs(x+n) − e2βYs(x)) = 1−npxe2α = 1−npxnpx.e2βYs(x+n) − e2βYs(x)α=12ln(1−npxnpx.e2βYs(x+n) − e2βYs(x))α
=12{ln(1−npx)−ln(npx.e2βYs(x+n) − e2βYs(x))} .$
Brass W. 1975. Methods for Estimating Fertility and Mortality from Limited and Defective Data. Chapel Hill: International Program of Laboratories for Population Statistics.
Brass W. 1985. Advances in Methods for Estimating Fertility and Mortality from Limited and Defective Data. London: London School of Hygiene & Tropical Medicine.
Brass W and EA Bamgboye. 1981. The Time Location of Reports of Survivorship: Estimates for Maternal and Paternal Orphanhood and the Ever-widowed. London: London School of Hygiene & Tropical Medicine.
Feeney G. 1980. "Estimating infant mortality trends from child survivorship data", Population Studies 34(1):109-128. doi: https://dx.doi.org/10.1080/00324728.1980.10412839
Feeney G. 1991. "Child survivorship estimation: Methods and data analysis", Asian and Pacific Population Forum 5(2-3):51-55, 76-87. https://hdl.handle.net/10125/3600.
Timæus IM. 1990. "Advances in the Measurement of Adult Mortality from Data on Orphanhood." Unpublished PhD thesis, London: University of London. https://doi.org/10.17037/PUBS.04653370
UN Population Division. 2013. World Population Prospects: The 2012 Revision. New York: United Nations, Department of Economic and Social Affairs. https://www.un.org/development/desa/pd/sites/
Timæus IM and Moultrie TA
Suggested citation
Timæus IM and Moultrie TA. 2013. Combining indirect estimates of child and adult mortality to produce a life table. In Moultrie TA, Dorrington RE, Hill AG, Hill K, Timæus IM and Zaba B (eds). Tools
for Demographic Estimation. Paris: International Union for the Scientific Study of Population. https://demographicestimation.iussp.org/content/
combining-indirect-estimates-child-and-adult-mortality-produce-life-table. Accessed 2024-11-12. | {"url":"https://demographicestimation.iussp.org/content/combining-indirect-estimates-child-and-adult-mortality-produce-life-table","timestamp":"2024-11-11T23:51:09Z","content_type":"text/html","content_length":"124852","record_id":"<urn:uuid:6082b234-dd39-467e-b38a-a73cf20caf3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00401.warc.gz"} |
Finding Probability - School Games
Finding Probability
Fullscreen Mode
A little About Probability Activity.
What probability is it? So you shouldn’t worry if you don’t. Cette activity is designed to help you understand probability.
Let us learn different types of probability and the calculation method.
Probability is a measure of the Probability that an event will occur expressed as a number between 0 and 1. Probability for an event is determined by number of favorable outcomes divided by total
number of possible outcomes.
Two types of probability exist: experimental probability theoretical probability and theoretical probability. Probability Theoretical is based on mathematical principles and assumes that all outcomes
are equally likely. The probability of rolling a six on a fair die is 1/6 since there is only one favorable outcome out of six possible outcomes. On the other hand Experimental probability based on
actual data involves performing an experiment or observing a real-life situation to determine the probability of an event.
The Probability of a given event can be expressed as a fraction decimal or percentage. 0 means that an event is impossible while 1 means that an event is certain to occur. If The probability of The
event is closer to 1 The more likely it is that The event will occur. In many areas of life Probability is used including business sports and gambling to make informed decisions based on the
likelihood of certain outcomes. | {"url":"https://www.schoolgames.io/edu/finding-probability/0254423/","timestamp":"2024-11-03T08:49:17Z","content_type":"text/html","content_length":"157077","record_id":"<urn:uuid:bef47bd0-ab55-49a0-9bfc-734835d26db4>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00519.warc.gz"} |
Ring-element analysis of layered orthotropic bodies
For the analysis of arbitrarily laminated circular bodies, a displacement-based ring-element is presented. The analysis is performed in a cylindrical coordinate system. The method of analysis
requires the boundary conditions as well as the external forces to be π-periodic. The element formulation accounts for a desired degree of approximation of the displacement field in the direction of
the circumference. This is done by a truncated Fourier expansion of the angular dependence of the displacements in terms of trigonometric functions. Thus the Fourier expansion coefficients are the
unknowns to be determined in the finite element analysis. The element chosen is an eight node isoparametric element of the serendipity family. The Fourier series show very high rate of convergence
for the problems solved. The investigation shows that the computational work is remarkably reduced in relation to that of solutions obtained by traditional 3D elements. A scheme for analytical
integration of the angular dependence of the stiffness matrix is given.
• Programområde 3: Energiressourcer
Dyk ned i forskningsemnerne om 'Ring-element analysis of layered orthotropic bodies'. Sammen danner de et unikt fingeraftryk. | {"url":"https://pub.geus.dk/da/publications/ring-element-analysis-of-layered-orthotropic-bodies","timestamp":"2024-11-03T10:01:50Z","content_type":"text/html","content_length":"53398","record_id":"<urn:uuid:2d6fcd19-6c5e-4796-9072-6b22041f114a>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00379.warc.gz"} |
Refinement associated with recombinant human being chemokine CCL2 within E. coli and its particular
To enlighten AM 095 the relevance of our results, we prove connected medical technology that an ensemble of emitters in a cavity are explained by an effective long-range Hamiltonian. The specific
instance of a disordered molecular wire put in an optical hole is talked about, showing that the DIT and DET regimes could be achieved with advanced experimental setups.The discrepancy between
findings from γ-ray astronomy of the ^Fe/^Al γ-ray flux ratio and recent calculations is an unresolved puzzle in nuclear astrophysics. The stellar β-decay rate of ^Fe is one of the major atomic
concerns impeding us from an exact prediction. The important Gamow-Teller strengths through the low-lying states in ^Fe to the ^Co ground condition tend to be measured for the first time utilising
the exclusive dimension associated with ^Co(t,^He+γ)^Fe charge-exchange effect. The brand new stellar decay rate of ^Fe is a factor of 3.5±1.1 bigger than the currently followed rate at T=1.2GK.
Stellar development calculations show that the ^Fe production yield of an 18 solar mass celebrity is diminished substantially by 40% when using the brand-new rate. Our outcome eliminates one of many
major nuclear concerns into the expected yield of ^Fe and alleviates the current discrepancy regarding the ^Fe/^Al ratio.It was established at the start of the twenty-first century that the important
settled shear stress of small-sized (diameter from 50 nm to 10μm) metallic crystals fabricated from bulk crystals increases significantly with decreasing specimen diameter. Dou and Derby [Scr.
Mater. 61, 524 (2009)SCMAF71359-646210.1016/j.scriptamat.2009.05.012] showed that, the critical shear stresses of small-sized single crystals of varied fcc metals obeyed a universal power law of
specimen size with an exponent of -0.66. In this study, we succeeded in reproducing practically completely the aforementioned universal relation without any flexible parameters, centered on a
deformation procedure controlled because of the procedure of single-ended dislocation resources.We identify the crazy period associated with the Bose-Hubbard Hamiltonian because of the
energy-resolved correlation between spectral features and architectural changes associated with the associated eigenstates as revealed by their particular general fractal proportions. The
eigenvectors are proven to come to be ergodic when you look at the thermodynamic limitation, in the configuration room Fock basis, in which arbitrary matrix theory offers an amazing description of
the typical structure. The distributions for the generalized fractal proportions, nevertheless, are a lot more distinguishable from random matrix theory because the Hilbert room dimension develops.We
report on a demonstration of Ramsey interferometry by three-dimensional movement with a trapped ^Yb^ ion. We used a momentum kick towards the ion in a direction diagonal towards the trap axes to
start three-dimensional movement using a mode-locked pulse laser. The interference sign ended up being analyzed theoretically to demonstrate three-dimensional matter-wave interference. This work
paves the best way to recognizing matter-wave interferometry using trapped ions.We report the existence of stable dissipative light bullets in Kerr cavities. These three-dimensional (3D) localized
structures consist of often an isolated light bullet (LB), bound together, or could happen in clusters forming well-defined 3D habits. They could be seen as fixed states in the guide frame going
utilizing the group velocity of light in the cavity. The sheer number of pounds and their distribution in 3D configurations are decided by the initial circumstances, while their maximum peak energy
remains continual for a fixed worth of the system variables. Their bifurcation drawing permits us to explain this occurrence Genetic resistance as a manifestation of homoclinic snaking for
dissipative light bullets. Nonetheless, if the energy of the inserted beam is increased, pounds lose their particular stability additionally the cavity area displays huge, short-living 3D pulses. The
statistical characterization of pulse amplitude reveals a long tail probability distribution, indicating the incident of severe occasions, usually called rogue waves.Employing impartial large-scale
time-dependent density-matrix renormalization-group simulations, we demonstrate the generation of a charge-current vortex via spin shot in the Rashba system. The spin current is polarized
perpendicular towards the system jet and injected from an attached antiferromagnetic spin sequence. We talk about the transformation between spin and orbital angular momentum in the current vortex
that develops because of this conservation associated with total angular energy and also the spin-orbit communication. This is in comparison to the spin Hall result, when the angular-momentum
conservation is violated. Eventually, we predict the electromagnetic area that accompanies the vortex with regard to possible future experiments.Higher-order topological insulators (HOTIs), a new
horizon of topological levels of matter, host lower-dimensional place or hinge says, providing crucial stepping-stones towards the realization of powerful topological waveguides in greater
dimensions. The nontrivial band topology that offers increase to the part or hinge states is generally enabled by particular crystalline symmetries. As an outcome, higher-order topological boundary
states tend to be associated with specific corners or hinges, lacking the flexibleness of switching and picking. Right here, we report the experimental realization of topologically switchable and
valley-selective place states in a two-dimensional sonic crystal. Such interesting properties tend to be enabled by exploiting the higher-order topology assisted with the valley degree of freedom.
For this specific purpose, we recognize a valley HOTI of second-order topology described as the nontrivial bulk polarization. Interestingly, the hosted corner says are located becoming valley reliant
and therefore enable flexible control and manipulation from the trend localization. Topological turn on or off and valley choice of the place says tend to be directly seen through spatial scanning
for the sound field. | {"url":"https://tgf-beta-inhibitor.com/index.php/refinement-associated-with-recombinant-human-being-chemokine-ccl2-within-e-coli-and-its-particular/","timestamp":"2024-11-06T20:44:38Z","content_type":"text/html","content_length":"18780","record_id":"<urn:uuid:60786e5d-8682-4d93-bec4-734032089aaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00495.warc.gz"} |
External Grid
External Grid
Create Function
For creating a single external grid:
For creating multiple external grids at once:
Component Table Data
Parameter Datatype Value Range Explanation
name string Name of the external grid
junction integer Index of connected junction
p_bar float \(>\) 0 Pressure set point [bar]
t_k float \(>\) 0 Temperature set point [K]
in_service boolean True / False Specifies if the external grid is in service.
Naming conventions:
“p” - for node with fixed pressure
type string Type variable to classify external grids
“t” - for node with fixed temperature
“pt” - for node with fixed pressure and temperature
Physical Model
An external grid is used to denote nodes with fixed values of pressure or temperature, that shall not be solved for anymore. In many cases, an external grid represents a connection to a higher-level
grid (e.g. representing the medium pressure level in a low pressure grid). Please note the type naming convention, stating that “p” means that the pressure is fixed, “t” means that the temperature is
fixed and “pt” means that both values are fixed at the connected junction. For nodes with fixed pressure, the mass flow into or out of the system is not known prior to calculation, but is a result of
the pipeflow calculation.
Also note that there has to be at least one fixed value of pressure for hydraulic calculations and one fixed value for temperature in heat transfer calculations for each separate part of the grid.
This is also checked for in the connectivity check.
Result Table Data
Parameter Datatype Explanation
mdot_kg_per_s float Mass flow at external grid node [kg/s] (negative if leaving the system) | {"url":"https://pandapipes.readthedocs.io/en/v0.9.0/components/ext_grid/ext_grid_component.html","timestamp":"2024-11-01T19:39:34Z","content_type":"text/html","content_length":"15389","record_id":"<urn:uuid:3d43ccca-8da0-4cac-850f-5f5b7b18c019>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00561.warc.gz"} |
Alexis Saurin (IRIF), Linear logic with fixed points and infinitary proofs, from straight threads to bouncing threads
• Oct. 23, 2017, 15:00 - 16:00
Infinitary and circular proofs are commonly used in fixed point logics. Being natural intermediate devices between semantics and traditional finitary proof systems, they are commonly found in
completeness arguments, automated deduction, verification, etc. However, their proof theory is surprisingly under-developed. In particular, little is known about the computational behavior of such
proofs through cut elimination. Taking such aspects into account has unlocked rich developments at the intersection of proof theory and programming language theory. One would hope that extending this
to infinitary calculi would lead, e.g., to a better understanding of recursion and corecursion in programming languages. Structural proof theory is notably based on two fundamental properties of a
proof system: cut elimination and focalization. In this talk, we consider the infinitary proof system μMALL∞ for multiplicative and additive linear logic extended with least and greatest fixed
points, and prove these two key results. We thus establish μMALL∞ as a satisfying computational proof system in itself, rather than just an intermediate device in the study of finitary proof systems.
In the last part of the talk, we will discuss some ongoing work on relaxing the validity condition for infinitary proofs and therefore accepting more proofs. This is motivated by the fact that usual
validity conditions only consider straight threads -- progressing from conclusion to premise. While natural in the cut-free setting, it is quite restrictive in presence of cut. This is joint work
with Baelde, Doumane and Jaber. | {"url":"https://linear-logic.org/en/events/general-meeting-2017/talks/saurin/","timestamp":"2024-11-06T11:33:51Z","content_type":"text/html","content_length":"3896","record_id":"<urn:uuid:c5121eed-9c30-427d-9d7f-d986fe43f6db>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00674.warc.gz"} |
Difference Between Exponent and Power
Power denotes the repeated multiplication of a factor and the number which is raised to that base factor is the exponent. This is the main difference between power and exponent. For example, 3^2 is
the power where 3 is the base and 2 is the exponent. Learn more about exponent and power here.
In mathematics, the little digit placed above and to the right of any number is known as a superscript. Large numbers are difficult to read, compare and operate because of which they are denoted in
the form of small numbers with the help of superscripts. This is done with the help of powers and exponents. Exponents represent the number of times a base number is multiplied. At the same time,
power is different from an exponent and consists of two parts known as the base number and the exponent.
Power and Exponent Definition
Power: In Mathematics, the term ‘power’ defines the raising a base number to the exponent. It denotes that the two basic elements of powers are “base number” and “exponent”. Base Number is defined
as a number which is multiplied by itself, whereas the exponent represents the number of times the base number is multiplied. In short, power is a number expressed using the exponents. It implies the
repeated multiplication of the same factor. Some special terms are used in the case of powers are:
When a number is:
• Squared – power is 2
• cubed – Power is 3
• “to the power of” – used for powers more than 3
Exponent: In mathematics, the exponent is defined as a small number, which is positioned at the up-right of the base number. An exponent can be constants, numbers or any variables. Exponents
represent how many times the base number has to be multiplied itself. Usually,l the large numbers are expressed using the exponents. This process is known as the raising to a power.
We can see exponents terms in many scientific notations to denote the large numbers as the powers of 10. Example: the distance between the earth and the sun is expressed in terms of an exponent
is 1.50 × 10^8 km. Also, there are some important rules while performing some arithmetic operations with exponents. They are:
• x^0 = 1
• (x^m)^n = x^mn
• x^m × y^m = (xy)^m
• x^m ÷ y^m = (x/y) ^m
• x^m × x^n = x^m+n
• x^m ÷ x^n = x^m-n
What are the Differences Between Exponent and Power?
Power Exponent
Refers to the whole expression representing the repeated multiplication of the same number Represents the number of times the base number is used as a factor in
multiplying itself
In \(2^{3}=2\times 2\times 2\), 2 is the base number which is to be multiplied by itself thrice and could be also In \(2^{3}=2\times 2\times 2\), 3 is the exponent which represents the number
called as “two to the power of three” or “two to the third power” of times 2 is to be multiplied by itself
When the numbers are expressed with an exponent, then it is said to be in the exponential form. From the differences between power and exponent provided here, we can say that an exponent is a little
digit placed above at the right of a given number, while the power represents the whole expression, containing the base number and the exponent.
Solved Examples
Q.1: Solve 5^2.5^3
Solution: Given,
Using exponent rule,
5^2.5^3 = 5^2+3
= 5^5
Q.2: Express in exponent form.
(i) 2x2x2x2x2
(ii) 3.3.3.3
(iii) 10.10.10
(i) 2x2x2x2x2 = 2^5
(ii) 3.3.3.3 = 3^4
(iii) 10.10.10 = 10^3
Q.3: Solve 5^5/5^2
Solution: By the law of exponent we know;
x^m/x^n = x^m-n
= 5^5-2
= 5^3
Frequently Asked Questions – FAQs
Are power and exponent same?
In mathematics, power defines a base number raised to the exponent, where base number is the factor which is multiplied by itself and exponent denotes the number of times the same base number is
What is the difference between power and degree?
The power represents a number which is raised to another number (called exponent) whereas degree represents the order of the polynomial.
What are the exponent rules?
The rules of exponents are:
x^0 = 1
(x^m)^n = x^mn
x^m × y^m = (xy)^m
x^m ÷ y^m = (x/y) ^m
x^m × x^n = x^m+n
x^m ÷ x^n = x^m-n
What does 4 raised to the power 3 equals to?
4 raised to the power 3 equals to is written as:
4^3 = 4 x 4 x 4 = 64 | {"url":"https://mathlake.com/Difference-Between-Exponent-and-Power","timestamp":"2024-11-13T07:50:51Z","content_type":"text/html","content_length":"14445","record_id":"<urn:uuid:00818ed9-72dd-4904-ad94-4447ce793a0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00585.warc.gz"} |
AmosWEB is Economics: Encyclonomic WEB*pedia
The change in total factor cost resulting from a change in the quantity of factor input employed by a perfectly competitive firm. Marginal factor cost, abbreviated MFC, indicates how total factor
cost changes with the employment of one more input. It is found by dividing the change in total factor cost by the change in the quantity of input used. Marginal factor cost is compared with
marginal revenue product to identify the profit-maximizing quantity of input to hire.
Marginal factor cost is the extra cost incurred when a perfectly competitive firm buys one more unit of an input. It plays THE key role in the study of factor markets and the profit maximizing
decision of a perfectly competitive firm relative to marginal revenue product. A perfectly competitive firm maximizes profit by equating marginal revenue product, the extra revenue generated from
using an input, with marginal factor cost, the extra cost incurred by using the input. If these two marginals are not equal, then profit can be increased by hiring more or less input.
The relation between marginal factor cost and quantity of input used depends on market structure. For a perfectly competitive firm, marginal factor cost is equal to factor price and average factor
cost, all of which are constant. For a monopsony, monopsonistically competitive, or oligopsony firm, marginal factor cost is greater than average factor cost and factor price, all of which increase
with larger quantities of input. The constant or increasing nature of marginal factor cost is a prime indication of the market control of a firm.
Whichever market structure is involved, marginal factor cost is calculated as the change in total factor cost divided by the change in the quantity of the factor purchased, as illustrated by this
marginal factor cost = change in total factor costchange in factor quantity
If the firm is hiring the factor in a perfectly competitive factor market, then the factor price is fixed or constant and so too is marginal factor cost. If the firm is hiring the factor in an
imperfectly competitive factor market, best illustrated by monopsony, then the factor price increases with larger factor quantities and so too does marginal factor cost.
Perfect competition is a market structure with a large number of small participants (buyers and sellers). The good exchanged in the market is identical, regardless of who sells or who buys.
Participants have perfect knowledge and perfect mobility into and out of the market. These conditions mean perfectly competitive buyers are price takers, they have no market control and must pay the
going market price for all input bought.
Marginal Factor Cost,
Perfect Competition
The table to the right summarizes the marginal factor cost incurred by a hypothetical buyer, Maggie's Macrame Shoppe, for hiring store clerks in a perfectly competitive labor market. Maggie's Macrame
Shoppe is one of thousands of small retail stores in the greater Shady Valley metropolitan area that hires labor with identical skills. As such, Maggie pays the going wage for labor.
The first column in the table is the quantity of workers hired, ranging from 0 to 10 workers. The second column is the price Maggie pays for hiring her workers, which is constant at $10 per worker.
The third column is the total factor cost Maggie incurs for hiring various numbers of workers. If Maggie hires only one worker, then she pays only $10. If she hires five workers, she pays $50. In
each case, total factor cost in column three is calculated as the quantity in the first column multiplied by the price in the second column.
The fourth column then presents the marginal factor cost incurred by Maggie at each level of employment. These numbers are found by dividing the change in total factor cost in the third column by the
change in factor quantity in the first column. For example, when Maggie increases employment from 4 workers to 5 workers, her total factor cost increases from $40 to $50, an increase of $10. As such,
the marginal factor cost of hiring the fifth worker is $10 (= $10/1). Each value in the fourth column is calculated in the same way.
The two key observations about marginal factor cost are:
• First, marginal factor cost is constant. Each level of employment results in the same $10 marginal factor cost.
• Second, marginal factor cost is equal to price (which is also average factor cost) at every level of employment. This results because Maggie is operating in a perfectly competitive labor market.
Marginal Factor Cost Curve,
Perfect Competition
Marginal factor cost is commonly represented by a marginal factor cost curve, such as the one labeled MFC and displayed in the exhibit to the right. This particular marginal factor cost curve is that
for labor hired by Maggie's Macrame Shoppe.
The vertical axis measures marginal factor cost and the horizontal axis measures the quantity of input (workers). Although quantity on this particular graph stops at 10 workers, the nature of perfect
competition indicates it could easily go higher.
First and foremost, the marginal factor cost curve is horizontal at the going factor price of $10. This indicates that if Maggie hires the first worker, then her total factor cost increases by $10.
Alternatively, if she hires the tenth worker, then her total factor cost increases by $10. Should she hire a hundredth worker, then she might move well beyond the graph, but her total factor cost
increases by $10.
The "curve" is actually a "straight line" because Maggie is a price taker in the labor market. She pays $10 for each worker whether she hires 1 worker or 10 workers or 100 workers. Her extra factor
cost of hiring an extra worker is always $10. The constant price is what makes Maggie's marginal factor cost curve a straight line, and which indicates that Maggie has no market control.
<= MARGINAL FACTOR COST, MONOPSONY MARGINAL PHYSICAL PRODUCT =>
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Features – Page 2 – The Aperiodical
We all know mathematicians are the coolest people on the planet. But it turns out that of all the people not on the planet, all of them are in fact either mathematicians, or have mathematical
backgrounds or training. Astronauts – and Russian cosmonauts – are all super mathsy people, and if they weren’t already awesome enough, this really seals the deal for me.
Carnival of Mathematics 145
145th Carnival of Mathematics, hosted here at The Aperiodical.
If you’re not familiar with the Carnival of Mathematics, it’s a monthly blog post, hosted on some kind volunteer’s maths blog, rounding up their favourite mathematical blog posts (and submissions
they’ve received through our form) from the past month, ish. If you think you’d like to host one on your blog, simply drop an email to katie@aperiodical.com and we can find an upcoming month you can
do. On to the Carnival!
“I own more maths books by Martin Gardner than by women, is that bad?”
Today is International Women’s Day, so we’ve taken a moment to think about the woman mathematicians in our lives.
We each have fairly sizeable collections of maths books, which prompted CLP to wonder how many of them are by female authors. A quick scan of our respective bookshelves later, here’s what we found.
The maths of the Grime Cube
five cubes named after him, internet maths phenomenon James Grime has now developed a new Rubik’s cube-style puzzle for internet maths joy merchants Maths Gear. I’ve been slightly involved in the
development process, so I thought I’d share some of the interesting maths behind it.
Another name for a Rubik’s cube is ‘the Magic Cube’ – and Dr James Grime wondered if you could make a Magic Cube which incorporates its 2D friend, the Magic Square.
A more equitable statement of the jealous husbands puzzle
Every time I use the jealous husbands river crossing problem, I prefix it with a waffly apology about its formulation. You’ll see what I mean; here’s a standard statement of the puzzle:
Three married couples want to cross a river in a boat that is capable of holding only two people at a time, with the constraint that no woman can be in the presence of another man unless her
(jealous) husband is also present. How should they cross the river with the least amount of rowing?
I’m planning to use this again next week. It’s a nice puzzle, good for exercises in problem-solving, particularly for Pólya’s “introduce suitable notation”. I wondered if there could be a better way
to formulate the puzzle – one that isn’t so poorly stated in terms of gender equality and sexuality.
Apéryodical: Roger Apéry’s Mathematical Story
This is a guest post by mathematician and maths communicator Ben Sparks.
Roger Apéry: 14th November 1916 – 18th December 1994
100 years ago (on 14^th November) was born a Frenchman called Roger Apéry. He died in 1994, is buried in Paris, and upon his tombstone is the cryptic inscription:
\[ 1 + \frac{1}{8} + \frac{1}{27} +\frac{1}{64} + \cdots \neq \frac{p}{q} \]
The centenary of Roger Apéry’s birth is an appropriate time to unpack something of this mathematical story.
Integer Sequence Reviews: A075771, A032799, A002717
It’s been almost two years since I last sat down with my friend David Cushing and did what God put us on this Earth to do: review integer sequences.
This week I lured David into my office with promises of tasty food and showed him some sequences I’d found. Thanks to (and also in spite of) my Windows 10 laptop, the whole thing was recorded for
your enjoyment. Here it is:
I can only apologise for the terrible quality of the video – I was only planning on using it as a reminder when I did a write-up, but once we’d finished I decided to just upload it to YouTube and be
done with it. | {"url":"https://aperiodical.com/category/main/features/page/2/","timestamp":"2024-11-05T22:24:30Z","content_type":"text/html","content_length":"43790","record_id":"<urn:uuid:59f7d1af-420a-444b-aa59-e6b149dbdc7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00713.warc.gz"} |
In the figure, the circle represents youth, the triangle represents fo
In the figure, the circle represents youth, the triangle represents footballers and the rectangle represents athletes.
Which letter(s) represent(s) athletes among youths who are not footballers?
Step by step video solution for In the figure, the circle represents youth, the triangle represents footballers and the rectangle represents athletes. Which letter(s) represent(s) athletes among
youths who are not footballers? by Reasoning experts to help you in doubts & scoring excellent marks in Class 12 exams.
Updated on:21/07/2023
Knowledge Check
• In the following figure, the triangle represents teachers, the circle represents students and the rectangle represents actors. Which number represents teachers who are also students and actors?
• In the following diagram, the triangle represents doctors, the circle represents players and the rectangle represents singers. Which region represents doctors who are singers but not players?
• In the following figure, rectangle represents Fashion designers, circle represents Equestrians, triangle represents Campers and square represents Golfers. Which set of letters represents
Equestrians who are not fashion designers? | {"url":"https://www.doubtnut.com/qna/649004513","timestamp":"2024-11-09T06:26:42Z","content_type":"text/html","content_length":"190386","record_id":"<urn:uuid:812ffde9-73e2-4d7f-b609-49f1978a7061>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00463.warc.gz"} |
Question - Question about the scale of the universe
How far can we go in our ambition to finding the smallest particle in the universe? And how far can we go in our ambition to know how big universe is
What are your opinions?
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Nov 27, 2019
A smallest thing might not exist in the universe.
Things could just keep getting smaller forever.
If a smallest thing does exist it opens a can of worms about (nothing) being between the smallest things.
Nothing properties= no space, no time and no ruler to the next thing.
Our understanding of reality will be totally wrong.
As far as we are capable of trying more and better, refining the theories and experiments. As long as we are alive.
With conditions:
1. There is the same interest: (scientific, economical, etc.) and the same trend of working on fundamental sustainability with genius/hard workers leaps.
2. Generation Z inherited or neglected ambitions.
'We'll keep on trying till the end of times..'
Our universe is all we could ever know. It grows every day. We know about how big it is. That doesn't mean that is all there is, though. So, can we ever know how "big" all there is... is?
I don't think so. I think that's a hard no for us. Higher dimensional beings could discover it and conveys it to us, maybe?
According to the Theosophical Society, matter is nothing at all, merely whirls, holes, disturbances in the ether that are the ultimate "atom". Before science gets there, it has to realize that there
is this matrix or ether underlying everything. In the beginning there was the Chaos, whatever that means, then there were those tiniest of all whatevers, then who knows what went next, maybe a Big
Bang or maybe not. Coming back to Earth, not everybody thinks that the Michelson-Morley experiment proved that there was no ether that Earth was moving through. Maybe none of the above makes any
sense, or maybe it does, but the space aliens refuse to explain. All they do is keep repeating that we're all just a bunch 'a radicals, worthless rough element, one step away from some sort of really
nasty Armageddon, which is all we deserve.
We can only go as far as our lifespan will allow.
Threescore years and ten and if by strength, fourscore and a bit more.
Mortality is something we have to accept as far as our natural lives.
Did you ever ponder as a child what lies at the end of the universe, and then what is beyond that? I did.
There is probably no limit to how far or how small.
Mankind likes to pigeonhole everything and use science and math to explain what he observes but he struggles with important questions such as what was there before the beginning, before t = 0? where
did we come from? What is infinity? because infinity cannot be contained.
So perhaps infinity is the only true reality. World without end. Only man is bound by time!
"There is probably no limit to how far or how small. " (Terra Austr.)
Others say there's a limit on the latter, the near end, which would be the Larssonian Theory.
That first comment of mine here was hasty, careless & regrettable. I was out in the streets, standing with the tablet next to the windows of a supermarket that has a strong Wi-Fi signal, far away
from my piles of notebooks. Now I'm there again, but with the pertinent data.
Some physicists think that the true, ultimate, structureless, & thus indivisible, atom is inaccessible, & that we will keep finding ever smaller subatomic particles, beyond the quark, others deny
that & believe what Thorbjorn Larsson replied at the UniverseToday website 10 yrs. ago when I said that at a lecture a physicist told us that there could be an infinite sequence of subatomic building
blocks, so that we would never reach the end.
"No, for whatever reason, likely to keep physics regulated from a UV-catastrophe, particles hit the Planck energy limit. That gives, observably, a discrete measure to entropy, and if you go through
the numbers there are only so many particles and so many interactions (fields)."
That was in a discussion following a report titled "Do galaxies recycle their material?"
The UV catastrophe was a theoretical difficulty that physics ran into & forced it to find a way out.
Larsson's scholarly comments were always on a professional level & hard to follow, & making things even worse was that, as he told us, he had never learned English on a formal basis & was learning it
right there, as he read our comments, so what we need here is the presence of resident Space.com physicist Hanneke Weitering, who will please be so kind as to decodify for us that piece of Larssonian
I'm using my real name so that if mein lieber Herr Professor Doktor Larsson ever comes around he'll remember the few discussions we had when we were somewhat younger & nattier.
That's not anything like how asteroids are bein
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... asteroids are being treated. An asteroid was given the name of the creator of the said website, a Mr. Fraser Cain. Several of them have the names of people on astronomy magazine staffs, but it's
nice to have one named after Anne Frank, if you've read her diary.
This matter of naming space rocks after people who are still alive is outrageously self-serving. In the U.S. no one who has not already gone back to dust & ashes can appear on a stamp, & a similar
rule should be applied to all celestial bodies. Will Space.com please do something about this? It would have to be a worldwide Campaign To Get Rid of Living People's Names On Asteroids.
Planck units
define the minimal sizes for time, length, etc. These units are the least one can get that is applicable within science. There is no evidence that something smaller might not exist but this seems to
be an area of supposition, not science.
Planck units
define the minimal sizes for time, length, etc. These units are the least one can get that is applicable within science. There is no evidence that something smaller might not exist but this seems
to be an area of supposition, not science.
Nice having someone around with the know-how. If that matter is not proven knowledge yet then maybe there are other schools of thought, besides the no-more-particles & the more-&-more-particles
crowds. We must find out where Hanneke stands concerning this puzzle.
In Quantum Field Theory what we call particles are really the smallest quanta of energy carried within a particular quantum field hence the definition of size becomes murky in fact there is evidence
to support that the very notion of space (or alternatively time though the latter seems less likely)will likely break down at some scale. There are two major experimental results the Bell's
inequality test and the quantum eraser delayed choice experiments which effectively show that either space or time is an emergent phenomenon that is not fundamental.
While we lack a final answer on this there is relevant work studying a type of quantum superconductor system that has been called "artificial atoms" due to the systems having well defined energy
states has shown that at least in complex many particle quantum systems described by a single wavefunction the change in state of a quantum system is continuous which lends weight towards time being
more fundamental than space. (In which case the construct of spacetime in GR emerges as a consequence of time acting as a generator of space through some mechanism)
What needs to be recognized is that at the quantum scale the notion of size at best becomes a matter of the wavelength of the local quantum wavefunction rather than any sort of physical object thus
the question you ask regarding finding the "smallest" particle becomes poorly defined In particular the Heisenberg uncertainty principal places upper limits on our ability to probe space at least
without paying in an uncertainty of momentum. The high uncertainty has the consequence that probing smaller and smaller length scales requires proportionally larger amounts of energy and thus
according to our current theories which we know are incomplete at some point the gravity of the system becomes important and in the conventional picture drives the formation of a micro black hole.
This is probably incomplete but without a theory of quantum gravity we can not say what happens below the plank length threshold or even if anything exists below that unit of length.
To go smaller we need a framework for quantum gravity there are a few prospective candidates but all of them have problems unifying General Relativity(GR) and Quantum Field Theory(QFT) particularly
in the form of renormalization terms which blow up to infinity.
String Theory is one such candidate though it rests on another model called Supersymmetry which is on increasingly shaky ground having failed in virtually all of its theoretical predictions with
remaining model variations having to make increasingly convoluted "corrections" that shift these predicted particles to higher and higher energies. Then there is also the issue string theory has with
requiring additional spatial dimensions that hasn't been resolved. Were it to be validated as a framework it predicts the existence of oscillating "strings" which exist below the plank scale and
generate familiar matter as various wave oscillations of these strings. In this model these strings thus would be the smallest unit.
Its main competitor Quantum Loop Gravity has similar issues having failed on a few potential predictions that required gravitational waves to travel slightly slower than light I don't know for sure
if that flaw has been worked around entirely but it doesn't seem to be completely dead as a concept.
There is an interesting prospective with Wolfram's physics project and their underlying rules for which a Turing machine must obey when operating on some existing network Notably it reproduces a
generalized for of the Einstein field equations where space is a type of emergent property that arises from network connectivity and the ability for updates to propagate through the network. It also
introduces a few other types of space that arise points towards other types of space based properties such as the ordering of updates driving entanglement and branchial space which forms the
framework for quantum mechanics when viewed as a projection. The last of these "spaces" is rulial space the space where the algorithm gets to vary. Point is again this predicts sub plank length
structure though its hard to say what and how those precise model rule dependent features would look.
In summary we don't know and future progress depends on the active theoretical development of quantum gravity.
In a likely vain attempt to convert some points within your nice post into
, I'll at least give it a try.
There are two major experimental results the Bell's inequality test and the quantum eraser delayed choice experiments which effectively show that either space or time is an emergent phenomenon
that is not fundamental.
Is this "emergence" what I read about where the sum of the parts may not equal the whole?
... which lends weight towards time being more fundamental than space. (In which case the construct of spacetime in GR emerges as a consequence of time acting as a generator of space through some
That's interesting. Is GR applicable only when the sum of the parts do equal the whole? [Assumes my prior statement is correct.]
...To go smaller [than the Plank scale] we need a framework for quantum gravity there are a few prospective candidates but all of them have problems unifying General Relativity(GR) and Quantum
Field Theory(QFT) particularly in the form of renormalization terms which blow up to infinity.
Is there a chance any such model could be experimentally tested directly, or just highly unlikely?
Aug 14, 2020
Small and large are as much a matter of distance as here and there in space and time. Toss out a fishing line. Toss it farther with more line. Farther and more line. Ever farther and ever more line.
What weight was the fishing line itself? What weight is the gain in fishing line itself now? It gains weight speedily the more fishing line is required to toss farther out. Eventually you reach a
point where the fishing line's weight -- you are trying to toss -- is infinity. You've reached a weighty horizon of collapse, to zero, in distance you can toss that totality of fishing line. At your
end the mass of the fishing line is almost indistinguishable from infinitesimal. At the other end the total mass of the fishing line in its total length is almost indistinguishable from infinite. At
that (closed systemic -- the fishing line) end, 'c' and the Planck Big Bang horizon.
You would have to move toward the distant to gain some room for your limited and limiting capability. It wouldn't help, not even if you moved an infinite distance. The collapsed horizon of farthest
possibility would remain exactly the same; would remain constant to you.... the same weight of infinity in the end and totality of fishing line, the same zeroing, then, to any attempt -- and all
capability -- to toss the hook beyond the horizon. A horizon then relatively constant to you. Real it may not be the Universe (U) you travel, but real it will always be to your reel of fishing
line... to your finite, local, relative, universe (u).
It would only be in movement, in travel (could only be in movement, in travel), that you would realize there to be more: For you to realize the possibility -- even the possibility -- of
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IMO there isn't something smaller than quarks (strings aside if that theory is correct). Anyway, even if something smaller existed, we can't discover a particle smaller than the plank lenght, as
already said in post #9.
Talking about the extremely high, I only know that the radius of the observable Universe is about 45 light years...
Talking about the extremely high, I only know that the radius of the observable Universe is about 45 light years...
I'm really sorry for this mistake... I didn't noticed it before.
Universe has a radius 45 BILLION light years.
vincenzosassone: I've made mistakes in posts at times on other sites, and I certainly got corrected and even ridiculed for it.
This was the reason I joined space.com
I like this forum because people aren't so ready to jump on you if you make a mistake.
Something I discovered is how much the observable universe has increased over time as instrumentation and technology improve.
Amazing thing isn't it that we are peeking so far back in time?
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I'm really sorry for this mistake... I didn't noticed it before.
The observable Universe has a radius 45 BILLION light years.
We knew it was only a nit and not a mistake in judgement.
Helio: that's what I love about this site. People are so kind!
"Science begets knowledge, opinion ignorance.
Feb 18, 2020
How far can we go in our ambition to finding the smallest particle in the universe? And how far can we go in our ambition to know how big universe is?
Smallest: You not only have to think about the smallest entities, but those which can be used to measure something very small. Of course, it is not some very small entity which must be use in
measurement. For example, you might use the wavelength of light. Thus, to follow this idea, you could use the smallest wavelength of the electromagnetic spectrum.
Universe. We are limited by what defines the observable Universe. Thus the expansion of the Universe and the speed of light are, together, the limiting factor(s).
Aug 14, 2020
Science is only a tool. I wish people would stop worshipping it as a divinity. I've read again and again historians like Will Durant being amazed how much people of 1000 CE and 2000 CE (approaching
(Will Durant) at the time) mirror each other regarding their religious zealotry. Only in 2000 CE the religion is "Science" and scientists are the priests.... man-gods to some.
Quanta Magazine has a lead article out today on the strong force. I enjoyed it very much. Particularly such words from the professional scientists as, "We don't know," and similar words to the same
effect. Such scientists attempt to map territories, they use science as tools like anyone else uses tools to work with in their profession, even their hobby and/or art, whatever. They do not claim
divinity for themselves or their profession, or their views of the Universe at large / small. Like the philosopher-physicist who would not deny the old lady's belief that the universe rested on the
back of a turtle, and that it was turtles all the way up and down (a really amazing similarity to 'planes' in chaos theory), they admit -- like Stephen Hawking -- there are things even in their own
fields they have their opinions on but really do not know for facts. They, these particular scientists at least, do not savage opinions... they had only opinions going in. Knowledge begins as
opinion, even fiction and fantasy. What was fiction and fantasy long ago (Jules Verne, for one opinionated writer), quite often is material fact today.
Discovery and/or invention all too often does not come from any professional scientist, but from an everyday common tool using nobody who simply had an idea... and an opinion.
Some professional scientists went in the other direction, the wrong direction, with their supposed great knowledge... attached to their opinions. Lord Kelvin's opinion that heavier than air ships
could never fly. Von Neumann's opinion that computers would eventually have to occupy block size warehouse-like buildings and more, and only the largest, richest, nations and corporations would ever
be able to afford one.
Science, and scientific method, is strictly a tool and should never be made into, and worshipped for, a religion. Again I wish people would stop pushing it for a religion; an elitist religion at
that. They really do nothing more than stiff it.
Feb 14, 2020
Sept 27, 2021
The measurables (Matter-Energy, ME) are related to E=mc2 (and QED), QCD, Planck limits (h), Gravitation (speed), Action at a distance (Quantum pairing), some in Standard Model and some Beyond SM,
etc. Smallest particles are commonly measured using these (including radiation probes).
What I believe is that these are all being explored very hard through research and experimentation (Cosmology observations, LHC, etc.) to arrive at the physical limits of Dark Matter. DM is what
creates spacetime-structure [Universal Geometry for free particles and radiation space and Affine Geometry where bound structures (nucleus and electron orbits) are found]. DM is where matter-energy
are created from and where these disappear.
I am still not clear about whether Dark Energy - DE is an equivalence to known phenomena or those not yet known hence I defer it from comment.
This puts limits on the smallest particle and DM creates non-ME particles in multidimensional space that eventually manifest as ME particles like those in SM.
Dr Ravi Sharma | {"url":"https://forums.space.com/threads/question-about-the-scale-of-the-universe.46998/","timestamp":"2024-11-05T16:43:00Z","content_type":"text/html","content_length":"295114","record_id":"<urn:uuid:6855636e-d92f-4a85-9bdf-c1b4b5312153>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00242.warc.gz"} |
Measure Features
Use the Measure tool to measure geometry features including a length, an angle, and the dimensions of a bounding box.
Location: all ribbons, Home group
Click the
• Measurements persist after you exit the tool, are listed in the Model Browser, and dynamically update when you make changes to the geometry.
• When you double-click a length measure in the modeling window, it gives you the x, y, z components of the distance vector in the global system.
Measure Box
Use the Measure Box tool to measure the dimensions of the bounding box for a selected part or generated shape.
1. From the Home tools, Measure tool group, click the Measure Box tool.
2. Select the part you want to measure.
The dimensions of the bounding box are displayed in the modeling window, and will remain until you exit the tool.
Measure Distance
Use the Measure Distance tool to measure the distance between two points.
1. From the Home tools, Measure tool group, click the Measure Length tool.
2. Click the model to define the start point and end point.
A line connecting the two points appears and displays a measurement. Dimensions persist after you exit the tool and are dynamically updated as you modify the geometry.
Measure Length
Use the Measure Length tool to measure not only the length of an edge, but also the radius of a circle or an arc, and the diameter or length of a cylinder.
1. From the Home tools, Measure tool group, click the Measure Length tool.
2. Hover over a feature to display the corresponding dimension.
□ Hover over a straight edge to display the length.
□ Hover over a circle or arc to display the radius.
□ Hover over a cylinder to display the diameter.
3. Click the feature to create a measurement. Dimensions persist after you exit the tool and are dynamically updated as you modify the geometry.
Measure Angle
Use the Measure Angle tool to measure an angle defined by three points.
The Measure Angle tool can be used to measure angles in the original model or an optimized shape.
1. From the Home tools, Measure tool group, click the Measure Angle tool.
2. Click the model to define three sequential points.
Note: Selectable points turn red when you hover over them in the modeling window.
A measurement appears and displays the angle between the three points. Point #2 is the vertex of the angle. Dimensions persist after you exit the tool and are dynamically updated as you modify
the geometry. | {"url":"https://2021.help.altair.com/2021.2/polyfoam/en_us/topics/shared/home/measure_tool_st_c.htm","timestamp":"2024-11-03T15:32:25Z","content_type":"application/xhtml+xml","content_length":"51627","record_id":"<urn:uuid:7925259a-995f-4583-b105-ea9ce8aa1a0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00531.warc.gz"} |
using sam - aligned marks??
Jan 13, 2004
Im currently in year 12 and tried using SAM. However when I enter the marks under the "aligned marks" coloumn, am I to enter the raw marks which i think i might get...say 50 out of 100 for 3u maths
or should I enter the scaled mark which for the raw mark of 50 out of 100 would probably be about 70 out of 100?
Also when entering 3u and 4u maths marks into Sam do you do it out of 50 or 100, and if I am doing 4u maths, do i need to also include 2u adv, ext1 and ext2 maths into SAM??
Thanks a lot,
Originally posted by Heks12
am I to enter the raw marks which i think i might get...say 50 out of 100 for 3u maths or should I enter the scaled mark which for the raw mark of 50 out of 100 would probably be about 70 out of
The latter. Though the right term to use is 'aligned mark'.
Originally posted by Heks12
Also when entering 3u and 4u maths marks into Sam do you do it out of 50 or 100, and if I am doing 4u maths, do i need to also include 2u adv, ext1 and ext2 maths into SAM??
You enter it as a mark out of 100. You only need to include Mathematics extension 1 and 2. Mathematics 2U is not required if that is what you meant by '2u adv'.
Oct 12, 2002
Jul 6, 2002
Chudy there are a number of inaccuracies and misleading statements in that explanation... you might want to consider amending it.
Oct 12, 2002
Originally posted by Lazarus
Chudy there are a number of inaccuracies and misleading statements in that explanation... you might want to consider amending it.
no way dude! it's perfect. if there's something dramatically wrong with it you'll have to tell me.
That first paragraph doesn't really have that much to do with alignment.
You may stuff up the HSC big time (but still comparatively well across the state). You may have also stuffed up school big-time. take english - your mark was 50 % at school but you came 1st. You
end up getting an HSC exam mark of 90 % (because you were in the top say...10% of the state). They'll therefore shoot that 50 % straight to 90. (or less, depending on what position you were at
school, and how your mark compared with that of your peers)
That's more to do with the moderation of assessment marks.
Alignment is adjusting the raw HSC mark to fit a set of performance standards to ensure that student achievement is reported in terms of the same standards each year.
How the subject aligns is determined by a set of judges who do the paper and decide what raw mark will correspond to what aligned mark. This process is carried out before and after the students have
sit their examinations.
Which means, say for example in 2U mathematics a student had a RAW mark of 60/100. The judges may decide that a 60/100 would be worth 70/100, and thus the student receives a 70/100 in their HSC
report which is the aligned mark.
Oct 12, 2002
Ahh you pedantics...
ok, so there is a slight difference between alignment and moderation. Never the less, both operate on the same principle.
Bah - can't be bothered changing it lol
Alignment and moderation are 2 different processes. Entirely.
But in this case, the thread creator would need to take account of assessment moderation in order to use the program.
Oct 12, 2002
Originally posted by Ragerunner
Alignment and moderation are 2 different processes. Entirely.
But in this case, the thread creator would need to take account of assessment moderation in order to use the program.
Well for the students themselves, I don't think it's very important to be aware of such a technicality. All they need to know is that there's a lot of fidgeting with the marks going on after you do
the exam and that occurs both with the HSC exam and your school marks. Therefore, it's not necessary to panic in the event you stuff up your exam or your school assessments -- it's all relative.
It's scaling they gotta be afraid of
Yes that is true, but I was referring to your post and why it doesn't have much to do with alignment
Feb 4, 2004
Hey I was just wondering if a lot more subjects will align "up" than scale up? I do chem, phys, eco, 4u maths and advanced english, and I know maths is the only subject likely to scale up, but will
aligning help with the others?
Nobody really knows the exact answers, but in most cases, it appears that most subjects do tend to 'align' up.
Jul 6, 2002
Just remember that 'aligning' has no effect on UAIs.
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st: about dummy variables
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
st: about dummy variables
From Tae Hun Kim <[email protected]>
To [email protected]
Subject st: about dummy variables
Date Wed, 16 Mar 2005 17:20:46 -0600
I would be greatfull if someone helps me to understand dummy variables.
For example,
Y= a+b*X+c*dum+d*(X*dum)+e
where Y : income, X : education(years), dum=dummy variable (dum=1 if
male, dum=0 if female)
If I estimate this equation using OLS, recalulated coefficents are the
same regardless of whether dum=1 if male (0 otherwise) or dum=1 if
female(0 otherwise).
i.e. if I define a dummy variable as dum=1 if male, o otherwise
male(1) : E(Y)= (a+c)+(b+d)X, female(1) : E(Y)=a+bX
if I define a dummy variable as dum=1 if female, o otherwise
male(2) : E(Y)=a+bX, female(2) : E(Y)= (a+c)+(b+d)X
=>the coefficients of male(1) and male(2) equation are the same.
1. But when I estimate this equation using GMM or IV, the coefficients
of male(1) and male(2) equation are not the same. Why?
2. If the coefficient are different, what is the criteria to define a
dummy variable(male=1 or female=1) ?
Thanks in advance
TaeHun Kim
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2005-03/msg00558.html","timestamp":"2024-11-07T03:21:41Z","content_type":"text/html","content_length":"7612","record_id":"<urn:uuid:97354b79-7787-4805-bdc4-0a2a2c014494>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00785.warc.gz"} |
Least Common Multiple of 12 and 10
What is the Least Common Multiple of 2 and 5?
The LCM (least common multiple) of the numbers 2 and 5 is60
How to find the Least Common Multiple of 2 and 5?
Other short text for homepage, lorem ipsum dolor.Bacon ipsum dolor amet chuck spare ribs shoulder turducken pork chop. Ribeye shank pig, brisket shankle leberkas .
Follow the steps below, and let's calculate the LCM of 2 and 5.
Method 1 - Prime factorization
Step 1: Create a list of all the prime factors of the numbers 2 and 5:
Prime factors of 12
The prime factors of 12 are 3 and 2. Prime factorization of 12 in exponential form is:
12 =2^2x3^1
Prime factors of 10
The prime factors of 10 are 5 and 2. Prime factorization of 10 in exponential form is:
10 =2^1x5^1
Step 2: Identify the highest power of each prime number from the above boxes:
Method 2 - List of Multiples
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
LCM(10,12) = 60 | {"url":"https://lcm-gcf.com/lcm-of-12-10","timestamp":"2024-11-06T17:23:37Z","content_type":"text/html","content_length":"19871","record_id":"<urn:uuid:164daf5f-4ea5-4be5-93f1-10ade24eae70>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00149.warc.gz"} |
For example, if the tax rate is 1% and you're financing a home with a loan amount of $,, the mortgage tax would be $2, However, the calculation of. How do you calculate a mortgage buydown? Enter the
number of years of your loan term, the total loan amount, and the interest rate percentage into the. This calculator is intended to help estimate a monthly payment, and understand the amount of
interest you will pay based on your loan amount, interest rate. Find out what your mortgage payment could be, and learn how you can save interest by changing your payment frequency and making
prepayments. The amount you expect to borrow from your financial institution. · Annual interest rate for this mortgage. · The number of years and months over which you will.
It will quickly estimate the monthly payment based on the home price (less downpayment), the loan term and the interest rate. There are also optional fields. Mortgage Calculator ; Home Value: $ ;
Down payment: $ % ; Loan Amount: $ ; Interest Rate: % ; Loan Term: years. This calculator helps you to determine what your adjustable mortgage payments may be. If you buy a home with a loan for $, at
percent your monthly payment on a year loan would be $, and you would pay $, in interest. Mortgage Calculator ; Purchase Price · Down Payment. $ ; Term · Interest Rate. % ; Property Tax · PMI. % ;
Property Insurance · Start Date. Therefore, a loan at 6%, with monthly payments and compounding simply requires using a rate of % per month (6%/12 = %). Unfortunately, mortgages are not. Just fill
out the information below for an estimate of your monthly mortgage payment, including principal, interest, taxes, and insurance. Breakdown; Schedule. For example, a year fixed-rate mortgage would
have 30 years x 12 months = payments. Put the values into the formula: Once you have the monthly interest. Use the RBC Royal Bank mortgage payment calculator to see how mortgage amount, interest
rate, and other factors can affect your payment. This is calculated by first multiplying the $, loan by the % interest rate, then dividing by If the mortgage closes on Jan. 25, you owe $ How to
Calculate Monthly Loan Payments · If your rate is %, divide by 12 to calculate your monthly interest rate. · Calculate the repayment term in.
This calculator uses your maximum PI payment to determine the mortgage amount that you could qualify for. Start interest rates at. The current interest rate you. Lenders multiply your outstanding
balance by your annual interest rate and divide by 12, to determine how much interest you pay each month. The Payment Calculator can determine the monthly payment amount or loan term for a fixed
interest loan. Multiply the factor shown by the number of thousands in your mortgage amount, and the result is your monthly principal and interest payment. For the total cost. Use this free mortgage
calculator to estimate your monthly mortgage payments and annual amortization. A mortgage payment calculator takes into account factors including home price, down payment, loan term and loan interest
rate in order to determine how much. To use our amortization calculator, type in a dollar figure under “Loan amount.” Adjust “Loan term,” “Interest rate” and “Loan start date” to customize the. Use
our mortgage payment calculator to estimate how much your payments could be. Calculate interest rates, amortization & how much home you could afford. To use our amortization calculator, type in a
dollar figure under “Loan amount.” Adjust “Loan term,” “Interest rate” and “Loan start date” to customize the.
Shop around for a lower interest rate. Different lenders offer varying interest rates. A lower rate equals a lower monthly mortgage payment. Lengthen the term. Free mortgage calculator to find
monthly payment, total home ownership cost, and amortization schedule with options for taxes, PMI, HOA, and early payoff. Use this calculator to determine the Annual Percentage Rate (APR) for your
mortgage. Press the report button for a full amortization schedule. Use our mortgage calculator to get an idea of your monthly payment by adjusting the interest rate, down payment, home price and
more. Your monthly payment is $1, under a year fixed-rate mortgage with a % interest rate. This calculation only includes principal and interest but does.
Calculate your monthly home loan payments, estimate how much interest you'll pay over time, and understand the cost of your mortgage insurance, taxes, and. Use the mortgage calculator to get an
estimate of your monthly mortgage payments. Interest Rate. %. Advanced View | Reset. $1, Monthly mortgage payment.
Mccoy Federal Credit Union Refinance Auto Loan | Crypto Fx Invest | {"url":"https://uviya.ru/overview/how-to-figure-out-mortgage-payment-with-interest-rate.php","timestamp":"2024-11-03T01:17:07Z","content_type":"text/html","content_length":"12303","record_id":"<urn:uuid:da2b3b16-5dab-4737-a62c-9bb836223ce2>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00091.warc.gz"} |
how to know if a course is successfull?
Hey guys,
First time here. Im interested in teaching a very specific topic and there´s only one "close enough" course. I think that knowing how many students that course has would be a good proxy of the demand
I might see for my course.
So here´s my question: Each course informs on its presentation page how many reviews it has and how many students it has (this one has about 10.000). Did all those students paid for the course? Is it
fair to assume that that course incomed about $70.000 (10.000 x 50% of $14)?
thanks in advance!
• You can't use enrollments to judge a course's success or money earned. I use the rule-of-thumb to multiply the number of reviews by 10 (normal percentage of reviews per student) and then again by
the amount you expect to earn per student (depends on the source of student). That will give you a fair estimate of how much money a course actually produces. All numbers are gross estimates.
Percent of reviews per student can vary widely.
• Not exactly some of them are referral , some of them came with free promotions however if you are talking about without promocode yes u can take them as exact number
• thanks a lot! very helpful answer
• thanks for your answer, very useful.
• There is high variability as there are students from free coupons that contribute $0 to instructor income, instructor coupon that contribute on average $8-9 per student, Udemy promotion that
brings in around $1-$5 per student, then there is Udemy for Business and Personal Subscription that brings $0.02 per minute consumed.
If the number of students is too high as compared to reviews, for example students is 10x reviews then there are a lot of free coupons but apart from that there is no way of determining.
The only sure way is to go to Marketplace Insights, where you can get idea of income of bestseller course, a general range.
• thanks for taking the time to asnwer!
• Forget the number of students. It means little. A reliable way to figure income is to take the number of reviews and multiply by $11. That will come very close. I'll be more precise. I have
85,872 reviews as of this moment. And I have earned a total of $931.020.26 as of this moment. Divide reviews into earnings at it will give you exactly $10.84.
• Valoro mucho sus orientaciones. Gracias amigos.
• Aconselhor que leia as políticas de preços da Udemy. Não é assim que funciona. Tem princípios bazares na plataforma, que desfazem por completo esta métrica. A Udemy oferta cupons de desconto que
podem zerar seu valor de curso, a Udemy pode não expor seus cursos na vitrine, eles podem ofertar cupons de desconto de até 100%, o que zera seu valor a receber. Portanto, @Lucas190
Lucas190, infelizmente não é assim que funciona. Também entrei bastante empolgado e como sou um velho que atua na docência há várias décadas, pensei que ia ficar rico. Com o tempo, você vai ver
como funciona. Boa sorte. | {"url":"https://community.udemy.com/en/discussion/comment/122269#Comment_122269","timestamp":"2024-11-11T07:16:12Z","content_type":"text/html","content_length":"285415","record_id":"<urn:uuid:c5808b10-8f05-4430-af2b-a3a3428166b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00611.warc.gz"} |
Building a REPL-Based Calculator: Requirements and Specifications
13.1.1 Building a REPL-Based Calculator: Requirements and Specifications
In this section, we will delve into the requirements and specifications for building a REPL-based calculator using Clojure. This project is designed to provide a hands-on experience with Clojure’s
REPL environment while reinforcing fundamental concepts of functional programming. The calculator will support basic arithmetic operations: addition, subtraction, multiplication, and division. This
exercise will not only enhance your understanding of Clojure syntax and functional paradigms but also demonstrate the power of interactive development with the REPL.
Introduction to the REPL-Based Calculator
The REPL (Read-Eval-Print Loop) is a powerful tool in the Clojure ecosystem, providing an interactive programming environment that allows for rapid development and testing of code. By building a
calculator within the REPL, you will gain practical experience in writing and evaluating Clojure expressions, managing state, and implementing functional logic.
Project Overview
The primary goal of this project is to create a simple yet functional calculator that operates within the Clojure REPL. The calculator will be capable of performing the following operations:
• Addition (+): Summing two or more numbers.
• Subtraction (-): Calculating the difference between two numbers.
• Multiplication (*): Computing the product of two or more numbers.
• Division (/): Dividing one number by another, with considerations for division by zero.
Functional Requirements
1. User Input and Interaction: The calculator should accept user input in the form of mathematical expressions. Users will input expressions directly into the REPL, and the calculator will evaluate
and return the result.
2. Basic Arithmetic Operations: Implement functions for each of the arithmetic operations. These functions should be pure, meaning they do not produce side effects and return consistent results for
the same inputs.
3. Error Handling: The calculator should gracefully handle errors such as division by zero and invalid input. Error messages should be clear and informative, guiding the user to correct their input.
4. Interactive Feedback: Upon evaluating an expression, the calculator should immediately display the result or an error message. This feedback loop is crucial for the interactive nature of the
5. Extensibility: The design should allow for easy addition of new operations or features in the future. This might include additional mathematical functions or support for more complex expressions.
Non-Functional Requirements
1. Performance: The calculator should efficiently handle typical arithmetic operations without noticeable delay. While performance is not a critical concern for this simple calculator, it should not
degrade significantly with increased input size.
2. Usability: The user interface, though text-based, should be intuitive and straightforward. Users should be able to enter expressions naturally and receive immediate feedback.
3. Reliability: The calculator should consistently produce correct results and handle errors gracefully. Reliability is key to ensuring a positive user experience.
4. Maintainability: The codebase should be clean, well-documented, and easy to understand. This will facilitate future enhancements and maintenance.
Technical Specifications
1. Clojure Version: The calculator will be developed using Clojure version 1.10 or later. This ensures compatibility with the latest features and improvements in the language.
2. Development Environment: The project will be developed and tested within the Clojure REPL, utilizing a suitable editor or IDE such as Emacs with CIDER, IntelliJ IDEA with Cursive, or VSCode with
3. Function Definitions: Each arithmetic operation will be implemented as a separate function. These functions will take numeric arguments and return the result of the operation.
4. Expression Parsing: User input will be parsed into a format that can be evaluated by the calculator functions. This may involve tokenizing the input string and converting it into a sequence of
operations and operands.
5. Error Handling Mechanisms: Implement error handling using Clojure’s try, catch, and throw constructs. This will allow the calculator to manage exceptions and provide informative error messages.
Implementation Plan
1. Set Up the Development Environment: Ensure that Clojure and the chosen editor or IDE are properly installed and configured. Familiarize yourself with the REPL workflow and basic Clojure syntax.
2. Define Arithmetic Functions: Implement pure functions for addition, subtraction, multiplication, and division. Each function should accept two or more numeric arguments and return the result of
the operation.
3. Implement User Input Handling: Develop a mechanism for reading and parsing user input. This will involve converting input strings into a format that can be processed by the arithmetic functions.
4. Integrate Error Handling: Add error handling logic to manage invalid input and division by zero. Ensure that error messages are clear and helpful.
5. Test and Validate: Thoroughly test the calculator with a variety of inputs to ensure accuracy and reliability. Use the REPL to interactively debug and refine the implementation.
6. Document the Code: Write comprehensive documentation for the codebase, including comments and usage instructions. This will aid in future maintenance and extension of the calculator.
Code Examples and Snippets
To illustrate the implementation of the REPL-based calculator, consider the following code snippets:
Addition Function
(defn add
"Returns the sum of all arguments."
[& numbers]
(reduce + numbers))
Subtraction Function
(defn subtract
"Subtracts all subsequent numbers from the first number."
[first & rest]
(reduce - first rest))
Multiplication Function
(defn multiply
"Returns the product of all arguments."
[& numbers]
(reduce * numbers))
Division Function
(defn divide
"Divides the first number by all subsequent numbers. Throws an error if division by zero is attempted."
[first & rest]
(reduce / first rest)
(catch ArithmeticException e
(println "Error: Division by zero is not allowed."))))
Best Practices and Optimization Tips
• Use Pure Functions: Ensure that all arithmetic functions are pure, enhancing testability and reliability.
• Leverage Clojure’s REPL: Use the REPL for iterative development and testing, taking advantage of its interactive capabilities.
• Handle Errors Gracefully: Implement robust error handling to improve user experience and prevent crashes.
• Keep the Codebase Clean: Write clear, concise code with appropriate comments and documentation.
Common Pitfalls
• Ignoring Edge Cases: Be mindful of edge cases such as division by zero and invalid input. Implement comprehensive error handling to address these scenarios.
• Overcomplicating the Design: Keep the design simple and focused on the core functionality. Avoid unnecessary complexity that could hinder maintainability.
• Neglecting Testing: Thoroughly test the calculator with a variety of inputs to ensure accuracy and robustness.
Building a REPL-based calculator in Clojure is an excellent way to deepen your understanding of functional programming and interactive development. By following the outlined requirements and
specifications, you will create a robust and extensible calculator that demonstrates the power and flexibility of Clojure. This project serves as a foundation for further exploration and
experimentation with Clojure’s rich ecosystem and functional paradigms.
Quiz Time!
### What is the primary goal of the REPL-based calculator project? - [x] To create a simple yet functional calculator that operates within the Clojure REPL. - [ ] To develop a graphical user
interface for a calculator. - [ ] To implement advanced mathematical functions like trigonometry. - [ ] To integrate the calculator with a database. > **Explanation:** The primary goal is to create a
simple functional calculator that operates within the Clojure REPL, focusing on basic arithmetic operations. ### Which arithmetic operations are supported by the REPL-based calculator? - [x]
Addition, Subtraction, Multiplication, Division - [ ] Addition, Subtraction, Exponentiation - [ ] Multiplication, Division, Modulus - [ ] Subtraction, Division, Square Root > **Explanation:** The
calculator supports basic arithmetic operations: addition, subtraction, multiplication, and division. ### What is a key non-functional requirement for the calculator? - [x] Usability - [ ] Support
for complex numbers - [ ] Integration with web services - [ ] Multi-language support > **Explanation:** Usability is a key non-functional requirement, ensuring the calculator is intuitive and
straightforward to use. ### How should the calculator handle division by zero? - [x] Gracefully handle the error and provide a clear message. - [ ] Ignore the error and continue processing. - [ ]
Terminate the program immediately. - [ ] Log the error without notifying the user. > **Explanation:** The calculator should gracefully handle division by zero and provide a clear error message to the
user. ### What is the purpose of using pure functions in the calculator? - [x] To enhance testability and reliability. - [ ] To improve performance. - [ ] To enable multi-threading. - [ ] To support
graphical output. > **Explanation:** Pure functions enhance testability and reliability by ensuring consistent results without side effects. ### Which Clojure construct is used for error handling in
the calculator? - [x] `try`, `catch`, `throw` - [ ] `if`, `else`, `cond` - [ ] `loop`, `recur` - [ ] `def`, `let` > **Explanation:** Clojure's `try`, `catch`, and `throw` constructs are used for
error handling. ### What is a common pitfall when developing the calculator? - [x] Ignoring edge cases like division by zero. - [ ] Overusing graphical elements. - [ ] Implementing too many features.
- [ ] Using a database for storing results. > **Explanation:** Ignoring edge cases like division by zero is a common pitfall that can lead to errors. ### How can the calculator be extended in the
future? - [x] By adding new operations or features. - [ ] By integrating with a web server. - [ ] By converting it to a mobile app. - [ ] By using a different programming language. > **Explanation:**
The calculator can be extended by adding new operations or features, enhancing its functionality. ### What is the role of the REPL in this project? - [x] To provide an interactive programming
environment for rapid development and testing. - [ ] To compile the Clojure code into machine language. - [ ] To manage database connections. - [ ] To render graphical user interfaces. >
**Explanation:** The REPL provides an interactive programming environment for rapid development and testing. ### True or False: The calculator's codebase should be clean and well-documented. - [x]
True - [ ] False > **Explanation:** A clean and well-documented codebase is essential for maintainability and future enhancements. | {"url":"https://clojureforjava.com/1/13/1/1/","timestamp":"2024-11-04T02:41:50Z","content_type":"text/html","content_length":"183159","record_id":"<urn:uuid:09633ff3-fb4a-4ec4-b0c1-57684fc281b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00286.warc.gz"} |
Digital signal processing
Digital signal processing Question Paper - Dec 16 - Electronics Engineering (Semester 7) - Mumbai University (MU)
written 5.2 years ago by • modified 2.9 years ago
1 Answer
1.a. Differentiate between Butterworth and chebyshev filter
(5 marks) 00
1.b. Explain the concept of pipelining in DSP processor
(5 marks) 00
1.c.Explain frequency warping effect in designing IIR filter using BLT method.
(5 marks) 00
1.d. Explain Quantization effect in computation of DFT
(5 marks) 00
1.e. State the relationship between DFS, DFT and Z Transform
(5 marks) 00
2.a.Compute IDFT of the following sequence using inverse FFT algorithm.
$ x(k)=\{3,0,3,0,3,0,3,0\} $
(10 marks) 00
2.b. Prove the Parseval's theorem for the sequence $x(n)=\{2,4,2,4\}$
(5 marks) 00
2.c. Find the circular convolution of the sequences $x(n)=\{1,2,1,2\}$ and $h(n)=\{4,0,4,0\}$
(5 marks) 00
3.a. Design an analog Butterworth filter that has $-2 \mathrm{dB}$ passband attenuation at frequency of 20 rad/sec and atleast $-10 \mathrm{dB}$ stopband attenuation at $30 \mathrm{rad} / \mathrm
{sec} .$
(10 marks) 00
3.b. Convert the following filters with system functions
(i) $H(s)=\frac{1}{(s+2)(s+0.6)}$
(ii) $H(s)=\frac{(s+0.1)}{(s+0.1)^{2}+9}$
into a digital filter by means of impulse invariant and BLT method.
(10 marks) 00
4.a.Explain the concept of linear phase in FIR filter. prove the following statement 'a filter is said to have linear phase response if its phase response is $\theta(w)=-\alpha w$
(10 marks) 00
4.b. Design a low pass FIR filter with 7 coefficients for the following specifications passband frequency $=0.25 \mathrm{khz}$ and sampling frequency $=1 \mathrm{khz} .$ Use hamming window in
(10 marks) 00
5.a. Draw neat architecture of TMS 320C67xx DSP processor and explain each block.
(10 marks) 00
5.b.Explain addressing modes of DSP processor with example.
(10 marks) 00
(7 marks) 00
6.b. Application of DSP processor to Radar signal processing
(7 marks) 00
6.c.Limit cycle oscillations
(6 marks) 00
6.d. Product quantization error and input error
(6 marks) 00
written 5.2 years ago by
Digital signal processing - Dec 16
Electronics Engineering (Semester 7)
Total marks: 80
Total time: 3 Hours INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
Q.1] Answer any four questions
Q6] Write short notes on (any three) | {"url":"https://www.ques10.com/p/49555/digital-signal-processing-question-paper-dec-16-el/","timestamp":"2024-11-03T09:37:56Z","content_type":"text/html","content_length":"30651","record_id":"<urn:uuid:9bc378dc-6819-44d9-8fcc-04423d9e3e57>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00551.warc.gz"} |
Non-parametric Bayesian density estimation for biological sequence space with applications to pre-mRNA splicing and the karyotypic diversity of human cancer
Chen, Wei-Chia, Zhou, Juannan, Sheltzer, Jason, Kinney, Justin, McCandlish, David (December 2020) Non-parametric Bayesian density estimation for biological sequence space with applications to
pre-mRNA splicing and the karyotypic diversity of human cancer. BioRxiv. (Unpublished)
Density estimation in sequence space is a fundamental problem in machine learning that is of great importance in computational biology. Due to the discrete nature and large dimensionality of sequence
space, how best to estimate such probability distributions from a sample of observed sequences remains unclear. One common strategy for addressing this problem is to estimate the probability
distribution using maximum entropy, i.e. calculating point estimates for some set of correlations based on the observed sequences and predicting the probability distribution that is as uniform as
possible while still matching these point estimates. Building on recent advances in Bayesian field-theoretic density estimation, we present a generalization of this maximum entropy approach that
provides greater expressivity in regions of sequence space where data is plentiful while still maintaining a conservative maximum entropy char-acter in regions of sequence space where data is sparse
or absent. In particular, we define a family of priors for probability distributions over sequence space with a single hyper-parameter that controls the expected magnitude of higher-order
correlations. This family of priors then results in a corresponding one-dimensional family of maximum a posteriori estimates that interpolate smoothly between the maximum entropy estimate and the
observed sample frequencies. To demonstrate the power of this method, we use it to explore the high-dimensional geometry of the distribution of 5′ splice sites found in the human genome and to
understand the accumulation of chromosomal abnormalities during cancer progression.
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Administrator's edit/view item | {"url":"https://repository.cshl.edu/id/eprint/40603/","timestamp":"2024-11-04T11:44:31Z","content_type":"application/xhtml+xml","content_length":"27463","record_id":"<urn:uuid:d1207e48-0667-4f72-9ee5-ca645acd5142>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00232.warc.gz"} |
Floyd's Algorithm
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What is Floyd's Algorithm
Floyd's Algorithm, also known as Floyd-Warshall Algorithm or Roy-Floyd Algorithm, is a popular approach for finding the shortest path between all pairs of vertices in a weighted graph. It is a
dynamic programming technique that effectively handles negative edge weights and detects negative cycles. Invented by Robert Floyd, a prominent computer scientist, this algorithm is designed for
graphs represented using adjacency matrices.
A weighted graph is a collection of vertices and edges where each edge has a weight or cost associated with it. A shortest path between two vertices is the path with the lowest total weight or cost.
Consider a graph with vertices A, B and C, where the weight of edge (A, B) is 3, edge (B, C) is 2 and edge (A, C) is 5. The shortest path from A to C is A -> B -> C, with a total cost of 3+2=5.
Importance of Floyd's Algorithm in Decision Mathematics
Floyd's Algorithm plays a significant role in various fields of decision mathematics, which is a branch of applied mathematics that deals with the construction and analysis of methods to make
informed decisions. Some of these applications include network routing, operations research, game theory and computer graphics. Let's take a closer look at some of the aspects that make Floyd's
Algorithm vital in decision mathematics:
• Optimal routing: The algorithm is widely used in network routing, as it finds the shortest path between all pairs of nodes, which is important for effective data transfer between devices in
distributed systems or computer networks.
• Efficient resource allocation: In operations research, the algorithm helps solve resource allocation problems by finding the most cost-effective path among all possible paths which minimizes the
overall cost thereby improving efficiency in various applications such as supply chain management and transportation logistics.
• Game theory: Floyd's Algorithm can be applied to game theory, particularly in two-player zero-sum games, where it can be used to find the optimal strategies for each player and analyse the game's
• Computational geometry: In computer graphics and computational geometry, the algorithm can help in simplifying polygonal models by approximating the optimal solution to the mesh simplification
There is always a trade-off between accuracy and algorithm efficiency. While Floyd's Algorithm is very useful for small graphs, its time complexity, of \(O(n^3)\) (where n is the number of vertices),
can lead to slow run times for very large or dense graphs. In these cases, alternative algorithms such as Dijkstra's or Bellman-Ford might be more efficient, depending on the specific problem
Floyd's Algorithm Example
To implement Floyd's Algorithm, follow these steps:
1. Create an adjacency matrix representation of the given weighted graph, where the value of each cell (i, j) represents the weight of the edge connecting vertex i to vertex j. If there is no edge
between the vertices, assume a very large positive value (infinity) as the weight.
2. Initialize a distance matrix with the same dimensions as the adjacency matrix. This matrix will store the shortest distances between all pairs of vertices. Copy the values from the adjacency
matrix to the distance matrix.
3. Loop through all vertices as intermediate points (k) to compare and update the shortest distances between a pair of vertices (i, j).
4. If the sum of distances from vertex i to k and k to j is less than the initial distance from i to j, update the cell (i, j) in the distance matrix with the new value.
5. Repeat step 3 for all vertices until the distance matrix converges, meaning no more updates are necessary.
6. The final distance matrix contains the shortest distance between any pair of vertices in the graph.
Consider the following adjacency matrix representing a weighted graph with 4 vertices:
0 5 ∞ 10
∞ 0 3 ∞
∞ ∞ 0 1
∞ ∞ ∞ 0
Here is a step-by-step illustration of Floyd's Algorithm using the example:
1. Initialize distance matrix: 0 5 ∞ 10∞ 0 3 ∞∞ ∞ 0 1∞ ∞ ∞ 0
2. Iterate over k = 1: 0 5 ∞ 10∞ 0 3 ∞∞ ∞ 0 1∞ ∞ ∞ 0
3. Iterate over k = 2: 0 5 8 10∞ 0 3 ∞∞ ∞ 0 1∞ ∞ ∞ 0
4. Iterate over k = 3: 0 5 8 9∞ 0 3 4∞ ∞ 0 1∞ ∞ ∞ 0
5. Iterate over k = 4: 0 5 8 9∞ 0 3 4∞ ∞ 0 1∞ ∞ ∞ 0
Solving Real-life Problems with Floyd's Algorithm Example
Floyd's Algorithm has numerous applications in solving practical problems that involve finding the shortest path between points. Here are a few real-life examples:
Transportation logistics: A company plans to ship goods between multiple cities. Using Floyd's Algorithm, they can determine the shortest path between any two cities, taking into account varying road
conditions, distances, and travel costs. This helps the company minimise transportation expenses and improve delivery time.
Social media analytics: In a social network, users are connected through a series of relationships. Floyd's Algorithm can be used to compute the shortest path between any two members in the network,
helping to analyse connectivity, discover hidden connections, and improve recommendations for new connections.
Micro-electronics: In micro-electronic circuits, the components need to be interconnected using the shortest possible paths to minimise signal loss and increase circuit performance. Designers can use
Floyd's Algorithm to identify the optimal routing for connections, based on constraints like wire resistance, capacitance, and inductance.
By understanding and leveraging the strengths of Floyd's Algorithm, you can efficiently solve complex decision problems that involve finding the shortest path in weighted graphs, enhancing your
ability to make well-informed choices and improving your performance in real-world applications.
Floyd's Algorithm Application
In further mathematics, Floyd's Algorithm has a variety of important use cases that demonstrate its versatility in solving complex problems. Some of the key areas where it is applied include:
• Graph theory: In graph theory, the algorithm finds the shortest path between all pairs of vertices in a weighted graph, a common and crucial task in many graph-related problems.
• Matrix analysis: As the algorithm exploits the adjacency matrix representation, it has applications in matrix analysis to compute the transitive closure, a broader concept capturing reachability
between graph vertices.
• Linear programming: Given its dynamic programming nature, Floyd's Algorithm is frequently employed to solve mathematical optimisation problems along with other linear programming techniques such
as simplex method and duality theory.
• Partial differential equations: By solving for the shortest path in discretized domains, the algorithm can be adapted to solve partial differential equations, contributing to the study of heat
transfer, fluid dynamics and more.
In addition to these use cases, Floyd's Algorithm can be incorporated into novel hybrid approaches, combining its strengths with other mathematical techniques to devise innovative problem-solving
Practical Applications of Floyd's Algorithm in Decision Mathematics
Decision mathematics is a branch of applied mathematics that encompasses various methods employed to make intelligent decisions. Floyd's Algorithm is an integral component across numerous practical
applications within this context:
• Urban planning: In designing and analysing transport systems, planners can utilise the algorithm to locate the optimal road connections among cities, minimizing travel time, congestion and
ensuring overall suitability.
• Travel industry: Airlines, railways, and car rental services may benefit from the algorithm as it aids in identifying the most efficient routes, fuel consumption, and cost savings, offering an
advantageous position in the competitive market.
• Telecommunication: The algorithm is essential in the design and management of telecommunication networks, optimising data transfer and routing paths, reducing latency and increasing overall
network performance.
• Finance: In portfolio management and risk analysis, financial institutions adapt Floyd's Algorithm to assess asset correlations and vulnerability to potential market shocks, enhancing investment
Floyd's Algorithm is an indispensable tool not only in theoretical mathematics but also in countless real-world applications where finding an optimal solution to complex problems is essential. As
decision-makers and mathematicians face ever-growing challenges, the relevance and significance of this versatile algorithm will only continue to expand.
Floyd's Algorithm vs. Dijkstra's Algorithm
Both Floyd's Algorithm and Dijkstra's Algorithm are commonly used for solving shortest path problems in weighted graphs. While they share similarities, they differ in significant ways as well.
Here is a comparison of the key aspects of both algorithms:
1. Purpose: Floyd's Algorithm is designed to find the shortest path between all pairs of vertices in a graph, whereas Dijkstra's Algorithm focuses on the shortest path from a single source vertex to
all other vertices in the graph.
2. Dynamic Programming: Floyd's Algorithm uses a dynamic programming approach, while Dijkstra's Algorithm utilises a greedy method.
3. Negative Weights: Floyd's Algorithm can handle negative weights and detect negative cycles, whereas Dijkstra's Algorithm is not suitable for graphs with negative edge weights.
4. Time Complexity: The time complexity of Floyd's Algorithm is \(O(n^3)\) (where n is the number of vertices), while Dijkstra's Algorithm has a time complexity of \(O(n^2)\) for dense graphs, which
can be reduced to \(O(n \log n)\) with the help of priority queues for sparse graphs.
5. Data Structure: Floyd's Algorithm uses a matrix as its primary data structure, while Dijkstra's Algorithm often employs priority queues and lists to maintain the data.
Benefits and Limitations of Floyd's Algorithm and Dijkstra's Algorithm
Each algorithm offers its own set of advantages and drawbacks, which should be considered when choosing the most appropriate one for a particular problem. Below are some benefits and limitations of
Floyd's Algorithm and Dijkstra's Algorithm:
Floyd's Algorithm Dijkstra's Algorithm
Benefits: Benefits:
• Handles negative edge weights effectively • Faster for single-source shortest path problems
• Can detect negative cycles • Handles positive edge weights efficiently
• Finds all-pairs shortest path • Able to handle large and dense graphs through priority queues
• Easy to implement with adjacency matrices • Widely applicable in various domains
Limitations: Limitations:
• Higher time complexity than Dijkstra's Algorithm for single-source problems • Cannot handle negative edge weights
• Not suitable for very large or dense graphs • Not designed for all-pairs shortest path problems
• Less efficient in handling sparse graphs without priority queues
Ultimately, the choice between Floyd's Algorithm and Dijkstra's Algorithm depends on the specific problem requirements and constraints. If the problem requires finding the shortest path between all
pairs of vertices and includes negative weights, Floyd's Algorithm is the more suitable option. On the other hand, if the task involves a single-source shortest path problem with positive weights,
Dijkstra's Algorithm is the preferred choice.
Floyd's Algorithm Cycle Detection
In addition to finding the shortest path in weighted graphs, Floyd's Algorithm can also be employed to detect cycles. A cycle is a sequence of vertices in a graph where the first and last vertices
are the same, and no vertex appears more than once (except for the first and last vertex).
To identify cycles in a graph using Floyd's Algorithm, follow these steps:
1. Execute Floyd's Algorithm to compute the shortest paths between all pairs of vertices. This process updates the distance matrix and checks for the presence of negative cycles.
2. Check the diagonal elements of the final distance matrix. If any diagonal element has a negative value, there is a negative cycle in the graph.
3. Identify vertices involved in the negative cycle using the updated distance matrix and trace back the path of the cycle.
For example, consider the following adjacency matrix for a directed graph with four vertices:
0 -1 4 ∞
8 0 3 ∞
4 ∞ 0 ∞
∞ 2 ∞ 0
After executing Floyd's Algorithm, the final distance matrix becomes:
0 -1 2 ∞
3 0 1 ∞
7 4 0 ∞
∞ 2 ∞ 0
As there are no negative values on the diagonal, this graph does not have any negative cycles.
The Role of Cycle Detection in Decision Mathematics
Detecting cycles, especially negative cycles, plays a critical role in various areas of decision mathematics, as it can provide valuable insights and impact decision-making processes. Here are some
examples of cycle detection applications in decision mathematics:
• Economic networks: Detecting cycles in economic and financial systems, such as international trade or currency exchange networks, can unveil patterns and vulnerabilities, enabling better economic
forecasting and policy decisions.
• Operations research: In resource allocation and scheduling problems, cycle detection helps identify and eliminate infeasible solutions or counterproductive operations, thereby improving the
overall efficiency of the system.
• Game theory: Cycles provide important insights in the analysis of games, dynamic systems, and iterative algorithms, such as the convergence or divergence behaviour, strategic equilibria, and
potential oscillatory patterns.
• Graph algorithms: Beyond Floyd's Algorithm, cycle detection is a crucial aspect in many graph algorithms, such as topological sorting, strongly connected components, and minimum spanning tree
Understanding how to detect cycles in graphs using Floyd's Algorithm equips you with a powerful tool to address complex decision-making problems across various domains. By being able to identify and
assess cycles, you can make more informed and insightful decisions, ultimately leading to better outcomes.
Floyd's Algorithm - Key takeaways
• Floyd's Algorithm: Algorithm for finding shortest path between all pairs of vertices in a weighted graph, can handle negative edge weights and detect negative cycles.
• Importance in Decision Mathematics: Applications in network routing, operations research, game theory and computer graphics.
• Implementation Steps: Create adjacency matrix, initialize distance matrix, loop through vertices, and update shortest distances accordingly.
• Comparison to Dijkstra's Algorithm: Floyd's Algorithm finds all-pairs shortest path and can handle negative weights, while Dijkstra's Algorithm finds single-source shortest path and cannot handle
negative weights.
• Cycle Detection: Floyd's Algorithm can be used to detect cycles in graphs by checking the final distance matrix for negative diagonal values.
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Frequently Asked Questions about Floyd's Algorithm
How Floyd Warshall algorithm works?
Floyd Warshall algorithm works by finding the shortest paths between all pairs of vertices in a weighted graph. It iterates through each vertex as an intermediary step, updating the shortest
distances between other pairs of vertices. The algorithm utilises dynamic programming to solve this problem, considering optimal substructure and overlapping subproblems.
Is Floyds algorithm greedy?
No, Floyd's algorithm is not greedy. It is a dynamic programming approach used to find the shortest paths between all pairs of vertices in a weighted graph. Unlike greedy algorithms, it considers all
possible paths before determining the optimal solution.
What is the difference between Floyd's and Dijkstra's algorithm?
Floyd's algorithm finds the shortest paths between all pairs of vertices in a weighted graph, while Dijkstra's algorithm focuses on finding the shortest path from a single source vertex to all other
vertices. Floyd's algorithm handles negative weight edges, whereas Dijkstra's algorithm does not.
What are the steps for Floyd algorithm?
Floyd's Algorithm involves the following steps: 1) Initialise a distance matrix with direct edge weights between nodes or infinity if no direct connection exists. 2) Iterate through each node as an
intermediate node. 3) Update the distance matrix by comparing the direct distance between two nodes with the distance using the intermediate node. 4) Continue until all nodes have been considered as
What does Floyd's algorithm do?
Floyd's algorithm, also known as Floyd-Warshall algorithm, is used for finding the shortest paths between all pairs of vertices in a weighted graph with positive or negative edge weights. It works by
comparing all possible paths through the graph and recording the minimum distance for each vertex pair.
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SchoolPoint Login / Howick College / Course Selection / Student Voice / Student Conferences / Inbox Design
Year 10 Mathematics (10MATH)
Course Description
Teacher in Charge: Mr Z. Irani, Mrs C. Jaffar.
Year 10 Mathematics and Statistics covers Number, Algebra, Geometry, and Statistics. Students work at levels 4 - 6 of the NZ curriculum with an emphasis on strengthening numeracy skills. They also
have the opportunity to attempt an NCEA standard at level 6 to help them prepare for the senior school in year 11.
Course Overview
Term 1
Geometric Reasoning
Pythagoras and Trigonometry
Term 2
Algebra - Equations and Expressions
Term 3
Algebra - Patterns and Graphs (Includes Parabolas)
Semester A
Project Work
Students are placed in the appropriate maths class based on the results of their topic tests, end-of-year exam and e-asTTle results from year 9.
Contributions and Equipment/Stationery
There is a $30 course fee that covers the subscription to Education Perfect, which is our e-learning platform for 10MAT NZ Grapher, 10 Ticks worksheets and Walker Maths Algebra workbooks.
Owing to teachers responding to individual students' needs, courses and NCEA standards taught in a subject maybe different to those displayed. | {"url":"https://howick.schoolpoint.co.nz/courses/course/10math","timestamp":"2024-11-07T06:51:36Z","content_type":"text/html","content_length":"43376","record_id":"<urn:uuid:4b018c81-3de1-4a83-a923-1afb01f0826b>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00704.warc.gz"} |
Measurements to determine practical losses in an ATU (Antenna Tuning Unit).
I built two identical tuners to go from 50 Ohms to a complex impedance and then back to 50 Ohms. This makes it easy to measure the Insertion Loss in a 50 Ohm system.
Schematic representation of the ATU. Thank you Eric ON1BEW for turning my sketch into a nice scheme.
I made three simulations. One pure resistor and two complex impedances.
First case: 220 Ohms pure resistance.
1) Simulation on Smith diagram:
We do this exercise with two tuners and then place the tuners with the 220 Ohm side facing each other. Then we start and end with 50 Ohms each time.
This is the 50 Ohms to 220 Ohms and back to 50 Ohms.
2) Practical measurements:
Resistor = 220 Ohms Start measurement: tuner bypassed
Tuner 1 and 2 tuning for 7.1 MHz
2 Tuners connected with the 220 Ohms side together.
A loss of 0,2 dB is measured for the 2 devices.
0,1 dB for 1 tuner means a power loss of 2,3 %
Result in Elsie: Theoretical approach:
A loss of 0,14 dB is calculated for the 2 devices.
0,07 dB for 1 tuner means a power loss of 1,6 %
Second case: 40 Ohms -J150 Ohms
A different approach is needed for complex impedances.
1) Simulation on Smith diagram:
If we want to go from 50 to 40 -j150 ohms, a different tuner configuration is needed.
If you use the network as a mirror, you end up in the opposite impedance (40-J150 --> 40 + J150). A different approach is therefore needed.
2) Practical measurements:
Resistor = 2 X 22 Ohms Capacitor = 150 pF
Start measurement: tuner bypassed
Tuner 1 tuning for 7.1 MHz
After adjusting the first tuner, the second tuner is connected and adjusted to get back to 50 Ohms.
A loss of 1 dB is measured for the 2 devices.
0,5 dB for 1 tuner means a power loss of 10,9 %
Third case: 33 Ohms -J80 Ohms
1) Simulation on Smith diagram:
2) Practical measurements:
Resistor = 33 Ohms Capacitor = 270 pF
Start measurement: tuner bypassed
Tuner 1 tuning for 7.1 MHz
After adjusting the first tuner, the second tuner is connected and adjusted to get back to 50 Ohms.
A loss of 1 dB is measured for the 2 devices.
0,5 dB for 1 tuner means a power loss of 10,9 % | {"url":"https://www.xtgaby.com/post/measurements-to-determine-practical-losses-in-an-atu-antenna-tuning-unit","timestamp":"2024-11-06T15:16:45Z","content_type":"text/html","content_length":"1050035","record_id":"<urn:uuid:85946780-6c37-4cc5-9caf-d5fc8c8c51a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00856.warc.gz"} |
Queue Data Structure - The Coding Shala
Home >> Data Structures >> Queue
Queue Data Structure
In this post, we will see the basics of the queue. The queue is a Data Structure, also known as the First-in-first-out data structure(FIFO). In a queue, there are two main operations: Enqueue and
Dequeue. The insert Operation is called enqueue and the new element is always added at the end of the queue. The delete operation is called dequeue and we only allowed to remove the first element.
Queue Java Program
The following is a simple implementation of the queue:
import java.util.ArrayList;
import java.util.List;
//Queue implementation
class MyQueue{
//to store data
private List<Integer> data;
private int q_start; //start point of queue
//initial state
data = new ArrayList<Integer>();
q_start = 0;
//Enqueue Operation
//add element at the end of queue are return true if successful
public boolean EnQueue(int element) {
return true;
//DeQueue operation
//remove element if there any
//move start pointer
public boolean DeQueue() {
if(isEmpty()) return false;
return true;
//check is empty or not
public boolean isEmpty() {
if(data.size() <= q_start) return true;
return false;
//return first element
public int Front(){
if(isEmpty()) return -1;
return data.get(q_start);
public class Main{
public static void main(String[] args) {
MyQueue queue = new MyQueue();
if(queue.isEmpty() == false) { //check if not empty
System.out.println("First Element is: "+queue.Front());
if(queue.isEmpty() == false) { //check if not empty
System.out.println("First Element is: "+queue.Front());
System.out.println("First Element is: "+queue.Front());
//now queue is empty will return -1
System.out.println("First Element is: "+queue.Front());
if(queue.isEmpty() == false) { //check if not empty
System.out.println("First Element is: "+queue.Front());
}else {
System.out.println("Queue is Empty");
First Element is: 4
First Element is: 5
First Element is: 6
First Element is: -1
Queue is Empty
Other Posts You May Like
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Guest Post: Quantum Computing Update
Guest Post by Steve Blank
In March 2022 I wrote a description of the Quantum Technology Ecosystem. I thought this would be a good time to check in on the progress of building a quantum computerand explain more of the basics.
Just as a reminder, Quantum technologies are used in three very different and distinct markets:
Quantum Computing, Quantum Communications and Quantum Sensing and Metrology. If you don’t know the difference between a qubit and cueball, (I didn’t) read the tutorial here.
Summary –
• There’s been incremental technical progress in making physical qubits
• There is no clear winner yet between the seven approaches in building qubits
• Reminder – why build a quantum computer?
• How many physical qubits do you need?
• Advances in materials science will drive down error rates
• Regional research consortiums
• Venture capital investment FOMO and financial engineering
We talk a lot about qubits in this post. As a reminder a qubit – is short for a quantum bit. It is a quantum computing element that leverages the principle of superposition (that quantum particles
can exist in many possible states at the same time) to encode information via one of four methods: spin, trapped atoms and ions, photons, or superconducting circuits.
Incremental Technical Progress
As of 2024 there are seven different approaches being explored to build physical qubits for a quantum computer. The most mature currently are Superconducting, Photonics, Cold Atoms, Trapped Ions.
Other approaches include Quantum Dots, Nitrogen Vacancy in Diamond Centers, and Topological. All these approaches have incrementally increased the number of physical qubits.
These multiple approaches are being tried, as there is no consensus to the best path to building logical qubits. Each company believes that their technology approach will lead them to a path to
scale to a working quantum computer.
Every company currently hypes the number of physical qubits they have working. By itself this is a meaningless number to indicate progress to a working quantum computer. What matters is the number of
logical qubits.
Reminder – Why Build a Quantum Computer?
One of the key misunderstandings about quantum computers is that they are faster than current classical computers on all applications. That’s wrong. They are not. They are faster on a small set of
specialized algorithms. These special algorithms are what make quantum computers potentially valuable. For example, running Grover’s algorithm on a quantum computer can search unstructured data
faster than a classical computer. Further, quantum computers are theoretically very good at minimization / optimizations /simulations…think optimizing complex supply chains, energy states to form
complex molecules, financial models (looking at you hedge funds,) etc.
It’s possible that quantum computers will be treated as “accelerators” to the overall compute workflows – much like GPUs today. In addition, several companies are betting that “algorithmic” qubits
(better than “noisy” but worse than “error-corrected”) may be sufficient to provide some incremental performance to workflows lie simulating physical systems. This potentially opens the door for
earlier cases of quantum advantage.
However, while all of these algorithms might have commercial potential one day, no one has yet to come up with a use for them that would radically transform any business or military application.
Except for one – and that one keeps people awake at night. It’s Shor’s algorithm for integer factorization – an algorithm that underlies much of existing public cryptography systems.
The security of today’s public key cryptography systems rests on the assumption that breaking into those keys with a thousand or more digits is practically impossible. It requires factoring large
prime numbers (e.g., RSA) or elliptic curve (e.g., ECDSA, ECDH) or finite fields (DSA) that can’t be done with any type of classic computer regardless of how large. Shor’s factorization algorithm can
crack these codes if run on a Universal Quantum Computer. This is why NIST has been encouraging the move to Post-Quantum / Quantum-Resistant Codes.
How many physical qubits do you need for one logical qubit?
Thousands of logical qubits are needed to create a quantum computer that can run these specialized applications. Each logical qubit is constructed out of many physicalqubits. The question is, how
many physical qubits are needed? Herein lies the problem.
Unlike traditional transistors in a microprocessor that once manufactured always work, qubits are unstable and fragile. They can pop out of a quantum state due to noise, decoherence (when a qubit
interacts with the environment,) crosstalk (when a qubit interacts with a physically adjacent qubit,) and imperfections in the materials making up the quantum gates. When that happens errors will
occur in quantum calculations. So to correct for those error you need lots of physical qubits to make one logical qubit.
So how do you figure out how many physical qubits you need?
You start with the algorithm you intend to run.
Different quantum algorithms require different numbers of qubits. Some algorithms (e.g., Shor’s prime factoring algorithm) may need >5,000 logical qubits (the number may turn out to be smaller as
researchers think of how to use fewer logical qubits to implement the algorithm.)
Other algorithms (e.g., Grover’s algorithm) require fewer logical qubits for trivial demos but need 1000’s of logical qubits to see an advantage over linear search running on a classical computer.
(See here, here and here for other quantum algorithms.)
Measure the physical qubit error rate.
Therefore, the number of physical qubits you need to make a single logical qubit starts by calculating the physical qubit error rate (gate error rates, coherence times, etc.) Different technical
approaches (superconducting, photonics, cold atoms, etc.) have different error rates and causes of errors unique to the underlying technology.
Current state-of-the-art quantum qubits have error rates that are typically in the range of 1% to 0.1%. This means that on average one out of every 100 to one out of 1000 quantum gate operations will
result in an error. System performance is limited by the worst 10% of the qubits.
Choose a quantum error correction code
To recover from the error prone physical qubits, quantum error correction encodes the quantum information into a larger set of physical qubits that are resilient to errors. Surface Codes is the most
commonly proposed error correction code. A practical surface code uses hundreds of physical qubits to create a logical qubit. Quantum error correction codes get more efficient the lower the error
rates of the physical qubits. When errors rise above a certain threshold, error correction fails, and the logical qubit becomes as error prone as the physical qubits.
The Math
To factor a 2048-bit number using Shor’s algorithm with a 10^-2 (1% per physical qubit) error rate:
• Assume we need ~5,000 logical qubits
• With an error rate of 1% the surface error correction code requires ~ 500 physical qubits required to encode one logical qubit. (The number of physical qubits required to encode one logical qubit
using the Surface Code depends on the error rate.)
• Physical cubits needed for Shor’s algorithm= 500 x 5,000 = 2.5 million
If you could reduce the error rate by a factor of 10 – to 10^-3 (0.1% per physical qubit,)
• Because of the lower error rate, the surface code would only need ~ 100 physical qubits to encode one logical qubit
• Physical cubits needed for Shor’s algorithm= 100 x 5,000 = 500 thousand
In reality there another 10% or so of ancillary physical bits needed for overhead. And no one yet knows the error rate in wiring multiple logical bits together via optical links or other
(One caveat to the math above. It assumes that every technical approach (Superconducting, Photonics, Cold Atoms, Trapped Ions, et al) will require each physical qubit to have hundreds of bits of
error correction to make a logical qubit. There is always a chance a breakthrough could create physical qubits that are inherently stable, and the number of error correction qubits needed drops
substantially. If that happens, the math changes dramatically for the better and quantum computing becomes much closer.)
Today, the best anyone has done is to create 1,000 physical qubits.
We have a ways to go.
Advances in materials science will drive down error rates
As seen by the math above, regardless of the technology in creating physical qubits (Superconducting, Photonics, Cold Atoms, Trapped Ions, et al.) reducing errors in qubits can have a dramatic effect
on how quickly a quantum computer can be built. The lower the physical qubit error rate, the fewer physical qubits needed in each logical qubit.
The key to this is materials engineering. To make a system of 100s of thousands of qubits work the qubits need to be uniform and reproducible. For example, decoherence errors are caused by defects in
the materials used to make the qubits. For superconducting qubits that requires uniform thickness, controlled grain size, and roughness. Other technologies require low loss, and uniformity. All of
the approaches to building a quantum computer require engineering exotic materials at the atomic level – resonators using tantalum on silicon, Josephson junctions built out of magnesium diboride,
transition-edge sensors, Superconducting Nanowire Single Photon Detectors, etc.
Materials engineering is also critical in packaging these qubits (whether it’s superconducting or conventional packaging) and to interconnect 100s of thousands of qubits, potentially with optical
links. Today, most of the qubits being made are on legacy 200mm or older technology in hand-crafted processes. To produce qubits at scale, modern 300mm semiconductor technology and equipment will be
required to create better defined structures, clean interfaces, and well-defined materials. There is an opportunity to engineer and build better fidelity qubits with the most advanced semiconductor
fabrication systems so path from R&D to high volume manufacturing is fast and seamless.
There are likely only a handful of companies on the planet that can fabricate these qubits at scale.
Regional research consortiums
Two U.S. states; Illinois and Colorado are vying to be the center of advanced quantum research.
Illinois Quantum and Microelectronics Park (IQMP)
Illinois has announced the Illinois Quantum and Microelectronics Park initiative, in collaboration with DARPA’s Quantum Proving Ground (QPG) program, to establish a national hub for quantum
technologies. The State approved $500M for a “Quantum Campus” and has received $140M+ from DARPA with the state of Illinois matching those dollars.
Elevate Quantum
Elevate Quantum is the quantum tech hub for Colorado, New Mexico, and Wyoming. The consortium was awarded $127m from the Federal and State Governments – $40.5 million from the Economic Development
Administration (part of the Department of Commerce) and $77m from the State of Colorado and $10m from the State of New Mexico.
(The U.S. has a National Quantum Initiative (NQI) to coordinate quantum activities across the entire government see here.)
Venture capital investment, FOMO, and financial engineering
Venture capital has poured billions of dollars into quantum computing, quantum sensors, quantum networking and quantum tools companies.
However, regardless of the amount of money raised, corporate hype, pr spin, press releases, public offerings, no company is remotely close to having a quantum computeror even being close to run any
commercial application substantively faster than on a classical computer.
So why all the investment in this area?
1. FOMO – Fear Of Missing Out. Quantum is a hot topic. This U.S. government has declared quantum of national interest. If you’re a deep tech investor and you don’t have one of these companies in
your portfolio it looks like you’re out of step.
2. It’s confusing. The possible technical approaches to creating a quantum computer – Superconducting, Photonics, Cold Atoms, Trapped Ions, Quantum Dots, Nitrogen Vacancy in Diamond Centers, and
Topological – create a swarm of confusing claims. And unless you or your staff are well versed in the area, it’s easy to fall prey to the company with the best slide deck.
3. Financial engineering. Outsiders confuse a successful venture investment with companies that generate lots of revenue and profit. That’s not always true.
Often, companies in a “hot space” (like quantum) can go public and sell shares to retail investors who have almost no knowledge of the space other than the buzzword. If the stock price can stay high
for 6 months the investors can sell their shares and make a pile of money regardless of what happens to the company.
The track record so far of quantum companies who have gone public is pretty dismal. Two of them are on the verge of being delisted.
Here are some simple questions to ask companies building quantum computers:
• What is their current error rates?
• What error correction code will they use?
• Given their current error rates, how many physical qubits are needed to build one logical qubit?
• How will they build and interconnect the number of physical qubits at scale?
• What number of qubits do they think is need to run Shor’s algorithm to factor 2048 bits.
• How will the computer be programmed? What are the software complexities?
• What are the physical specs – unique hardware needed (dilution cryostats, et al) power required, connectivity, etc.
Lessons Learned
□ Lots of companies
□ Lots of investment
□ Great engineering occurring
□ Improvements in quantum algorithms may add as much (or more) to quantum computing performance as hardware improvements
□ The winners will be the one who master material engineering and interconnects
□ Jury is still out on all bets
Update: the kind folks at Applied Materials pointed me to the original 2012 Surface Codes paper. They pointed out that the math should look more like:
• To factor a 2048-bit number using Shor’s algorithm with a 0.3% error rate (Google’s current quantum processor error rate)
• Assume we need ~ 2,000 (not 5,000) logical qubits to run Shor’s algorithm.
• With an error rate of 0.3% the surface error correction code requires ~ 10 thousand physical qubits to encode one logical qubit to achieve 10^-10 logical qubit error rate.
• Physical cubits needed for Shor’s algorithm= 10,000 x 2,000 = 20 million
Still pretty far away from the 1,000 qubits we currently can achieve.
For those so inclined…
The logical qubit error rate P_L is P_L = 0.03 (p/p_th)^((d+1)/2), where p_th ~ 0.6% is the error rate threshold for surface codes, p the physical qubit error rate, and d is the size of the code,
which is related to the number of the physical qubits: N = (2d – 1)^2.
See the plot below for P_L versus N for different physical qubit error rate for reference.
Steve Blank is an adjunct professor at Stanford and co-founder of Stanford’s Gordian Knot Center for National Security Innovation. | {"url":"https://thequantuminsider.com/2024/10/26/guest-post-quantum-computing-update/","timestamp":"2024-11-12T19:57:18Z","content_type":"text/html","content_length":"364369","record_id":"<urn:uuid:a76281a4-b9e8-4b54-9c13-4ef9a5648606>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00161.warc.gz"} |
DBSCAN clustering with Haversine metric
Hi there,
I have an array of points (latitude, longitude) in radians and I am trying to use the package Clustering.jl to cluster my points. In the documentation it seems like a metric can be given, the default
being: Euclidean() however I need to use the Haversine metric and I dont seem to find the way of using such metric. Any ideas?
Thanks a lot!
Perhaps you can try the clustering API in GeoStats.jl. I believe we have all Clustering.jl models over there too. The latlon will be taken care of for you internally. But it is been a while since the
last time we tested DBSCAN
Looking at the documentation it seems that only DBScan takes a metric keyword argument:
You should be able to pass Haversine() to it instead of Euclidean().
Some other (not all) methods (e.g. hierarchical, k-mediods), take a distance matrix, instead of a data matrix, so you could compute all pairwise distances yourself with Haversine() first.
1 Like
I tried and passing Haversine() did not work (or other transformations of the same command). Yep, I could do the pairwise distance but as the documentation says, that method results in efficiency
losses so I was wondering whether it could be avoided. After all, passing some haversine version of Euclidean() seems like it should be there.
1 Like
Alternative: project the geogs into a Cartesian system and use euclidean distances. And at the end convert back to geogs. | {"url":"https://discourse.julialang.org/t/dbscan-clustering-with-haversine-metric/121685","timestamp":"2024-11-14T07:48:58Z","content_type":"text/html","content_length":"26270","record_id":"<urn:uuid:f661dea9-b89f-480d-8eed-a2f8de609300>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00570.warc.gz"} |
Wavelets and MERA | Tensors.net
top of page
New research: Wavelets and MERA
Recently a precise connection between discrete wavelets transformations (DWT's) and MERA tensor networks has been established. This connection has opened the possibility for using techniques
developed in the context of tensor networks to design new wavelet transformations and also for the use of wavelet methods to gain a better analytic understanding of MERA.
Here we demonstrate the use of a new wavelet design (TypeII, edge-centered symmetric wavelet from Fig.8(b) of this reference) for the purpose of image compression. We choose this wavelet design for
the demonstration as it seems to be a good compromise between the accuracy (typically slightly improving over that of the CDF-9/7 wavelets used in JPEG2000 at the same compression ratio) and the
computational cost of transformation (with only a slight cost increase over the CDF-9/7 transformation). In addition, these new wavelets have the advantage of being fully orthogonal, whereas the CDF
wavelets are not.
This transformation is based on a depth-3 ternary unitary circuit (fully characterized by 3 rotation angles theta as given below), with a scaling function and two wavelet functions at each scale, as
depicted below. During the transformation, two copies of this circuit are applied (one to the even sublattice and the other to the odd sublattice), whereupon symmetric / anti-symmetric scaling
functions and wavelets are formed by taking the appropriate linear combinations from each circuit. The example code has the following features:
• Boundaries are treated using a symmetric extension
• Compatible with grey-scale or color images of any dimension
• Compression ratio can be specified
• The symmetric wavelets have 4 vanishing moments, while the anti-symmetric wavelets have 3 vanishing moments
Circuit structure:
Unitary gates:
Scaling sequences:
Scaling functions and wavelets
Example usage: image compression
Original image
wavelet transform
Recovered image, PSNR 31.27
inverse transform
Truncate (retain 3% of coefficients)
Lifting / lowering step of DWT (MATLAB function):
Image compression demonstration (MATLAB script):
bottom of page | {"url":"https://www.tensors.net/wavelets-mera","timestamp":"2024-11-04T23:02:45Z","content_type":"text/html","content_length":"641320","record_id":"<urn:uuid:7001b437-a08d-4017-b9f3-e94c5344f1fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00614.warc.gz"} |
An Essay on the Principle of Population
Thomas Malthus (1798)
An Essay on the Principle of Population
CHAPTER 15
Models too perfect may sometimes rather impede than promote improvement - Mr Godwin's essay on 'Avarice and Profusion' - Impossibility of dividing the necessary labour of a society amicably among all
-Invectives against labour may produce present evil, with little or no chance of producing future good - An accession to the mass of agricultural labour must always be an advantage to the labourer.
MR GODWIN in the preface to his Enquirer, drops a few expressions which seem to hint at some change in his opinions since he wrote the Political Justice; and as this is a work now of some years
standing, I should certainly think that I had been arguing against opinions which the author had himself seen reason to alter, but that in some of the essays of the Enquirer, Mr Godwin's peculiar
mode of thinking appears in as striking a light as ever.
It has been frequently observed that though we cannot hope to reach perfection in any thing, yet that it must always be advantageous to us to place before our eyes the most perfect models. This
observation has a plausible appearance, but is very far from being generally true. I even doubt its truth in one of the most obvious exemplifications that would occur. I doubt whether a very young
painter would receive so much benefit, from an attempt to copy a highly finished and perfect picture, as from copying one where the outlines were more strongly marked and the manner of laying on the
colours was more easily discoverable. But in cases where the perfection of the model is a perfection of a different and superior nature from that towards which we should naturally advance, we shall
not always fail in making any progress towards it, but we shall in all probability impede the progress which we might have expected to make had we not fixed our eyes upon so perfect a model. A highly
intellectual being,.exempt from the infirm calls of hunger or sleep, is undoubtedly a much more perfect existence than man, but were man to attempt to copy such a model, he would not only fail in
making any advances towards it; but by unwisely straining to imitate what was inimitable, he would probably destroy the little intellect which he was endeavouring to improve.
The form and structure of society which Mr Godwin describes is as essentially distinct from any forms of society which have hitherto prevailed in the world as a being that can live without food or
sleep is from a man. By improving society in its present form, we are making no more advances towards such a state of things as he pictures than we should make approaches towards a line, with regard
to which we were walking parallel. The question, therefore, is whether, by looking to such a form of society as our polar star, we are likely to advance or retard the improvement of the human
species? Mr Godwin appears to me to have decided this question against himself in his essay on 'Avarice and Profusion' in the Enquirer.
Dr Adam Smith has very justly observed that nations as well as individuals grow rich by parsimony and poor by profusion, and that, therefore, every frugal man was a friend and every spendthrift an
enemy to his country. The reason he gives is that what is saved from revenue is always added to stock, and is therefore taken from the maintenance of labour that is generally unproductive and
employed in the maintenance of labour that realizes itself in valuable commodities. No observation can be more evidently just. The subject of Mr Godwin's essay is a little similar in its first
appearance, but in essence is as distinct as possible. He considers the mischief of profusion as an acknowledged truth, and therefore makes his comparison between the avaricious man, and the man who
spends his income. But the avaricious man of Mr Godwin is totally a distinct character, at least with regard to his effect upon the prosperity of the state, from the frugal man of Dr Adam Smith. The
frugal man in order to make more money saves from his income and adds to his capital, and this capital he either employs himself in the maintenance of productive labour, or he lends it to some other
person who wil1 probably employ it in this way. He benefits the state because he adds to its general capital, and because wealth employed as capital not only sets in motion more labour than when
spent as income, but the labour is besides of a more valuable kind. But the avaricious man of Mr Godwin locks up his wealth in a chest and sets in motion no labour of any kind, either productive or
unproductive. This is so essential a difference that Mr Godwin's decision in his essay appears at once as evidently false as Dr Adam Smith's position is evidently true. It could not, indeed, but
occur to Mr Godwin that some present inconvenience might arise to the poor from thus locking up the funds destined for the maintenance of labour. The only way, therefore, he had of weakening this
objection was to compare the two characters chiefly with regard to their tendency to accelerate the approach of that happy state of cultivated equality, on which he says we ought always to fix our
eyes as our polar star.
I think it has been proved in the former parts of this essay that such a state of society is absolutely impracticable. What consequences then are we to expect from looking to such a point as our
guide and polar star in the great sea of political discovery? Reason would teach us to expect no other than winds perpetually adverse, constant but fruitless toil, frequent shipwreck, and certain
misery. We shall not only fail in making the smallest real approach towards such a perfect form of society; but by wasting our strength of mind and body, in a direction in which it is impossible to
proceed, and by the frequent distress which we must necessarily occasion by our repeated failures, we shall evidently impede that degree of improvement in society, which is really attainable.
It has appeared that a society constituted according to Mr Godwin's system must, from the inevitable laws of our nature, degenerate into a class of proprietors and a class of labourers, and that the
substitution of benevolence for self-love as the moving principle of society, instead of producing the happy effects that might be expected from so fair a name, would cause the same pressure of want
to be felt by the whole of society, which is now felt only by a part. It is to the established administration of property and to the apparently narrow principle of self-love that we are indebted for
all the noblest exertions of human genius, all the finer and more delicate emotions of the soul, for everything, indeed, that distinguishes the civilized from the savage state; and no sufficient
change has as yet taken place in the nature of civilized man to enable us to say that he either is, or ever will be, in a state when he may safely throw down the ladder by which he has risen to this
If in every society that has advanced beyond the savage state, a class of proprietors and a class of labourers must necessarily exist, it is evident that, as labour is the only property of the class
of labourers, every thing that tends to diminish the value of this property must tend to diminish the possession of this part of society. The only way that a poor man has of supporting himself in
independence is by the exertion of his bodily strength. This is the only commodity he has to give in exchange for the necessaries of life. It would hardly appear then that you benefit him by
narrowing the market for this commodity, by decreasing the demand for labour, and lessening the value of the only property that he possesses.
It should be observed that the principal argument of this Essay only goes to prove the necessity of a class of proprietors, and a class of labourers, but by no means infers that the present great
inequality of property is either necessary or useful to society. On the contrary, it must certainly be considered as an evil, and every institution that promotes it is essentially bad and impolitic.
But whether a government could with advantage to society actively interfere to repress inequality of fortunes may be a matter of doubt. Perhaps the generous system of perfect liberty adopted by Dr
Adam Smith and the French economists would be ill exchanged for any system of restraint.
Mr Godwin would perhaps say that the whole system of barter and exchange is a vile and iniquitous traffic. If you would essentially relieve the poor man, you should take a part of his labour upon
yourself, or give him your money, without exacting so severe a return for it. In answer to the first method proposed, it may be observed, that even if the rich could be persuaded to assist the poor
in this way, the value of the assistance would be comparatively trifling. the rich, though they think themselves of great importance, bear but a small proportion in point of numbers to the poor, and
would, therefore, relieve them but of a small part of their burdens by taking a share. Were all those that are employed in the labours of luxuries added to the number of those employed in producing
necessaries, and could these necessary labours be amicably divided among all, each man's share might indeed be comparatively light; but desirable as such an amicable division would undoubtedly be, I
cannot conceive any practical principle according to which it could take place. It has been shewn, that the spirit of benevolence, guided by the strict impartial justice that Mr Godwin describes,
would, if vigorously acted upon, depress in want and misery the whole human race. Let us examine what would be the consequence, if the proprietor were to retain a decent share for himself, but to
give the rest away to the poor, without exacting a task from them in return. Not to mention the idleness and the vice that such a proceeding, if general, would probably create in the present state of
society, and the great. risk there would be, of diminishing the produce of land, as well as the labours of luxury, another objection yet remains.
Mr Godwin seems to have but little respect for practical principles; but I own it appears to me, that he is a much greater benefactor to mankind, who points out how an inferior good may be attained,
than he who merely expatiates on the deformity of the present state of society, and the beauty of a different state, without pointing out a practical method, that might be immediately applied, of
accelerating our advances from the one, to the other.
It has appeared that from the principle of population more will always be in want than can be adequately supplied. The surplus of the rich man might be sufficient for three, but four will be desirous
to obtain it. He cannot make this selection. of three out of the four without conferring a great favour on those that are the objects of his choice. These persons must consider themselves as under a
great obligation to him and as dependent upon him for their support. The rich man would feel his power and the poor man his dependence, and the evil effects of these two impressions on the human
heart are well known. Though I perfectly agree with Mr Godwin therefore in the evil of hard labour, yet I still think it a less evil, and less calculated to debase the human mind, than dependence,
and every history of man that we have ever read places in a strong. point of view the danger to which that mind is exposed which is entrusted with constant power.
In the present state of things, and particularly when labour is in request, the man who does a day's work for me confers full as great an obligation upon me as I do upon him. I possess what he wants,
he possesses what I want. We make an amicable exchange. The poor man walks erect in conscious independence; and the mind of his employer is not vitiated by a sense of power.
Three or four hundred years ago there was undoubtedly much less labour in England, in proportion to the population, than at present, but there was much more dependence, and we probably should not now
enjoy our present degree of civil liberty if the poor, by the introduction of manufactures, had not been enabled to give something in exchange for the provisions of the great Lords, instead of being
dependent upon their bounty. Even the greatest enemies of trade and manufactures, and I do not reckon myself a very determined friend to them, must allow that when they were introduced into England,
liberty came in their train.
Nothing that has been said tends in the most remote degree to undervalue the principle of benevolence. It is one of the noblest and most godlike qualities of the human heart, generated, perhaps,
slowly and gradually from self-love, and afterwards intended to act as a general law, whose kind office it should be, to soften the partial deformities, to correct the asperities, and to smooth the
wrinkles of its parent: and this seems to be the analog of all nature. Perhaps there is no one general law of nature that will not appear, to us at least, to produce partial evil; and we frequently
observe at the same time, some bountiful provision which, acting as another general law, corrects the inequalities of the first.
The proper office of benevolence is to soften the partial evils. arising from self-love, but it can never be substituted in its place. If no man were to allow himself to act till he had completely
determined that the action he was about to perform was more conducive than any other to the general good, the most enlightened minds would hesitate in perplexity and amazement; and the unenlightened
would be continually committing the grossest mistakes.
As Mr Godwin, therefore, has not laid down any practical principle according to which the necessary labours of agriculture might be amicably shared among the whole class of labourers, by general
invectives against employing the poor he appears to pursue an unattainable good through much present evil. For if every man who employs the poor ought to be considered as their enemy, and as adding
to the weight of their oppressions, and if the miser is for this reason to be preferred to the man who spends his income, it follows that any number of men who now spend their incomes might, to the
advantage of society, be converted into misers. Suppose then that a hundred thousand persons who now employ ten men each were to lock up their wealth from general use, it is evident, that a million
of working men of different kinds would be completely thrown out of all employment. The extensive misery that such an event would produce in the present state of society Mr Godwin himself could
hardly refuse to acknowledge, and I question whether he might not find some difficulty in proving that a conduct of this kind tended more than the conduct of those who spend their incomes to 'place
human beings in the condition in which they ought to be placed.' But Mr Godwin says that the miser really locks up nothing, that the point has not been rightly understood, and that the true
development and definition of the nature of wealth have not been applied to illustrate it. Having defined therefore wealth, very justly, to be the commodities raised and fostered by human labour, he
observes that the miser locks up neither corn, nor oxen, nor clothes, nor houses. Undoubtedly he does not really lock up these articles, but he locks up the power of producing them, which is
virtually the same. These things are certainly used and consumed by his contemporaries, as truly, and to as great an extent, as if he were a beggar; but not to as great an extent as if he had
employed his wealth in turning up more land, in breeding more oxen, in employing more tailors, and in building more houses. But supposing, for a moment, that the conduct of the miser did not tend to
check any really useful produce, how are all those who are thrown out of employment to obtain patents which they may shew in order to be awarded a proper share of the food and raiment produced by the
society? This is the unconquerable difficulty.
I am perfectly willing to concede to Mr Godwin that there is much more labour in the world than is really necessary, and that, if the lower classes of society could agree among themselves never to
work more than six or seven hours in the day, the commodities essential to human happiness might still be produced in as great abundance as at present. But it is almost impossible to conceive that
such an agreement could be adhered to. From the principle of population, some would necessarily be more in want than others. Those that had large families would naturally be desirous of exchanging
two hours more of their labour for an ampler quantity of subsistence. How are they to be prevented from making this exchange? it would be a violation of the first and most sacred property that a man
possesses to attempt, by positive institutions, to interfere with his command over his own labour.
Till Mr Godwin, therefore, can point out some practical plan according to which the necessary labour in a society might be equitably divided, his invectives against labour, if they were attended to,
would certainly produce much present evil without approximating us to that state of cultivated equality to which he looks forward as his polar star, and which, he seems to think, should at present be
our guide in determining the nature and tendency of human actions. A mariner guided by such a polar star is in danger of shipwreck.
Perhaps there is no possible way in which wealth could in general be employed so beneficially to a state, and particularly to the lower orders of it, as by improving and rendering productive that
land which to a farmer would not answer the expense of cultivation. Had Mr Godwin exerted his energetic eloquence in painting the superior worth and usefulness of the character who employed the poor
in this way, to him who employed them in narrow luxuries, every enlightened man must have applauded his efforts. The increasing demand for agricultural labour must always tend to better the condition
of the poor; and if the accession of work be of this kind, so far is it from being true that the poor would be obliged to work ten hours for the same price that they before worked eight, that the
very reverse would be the fact; and a labourer might then support his wife and family as well by the labour of six hours as he could before by the labour of eight.
The labour created by luxuries, though useful in distributing the produce of the country, without vitiating the proprietor by power, or debasing the labourer by dependence, has not, indeed, the same
beneficial effects on the state of the poor. A great accession of work from manufacturers, though it may raise the price of labour even more than an increasing demand for agricultural labour, yet, as
in this case the quantity of food in the country may not be proportionably increasing, the advantage to the poor will be but temporary, as the price of provisions must necessarily rise in proportion
to the price of labour. Relative to this subject, I cannot avoid venturing a few remarks on a part of Dr Adam Smith's Wealth of Nations, speaking at the same time with that diffidence which I ought
certainly to feel in differing from a person so justly celebrated in the political world.
Contents | next chapter | Political Economy Archive | {"url":"https://www.marxists.org/reference/subject/economics/malthus/ch15.htm","timestamp":"2024-11-04T23:07:36Z","content_type":"text/html","content_length":"21427","record_id":"<urn:uuid:e4c6b223-d262-42e0-8994-74bf19830d7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00000.warc.gz"} |
We consider a family of exclusion processes defined on the discrete interval with weak boundary interaction that allows the creation and annihilation of particles on a neighborhood of radius L of the
boundary under very general rates. We prove that the hydrodynamic equation is the heat equation with non-linear Robin boundary conditions. We present a particular choice of boundary rates for which
we have multiple stationary solutions but for which it still holds the uniqueness of the solution of its hydrodynamic equation. We also prove the associated dynamical large deviations principle.
Joint work with Claudio LandimĀ andĀ Beatriz Salvador. | {"url":"https://events.dimai.unifi.it/news/api/190/summary","timestamp":"2024-11-03T19:30:55Z","content_type":"text/html","content_length":"2387","record_id":"<urn:uuid:24938d12-11f7-4fe1-af40-2c7924a2a0ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00728.warc.gz"} |
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Auto Loan Payment Calculator
An Auto Loan calculator helps you understand the periodic cash flow outflows in the form of equal installments for your Auto Loans.
The formula for calculating Auto Loan payment is as per below:
• L is the loan amount
• r is the rate of interest per annum
• n is the number of periods or frequency wherein the loan amount is to be paid
About Auto Loan Calculator
Purchasing a new car or any two-wheeler vehicle nowadays is mostly financed through the bank as, with time, the prices have been rising, and for most people, it becomes over budget if they spend the
entire amount in one go. Therefore, it becomes necessary for the borrower to know what amount of periodical installment he would be paying on the same and what extra amount he would be paying in the
form of interest, and whether it's viable for him to pay that much extra. Also, the borrowers can compare the installment amount across terms and money lenders, and whichever suits them best, they
will opt for the same.
In the case of an auto loan payment, most of the loans are financed up to 80% or a maximum of up to 90%; there are very few cases where the vehicle purchase is financed fully. In some cases, the bank
charges processing fees upfront, or any dealer charges, any rebate available, etc. All these need to be taken into account before calculating the installment amount.
How to Calculate using the Auto Loan Payment Calculator?
One needs to follow the below steps to calculate the monthly installment amounts of auto loans.
1. First of all, determine the loan amount required. Banks usually provide more loan amounts to those with a good credit score and fewer to those with a lower credit score.
First, we shall enter the principal amount:
2. Multiply the loan amount by a rate of interest.
3. Now, we need to compound the same by rate until the loan period.
4. We now need to discount the above result obtained in step 3 by the following:
5. After entering the above formula in excel, we shall obtain periodically for Auto Loan installments.
Auto Loan Calculator
Example #1
Mr. Sachin, who stays in New York, is looking to purchase a luxury car costing around $62,500 ex-showroom. On inquiry with the dealer, Mr. Sachin came to know that he is supposed to pay Insurance on
the same, which costs around 4% of the cost price, then he needs to pay road tax, which is around 2% of the cost price. Since the cost, including all the charges, exceeds his budget, he decides to
get the vehicle financed by a renowned bank. The bank will finance it 85% as his credit score was excellent, and the rest of the amount he has to pay in cash or cheque. He opts for 24 months of equal
installment. The interest rate would be 8%, and the bank shall also charge him upfront 1% processing fees.
Based on the above information, you must calculate the monthly installment amount and what extra amount he would be paying had he not opted for the loan.
We need to calculate the cost price of the vehicle first.
• Now, we shall calculate the loan amount, which is 66,250 *85%, which is 56,312.50.
• There are Processing charges, which are 1% of the loan amount, which has to be paid upfront, that is $563.13.
• Further, the down payment will be 66,250 – 56,312.50, which is $9,937.50
• The period it is required to be paid is 24 equally installments, and lastly, the rate of interest is 8.00% fixed, which shall be calculated monthly, which is 8.00%/12, which is 0.67%.
• Now we shall use the below formula to calculate the EMI amount.
EMI = /
= /
= 2,546.86
• Therefore, the EMI amount for Mr. Sachin for two years on the loan amount of 66,250 shall be 2,546.86, and when multiplied same with 24, the total amount paid will be 61,124.68 less 56,312.50
that, is equal to 4,812.18
• And total outgo due to the loan will be 4,812.18 + 563.13, which is 5,375.31
Example #2
Mrs. Shivani is looking to buy a new two-wheeler by financing it through the bank. The total price of the vehicle is $15,900, including all charges. The Bank will finance 75%, and the rest shall be
paid upfront. However, she is confused with the tenure as she wants to keep her monthly installment lower and doesn’t want to pay interest on the loan period by more than $1,500. The rate of interest
the bank is charging her is 9.5%. The Bank has offered her either to take installments for 24 months or 36 months. You must advise which installment term she should choose that will fit her
We need to calculate the EMI amount for that first, and we shall calculate the loan amount, which is 15,900*70% which is $11,925.00. The interest rate is 9.5% fixed, which shall be calculated
monthly, which is 9.5%/12, which is 0.79%.
• Now we shall use the below formula to calculate the EMI amount for 2 years, which is 24 months.
EMI= /
= /
= $547.53
• Total interest outgo equals to ($547.53*24)-$11,925 which is $1,215.73
• Now we shall use the below formula to calculate the EMI amount for 3 years which is 36 months.
EMI = /
= /
= $381.99
• Total interest outgo equals to ($381.99*36)-$11,925 which is $1,826.75
• Her preference was to keep a lower EMI, which is matched with a 3-year option, but the second requirement is interest outgo should not be more than $1,500, which is not met, and hence she would
opt for 24 months tenure.
Auto Loan Payment Calculator nowadays is quite common these days since it makes it comfortable for the borrower to pay for the vehicle without any large outgo. The higher the credit score, the higher
the loan amount and the lesser the down payment the borrower would be required to make.
Recommended Articles
This has been a guide to an Auto Loan Payment Calculator. Here we provide the calculator used to calculate the monthly installment or EMI amount, with examples. You may also take a look at the
following useful articles - | {"url":"https://www.wallstreetmojo.com/auto-loan-calculator/","timestamp":"2024-11-09T00:59:13Z","content_type":"text/html","content_length":"333649","record_id":"<urn:uuid:c550a6b3-ea0c-4219-b1ca-c5c6ac834f8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00133.warc.gz"} |
High-energy photon emission & radiation reaction
High-energy photon emission & radiation reaction¶
Accelerated charges emit electromagnetic radiation, and by doing so, lose some of their energy and momentum. This process is particularly important for high-energy particles traveling in strong
electromagnetic fields where it can strongly influence the dynamics of the radiating charges, a process known as radiation reaction.
In Smilei, different modules treating high-energy photon emission & its back-reaction have been implemented. We first give a short overview of the physics (and assumptions) underlying these modules,
before giving more pratical information on what each module does. We then give examples & benchmarks, while the last two sections give additional information on the implementation of the various
modules and their performances.
The processes discussed in this section bring into play a characteristic length [the classical radius of the electron \(r_e = e^2/(4\pi \epsilon_0 m_e c^2)\) in classical electrodynamics (CED) or the
standard Compton wavelength \(\lambda_C=\hbar/(m_e c)\) in quantum electrodynamics (QED)]. As a result, a simulation will require the user to define the absolute scale of the system by defining the
reference_angular_frequency_SI parameter (see Units for more details).
Also note that, unless specified otherwise, SI units are used throughout this section, and we use standard notations with \(m_e\), \(e\), \(c\) and \(\hbar\) the electron mass, elementary charge,
speed of light and reduced Planck constant, respectively, and \(\epsilon_0\) the permittivity of vacuum.
Inverse Compton scattering¶
This paragraph describes the physical model and assumptions behind the different modules for high-energy photon emission & radiation reaction that have been implemented in Smilei. The presentation is
based on the work [Niel2018a].
All the modules developed so far in Smilei assume that:
• the radiating particles (either electrons or positrons) are ultra-relativistic (their Lorentz factor \(\gamma \gg 1\)), hence radiation is emitted in the direction given by the radiating particle
• the electromagnetic field varies slowly over the formation time of the emitted photon, which requires relativistic field strengths [i.e., the field vector potential is \(e\vert A^{\mu}\vert/(mc^
2) \gg 1\)], and allows to use quasi-static models for high-energy photon emission (locally-constant cross-field approximation),
• the electromagnetic fields are small with respect to the critical field of Quantum Electrodynamics (QED), more precisely both field invariants \(\sqrt{c^2{\bf B}^2-{\bf E}^2}\) and \(\sqrt{c{\bf
B}\cdot{\bf E}}\) are small with respect to the Schwinger field \(E_s = m^2 c^3 / (\hbar e) \simeq 1.3 \times 10^{18}\ \mathrm{V/m}\),
• all (real) particles radiate independently of their neighbors (incoherent emission), which requires the emitted radiation wavelength to be much shorter than the typical distance between (real)
particles \(\propto n_e^{-1/3}\).
Rate of photon emission and associated quantities¶
Under these assumptions, high-energy photon emission reduces to the incoherent process of nonlinear inverse Compton scattering. The corresponding rate of high-energy photon emission is given by
\[\frac{d^2 N_{\gamma}}{d\tau d\chi_{\gamma}} = \frac{2}{3}\frac{\alpha^2}{\tau_e}\,\frac{S(\chi,\chi_{\gamma}/\chi)}{\chi_{\gamma}}\]
with \(\tau_e = r_e/c\) the time for light to cross the classical radius of the electron, and \(\alpha\) the fine-structure constant. This rate depends on two Lorentz invariants, the electron quantum
\[\chi = \frac{\gamma}{E_s} \sqrt{ \left({\bf E} + {\bf v} \times {\bf B}\right)^2 - ({\bf v }\cdot{\bf E})^2/c^2 }\]
and the photon quantum parameter (at the time of photon emission):
\[\chi_{\gamma} = \frac{\gamma_{\gamma}}{E_s} \sqrt{ \left({\bf E} + {\bf c} \times {\bf B}\right)^2 - ({\bf c }\cdot{\bf E})^2/c^2 }\]
where \(\gamma = \varepsilon / (m_e c^2)\) and \(\gamma_{\gamma} = \varepsilon_{\gamma} / (m_e c^2)\) are the normalized energies of the radiating particle and emitted photon, respectively, and \({\
bf v}\) and \({\bf c}\) their respective velocities.
Note that considering ultra-relativistic (radiating) particles, both parameters are related by:
\[\xi = \frac{\chi_{\gamma}}{\chi} = \frac{\gamma_{\gamma}}{\gamma}\,.\]
In the photon production rate Eq. (15) appears the quantum emissivity:
\[S(\chi,\xi) = \frac{\sqrt{3}}{2\pi}\,\xi\,\left[\int_{\nu}^{+\infty} {\rm K}_{5/3}(y) dy + \frac{\xi^2}{1-\xi}\,{\rm K}_{2/3}(\nu)\right]\,,\]
with \(\nu = 2\xi/[3\chi(1-\xi)]\).
Finally, the instantaneous radiated power energy-spectrum reads:
\[\frac{dP_{\rm inst}}{d\gamma_{\gamma}} = P_{\alpha}\,\gamma^{-1}\,S(\chi,\chi_{\gamma}/\chi)\,,\]
with \(P_{\alpha}=2\alpha^2 m_e c^2/(3\tau_e)\), and the instantaneous radiated power:
\[P_{\rm inst} = P_{\alpha}\,\chi^2\,g(\chi)\,,\]
with \(g(\chi)\) the so-called quantum correction:
\[g(\chi) = \frac{9 \sqrt{3} }{8 \pi} \int_0^{+\infty}{d\nu \left[ \frac{2\nu^2 }{\left( 2 + 3 \nu \chi \right) ^2}K_{5/3}(\nu) + \frac{4 \nu \left( 3 \nu \chi\right)^2 }{\left( 2 + 3 \nu \chi \
right)^4}K_{2/3}(\nu) \right]}\,.\]
Regimes of radiation reaction¶
Knowing exactly which model of radiation reaction is best to describe a given situation is not always easy, and the domain of application of each model is still discussed in the recent literature
(again see [Niel2018a] for more details). However, the typical value of the electron quantum parameter \(\chi\) in a simulation can be used as a way to assess which model is most suitable. We adopt
this simple (yet sometimes not completely satisfactory) point of view below to describe the three main approaches used in Smilei to account for high-energy photon emission and its back-reaction on
the electron dynamics.
For arbitrary values of the electron quantum parameter \(\chi\) (but mandatory in the quantum regime \(\chi \gtrsim 1\))¶
The model of high-energy photon emission described above is generic, and applies for any value of the electron quantum parameter \(\chi\) (of course as long as the assumptions listed above hold!). In
particular, it gives a correct description of high-energy photon emission and its back-reaction on the particle (electron or positron) dynamics in the quantum regime \(\chi \gtrsim 1\). In this
regime, photons with energies of the order of the energy of the emitting particle can be produced. As a result, the particle energy/velocity can exhibit abrupt jumps, and the stochastic nature of
high-energy photon emission is important. Under such conditions, a Monte-Carlo description of discrete high-energy photon emission (and their feedback on the radiating particle dynamics) is usually
used (see [Timokhin2010], [Elkina2011], [Duclous2011], and [Lobet2013]). More details on the implementation are given below.
In Smilei the corresponding description is accessible for an electron species by defining radiation_model = "Monte-Carlo" or "MC" in the Species() block (see Write a namelist for details).
Intermediate, moderately quantum regime \(\chi \lesssim 1\)¶
In the intermediate regime (\(\chi \lesssim 1\)), the energy of the emitted photons remains small with respect to that of the emitting electrons. Yet, the stochastic nature of photon emission cannot
be neglected. The electron dynamics can then be described by a stochastic differential equation derived from a Fokker-Planck expansion of the full quantum (Monte-Carlo) model described above
In particular, the change in electron momentum during a time interval \(dt\) reads:
\[d{\bf p} = {\bf F}_{\rm L} dt + {\bf F}_{\rm rad} dt + mc^2 \sqrt{R\left( \chi, \gamma \right)} dW \mathbf{u} / \left( \mathbf{u}^2 c\right)\]
where we recognize 3 terms:
• the Lorentz force \({\bf F}_{\rm L} = \pm e ({\bf E} + {\bf v}\times{\bf B})\) (with \(\pm e\) the particle’s charge),
• a deterministic force term \({\bf F}_{\rm rad}\) (see below for its expression), so-called drift term, which is nothing but the leading term of the Landau-Lifshitz radiation reaction force with
the quantum correction \(g(\chi)\),
• a stochastic force term, so-called diffusion term, proportional to \(dW\), a Wiener process of variance \(dt\). This last term allows to account for the stochastic nature of high-energy photon
emission, and it depends on functions which are derived from the stochastic model of radiation emission presented above:
\[ R\left( \chi, \gamma \right) = \frac{2}{3} \frac{\alpha^2}{\tau_e} \gamma h \left( \chi \right)\]
\[ h \left( \chi \right) = \frac{9 \sqrt{3}}{4 \pi} \int_0^{+\infty}{d\nu \left[ \frac{2\chi^3 \nu^3}{\left( 2 + 3\nu\chi \right)^3} K_{5/3}(\nu) + \frac{54 \chi^5 \nu^4}{\left( 2 + 3 \nu \chi \
right)^5} K_{2/3}(\nu) \right]}\]
In Smilei the corresponding description is accessible for an electron species by defining radiation_model = "Niel" in the Species() block (see Write a namelist for details).
The classical regime \(\chi \ll 1\)¶
Quantum electrodynamics (QED) effects are negligible (classical regime) when \(\chi \ll 1\). Radiation reaction follows from the cummulative effect of incoherent photon emission. It can be treated as
a continuous friction force acting on the particles. Several models for the radiation friction force have been proposed (see [DiPiazza2012]). The ones used in Smilei are based on the Landau-Lifshitz
(LL) model [Landau1947] approximated for high Lorentz factors (\(\gamma \gg 1\)). Indeed, as shown in [Niel2018a], the LL force with the quantum correction \(g(\chi)\) naturaly emerges from the full
quantum description given above. This can easily be seen from Eq. (23), in which the diffusion term vanishes in the limit \(\chi \ll 1\) so that one obtains for the deterministic equation of motion
for the electron:
\[\frac{d{\bf p}}{dt} = {\bf F}_{\rm L} + {\bf F}_{\rm rad}\]
\[{\bf F}_{\rm rad} = -P_{\alpha} \chi^2 g(\chi)\,\mathbf{u} / \left( \mathbf{u}^2 c\right)\]
In Smilei the corresponding description is accessible for an electron species by defining radiation_model = "corrected-Landau-Lifshitz" or "cLL" in the Species() block (see Write a namelist for
• for \(\chi \rightarrow 0\), the quantum correction \(g(\chi) \rightarrow 1\), \(P_{\rm inst} \rightarrow P_{\alpha}\,\chi^2\) (which is the Larmor power) and \(dP_{\rm inst}/d\gamma_{\gamma}\)
[Eq. (20)] reduces to the classical spectrum of synchrotron radiation.
• the purely classical (not quantum-corrected) LL radiation friction is also accessible in Smilei, using radiation_model = "Landau-Lifshitz" or "LL" in the Species().
Choosing the good model for your simulation¶
The next sections describe in more details the different models implemented in Smilei. For the user convenience, Table 3 briefly summarises the models and how to choose the most appropriate radiation
reaction model for your simulation.
In [Niel2018a], an extensive study of the links between the different models for radiation reaction and their domain of applicability is presented. The following table is mainly informative.
Regime \(\chi\) value Description Models
Classical radiation emission \(\chi \sim 10^{-3}\) \(\gamma_\gamma \ll \gamma\), radiated energy overestimated for \(\chi > 10^{-2}\) Landau-Lifshitz
Semi-classical radiation emission \(\chi \sim 10^{-2}\) \(\gamma_\gamma \ll \gamma\), no stochastic effects Corrected Landau-Lifshitz
Weak quantum regime \(\chi \sim 10^{-1}\) \(\gamma_\gamma < \gamma\), \(\gamma_\gamma \gg mc^2\) Stochastic model of Niel et al / Monte-Carlo
Quantum regime \(\chi \sim 1\) \(\gamma_\gamma \gtrsim \gamma\) Monte-Carlo
C++ classes for the radiation processes are located in the directory src/Radiation. In Smilei, the radiative processes are not incorporated in the pusher in order to preserve the vector performance
of the pusher when using non-vectorizable radiation models such as the Monte-Carlo process.
Description of the files:
• Class RadiationTable: useful tools, parameters and the tables.
• Class Radiation: the generic class from which will inherit specific classes for each model.
• Class RadiationFactory: manages the choice of the radiation model among the following.
• Class RadiationLandauLifshitz: classical Landau-Lifshitz radiation process.
• Class RadiationCorrLandauLifshitz: corrected Landau-Lifshitz radiation process.
• Class RadiationNiel: stochastic diffusive model of [Niel2018a].
• Class RadiationMonteCarlo: Monte-Carlo model.
As explained below, many functions have been tabulated because of the cost of their computation for each particle. Tables can be generated by the external tool smilei_tables. More information can be
found in Generation of the external tables.
Continuous, Landau-Lifshitz-like models¶
Two models of continuous radiation friction force are available in Smilei: (i) the approximation for high-math:gamma of the Landau-Lifshitz equation (taking \(g(\chi)=1\) in Eq. (26)), and (ii) the
corrected Landau-Lifshitz equation Eq. (26). The modelS are accessible in the species configuration under the name Landau-Lifshitz (equiv. LL) and corrected-Landau-Lifshitz (equiv. ‘cLL’).
The implementation of these continuous radiation friction forces consists in a modification of the particle pusher, and follows the simple splitting technique proposed in [Tamburini2010]. Note that
for the quantum correction, we use a fit of the function \(g(\chi)\) given by
\[g \left( \chi_{\pm} \right) = \left[ 1 + 4.8 \left( 1 + \chi_{\pm} \right) \log \left( 1 + 1.7 \chi_{\pm} \right) + 2.44 \chi_{\pm}^2 \right]^{-2/3}\]
This fit enables to keep the vectorization of the particle loop.
Fokker-Planck stochastic model of Niel et al.¶
Equation (23) is implemented in Smilei using a simple explicit scheme, see [Niel2018a] Sec. VI.B for more details. This stochastic diffusive model is accessible in the species configuration under the
name Niel.
The direct computation of Eq. (25) during the emission process is too expensive. For performance issues, Smilei uses tabulated values or fit functions.
Concerning the tabulation, Smilei first checks the presence of an external table at the specified path. If the latter does not exist at the specified path, the table is computed at initialization.
The new table is outputed on disk in the current simulation directory. It is recommended to use existing external tables to save simulation time. The computation of h during the simulation can slow
down the initialization and represents an important part of the total simulation. The parameters such as the \(\chi\) range and the discretization can be given in RadiationReaction.
Polynomial fits of this integral can be obtained in log-log or log10-log10 domain. However, high accuracy requires high-order polynomials (order 20 for an accuracy around \(10^{-10}\) for instance).
In Smilei, an order 5 (see Eq. (28)) and 10 polynomial fits are implemented. They are valid for quantum parameters \(\chi\) between \(10^{-3}\) and 10.
\[\begin{split}h_{o5}(\chi) = \exp{ \left(1.399937206900322 \times 10^{-4} \log(\chi)^5 \\ + 3.123718241260330 \times 10^{-3} \log{(\chi)}^4 \\ + 1.096559086628964 \times 10^{-2} \log(\chi)^3 \\
-1.733977278199592 \times 10^{-1} \log(\chi)^2 \\ + 1.492675770100125 \log(\chi) \\ -2.748991631516466 \right) }\end{split}\]
An additional fit from [Ridgers2017] has been implemented and the formula is given in Eq. (29).
\[h_{Ridgers}(\chi) = \chi^3 \frac{165}{48 \sqrt{3}} \left(1. + (1. + 4.528 \chi) \log(1.+12.29 \chi) + 4.632 \chi^2 \right)^{-7/6}\]
Monte-Carlo full-quantum model¶
The Monte-Carlo treatment of the emission is more complex process than the previous ones and can be divided into several steps ([Duclous2011], [Lobet2013], [Lobet2015]):
1. An incremental optical depth \(\tau\), initially set to 0, is assigned to the particle. Emission occurs when it reaches the final optical depth \(\tau_f\) sampled from \(\tau_f = -\log{\xi}\)
where \(\xi\) is a random number in \(\left]0,1\right]\).
2. The optical depth \(\tau\) evolves according to the field and particle energy variations following this integral:
\[ \frac{d\tau}{dt} = \int_0^{\chi_{\pm}}{ \frac{d^2N}{d\chi dt} d\chi } = \frac{2}{3} \frac{\alpha^2}{\tau_e} \int_0^{\chi_{\pm}}{ \frac{S(\chi_\pm, \chi/\chi_{\pm})}{\chi} d\chi } \equiv \frac
{2}{3} \frac{\alpha^2}{\tau_e} K (\chi_\pm)\]
that simply is the production rate of photons (computed from Eq. (15)). Here, \(\chi_{\pm}\) is the emitting electron (or positron) quantum parameter and \(\chi\) the integration variable.
3. The emitted photon’s quantum parameter \(\chi_{\gamma}\) is computed by inverting the cumulative distribution function:
\[ \xi = P(\chi_\pm,\chi_{\gamma}) = \frac{\displaystyle{\int_0^{\chi_\gamma}{ d\chi S(\chi_\pm, \chi/\chi_{\pm}) / \chi }}}{\displaystyle{\int_0^{\chi_\pm}{d\chi S(\chi_\pm, \chi/\chi_{\pm}) / \
chi }}}.\]
The inversion of \(\xi = P(\chi_\pm,\chi_{\gamma})\) is done after drawing a second random number \(\phi \in \left[ 0,1\right]\) to find \(\chi_{\gamma}\) by solving :
\[\xi^{-1} = P^{-1}(\chi_\pm, \chi_{\gamma}) = \phi\]
4. The energy of the emitted photon is then computed: \(\varepsilon_\gamma = mc^2 \gamma_\gamma = mc^2 \gamma_\pm \chi_\gamma / \chi_\pm\).
5. The particle momentum is then updated using momentum conservation and considering forward emission (valid when \(\gamma_\pm \gg 1\)).
\[ d{\bf p} = - \frac{\varepsilon_\gamma}{c} \frac{\mathbf{p_\pm}}{\| \mathbf{p_\pm} \|}\]
The resulting force follows from the recoil induced by the photon emission. Radiation reaction is therefore a discrete process. Note that momentum conservation does not exactly conserve energy.
It can be shown that the error \(\epsilon\) tends to 0 when the particle energy tends to infinity [Lobet2015] and that the error is small when \(\varepsilon_\pm \gg 1\) and \(\varepsilon_\gamma \
ll \varepsilon_\pm\). Between emission events, the electron dynamics is still governed by the Lorentz force.
If the photon is emitted as a macro-photon, its initial position is the same as for the emitting particle. The (numerical) weight is also conserved.
The computation of Eq. (30) would be too expensive for every single particles. Instead, the integral of the function \(S(\chi_\pm, \chi/\chi_{\pm}) / \chi\) also referred to as \(K(\chi_\pm)\) is
This table is named integfochi Related parameters are stored in the structure integfochi in the code.
Similarly, Eq. (31) is tabulated (named xi in the code). The only difference is that a minimum photon quantum parameter \(\chi_{\gamma,\min}\) is computed before for the integration so that:
\[ \frac{\displaystyle{\int_{0}^{\chi_{\gamma,\min}}{d\chi S(\chi_\pm, \chi/\chi_{\pm}) / \chi}}} {\displaystyle{\int_0^{\chi_\pm}{d\chi S(\chi_\pm, \chi/\chi_{\pm}) / \chi}}} < \epsilon\]
This enables to find a lower bound to the \(\chi_\gamma\) range (discretization in the log domain) so that the remaining part is negligible in term of radiated energy. The parameter \(\epsilon\) is
called xi_threshold in RadiationReaction and the tool smilei_tables (Generation of the external tables.).
The Monte-Carlo model is accessible in the species configuration under the name Monte-Carlo or mc.
Radiation emission by ultra-relativistic electrons in a constant magnetic field¶
This benchmark closely follows benchmark/tst1d_18_radiation_spectrum_chi0.1.py. It considers a bunch of electrons with initial Lorentz factor \(\gamma=10^3\) radiating in a constant magnetic field.
The magnetic field is perpendicular to the initial electrons’ velocity, and its strength is adjusted so that the electron quantum parameter is either \(\chi=0.1\) or \(\chi=1\). In both cases, the
simulation is run over a single gyration time of the electron (computed neglecting radiation losses), and 5 electron species are considered (one neglecting all radiation losses, the other four each
corresponding to a different radiation model: LL, cLL, FP and MC).
In this benchmark, we focus on the differences obtained on the energy spectrum of the emitted radiation considering different models of radiation reaction. When the Monte-Carlo model is used, the
emitted radiation spectrum is obtained by applying a ParticleBinning diagnostic on the photon species. When other models are considered, the emitted radiation spectrum is reconstructed using a
RadiationSpectrum diagnostic, as discussed in RadiationSpectrum diagnostics, and given by Eq. (20) (see also [Niel2018b]). Fig. 28 presents for both values of the initial quantum parameter \(\chi=0.1
\) and \(\chi=1\) the resulting power spectra obtained from the different models, focusing of the (continuous) corrected-Landau-Lifshitz (cLL), (stochastic) Fokker-Planck (Niel) and Monte-Carlo (MC)
models. At \(\chi=0.1\), all three descriptions give the same results, which is consistent with the idea that at small quantum parameters, the three descriptions are equivalent. In contrast, for \(\
chi=1\), the stochastic nature of high-energy photon emission (not accounted for in the continuous cLL model) plays an important role on the electron dynamics, and in turns on the photon emission.
Hence only the two stochastic model give a satisfactory description of the photon emitted spectra. More details on the impact of the model on both the electron and photon distribution are given in
Counter-propagating plane wave, 1D¶
In the benchmark benchmark/tst1d_09_rad_electron_laser_collision.py, a GeV electron bunch is initialized near the right domain boundary and propagates towards the left boundary from which a plane
wave is injected. The laser has an amplitude of \(a_0 = 270\) corresponding to an intensity of \(10^{23}\ \mathrm{Wcm^{-2}}\) at \(\lambda = 1\ \mathrm{\mu m}\). The laser has a Gaussian profile of
full-with at half maxium of \(20 \pi \omega_r^{-1}\) (10 laser periods). The maximal quantum parameter \(\chi\) value reached during the simulation is around 0.5.
Fig. 29 shows that the Monte-Carlo, the Niel and the corrected Landau-Lifshitz models exhibit very similar results in term of the total radiated and kinetic energy evolution with a final radiation
rate of 80% the initial kinetic energy. The relative error on the total energy is small (\(\sim 3\times10^{-3}\)). As expected, the Landau-Lifshitz model overestimates the radiated energy because the
interaction happens mainly in the quantum regime.
Fig. 30 shows that the Monte-Carlo and the Niel models reproduce the stochastic nature of the trajectories as opposed to the continuous approaches (corrected Landau-Lifshitz and Landau-Lifshitz). In
the latter, every particles initially located at the same position will follow the same trajectories. The stochastic nature of the emission for high \(\chi\) values can have consequences in term of
final spatial and energy distributions. Not shown here, the Niel stochastic model does not reproduce correctly the moment of order 3 as explained in [Niel2018a].
Synchrotron, 2D¶
A bunch of electrons of initial momentum \(p_{-,0}\) evolves in a constant magnetic field \(B\) orthogonal to their initial propagation direction. In such a configuration, the electron bunch is
supposed to rotate endlessly with the same radius \(R = p_{-,0} /e B\) without radiation energy loss. Here, the magnetic field is so strong that the electrons radiate their energy as in a synchrotron
facility. In this setup, each electron quantum parameter depends on their Lorentz factors \(\gamma_{-}\) according to \(\chi_{-} = \gamma_{-} B /m_e E_s\). The quantum parameter is maximum at the
beginning of the interaction. The strongest radiation loss are therefore observed at the beginning too. As energy decreases, radiation loss becomes less and less important so that the emission regime
progressively move from the quantum to the classical regime.
Similar simulation configuration can be found in the benchmarks. It corresponds to two different input files in the benchmark folder:
• tst2d_08_synchrotron_chi1.py: tests and compares the corrected Landau-Lifshitz and the Monte-Carlo model for an initial \(\chi = 1\).
• tst2d_09_synchrotron_chi0.1.py: tests and compares the corrected Landau-Lifshitz and the Niel model for an initial \(\chi = 0.1\).
In this section, we focus on the case with initial quantum parameter \(\chi = 0.1\). The magnetic field amplitude is \(B = 90 m \omega_r / e\). The initial electron Lorentz factor is \(\gamma_{-,0} =
\varepsilon_{-,0}/mc^2 = 450\). Electrons are initialized with a Maxwell-Juttner distribution of temperature \(0.1 m_e c^2\).
Fig. 31 shows the time evolution of the particle kinetic energy, the radiated energy and the total energy. All radiation models provide similar evolution of these integrated quantities. The relative
error on the total energy is between \(2 \times 10^{-9}\) and \(3 \times 10^{-9}\).
The main difference between models can be understood by studying the particle trajectories and phase spaces. For this purpose, the local kinetic energy spatial-distribution at \(25 \omega_r^{-1}\) is
shown in Fig. 32 for the different models. With continuous radiation energy loss (corrected Landau-Lifshitz case), each electron of the bunch rotates with a decreasing radius but the bunch. Each
electron of similar initial energies have the same trajectories. In the case of a cold bunch (null initial temperature), the bunch would have kept its original shape. The radiation with this model
only acts as a cooling mechanism. In the cases of the Niel and the Monte-Carlo radiation models, stochastic effects come into play and lead the bunch to spread spatially. Each individual electron of
the bunch, even with similar initial energies, have different trajectories depending on their emission history. Stochastic effects are particularly strong at the beginning with the highest \(\chi\)
values when the radiation recoil is the most important.
Fig. 33 shows the time evolution of the electron Lorentz factor distribution (normalized energy) for different radiation models. At the beginning, the distribution is extremely broad due to the
Maxwell-Juttner parameters. The average energy is well around \(\gamma_{-,0} = \varepsilon_{-,0}/mc^2 = 450\) with maximal energies above \(\gamma_{-} = 450\).
In the case of a initially-cold electron beam, stochastic effects would have lead the bunch to spread energetically with the Monte-Carlo and the Niel stochastic models at the beginning of the
simulation. This effect is hidden since electron energy is already highly spread at the beginning of the interaction. This effect is the strongest when the quantum parameter is high in the quantum
In the Monte-Carlo case, some electrons have lost all their energy almost immediately as shown by the lower part of the distribution below \(\gamma_{-} = 50\) after comparison with the Niel model.
Then, as the particles cool down, the interaction enters the semi-classical regime where energy jumps are smaller. In the classical regime, radiation loss acts oppositely to the quantum regime. It
reduces the spread in energy and space. In the Landau-Lifshitz case, this effect starts at the beginning even in the quantum regime due to the nature of the model. For a initially-cold electron
bunch, there would not have been energy spread at the beginning of the simulation. All electron would have lost their energy in a similar fashion (superimposed behavior). This model can be seen as
the average behavior of the stochastic ones of electron groups having the same initial energy.
Thin foil, 2D¶
This case is not in the list of available benchmarks but we decided to present these results here as an example of simulation study. An extremely intense plane wave in 2D interacts with a thin,
fully-ionized carbon foil. The foil is located 4 µm from the left border (\(x_{min}\)). It starts with 1 µm of linear pre-plasma density, followed by 3 µm of uniform plasma of density 492 times
critical. The target is irradiated by a gaussian plane wave of peak intensity \(a_0 = 270\) (corresponding to \(10^{23}\ \mathrm{Wcm^{-2}}\)) and of FWHM duration 50 fs. The domain has a
discretization of 64 cells per µm in both directions x and y, with 64 particles per cell. The same simulation has been performed with the different radiation models.
Electrons can be accelerated and injected in the target along the density gradient through the combined action of the transverse electric and the magnetic fields (ponderomotive effects). In the
relativistic regime and linear polarization, this leads to the injection of bunches of hot electrons every half laser period that contribute to heat the bulk. When these electrons reach the rear
surface, they start to expand in the vacuum, and, being separated from the slow ion, create a longitudinal charge-separation field. This field, along the surface normal, has two main effects:
• It acts as a reflecting barrier for electrons of moderate energy (refluxing electrons).
• It accelerates ions located at the surface (target normal sheath acceleration, TNSA).
At the front side, a charge separation cavity appears between the electron layer pushed forward by the ponderomotive force and ions left-behind that causes ions to be consequently accelerated. This
strong ion-acceleration mechanism is known as the radiation pressure acceleration (RPA) or laser piston.
Under the action of an extremely intense laser pulse, electrons accelerated at the target front radiate. It is confirmed in Fig. 34 showing the distribution of the quantum parameter \(\chi\) along
the x axis for the Monte-Carlo, the Niel and the corrected Landau-Lifshitz (CLL) radiation models. The maximum values can be seen at the front where the electrons interact with the laser. Radiation
occurs in the quantum regime \(\chi > 0.1\). Note that there is a second peak for \(\chi\) at the rear where electrons interact with the target normal sheath field. The radiation reaction can affect
electron energy absorption and therefore the ion acceleration mechanisms.
The time evolutions of the electron kinetic energy, the carbon ion kinetic energy, the radiated energy and the total absorbed energy are shown in Fig. 35. The corrected-Landau-Lifshitz, the Niel and
the Monte-Carlo models present very similar behaviors. The absorbed electron energy is only slightly lower in the Niel model. This difference depends on the random seeds and the simulation
parameters. The radiated energy represents around 14% of the total laser energy. The classical Landau-Lifshitz model overestimates the radiated energy; the energy absorbed by electrons and ions is
therefore slightly lower. In all cases, radiation reaction strongly impacts the overall particle energy absorption showing a difference close to 20% with the non-radiative run.
The differences between electron \(p_x\) distributions are shown in Fig. 36. Without radiation reaction, electrons refluxing at the target front can travel farther in vacuum (negative \(p_x\)) before
being injected back to the target. With radiation reaction, these electrons are rapidly slowed down and newly accelerated by the ponderotive force. Inside the target, accelerated bunches of hot
electrons correspond to the regular positive spikes in \(p_x\) (oscillation at \(\lambda /2\)). The maximum electron energy is almost twice lower with radiation reaction.
The cost of the different models is summarized in Table 4. Reported times are for the field projection, the particle pusher and the radiation reaction together. Percentages correspond to the overhead
induced by the radiation module in comparison to the standard PIC pusher.
All presented numbers are not generalizable and are only indicated to give an idea of the model costs. The creation of macro-photons is not enabled for the Monte-Carlo radiation process.
Radiation model None LL CLL Niel MC
Counter-propagating Plane Wave 1D Haswell (Jureca) 0.2s 0.23s 0.24s 0.26s 0.3s
Synchrotron 2D Haswell (Jureca) \(\chi=0.05\), \(B=100\) 10s 11s 12s 14s 15s
Synchrotron 2D Haswell (Jureca) \(\chi=0.5\), \(B=100\) 10s 11s 12s 14s 22s
Synchrotron 2D KNL (Frioul) \(\chi=0.5\), \(B=100\) 21s 23s 23s 73s 47s
Interaction with a carbon thin foil 2D Sandy Bridge (Poincare) 6.5s 6.5s 6.6s 6.8s 6.8s
Descriptions of the cases:
• Counter-propagating Plane Wave 1D: run on a single node of Jureca with 2 MPI ranks and 12 OpenMP threads per rank.
• Synchrotron 2D: The domain has a dimension of 496x496 cells with 16 particles per cell and 8x8 patches. A 4th order B-spline shape factor is used for the projection. The first case has been run
on a single Haswell node of Jureca with 2 MPI ranks and 12 OpenMP threads per rank. the second one has been run on a single KNL node of Frioul configured in quadrant cache using 1 MPI rank and 64
OpenMP threads. On KNL, the KMP_AFFINITY is set to fine and scatter.
Only the Niel model provides better performance with a compact affinity.
• Thin foil 2D: The domain has a discretization of 64 cells per \(\mu\mathrm{m}\) in both directions, with 64 particles per cell. The case is run on 16 nodes of Poincare with 2 MPI ranks and 8
OpenMP threads per rank.
The LL and CLL models are vectorized efficiently. These radiation reaction models represent a small overhead to the particle pusher.
The Niel model implementation is split into several loops to be partially vectorized. The table lookup is the only phase that can not be vectorized. Using a fit function enables to have a fully
vectorized process. The gain depends on the order of the fit. The radiation process with the Niel model is dominated by the normal distribution random draw.
The Monte-Carlo pusher is not vectorized because the Monte-Carlo loop has not predictable end and contains many if-statements. When using the Monte-Carlo radiation model, code performance is likely
to be more impacted running on SIMD architecture with large vector registers such as Intel Xeon Phi processors. This can be seen in Table 4 in the synchrotron case run on KNL. | {"url":"https://smileipic.github.io/Smilei/Understand/radiation_loss.html","timestamp":"2024-11-10T20:28:17Z","content_type":"text/html","content_length":"96046","record_id":"<urn:uuid:631c0102-ecb2-42df-84b6-0741f3455381>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00474.warc.gz"} |
Distributome Blog
In March 2012, we expanded the collection of Distributome calculators, simulators and experiments.
In addition we introduced a new mechanism to infer the unique URLs (HTML wrappers) for the Distributome Tools (calculators, simulators and experiments). When users navigate the Distributome universe,
the second panel on the right side contains dynamic links to Sim, Calc and Exp that are specific for the user-selected distribution. These links need to be dynamically generated using the name of the
Distribution. In distributome.js, this dynamic linking is accomplished via:
• document.getElementById(‘distributome.calculator’).href = ‘./calc/’+firstChar+nodeName+’Calculator.html‘;
• document.getElementById(‘distributome.experiment’).href = ‘./exp/’+firstChar+nodeName+’Experiment.html‘;
• document.getElementById(‘distributome.simulation’).href = ‘./sim/’+firstChar+nodeName+’Simulation.html‘;
In Distributome Navigator these 3 HTML tags/fields are defined as:
• <li><a href=“./calc/NormalCalculator.html” target=“_blank” title=“Interactive Distribution Calculator” id=“distributome.calculator“>Calculator</a></li>
• <li><a href=“./exp/PoissonExperiment.html” target=“_blank” title=“Run Virtual Distribution Experiment” id=“distributome.experiment“>Experiment</a></li>
• <li><a href=“./sim/NormalSimulation.html” target=“_blank” title=“Distribution Sampling and Simulation” id=“distributome.simulation“>Simulation</a></li>
We are looking for the best (most-efficient, most-reliable, most-scalable, most-extensible) solution to this mapping issue.
Distributome Web-applications Updates
We expanded the Distributome experiments (13), simulators (19), and calculators (19). We are continuing to expand all collections of web-apps, enhance their functionality, integrate with
other Distributome resources (e.g., activities) and improve the user experiences.
Distributome Data & Activity: Horse Kicks
In 1898, the Polish statistician and economist Ladislaus von Bortkiewicz published his famous book “Das Gesetz der kleinen Zahlen” (translation: The Law of Small Numbers). The book contained his
analysis of some fascinating data sets on the occurrence of rare events in large populations. In one case Bortkiewicz analyzed the number of soldiers in each corps of the Prussian cavalry who were
killed by being kicked by horses between the years 1875 and 1894. There were fourteen different corps examined and the data are available below. Ten of the fourteen corps had twenty squadrons with
soldiers in similar positions while the other four had features indicating substantive differences in their populations. Thus, Bortkiewicz argued that these four corps might be excluded from
analyses of the data. He writes (as translated by C.P. Winsor, 1947: Human Biology 19:154-161):
The Guard Corps contains, apart from artillery, engineers and trainees, 134 infantry companies and 40 cavalry squadrons; the XI corps has three divisions; the I corps has 30 and the VI corps has
25 squadrons, against a norm of 20 squadrons.
Problem 1: Explain why the number of soldiers in any one of the fourteen Prussian cavalry corps killed by horse kicks might be reasonably modeled by a Poisson distribution.
Problem 2: Consider the total number of soldiers killed by horse kicks in the fourteen corps put together (even including the four identified by Bortkiewicz as being different). What distribution
would provide a good model for those data?
Problem 3: Let’s compare the number of soldiers killed by horse kicks in the data to what would be expected under the Poisson probability model.
1. How well does the data fit the model if you suppose the rate of being killed by a horse kick is the same from corps to corps and year-to-year for the ten corps Bortkiewicz believes are similar?
2. How well does the data fit the model if you suppose the rate of being killed by a horse kick is the same from corps to corps and year-to-year for all fourteen corps in the data set?
3. Does allowing each corps to have its own rate of horse-kick deaths improve the fit of the model? Does allowing for different years to have different rates improve the fit of the model?
4. Researchers Preece, Ross, and Kirby suggest that corps-to-corps and year-to-year differences in average rates may be modeled as random draws from a Gamma distribution. If their idea is true,
what would be an appropriate model for the number of deaths by horse-kicks?
Data Description
These data indicate the number of deaths by horse-kicks in the Prussian Army from 1875 to 1894 for 14 army corps. The data are derived from Andrews and Herzberg’s book(1985, p. 18). Originally
published in the 1898 book “The Law of Small Numbers” by the Polish statistician and economist Ladislaus von Bortkiewicz. Ten of the corps have a similar structure of 20 squadrons each and performed
similar duties. The Guard Corps, Corps I, Corps VI, and Corps XI have different structures and performed somewhat different tasks then the others.
Data Download
Text Raw data: Distributome Data: Horse Kicks (*.txt file)
HTML Data Table
│Year │Guard.corps │corpsI │corpsII │corpsIII │corpsIV │corpsV │corpsVI │corpsVII │corpsVIII │corpsIX │corpsX │corpsXI │corpsXIV │corpsXV │
│1875 │0 │0 │0 │0 │0 │0 │0 │1 │1 │0 │0 │0 │1 │0 │
│1876 │2 │0 │0 │0 │1 │0 │0 │0 │0 │0 │0 │0 │1 │1 │
│1877 │2 │0 │0 │0 │0 │0 │1 │1 │0 │0 │1 │0 │2 │0 │
│1878 │1 │2 │2 │1 │1 │0 │0 │0 │0 │0 │1 │0 │1 │0 │
│1879 │0 │0 │0 │1 │1 │2 │2 │0 │1 │0 │0 │2 │1 │0 │
│1880 │0 │3 │2 │1 │1 │1 │0 │0 │0 │2 │1 │4 │3 │0 │
│1881 │1 │0 │0 │2 │1 │0 │0 │1 │0 │1 │0 │0 │0 │0 │
│1882 │1 │2 │0 │0 │0 │0 │1 │0 │1 │1 │2 │1 │4 │1 │
│1883 │0 │0 │1 │2 │0 │1 │2 │1 │0 │1 │0 │3 │0 │0 │
│1884 │3 │0 │1 │0 │0 │0 │0 │1 │0 │0 │2 │0 │1 │1 │
│1885 │0 │0 │0 │0 │0 │0 │1 │0 │0 │2 │0 │1 │0 │1 │
│1886 │2 │1 │0 │0 │1 │1 │1 │0 │0 │1 │0 │1 │3 │0 │
│1887 │1 │1 │2 │1 │0 │0 │3 │2 │1 │1 │0 │1 │2 │0 │
│1888 │0 │1 │1 │0 │0 │1 │1 │0 │0 │0 │0 │1 │1 │0 │
│1889 │0 │0 │1 │1 │0 │1 │1 │0 │0 │1 │2 │2 │0 │2 │
│1890 │1 │2 │0 │2 │0 │1 │1 │2 │0 │2 │1 │1 │2 │2 │
│1891 │0 │0 │0 │1 │1 │1 │0 │1 │1 │0 │3 │3 │1 │0 │
│1892 │1 │3 │2 │0 │1 │1 │3 │0 │1 │1 │0 │1 │1 │0 │
│1893 │0 │1 │0 │0 │0 │1 │0 │2 │0 │0 │1 │3 │0 │0 │
│1894 │1 │0 │0 │0 │0 │0 │0 │0 │1 │0 │1 │1 │0 │0 │
Sample Sizes and the Accuracy of Polls
Distributome Activity on Sample Sizes and the Accuracy of Polls
Surveys about public opinions on controversial social issues are becoming increasingly frequent as topics such as the legalization of marijuana, abortion policy, marriage rights for homosexuals, and
immigration policy are hotly debated in the media.
For example, both the Opinion Research Corporation (polling for CNN) and the Pew Research Center for The People and The Press conducted surveys of American adults in spring of 2011 to estimate the
percentage of the public that favors the legalization of marijuana. The sample sizes in the two polls were 824 for the Opinion Research Corporation poll and 1504 for the Pew poll.
This activity illustrates the inter-distribution relationships between Cauchy, Student’s T and Standard Normal (Gaussian) distributions. These relationships are used to provide a better
understanding of how strongly sample sizes are related to the accuracy of polls.
Hands-on Activity
In this activity you may assume that both of these pollsters use similar techniques that involve telephone interviews and weighting the answers given by individuals to align the respondents
demographics with population values and finally averaging to produce unbiased and essentially normally distributed estimates. Below are 4 related, but complementary, problems regarding this study.
Note: The problems below may be appropriate for an undergraduate course in probability. The last part (Problem 4) would be more appropriate for masters level course (and should have a General Cauchy
distribution tag and a tag for its relationship to the bivariate normal).
Specific Problems and their Solutions
Problem 1: Difference in Poll Accuracies?
The Pew poll had almost twice the sample size of the Opinion Research Corporation poll. What is the chance that it was more accurate than that poll for estimating p = the percentage of American
adults that favored the legalization of marijuana in spring, 2011? Be sure to clearly define how you are interpreting “more accurate.” Also state and justify any assumptions you make in solving for
this probability.
Alternative approaches
Alternative 1: Ratio of bivariate Normal variables.
See a Solution to First Problem: Alternative 1
Alternative 2: Direct calculation of the marginal distribution
See a Solution to First Problem: Alternative 2
Problem 2: Pooling Data across Polls?
Describe how you would combine the data from these two polls to form a single estimate of \(p\).
The obvious choice is to propose a linear combination of the two estimates weighting inversely proportional to the variances to get the smallest overall variance amongst such linear combinations.
Problem 3: Are these probability estimates correlated?
What is the correlation between your estimate above and the individual estimate produced by the Pew poll?
Note that \(Cov(\hat{p},\hat{p_1})=Cov(\frac{1504}{2328}\hat{p_1}+\frac{824}{2328}\hat{p_2}, \hat{p_1})=\frac{1504}{2328}\sigma_1^2=\frac{\sigma^2}{2328}\).
Problem 4: Accuracy of probability estimates?
What is the probability that your combined estimate (from the second problem) is more accurate than the estimate based only on the Pew poll?
We want \(P[|\hat{p}-p| < |\hat{p_1}-p| ]\).
Generalized Cauchy distribution CDF derivation
To derive the generalized Cauchy distribution CDF directly, we start with the bivariate normal distribution of ”X” and ”Y”:
See the Derivation of the Cauchy CDF
Cauchy, Student’s T and Gaussian distribution interrelations
The Student’s T-distribution represents a one-parameter homotopy path connecting Cauchy and Gaussian Distribution:
• \(Cauchy=T_{(df=1)} \longrightarrow T_{(df)}\longrightarrow N(0,1)=T_{(df=\infty)}\).
• See the Distributome Navigator, Cauchy, Student’s T and Gaussian distribution calculators.
• Explore the relations between these and many other probability distributions using the interactive graphical Distributome Navigator.
Increasing the sample size may help significantly in certain situations – but not as much as intuition often suggests.
Homicides Trend Activity
Distributome Homicide Trends Activity
A Columbus Dispatch newspaper story on Friday January 1, 2010 discussed a drop in the number of homicides in the city the previous year. Here are the first few paragraphs from the article:
• Homicides take big drop in city: Trend also being seen nationally, but why is a mystery.
• The number of homicides in Columbus dropped 25 percent last year after spiking in 2008. As of last night, the city was expected to close out 2009 with 83 homicides, 27 fewer than in 2008,
according to records kept by police and The Dispatch. In 2007, 79 people were slain in Columbus. “I don’t know that there’s one reason for homicides going up or down,” said Lt. David Watkins,
supervisor of the Police Division’s homicide unit.
• Why one year do we have 130, and then the next year we have 80?
• “You just can’t explain it,” Sgt. Dana Norman said. He supervises the third-shift squad that investigated 44 of last year’s homicides, which occurred at a rate of 11.1 for every 100,000 people in
Columbus, based on recent population estimates …
A table appearing with the article showed that there were 568 homicides in the previous 6 years.
Hands-on Activity
Sargent Norman’s statement that “”You just can’t explain it”” presents an intriguing probability question – Is it possible that natural random fluctuation might be a good explanation? Let’s consider
probability models for the number of observed crimes and how they might fluctuate to see if the data mentioned in the article is unusual.
• If homicides are rare events that might be independently perpetrated by individuals in a large population – what distribution would approximately describe the number of murders in a year?
A reasonable model would be the Poisson distribution (since the mean is quite large, a normal model with equal mean and variance would be an alternative approximation).
• Suppose the expected annual number of homicides in the city is denoted by \(\lambda\) and that the number of homicides is independent from year to year. The article notes that 2008 saw a “spike”
in the number of homicides and in fact that was the highest number in the last six years. If nothing is going on except random fluctuations – we want to know if observing 27 fewer homicides in
2009 after the peak year is unusual (peak here meaning the highest in the last 6 years).
Use the Distributome Poisson simulator for the model you specified above to examine the distribution of the change in the number of homicides you would see following a peak of a six year stretch.
Does the 27 murder drop seem unusual? Explain.
See a Hint
To get started, you will need
i) to find an estimate of \(\lambda\) to use in your simulations, and
ii) to examine groups of 7 years of simulated homicide data and isolate those cases that satisfy the conditions of the problem.
See the First part of the Answer
There were 568 homicides in the preceding six years so a reasonable estimate of \(\lambda\) would be \(\lambda \equiv \frac{568}{6} \equiv \frac{284}{3} \equiv 82.67\). From a simulation of 100,000
sets of six independent Poisson variables, we find the maximum would have a distribution with a histogram that looks like this image.
See the Second part of the Answer
The difference between this maximum and another independent \(Poisson(\lambda \equiv 82.67)\) variable would have a histogram that looks like the following:
The shaded region corresponds to values of at least 27, which happens about 12% of the time so the drop of homicides in Columbus would not be particularly unusual when nothing is happening but
regular random fluctuations.
Alternative Approach
This problem might also be viewed as an example of the regression effect where you should expect a regression to the mean following a very high observed value.
When viewing a random process over time it is the extremes that make the headlines – so the probability models we should use to answer the question “What is unusual?” should be probability models
about extremes.
Colorblindness Activity
Distributome Colorblindness Activity
Colorblindness – Can you see the number in this image?
This Distributome Activity illustrates an application of probability theory to study Colorblindness, typically a genetic disorder which results from an abnormality on the X chromosome. The condition
is thus rarer in women since a woman would need to have the abnormality on both of her X chromosomes in order to be colorblind (whether a woman has the abnormality on one X chromosome is essentially
independent of having it on the other).
The goal of this activity is to demonstrate an efficient protocol of estimating the probability that a randomly chosen individual may be colorblind.
Hands-on Activity
Suppose that \(p\) is the probability that a randomly selected ”man” is colorblind.
Alternate approach
You can also use the delta method to find the approximate variance for the estimator above.
In practice, it may difficult to obtain reliable parameter estimates when the event at hand is very rare (such as with colorblindness in women). The use of a valid probability model such as the
relationship between the chance of colorblindness in men and the chance in women may improve these estimates.
Distributome Blog allows LaTeX Post Editing Using MathJax
The Distributome Blog now allows editing using MathJax-based math typography. For example:
• Typing \\( \\int\_{\\pi}^{\\infty}{\\ln (x) dx} \\), replace the “\\” by “\” to render the formula in the blog page,
• Would generate this: \( \int_{\pi}^{\infty}{\ln (x) dx} \)
For hidden fields you need to use the following alternatives as analogues of commonly used TeX/LaTeX syntax (as the JavaScript code behind the MathJax and the Hidden-answers plug-ins are
• For equal-sign “=”, use the \( \\equiv \) symbol (\(\equiv \))
• For vertical bar “|”, use \( \\vert \) symbol (\(\vert \))
• There may be other LaTeX/TeX alternative symbols that may need to be used for MathJax math typesetting in hidden fields!
Distributome Navigator: Ontology/Hierarchical Graph Display
We introduced a new Distributome.xml.pref file, which allows customization of the look-and-feel of the Distributome Navigator and Editor.
One example of this customization is the ability to display hierarchically the Ontology of the collection of distributions contain in the Distributome XML DB. That is we have a mechanism to render
the nodes and edges as 3 levels: Top, Middle or All/Complete levels (affording to the pref file). The figure below illustrates this new hierarchical Distributome Navigator display.
Distributome BibTeX citation manager
We have finalized the new format for the Distributome meta-data about (XML) distributions and relations (Distributome.xml) and (BiBTeX) bibliographical citations (Distributome.bib).
There is a new Distributome DB/Meta-data HTML validator which renders the entire database, including references and citation URL links into a dynamic HTML webpage.
Background/Initial Proposal
The Distributome meta-data editor will provide an interactive bibliography BibTeX citation manager. This prototype contains an example demonstrating how we can elegantly handle Citations/references
(parsing, editing, writing, etc.) using pure HTML5/JavaScript:
1. Background: Using BibTex-js project
2. Example HTML (Distributome_BibTeX.html) that interactively consumes raw BiBTeX source files, converts them to JS/JSON and displays the references in HTML page.
3. An example of raw BibTeX source file (BibTeX_ExampleCitations.bib). These BibTeX sources can easily be obtained by users from the “Citation Download” or “Export Citation” links on most
publisher’s web-sites (See this example). So, these BiB sources are very easy to copy-paste into the Distributome References Editor Panel from another web-browser-window.
4. A JavaScript library (bibtex_js.js), which we may need to extend, that allows parsing BiB files and generating the JSON constructs hat are then rendered in the HTML (example, during editing in
the Bibliography/References panel of the Distributome Editor, or in the References tab of the Distributome Navigator, during
We decouple the references Section (Distributome_BibTeX.bib) from the main Distributome.xml DB, as the Distributome BiBTeX reference may increase to become very large (error prone). Thus we’ll need
to have a way to reference publications (from Distributome_BibTeX.bib) into theDistributome.xml. This can be achieved by the DOI (Unique Digital Object Identifier) or the URL that every publication
has. So inside the <cite>Publication_DOI_or_URL</cite> tag of distribution-nodes or relation-edges in the main Distributome.xml DB, we’ll just have pointers to unique DOI’s – the same unique DOI/URL
will be available in the Distributome_BibTeX.bib source file. Hence we can pair the references by their unique DOIs/URLs.
Then inside the Distributome_BibTeX.bib, each reference will have its unique DOI – this will enable linking of Bibliographic/references meta-data contained in Distributome_BibTeX.bib (on-demand) from
inside the Distributome.xml and the Navigator, itself. This may be an easy, clean and scalable approach.
There are many examples of resources for retrieving BibTeX publication references (bibliographic citations):
Below are examples of the second generation (V2) of the Distributome XML meta-data – this version decouples the meta information about distirbutions (nodes) and their relation (edges) from the
reference citation management (using BibTeX):
• Distributome.bib archive
Both the XML and the BibTeX files still need to be expanded, but they illustrate the integration (mapping) between distributions(nodes)/relations(edges) and the corresponding bibliographical
references (citations) using the unique Digital Object Identifiers (DOI) or URL addresses, specific to each publication/citation. See
• <cite>DOI</cite> tags in the DistributomeV2.xml source, and
• the standard Bib/TeX syntax encoding the bibliographical references in the BiBTeX source.
Distributome Update: December 07, 2011
• Distributome Human XML DB Search. We modified the search functionality to include:
□ Boolean expressions (AND, OR and NOT, all documented here)
□ Explicit URL addresses for all DB search queries that can be referenced and bookmarked (e.g., http://www.distributome.org/js/DistributomeDBSearch.xml.php?s=Normal+AND+Cauchy+NOT+nonlinear)
□ The Hierarchical Formatting of the search query results is done using JQuery Accordion mechanism.
□ Working on an integrated “Distributome Search” functionality (with standard or advanced search features) that allows complete searching the entire Distributome Site, Wiki resources (e.g.,
Activities) and DB.
• Bibliography (Reference Manager) – we are still exploring BibJSON. Jim’s Distributome BibJSON construct looks great. We just need to figure out how to computationally consume (parse, read, load)
and produce (revise, update, modify, save) these references programmatically. Looking for specific tools and examples of how we can accomplish these 2 operations from the Distributome site/
webapps? Are there open-source HTML5/JS parsers for BibJSON and how to tie these with the Distributome.xml DB?
• Resource Debugging: For technical users, we’ve introduced an optional debugging functionality documented here. This is the infrastructure that we’ll be now populating.
• Distributome Editor: We are in the process of implementing the user-interface for editing the XML DB meta-data in the browser. Aiming to complete this in the next 1-2 months.
• Distributome Navigator Layered/Multiscale View: Trying to simplify the Navigator view by introducing hierarchical multiscale rendering of the Distributome DB (nodes and edges). We’re working on
this and the approach is to employ a new Distributome.xml.pref (preferences file) that allows specifying diverse run-time Navigator behaviors, incl. the hierarchy of Distributions/Relations to
display. Please have a look at the current (15+15+All) list of 3-level hierarchy and let us know if we need modifications. This functionality will be live in the next few weeks. | {"url":"http://distributome.org/blog/?author=1&paged=2","timestamp":"2024-11-08T09:27:28Z","content_type":"text/html","content_length":"80265","record_id":"<urn:uuid:1d61f31c-9dd5-43da-9af8-ba84f0f146a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00410.warc.gz"} |
Molecular Simulation/The Lennard-Jones Potential - Wikibooks, open books for an open world
The Lennard-Jones potential describes the interactions of two neutral particles using a relatively simple mathematical model. Two neutral molecules feel both attractive and repulsive forces based on
their relative proximity and polarizability. The sum of these forces gives rise to the Lennard-Jones potential, as seen below:
The Lennard-Jones 6-12 potential approximates the intermolecular interactions of two atoms due to Pauli repulsion and London dispersion attraction. The potential is defined in terms of the well-depth
(${\displaystyle \epsilon }$) and the intercept (${\displaystyle \sigma }$). Other formulations use the radius where the minimum occurs, ${\displaystyle R_{min}}$, instead of ${\displaystyle \sigma }
${\displaystyle {\mathcal {V}}\left(r\right)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]=\varepsilon \left[\left({\frac {R_{min}}{r}}\right)
^{12}-2\left({\frac {R_{min}}{r}}\right)^{6}\right]}$
where ε is the potential well depth, σ is the distance where the potential equals zero (also double the van der Waals radius of the atom), and R[min] is the distance where the potential reaches a
minimum, i.e. the equilibrium position of the two particles.
The relationship between the ${\displaystyle \sigma }$ and ${\displaystyle R_{\min }}$ is ${\displaystyle R_{min}={\sqrt[{6}]{2}}\sigma }$
The first half of the Lennard-Jones potential is Pauli-Repulsion. This occurs when two closed shell atoms of molecules come in close proximity to each other and their electron density distributions
overlap. This causes high inter-electron repulsion and extremely short distances, inter-nuclei repulsion. This repulsion follows an exponential distribution of electron density:
${\displaystyle {\mathcal {V}}_{rep}\left(r\right)=Ae^{-cr}}$
where A and c are constants and r is the intermolecular distance. In a liquid, however, it is very unlikely that two particles will be at highly repulsive distances, and thus a simplified expression
can be used by assuming the potential has a r^-12 dependence (note that this high exponent means that the energy of repulsion drops off extremely fast as molecules separate). The resulting simple
polynomial is as follows:
${\displaystyle {\mathcal {V}}\left(r\right)={\frac {C_{12}}{r^{12}}}}$
Where the C[12] coefficient is defined as:
${\displaystyle C_{12}=4\varepsilon \sigma ^{12}=\varepsilon R_{min}^{12}}$
${\displaystyle \Phi _{12}(r)=A\exp \left(-Br\right)-{\frac {C}{r^{6}}}}$
The Second half of the Lennard-Jones potential is known as London dispersion, or induced Dipole-Dipole interaction. While a particular molecule may not normally have a dipole moment, at any one
instant in time its electrons may be asymmetrically distributed, giving an instantaneous dipole. The strength of these instantaneous dipoles and thus the strength of the attractive force depends on
the polarizability and ionization potential of the molecules. The ionization potential measures how strongly outer electrons are held to the atoms. The more polarizable the molecule, the more its
electron density can distort, creating larger instantaneous dipoles. Much like with Pauli Repulsion, this force is dependent on a coefficient, C[6], and also decays as the molecules move further
apart. In this case, the dependence is r^-6
${\displaystyle {\mathcal {V}}\left(r\right)={\frac {-C_{6}}{r^{6}}}}$
${\displaystyle C_{6}={\frac {3}{2}}\alpha _{1}^{'}\alpha _{2}^{'}{\frac {I_{1}I_{2}}{I_{1}+I_{2}}}}$
Where α' is the polarizability, usually given as a volume and I is the ionization energy, usually given as electron volts. Lastly, the C[6] constant can also be expressed by the variables as seen in
the Lennard-Jones equation:
${\displaystyle C_{6}=4\varepsilon \sigma ^{6}=2\varepsilon R_{min}^{6}}$
The estimation of Lennard-Jones potential parameters for mixed pairs of atoms (i.e., AB) using the Lorentz-Berthelot combination rule.
In the case of two separate molecules interacting, a combination rule called the Lorentz-Berthelot combination rule can be imposed to create new σ and ε values. These values are arithmetic and
geometric means respectively. For example, an Ar-Xe L-J plot will have intermediate σ and ε values between Ar-Ar and Xe-Xe. An example of this combination rule can be seen in the figure to the right.
${\displaystyle \sigma _{AB}={\frac {\sigma _{AA}+\sigma _{BB}}{2}}}$
${\displaystyle \varepsilon _{AB}={\sqrt {\varepsilon _{AA}\varepsilon _{BB}}}}$
Calculate the intermolecular potential between two Argon (Ar) atoms separated by a distance of 4.0 Å (use ϵ=0.997 kJ/mol and σ=3.40 Å).
${\displaystyle {\mathcal {V}}\left(r\right)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]}$
${\displaystyle {\mathcal {V}}\left(r\right)=4(0.997~{\text{kJ/mol}})\left[\left({\frac {3.40}{4.00}}\right)^{12}-\left({\frac {3.40}{4.00}}\right)^{6}\right]}$
${\displaystyle {\mathcal {V}}\left(r\right)=3.988(0.142242-0.377150)}$
${\displaystyle {\mathcal {V}}\left(r\right)=-0.94~{\text{kJ/mol}}}$
1. ↑ Lennard-Jones, J. E. (1924), "On the Determination of Molecular Fields", Proc. R. Soc. Lond. A, 106 (738): 463–477, Bibcode:1924RSPSA.106..463J, doi:10.1098/rspa.1924.0082.
2. ↑ R. A. Buckingham, The Classical Equation of State of Gaseous Helium, Neon and Argon, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 168 pp. 264-283 | {"url":"https://en.m.wikibooks.org/wiki/Molecular_Simulation/The_Lennard-Jones_Potential","timestamp":"2024-11-11T11:18:42Z","content_type":"text/html","content_length":"79508","record_id":"<urn:uuid:eeb4f772-9c0c-40ab-ae3e-5860d687852e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00502.warc.gz"} |
Python Numpy Introduction: tutorials 1 | Better4Code - Better4Code
Python Numpy Introduction: tutorials 1 | Better4Code
NumPy Introduction: Python is a versatile programming language that has become increasingly popular in the field of data science and machine learning. One of the most widely used libraries in these
fields is NumPy, short for Numerical Python. NumPy provides a powerful array-processing package that allows developers to work with large datasets efficiently.
In this article, we will introduce the basics of NumPy, including its key features and functionalities, and provide an example of how to use it in Python.
Features of NumPy
NumPy offers a range of features that make it a valuable tool for data analysis, such as:
N-dimensional arrays: NumPy arrays are a collection of values of the same data type, arranged in a grid. They can have any number of dimensions, making them highly flexible and efficient.
Broadcasting: NumPy allows arrays of different shapes and sizes to be used in mathematical operations, which makes it possible to perform complex computations with ease.
Vectorized operations: NumPy enables users to perform mathematical operations on entire arrays, rather than having to iterate through each element of the array. This makes it much faster and more
Numerical methods: NumPy provides a range of functions for performing numerical analysis, such as linear algebra, Fourier transforms, and random number generation.
Using NumPy in Python
To use NumPy in Python, we first need to install it. This can be done using pip, the package installer for Python:
pip install numpy
Once installed, we can import NumPy into our Python code:
import numpy as np
Here, we’ve imported NumPy and given it an alias of “np” to make it easier to use.
Example: Creating a NumPy Array
Let’s take a look at how we can create a NumPy array. In this example, we’ll create a 2-dimensional array:
import numpy as np
# create a 2-dimensional array
my_array = np.array([[1, 2, 3], [4, 5, 6]])
# print the array
This will output the following:
array([[1, 2, 3],
[4, 5, 6]])
In this example, we created a 2-dimensional array using the np.array() function. We passed in a list of lists, where each inner list represents a row of the array. We then printed the array using the
print() function.
NumPy is a powerful tool for data analysis and scientific computing in Python. Its array-processing capabilities and numerical methods make it a must-have library for anyone working with large
datasets. In this article, we’ve introduced the key features of NumPy and provided an example of how to use it in Python. We hope this article has been helpful in getting you started with NumPy!
Read All Articles of NUMPY:
2. Convert list/tuples into NumPy array
3. How NumPy allows multiple arrays without for loop | {"url":"https://www.better4code.com/numpy-introduction-python-numpy-tutorials-1/","timestamp":"2024-11-06T04:12:51Z","content_type":"text/html","content_length":"91403","record_id":"<urn:uuid:8e089f2c-9026-4dd9-be3b-c66f8bf4f13e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00618.warc.gz"} |
other hand, asks us to assign exactly one label to each vertex so to maximize the
total number of covered edges.
The Min-Rep problem, first introduced in [74], can be viewed as a variant of
LabelCover-Min that has shown itself to be a simpler starting point for many
hardness results, e.g. [24, 33, 37, 47]. One formalization of the problem is as follows.
The first part of the input consists of a bipartite graph with vertex set V and edge
set E. Within each bipartition, vertices are further partitioned into equal-sized sub-
sets, which are called supervertices. The set of all supervertices is denoted
V . We
then establish an equivalence relation between edges based on whether or not their
endpoints fall in exactly the same pair of supervertices. The non-empty equivalence
classes are called the graph’s superedges, the set whereof is denoted
E. Thus, an
instance of Min-Rep can be defined as a 4-tuple (V, E,
V ,
E), defining the vertices,
edges, supervertices, and superedges of the instance.
For a subset V
⊆ V of vertices, we say that V
covers a superedge ˆe ∈
E if there
exists an edge e ∈ ˆe such that both endpoints of e are in V
. Finally, the set of
feasible solutions for a given Min-Rep instance are exactly those vertex subsets that
cover every superedge in
E. As the name Min-Rep suggests, the objective is to find
a feasible solution V
⊆ V of minimal cardinality.
Problem 1.4 (Min-Rep): Given a 4-tuple (V, E,
V ,
E) representing a bipartite
graph G = (V, E) with supervertex set
V and superedge set
E, find a minimal-
cardinality subset V
⊆ V such that for each superedge ˆe ∈
E, both endpoints of
some e ∈
E are included in V
As shown in [74], Min-Rep has no 2
approximation algorithm unless NP ⊆
]. LabelCover-Max has since been shown to admit a (slightly
stronger) hardness of 2
1−1/ log log
under similar complexity conjectures [49]. To
Although the set of superedges is already implied by V , E, and
V , in this thesis we treat it as
part of the input for convenience. | {"url":"https://yonatan.us/thesis.html","timestamp":"2024-11-13T21:20:22Z","content_type":"application/xhtml+xml","content_length":"1048882","record_id":"<urn:uuid:291d57c2-84f9-4dc1-8014-c080510cbe06>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00178.warc.gz"} |
Tackling Google Sheets Formulas. 3 Steps to Resolve #ERRORS
Hey, it’s Max Makhrov! 👋
I’m a professional Sheets & Script developer, and I’m here to help you solve all your Spreadsheet headaches.
Photo by Sam Field on Unsplash
In this article, we’ll cover a few common problems that you might be facing:
• Finding it difficult to audit your formulas for errors or inconsistencies
• Frustrated with inconsistent results due to formatting issues
And as always, if you have any questions or comments, drop them below.
Errors. I make 5 to 10 times more mistakes than the average Google Sheets user. However, I’ve developed a workflow that helps me effectively deal with these errors until I resolve them.
I have prepared for you three steps to handle formula errors. Let’s tackle these errors together! 😎
1️⃣ Read Red-Triangle-Message
Can you spot the error in this formula causing a REF error?
Hint: The values in cells A1 and B1 are not formulas. The answer is behind the red triangle...
A critical step in resolving formula errors is to read the error message in the red triangle in the cell with the error. Three examples of why it’s helpful to read the error message:
1. “Division by zero” — adjust your formula to avoid dividing by zero.
2. “Invalid argument” — look at the argument you’ve provided to the formula, rather than searching for a problem with the formula itself.
3. “Array result was not expanded because…”. Some error messages are triggered by common mistakes that are easy to overlook, such as a typo in a cell reference or a missing quotation mark.
Hover your mouse over the cell with the red triangle.
As you may have guessed, I have a common error:
Array result was not expanded because it would overwrite data in C2.
It was a space in C2.
By reading error messages when you encounter formula errors, you can often save time and quickly identify the source of the problem. In my case, to fix the formula, I need to delete space from C2.
2️⃣ Try Formula Smaller Parts
Sometimes, you have a big formula with a few nested functions. The error message tells nothing or even misleads you. Here’s one sample error I get a lot during my work:
I really cannot fix it, reading the error text:
In ARRAY_LITERAL, an Array Literal was missing values for one or more rows.
It is almost the same as saying:
Your formula is wrong, try something else
We can still easily crack the formula. The trick is to mentally simplify the formula, and see what the formula parts are.
Here you see the formula becomes simpler all the time. The final version is an array literal, consisting of 2 parts: {someHeaders ; importRangeResults}. We can try these formula parts separately.
This formula will give “someHeaders”:
And it works just fine. No error here.
This formula will calculate “importRangeResults”:
We put it in a separate cell, and run it with an “=” sign. And here it is, the real source of our error:
Now the error text becomes super helpful and straightforward:
You need to connect these spreadsheets. The first time the destination spreadsheet pulls data from a new source spreadsheet, permission is needed to be granted.
The error text design does not look like an error. The message invites us to tap the green button “Allow access” and resolve the error.
💡Pro Tip: use “VSTACK” function instead of braces “{}”:
“VSTACK” works much better for stacking arrays because it shows individual errors for each part.
#3. Check Number Formats
Please take a look at this “#N/A” error in F2 cells:
Why does my VLOOKUP formula return ‘#N/A’ even when everything seems correct? Here’s a hint for you:
notice that “date” in B2 is aligned left, while date in E2 is aligned to the right.
In spreadsheets, numbers are aligned to the right by default. Text is aligned to the left by default:
If you work with LOOKUP functions, this fact matters. The thing is Spreadsheets have to follow the convention of all computers. This convention is using of a binary system. The binary system makes
calculations over numbers fast, and calculations over text slow:
Compare the text “Zero” and the number “0”
The number “0” can be represented using a single bit, which can take on a value of either 0 or 1. In most computer systems, the bit that represents 0 is set to 0, and the bit that represents 1 is
set to 1.
The word “zero” is a sequence of characters or symbols and its representation in terms of bits depends on the character encoding used. In ASCII encoding, which is a widely used character encoding
system, the word “zero” would be represented using 5 bytes or 40 bits, with each character being represented by 8 bits. However, in other character encoding systems, such as Unicode, the number
of bits used to represent the word “zero” may be different.
The word “Zero” is at least 40 times heavier than the number “0”. This fact forced creators of Spreadsheets not to use text if possible. Here’s what we have:
• numbers,
• dates (calculated as numbers),
• booleans (calculated as numbers: 0 or 1),
• and text.
In Google Sheets, cells can also contain objects, like images, spark charts, and smart chips.
💡Pro Tip: you can create a named function to determine what data type is in cells:
To use this formula, copy my sample Spreadsheet here
Let’s now go back to my error:
Can you get the reason for this error? Data types are different in cells B2 and E2. The “VLOOKUP” formula uses E2 as a key and tries to find that key in column B. Those values look the same for a
user. The computer treats them differently:
• B2 = text,
• E2 = date (saved as a number).
The computer cannot match texts with dates. This is why the formula shows “#NA”. To fix this error, I’ll need to change the values in the column “B” to dates.
💡Pro tip: covert both parts into a text using “TO_TEXT” function.
My sample is short, but it should show you the importance of double-checking your data. Knowing this will save you hours one day!
Final thoughts
Great to see you here! 👋
I just shared three steps that help you break through your formula errors! If you follow these steps in the order I gave you, you’ll have plenty of time to think and learn.
Let me know what you think in the comments below! And stay tuned for the next part of the series, where we’ll dive into the final step not covered here:
• Step #4. Ask for the Help: AI, Google, and Community
Talk soon! 👋 | {"url":"https://max-makhrov.medium.com/tackling-google-sheets-formulas-3-steps-to-resolve-errors-629a359f6111?source=author_recirc-----4415398bac7d----3---------------------eec06af6_7d81_4a3a_956b_1cd198cb5090-------","timestamp":"2024-11-04T05:38:33Z","content_type":"text/html","content_length":"166566","record_id":"<urn:uuid:c0f63697-ca47-4f4c-b42c-e99062fe8635>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00324.warc.gz"} |
New perfect number discovered
A positive integer N is described as a perfect number if the sum of all of its proper divisors (that is to say, factors of N other than N) is equal to itself. For instance, the proper divisors of 28
are 1, 2, 4, 7 and 14, which sum to 28.
The first few perfect numbers are {6, 28, 496, 8128, 33550336, …}. Since 2008, the largest known perfect number was $2^{86225217} - 2^{43112608}$. This record has now been broken, with the discovery
of the perfect number $2^{115770321} - 2^{57885160}$.
It is unknown as to whether there are any odd perfect numbers. An even number is perfect if and only if it is of the form $(2^p - 1)2^{p-1}$, where $2^p - 1$ is a (Mersenne) prime. Mersenne primes
are much easier to verify than ordinary primes of similar size; the Mersenne prime $2^p- 1$ can be verified in time $O(p^2 log(p) log(log(p)))$ by using the Lucas-Lehmer test together with the
Schönhage-Strassen algorithm for multiplying large integers. By comparison, a non-Mersenne prime of a similar size takes time $O(p^6)$ using the best known algorithm (AKS) for primality testing.
As of today, there are now 48 known perfect numbers. We don’t know whether there are any more, although it has been conjectured that there are infinitely many.
0 Responses to New perfect number discovered
1. Best post I read about the subject in the last week, since the discovery! All the other who did coverage for the finding of the new mersenne concentrated on that, writing a lot of incorrect math
around it. Concentrating on the perfect number instead, which is bigger, and sounds nicer, this was a good move. However, you should talk a bit more about how the number was find, say something
about Cooper, GIMPS or mersenneforum.com. 😛
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How To Figure Out Draw Length
How To Figure Out Draw Length - How to measure draw length for recurve bow. Draw a fbd of the structure. Web draw length formula. Web so, how to measure draw length? Shooter tip on how to determine
both your proper draw length and draw weight. Learn proper form for drawing a bow. This number is your draw length. What is the bow draw length? Web if you’re not sure, don’t worry, because the basic
procedure for how to measure draw length is super simple: Check with your local bow shop to help you fine tune.
How To Measure Draw Length knowhowaprendizagem
This is the arm span divided by 2.5 method, which we’ll cover in more detail below. For example, if you have a wingspan of 70 inches, your draw length would be 28 inches. The resulting value is your
draw length. Web measuring your draw length at home. Web how to measure and calculate draw length.
How to Measure Draw Length YouTube
You’re basically forming the letter “t” with your body. This is the arm span divided by 2.5 method, which we’ll cover in more detail below. Web so, how to measure draw length? Where dl is the draw
length (in) and ws is your wingspan. Web how to measure draw length.
How to measure your draw length WITH PICTURES
You’re basically forming the letter “t” with your body. In general, this is an excellent way of measuring arrow length when shooting a recurve bow or any other type of traditional bow. Web follow
along as john dudley shows you how to find your proper draw length for your compound bow.for more archery coaching and tips visit the school of.
How to Determine the Draw Length Blog
Now measure the distance from the tip of one middle finger to the tip of your other middle finger. A bow’s draw serves as the foundation of a good shot. Importance of correct draw length. And it is
the length of the distance from anchor point on the bow string to the pivot point on the bow grip. Learn how.
How to measure your draw length WITH PICTURES
Web how to measure draw length. For instance, in this example the calculated draw length will be 21 inches. Do some old school calculating. Web draw length formula. Divide this length by 2.5.
How To Measure Draw Length (With Calculator) Archery for Beginners
This will provide you with a fairly reliable estimate of your appropriate draw length. Web to determine your draw length, simply measure your armspan from the middle finger of one hand, to the middle
finger of your opposite hand. That’s not the only way to measure draw. Remember, your calculated draw length is enough information for a new archer to.
How to Measure Draw Length YouTube
The shooter draws the bow and the draw length is indicated by the markings on the arrow. Web how to measure and calculate draw length. Measure the wingspan from the tip of your middle finger to the
tip of your other middle finger. Web to calculate draw length, stretch your arms to the side, so that your body forms a.
Get it RIGHT! SEE How to Measure Your DRAW LENGTH Accurately YouTube
Do some old school calculating. The resulting value is your draw length. Web to determine your draw length, simply measure your armspan from the middle finger of one hand, to the middle finger of
your opposite hand. And it is the length of the distance from anchor point on the bow string to the pivot point on the bow grip..
3 Ways to Measure Draw Length for a Bow wikiHow
Now measure the distance from the tip of one middle finger to the tip of your other middle finger. Last updated on january 15, 2023 by archery care. For example, if you have a wingspan of 70 inches,
your draw length would be 28 inches. And it is the length of the distance from anchor point on the bow string.
How to measure your draw length WITH PICTURES
The resulting value is your draw length. Have a friend help to measure. Web the first involves you standing with your arms spread straight out to either side of your body at shoulder height. This is
your approximately draw length. Now take that measurement and divide by 2.5 to get your draw length.
Divide This Length By 2.5.
Web draw the shear force and bending moment diagrams for the cantilever beam supporting a concentrated load of 5 lb at the free end 3 ft from the wall. Web to calculate draw length, stretch your arms
to the side, so that your body forms a “t”. This measurement, minus 15 then divided by 2, is your draw length. Stand up tall with your back against a wall and stretch your arms out by your side in a
t formation, ensuring that your fingers are also outstretched.
You’re Basically Forming The Letter “T” With Your Body.
Web how to measure and calculate draw length. Now take that measurement and divide by 2.5 to get your draw length. Most archery shops will have a draw length check bow that has a faux arrow with
measurements marked. Where dl is the draw length (in) and ws is your wingspan.
Web Measure The Distance From The End Of Your Middle Finger To The End Of Your Other Middle Finger, Basically The Length Of Both Arms, Hands And Chest.
This number is your draw length. Check with your local bow shop to help you fine tune. Have a friend help to measure. That’s not the only way to measure draw.
Last Updated On January 15, 2023 By Archery Care.
That's all there is to it folks! Web the first involves you standing with your arms spread straight out to either side of your body at shoulder height. What is the bow draw length? How to measure
draw length for recurve bow.
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Ly out of chaos. This is what chaos theory is about. | www.adenosine-kinase.com
Ly out of chaos. This is what chaos theory is about.
Ly out of chaos. This is what chaos theory is about. All we have to complete to observe spontaneous selfordering would be to pull the stopper out of our bathtub drain. Water molecules quickly
selforder into a swirla vortexfrom purely physicodymic complex causation. We mistakenly contact this selfordering “selforganization,” but the vortex is not inside the least bit organized. It can be
only selfordered. What is the distinction No selection nodes are required for any bathtub swirl to selforder out of seemingly random Brownian motion. Proficient programming choices are usually not
expected for heat agitation of water molecules to selforder into a vortex. No configurable switches have to be purposefully set, each in a certain way, to achieve selfordering. No pursuit of a
objective is involved. No algorithmic optimization is essential. In addition, Prigogine’s dissipative structures don’t DO something formally productive. They 2-Cl-IB-MECA possess no capacity to
attain computatiol results.Life,They do not construct sophisticated Sustained Functiol Systems (SFS). Dissipative structures are momentary. They only seem sustained (e.g a candle flame) mainly
because of we observe by way of time a long string of momentary dissipative events or structures. That is exactly where their me comes from. They can not generate a sustained functiol machine or
program with optimized functiolity. Neither chaos nor the edge of chaos can make a Calculus Algorithm System that achieves computatiol accomplishment Organizer of formal function Bo fide systemChaos
is capable of creating incredibly complex physicodymic behavior. We need to in no way confuse this complexity with formal function, nevertheless. Order spontaneously appears out of disorder in the
comprehensive absence of any formal creative input or cybernetic magement. But, no algorithmic organization is produced by a candle flame. What seems to be a completely random environment is in truth
a caldron PubMed ID:http://jpet.aspetjournals.org/content/16/4/273 Shannon transmission engineering to explain intuitive details, meaning and function. Shannon’s equations define damaging
“uncertainty,” not optimistic “surprisal”. Functiol “surprisal” requires the acquisition of good precise semantic facts. Just as we can not Phillygenol chemical information clarify and measure
“intuitive information” utilizing Shannon combitorial uncertainty, we can’t clarify a actually organized technique appealing to absolutely nothing but a mystical “edge of chaos”. Lowered uncertainty
(“mutual entropy”) in Shannon theory comes closer to semantic information and facts. To attain this, having said that, we have to mix in the formal components of human information gained by
mathematical s.Ly out of chaos. That is what chaos theory is about. All we have to perform to observe spontaneous selfordering would be to pull the stopper out of our bathtub drain. Water molecules
speedily selforder into a swirla vortexfrom purely physicodymic complex causation. We mistakenly contact this selfordering “selforganization,” however the vortex is just not in the least bit
organized. It can be only selfordered. What is the distinction No decision nodes are required for a bathtub swirl to selforder out of seemingly random Brownian motion. Proficient programming options
will not be expected for heat agitation of water molecules to selforder into a vortex. No configurable switches have to be purposefully set, each in a specific way, to attain selfordering. No pursuit
of a goal is involved. No algorithmic optimization is needed. Also, Prigogine’s dissipative structures usually do not DO anything formally productive. They possess no capability to achieve
computatiol results.Life,They usually do not construct sophisticated Sustained Functiol Systems (SFS). Dissipative structures are momentary. They only seem sustained (e.g a candle flame) mainly
because of we observe by way of time a lengthy string of momentary dissipative events or structures. This can be exactly where their me comes from. They cannot create a sustained functiol machine or
program with optimized functiolity. Neither chaos nor the edge of chaos can make a Calculus Algorithm System that achieves computatiol results Organizer of formal function Bo fide systemChaos is
capable of generating incredibly complicated physicodymic behavior. We should by no means confuse this complexity with formal function, having said that. Order spontaneously seems out of disorder in
the comprehensive absence of any formal inventive input or cybernetic magement. But, no algorithmic organization is produced by a candle flame. What seems to become a entirely random atmosphere is
the truth is a caldron of complex interaction of numerous force fields. The complexity of interactive causation can generate the illusion of PubMed ID:http://jpet.aspetjournals.org/content/16/4/273
Shannon transmission engineering to explain intuitive data, meaning and function. Shannon’s equations define damaging “uncertainty,” not positive “surprisal”. Functiol “surprisal” needs the
acquisition of positive particular semantic data. Just as we can not clarify and measure “intuitive information” utilizing Shannon combitorial uncertainty, we cannot explain a truly organized system
attractive to practically nothing but a mystical “edge of chaos”. Lowered uncertainty (“mutual entropy”) in Shannon theory comes closer to semantic details. To achieve this, on the other hand, we
have to mix in the formal elements of human know-how gained by mathematical s. | {"url":"https://www.adenosine-kinase.com/2018/01/30/ly-out-of-chaos-this-is-what-chaos-theory-is-about/","timestamp":"2024-11-15T01:06:46Z","content_type":"text/html","content_length":"59330","record_id":"<urn:uuid:10844097-1d5b-454f-9bf2-6923ea276290>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00574.warc.gz"} |
American Mathematical Society
We investigate relationships between Mycielski ideals in ${2^\omega }$ generated by different systems. For a fixed Mycielski ideal $\mathfrak {M}$ we study properties of its compact members. For a
perfect Polish space $X$ and certain sets $A \subseteq X \times {2^\omega }$, the positions of $\{ x \in X:{A_X} \notin \mathfrak {M}\}$ in the Borel and projective hierarchies are established and
other section properties are observed. References
—, On $\sigma$-ideals having perfect members in all perfect sets, preprint. L. Larson, Typical compact sets in the Hausdorff metric are porous, Real Anal. Exchange 13 (1987-88), 116-118.
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Bibliographic Information
• © Copyright 1990 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 110 (1990), 243-250
• MSC: Primary 04A15; Secondary 54H05
• DOI: https://doi.org/10.1090/S0002-9939-1990-1007486-6
• MathSciNet review: 1007486 | {"url":"https://www.ams.org/journals/proc/1990-110-01/S0002-9939-1990-1007486-6/?active=current","timestamp":"2024-11-02T02:14:32Z","content_type":"text/html","content_length":"65641","record_id":"<urn:uuid:5e3b8e57-28fa-4ab6-9b5a-e8c5cee31cef>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00822.warc.gz"} |
non significant results discussion example
i asked first comebacks
non significant results discussion example
Nonetheless, single replications should not be seen as the definitive result, considering that these results indicate there remains much uncertainty about whether a nonsignificant result is a true
negative or a false negative. [2] Albert J. Yep. been tempered. Create an account to follow your favorite communities and start taking part in conversations. Although the emphasis on precision and
the meta-analytic approach is fruitful in theory, we should realize that publication bias will result in precise but biased (overestimated) effect size estimation of meta-analyses (Nuijten, van
Assen, Veldkamp, & Wicherts, 2015). Probability density distributions of the p-values for gender effects, split for nonsignificant and significant results. If researchers reported such a qualifier,
we assumed they correctly represented these expectations with respect to the statistical significance of the result. It depends what you are concluding. In a study of 50 reviews that employed
comprehensive literature searches and included both English and non-English-language trials, Jni et al reported that non-English trials were more likely to produce significant results at P<0.05,
while estimates of intervention effects were, on average, 16% (95% CI 3% to 26%) more beneficial in non . By continuing to use our website, you are agreeing to. The statistical analysis shows that a
difference as large or larger than the one obtained in the experiment would occur \(11\%\) of the time even if there were no true difference between the treatments. As such, the Fisher test is
primarily useful to test a set of potentially underpowered results in a more powerful manner, albeit that the result then applies to the complete set. The Reproducibility Project Psychology (RPP),
which replicated 100 effects reported in prominent psychology journals in 2008, found that only 36% of these effects were statistically significant in the replication (Open Science Collaboration,
2015). The effect of both these variables interacting together was found to be insignificant. For example, you might do a power analysis and find that your sample of 2000 people allows you to reach
conclusions about effects as small as, say, r = .11. In a precision mode, the large study provides a more certain estimate and therefore is deemed more informative and provides the best estimate. The
concern for false positives has overshadowed the concern for false negatives in the recent debate, which seems unwarranted. Published on 21 March 2019 by Shona McCombes. where pi is the reported
nonsignificant p-value, is the selected significance cut-off (i.e., = .05), and pi* the transformed p-value. The author(s) of this paper chose the Open Review option, and the peer review comments are
available at: http://doi.org/10.1525/collabra.71.pr. More specifically, when H0 is true in the population, but H1 is accepted (H1), a Type I error is made (); a false positive (lower left cell).
Columns indicate the true situation in the population, rows indicate the decision based on a statistical test. Simply: you use the same language as you would to report a significant result, altering
as necessary. APA style is defined as the format where the type of test statistic is reported, followed by the degrees of freedom (if applicable), the observed test value, and the p-value (e.g., t
(85) = 2.86, p = .005; American Psychological Association, 2010). Table 4 shows the number of papers with evidence for false negatives, specified per journal and per k number of nonsignificant test
results. More technically, we inspected whether p-values within a paper deviate from what can be expected under the H0 (i.e., uniformity). Third, we applied the Fisher test to the nonsignificant
results in 14,765 psychology papers from these eight flagship psychology journals to inspect how many papers show evidence of at least one false negative result. For example, for small true effect
sizes ( = .1), 25 nonsignificant results from medium samples result in 85% power (7 nonsignificant results from large samples yield 83% power). For the discussion, there are a million reasons you
might not have replicated a published or even just expected result. Hi everyone, i have been studying Psychology for a while now and throughout my studies haven't really done much standalone studies,
generally we do studies that lecturers have already made up and where you basically know what the findings are or should be. that do not fit the overall message. Since the test we apply is based on
nonsignificant p-values, it requires random variables distributed between 0 and 1. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open
Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We do not know whether these
marginally significant p-values were interpreted as evidence in favor of a finding (or not) and how these interpretations changed over time. I say I found evidence that the null hypothesis is
incorrect, or I failed to find such evidence. Assuming X medium or strong true effects underlying the nonsignificant results from RPP yields confidence intervals 021 (033.3%) and 013 (020.6%),
respectively. For large effects ( = .4), two nonsignificant results from small samples already almost always detects the existence of false negatives (not shown in Table 2). More precisely, we
investigate whether evidential value depends on whether or not the result is statistically significant, and whether or not the results were in line with expectations expressed in the paper. Overall
results (last row) indicate that 47.1% of all articles show evidence of false negatives (i.e. When H1 is true in the population and H0 is accepted (H0), a Type II error is made (); a false negative
(upper right cell). Results Section The Results section should set out your key experimental results, including any statistical analysis and whether or not the results of these are significant. And
then focus on how/why/what may have gone wrong/right. Strikingly, though In the discussion of your findings you have an opportunity to develop the story you found in the data, making connections
between the results of your analysis and existing theory and research. Potentially neglecting effects due to a lack of statistical power can lead to a waste of research resources and stifle the
scientific discovery process. Collabra: Psychology 1 January 2017; 3 (1): 9. doi: https://doi.org/10.1525/collabra.71. Bond and found he was correct \(49\) times out of \(100\) tries. the Premier
League. Using the data at hand, we cannot distinguish between the two explanations. Then using SF Rule 3 shows that ln k 2 /k 1 should have 2 significant The results suggest that 7 out of 10
correlations were statistically significant and were greater or equal to r(78) = +.35, p < .05, two-tailed. Funny Basketball Slang, I surveyed 70 gamers on whether or not they played violent games
(anything over teen = violent), their gender, and their levels of aggression based on questions from the buss perry aggression test. When researchers fail to find a statistically significant result,
it's often treated as exactly that - a failure. significant wine persists. Therefore caution is warranted when wishing to draw conclusions on the presence of an effect in individual studies (original
or replication; Open Science Collaboration, 2015; Gilbert, King, Pettigrew, & Wilson, 2016; Anderson, et al. another example of how to deal with statistically non-significant results English football
team because it has won the Champions League 5 times No competing interests, Chief Scientist, Matrix45; Professor, College of Pharmacy, University of Arizona, Christopher S. Lee (Matrix45 &
University of Arizona), and Karen M. MacDonald (Matrix45), Copyright 2023 BMJ Publishing Group Ltd, Womens, childrens & adolescents health, Non-statistically significant results, or how to make
statistically non-significant results sound significant and fit the overall message. JPSP has a higher probability of being a false negative than one in another journal. Due to its probabilistic
nature, Null Hypothesis Significance Testing (NHST) is subject to decision errors. Contact Us Today! Within the theoretical framework of scientific hypothesis testing, accepting or rejecting a
hypothesis is unequivocal, because the hypothesis is either true or false. The preliminary results revealed significant differences between the two groups, which suggests that the groups are
independent and require separate analyses. For example, for small true effect sizes ( = .1), 25 nonsignificant results from medium samples result in 85% power (7 nonsignificant results from large
samples yield 83% power). F and t-values were converted to effect sizes by, Where F = t2 and df1 = 1 for t-values. Statistical significance was determined using = .05, two-tailed test. Consider the
following hypothetical example. Besides in psychology, reproducibility problems have also been indicated in economics (Camerer, et al., 2016) and medicine (Begley, & Ellis, 2012). Rest assured, your
dissertation committee will not (or at least SHOULD not) refuse to pass you for having non-significant results. non-significant result that runs counter to their clinically hypothesized Therefore we
examined the specificity and sensitivity of the Fisher test to test for false negatives, with a simulation study of the one sample t-test. The simulation procedure was carried out for conditions in a
three-factor design, where power of the Fisher test was simulated as a function of sample size N, effect size , and k test results. Another potential explanation is that the effect sizes being
studied have become smaller over time (mean correlation effect r = 0.257 in 1985, 0.187 in 2013), which results in both higher p-values over time and lower power of the Fisher test. For each dataset
we: Randomly selected X out of 63 effects which are supposed to be generated by true nonzero effects, with the remaining 63 X supposed to be generated by true zero effects; Given the degrees of
freedom of the effects, we randomly generated p-values under the H0 using the central distributions and non-central distributions (for the 63 X and X effects selected in step 1, respectively); The
Fisher statistic Y was computed by applying Equation 2 to the transformed p-values (see Equation 1) of step 2. For example, suppose an experiment tested the effectiveness of a treatment for insomnia.
Table 4 also shows evidence of false negatives for each of the eight journals. When a significance test results in a high probability value, it means that the data provide little or no evidence that
the null hypothesis is false. Effect sizes and F ratios < 1.0: Sense or nonsense? At least partly because of mistakes like this, many researchers ignore the possibility of false negatives and false
positives and they remain pervasive in the literature. status page at https://status.libretexts.org, Explain why the null hypothesis should not be accepted, Discuss the problems of affirming a
negative conclusion. These errors may have affected the results of our analyses. This is done by computing a confidence interval. When there is discordance between the true- and decided hypothesis, a
decision error is made. funfetti pancake mix cookies non significant results discussion example. The analyses reported in this paper use the recalculated p-values to eliminate potential errors in the
reported p-values (Nuijten, Hartgerink, van Assen, Epskamp, & Wicherts, 2015; Bakker, & Wicherts, 2011). Significance was coded based on the reported p-value, where .05 was used as the decision
criterion to determine significance (Nuijten, Hartgerink, van Assen, Epskamp, & Wicherts, 2015). article. :(. Background Previous studies reported that autistic adolescents and adults tend to exhibit
extensive choice switching in repeated experiential tasks. We then used the inversion method (Casella, & Berger, 2002) to compute confidence intervals of X, the number of nonzero effects. The two
sub-aims - the first to compare the acquisition The following example shows how to report the results of a one-way ANOVA in practice. Results were similar when the nonsignificant effects were
considered separately for the eight journals, although deviations were smaller for the Journal of Applied Psychology (see Figure S1 for results per journal). {
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%2F11%253A_Logic_of_Hypothesis_Testing%2F11.06%253A_Non-Significant_Results, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-
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\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\
langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\
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newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\). Determining the effect of a program through an impact assessment
involves running a statistical test to calculate the probability that the effect, or the difference between treatment and control groups, is a . There are lots of ways to talk about negative
results.identify trends.compare to other studies.identify flaws.etc. Application 1: Evidence of false negatives in articles across eight major psychology journals, Application 2: Evidence of false
negative gender effects in eight major psychology journals, Application 3: Reproducibility Project Psychology, Section: Methodology and Research Practice, Nuijten, Hartgerink, van Assen, Epskamp, &
Wicherts, 2015, Marszalek, Barber, Kohlhart, & Holmes, 2011, Borenstein, Hedges, Higgins, & Rothstein, 2009, Hartgerink, van Aert, Nuijten, Wicherts, & van Assen, 2016, Wagenmakers, Wetzels,
Borsboom, van der Maas, & Kievit, 2012, Bakker, Hartgerink, Wicherts, & van der Maas, 2016, Nuijten, van Assen, Veldkamp, & Wicherts, 2015, Ivarsson, Andersen, Johnson, & Lindwall, 2013, http://
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10.3758/s13428-011-0089-5, http://books.google.nl/books/about/Introduction_to_Meta_Analysis.html?hl=&id=JQg9jdrq26wC, https://cran.r-project.org/web/packages/statcheck/index.html, https://doi.org/
10.1371/journal.pone.0149794, https://doi.org/10.1007/s11192-011-0494-7, http://link.springer.com/article/10.1007/s11192-011-0494-7, https://doi.org/10.1371/journal.pone.0109019, https://doi.org/
10.3758/s13423-012-0227-9, https://doi.org/10.1016/j.paid.2016.06.069, http://www.sciencedirect.com/science/article/pii/S0191886916308194, https://doi.org/10.1053/j.seminhematol.2008.04.003, http://
www.sciencedirect.com/science/article/pii/S0037196308000620, http://psycnet.apa.org/journals/bul/82/1/1, https://doi.org/10.1037/0003-066X.60.6.581, https://doi.org/10.1371/journal.pmed.0020124,
http://journals.plos.org/plosmedicine/article/asset?id=10.1371/journal.pmed.0020124.PDF, https://doi.org/10.1016/j.psychsport.2012.07.007, http://www.sciencedirect.com/science/article/pii/
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10.1037/gpr0000034, https://doi.org/10.1037/0033-2909.86.3.638, http://psycnet.apa.org/journals/bul/86/3/638, https://doi.org/10.1037/0033-2909.105.2.309, https://doi.org/10.1177/00131640121971392,
http://epm.sagepub.com/content/61/4/605.abstract, https://books.google.com/books?hl=en&lr=&id=5cLeAQAAQBAJ&oi=fnd&pg=PA221&dq=Steiger+%26+Fouladi,+1997&ots=oLcsJBxNuP&sig=iaMsFz0slBW2FG198jWnB4T9g0c,
https://doi.org/10.1080/01621459.1959.10501497, https://doi.org/10.1080/00031305.1995.10476125, https://doi.org/10.1016/S0895-4356(00)00242-0, http://www.ncbi.nlm.nih.gov/pubmed/11106885, https://
doi.org/10.1037/0003-066X.54.8.594, https://www.apa.org/pubs/journals/releases/amp-54-8-594.pdf, http://creativecommons.org/licenses/by/4.0/, What Diverse Samples Can Teach Us About Cognitive
Vulnerability to Depression, Disentangling the Contributions of Repeating Targets, Distractors, and Stimulus Positions to Practice Benefits in D2-Like Tests of Attention, Prespecification of
Structure for the Optimization of Data Collection and Analysis, Binge Eating and Health Behaviors During Times of High and Low Stress Among First-year University Students, Psychometric Properties of
the Spanish Version of the Complex Postformal Thought Questionnaire: Developmental Pattern and Significance and Its Relationship With Cognitive and Personality Measures, Journal of Consulting and
Clinical Psychology (JCCP), Journal of Experimental Psychology: General (JEPG), Journal of Personality and Social Psychology (JPSP). It does not have to include everything you did, particularly for a
doctorate dissertation. The results indicate that the Fisher test is a powerful method to test for a false negative among nonsignificant results. You will also want to discuss the implications of
your non-significant findings to your area of research. ), Department of Methodology and Statistics, Tilburg University, NL. Hipsters are more likely than non-hipsters to own an IPhone, X 2 (1, N =
54) = 6.7, p < .01. profit facilities delivered higher quality of care than did for-profit depending on how far left or how far right one goes on the confidence Was your rationale solid? Basically he
wants me to "prove" my study was not underpowered. This means that the probability value is \(0.62\), a value very much higher than the conventional significance level of \(0.05\). Discussion.
Restructuring incentives and practices to promote truth over publishability, The prevalence of statistical reporting errors in psychology (19852013), The replication paradox: Combining studies can
decrease accuracy of effect size estimates, Review of general psychology: journal of Division 1, of the American Psychological Association, Estimating the reproducibility of psychological science,
The file drawer problem and tolerance for null results, The ironic effect of significant results on the credibility of multiple-study articles. By Posted jordan schnitzer house In strengths and
weaknesses of a volleyball player Results and Discussion. Corpus ID: 20634485 [Non-significant in univariate but significant in multivariate analysis: a discussion with examples]. For r-values the
adjusted effect sizes were computed as (Ivarsson, Andersen, Johnson, & Lindwall, 2013), Where v is the number of predictors. The Fisher test was applied to the nonsignificant test results of each of
the 14,765 papers separately, to inspect for evidence of false negatives. Examples are really helpful to me to understand how something is done. It is generally impossible to prove a negative. As
opposed to Etz and Vandekerckhove (2016), Van Aert and Van Assen (2017; 2017) use a statistically significant original and a replication study to evaluate the common true underlying effect size,
adjusting for publication bias. For example, if the text stated as expected no evidence for an effect was found, t(12) = 1, p = .337 we assumed the authors expected a nonsignificant result.
suggesting that studies in psychology are typically not powerful enough to distinguish zero from nonzero true findings. Do not accept the null hypothesis when you do not reject it. , suppose Mr.
deficiencies might be higher or lower in either for-profit or not-for- So, if Experimenter Jones had concluded that the null hypothesis was true based on the statistical analysis, he or she would
have been mistaken. values are well above Fishers commonly accepted alpha criterion of 0.05 Fourth, discrepant codings were resolved by discussion (25 cases [13.9%]; two cases remained unresolved and
were dropped). Your discussion can include potential reasons why your results defied expectations. So, in some sense, you should think of statistical significance as a "spectrum" rather than a
black-or-white subject. I usually follow some sort of formula like "Contrary to my hypothesis, there was no significant difference in aggression scores between men (M = 7.56) and women (M = 7.22), t
(df) = 1.2, p = .50." The levels for sample size were determined based on the 25th, 50th, and 75th percentile for the degrees of freedom (df2) in the observed dataset for Application 1. Finally, we
computed the p-value for this t-value under the null distribution. [Article in Chinese] . Consequently, our results and conclusions may not be generalizable to all results reported in articles. Table
2 summarizes the results for the simulations of the Fisher test when the nonsignificant p-values are generated by either small- or medium population effect sizes.
Figure1.Powerofanindependentsamplest-testwithn=50per Proin interdum a tortor sit amet mollis. Journals differed in the proportion of papers that showed evidence of false negatives, but this was
largely due to differences in the number of nonsignificant results reported in these papers. Denote the value of this Fisher test by Y; note that under the H0 of no evidential value Y is
2-distributed with 126 degrees of freedom. If your p-value is over .10, you can say your results revealed a non-significant trend in the predicted direction. Although these studies suggest
substantial evidence of false positives in these fields, replications show considerable variability in resulting effect size estimates (Klein, et al., 2014; Stanley, & Spence, 2014). Moreover,
Fiedler, Kutzner, and Krueger (2012) expressed the concern that an increased focus on false positives is too shortsighted because false negatives are more difficult to detect than false positives.
13825339d2d51533e227f5c8ca08f6d3601f a valid real estate contract requires all except,
Famous Scandals Of The 1920s, Articles N
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Stochastic Simulation of Chemical ReactionsStochastic Simulation of Chemical Reactions
Stochastic Simulation of Chemical Reactions
Verifying a theoretical steady state concentration via stochastic simulation
In the previous module, we saw that we could avoid tracking the positions of individual particles if we assume that the particles are well-mixed, i.e., uniformly distributed throughout their
environment. We will apply this assumption in our current work as well, in part because the E. coli cell is so small. As a proof of concept, we will see if a well-mixed simulation replicates a
reversible reaction’s equilibrium concentrations of particles that we found in the previous lesson.
Even though we can calculate steady state concentrations manually, a particle-free simulation will be useful for two reasons. First, this simulation will give us snapshots of the concentrations of
particles in the system over multiple time points and allow us to see how quickly the concentrations reach equilibrium. Second, we will soon expand our model of chemotaxis to have many particles and
reactions that depend on each other, and direct mathematical analysis of the system will become impossible.
Note: The difficulty posed to precise analysis of systems with multiple chemical reactions is comparable to the famed “n-body problem” in physics. Predicting the motions of two celestial objects
interacting due to gravity can be done exactly, but once we add more bodies to the system, no exact solution exists, and we must rely on simulation.
Our particle-free model will apply an approach called Gillespie’s stochastic simulation algorithm, which is often called the Gillespie algorithm or just SSA for short. Before we explain how this
algorithm works, we will take a short detour to provide some needed probabilistic context.
The Poisson and exponential distributions
Imagine that you own a store and have noticed that on average, λ customers enter your store in a single hour. Let X denote the number of customers entering the store in the next hour; X is an example
of a random variable because its value may change depending on random chance. If we assume that customers are independent actors, then X follows a Poisson distribution. It can be shown that for a
Poisson distribution, the probability that exactly n customers arrive in the next hour is
\[\mathrm{Pr}(X = n) = \dfrac{\lambda^n e^{-\lambda}}{n!}\,,\]
where e is the mathematical constant known as Euler’s number and is equal to 2.7182818284…
Note: A derivation of the above formula for Pr(X = n) is beyond the scope of our work here, but if you are interested in one, please check out this article by Andrew Chamberlain.
Furthermore, the probability of observing exactly n customers in t hours, where t is an arbitrary positive number, is
\[\dfrac{(\lambda t)^n e^{-\lambda t}}{n!}\,.\]
We can also ask how long we will typically have to wait for the next customer to arrive. Specifically, what are the chances that this customer will arrive after t hours? If we let T be the random
variable corresponding to the wait time on the next customer, then the probability of T being at least t is the probability of seeing zero customers in t hours:
\[\mathrm{Pr}(T > t) = \mathrm{Pr}(X = 0) = \dfrac{(\lambda t)^0 e^{-\lambda t}}{0!} = e^{-\lambda t}\,.\]
In other words, the probability Pr(T > t) that the wait time is longer than time t decays exponentially as t increases. For this reason, the random variable T is said to follow an exponential
distribution. It can be shown that the expected value of the exponential distribution (i.e., the average amount of time that we will need to wait for the next event to occur) is 1/λ.
STOP: What is the probability Pr(T < t)?
The Gillespie algorithm
We now return to explain the Gillespie algorithm for simulating multiple chemical reactions in a well-mixed environment. The engine of this algorithm runs on a single question: given a well-mixed
environment of particles and a reaction involving those particles taking place at some average rate, how long should we expect to wait before this reaction occurs somewhere in the environment?
This is the same question that we asked in the previous discussion; we have simply replaced customers entering a store with instances of a chemical reaction. The average number λ of occurrences of
the reaction in a unit time period is the rate r at which the reaction occurs. Therefore, an exponential distribution with average wait time 1/r can be used to model the time between instances of the
Next, say that we have two reactions proceeding independently of each other and occurring at rates r[1] and r[2]. The combined average rates of the two reactions is r[1] + r[2], which is also a
Poisson distribution. Therefore, the wait time required to wait for either of the two reactions is exponentially distributed, with an average wait time equal to 1/(r[1] + r[2]).
Numerical methods allow us to generate a random number simulating the wait time of an exponential distribution. By repeatedly generating these numbers, we can obtain a series of wait times between
consecutive reaction occurrences.
Once we have generated a wait time, we should determine the reaction to which it corresponds. If the rates of the two reactions are equal, then we simply choose one of the two reactions randomly with
equal probability. But if the rates of these reactions are different, then we should choose one of the reactions via a probability that is weighted in direct proportion to the rate of the reaction;
that is, the larger the rate of the reaction, the more likely that this reaction corresponds to the current event.^1 To do so, we select the first reaction with probability r[1]/(λ[1] + r[2]) and the
second reaction with probability r[2]/(r[1] + r[2]). (Note that these two probabilities sum to 1.)
As illustrated in the figure below, we will demonstrate the Gillespie algorithm by returning to our ongoing example, in which we are modeling the forward and reverse reactions of ligand-receptor
binding and dissociation. These reactions have rates r[bind] = k[bind] · [L] · [T] and r[dissociate] = k[dissociate] · [LT], respectively.
First, we choose a wait time according to an exponential distribution with mean value 1/(r[bind] + r[dissociate]). Then, the probability that the event corresponds to a binding reaction is given by
Pr(L + T → LT) = r[bind]/(r[bind] + r[dissociate]),
and the probability that it corresponds to a dissociation reaction is
Pr(LT → L + T) = r[dissociate]/(r[bind] + r[dissociate]).
A visualization of a single reaction event used by the Gillespie algorithm for ligand-receptor binding and dissociation. Red circles represent ligands (L), and orange wedges represent receptors (T).
The wait time for the next reaction is drawn from an exponential distribution with mean 1/(k[bind] + k[dissociate]). The probability of this event corresponding to a binding or dissociation reaction
is proportional to the rate of the respective reaction.
To generalize the Gillespie algorithm to n reactions occurring at rates r[1], r[2], …, r[n], the wait time between reactions will be exponentially distributed with average 1/(r[1] + r[2] + … + r[n]).
Once we select the next reaction to occur, the likelihood that it is the i-th reaction is equal to
r[i ]/(r[1] + r[2] + … + r[n]).
Throughout this module, we will employ BioNetGen to apply the Gillespie algorithm to well-mixed models of chemical reactions. We will use our ongoing example of ligand-receptor binding and
dissociation to introduce the way in which BioNetGen represents molecules and reactions involving them. The following tutorial shows how to implement this rule in BioNetGen and use the Gillespie
algorithm to determine the equilibrium of a reversible ligand-receptor binding reaction.
Does the Gillespie algorithm confirm our steady state calculations?
In the previous lesson, we showed an example in which a system with 10,000 free ligand molecules and 7,000 free receptor molecules produced the following steady state concentrations using the
experimentally verified binding rate of k[bind] = 0.0146((molecules/µm^3)^-1)s^-1 and dissociation rate of k[dissociate] = 35s^-1:
• [LT] = 4,793 molecules/µm^3;
• [L] = 5,207 molecules/µm^3;
• [T] = 2,207 molecules/µm^3.
Our model uses the same number of initial molecules and the same reaction rates. The system evolves via the Gillespie algorithm, and we track the concentration of free ligand molecules, ligand
molecules bound to receptor molecules, and free receptor molecules over time.
The figure below demonstrates that the Gillespie algorithm quickly converges quickly to the same values calculated just above. Furthermore, the system reaches steady state in a fraction of a second.
A concentration plot over time for ligand-receptor dynamics via a BioNetGen simulation employing the Gillespie algorithm. Time is shown (in seconds) on the x-axis, and concentration is shown (in
molecules/µm^3) on the y-axis. The molecules quickly reach steady state concentrations that match those identified by hand.
This simple ligand-receptor model is just the beginning of our study of chemotaxis. In the next section, we will delve into the complex biochemical details of chemotaxis. Furthermore, we will see
that the Gillespie algorithm for stochastic simulations will scale easily as our model of this system grows more complex.
1. Schwartz R. Biological Modeling and Simulaton: A Survey of Practical Models, Algorithms, and Numerical Methods. Chapter 17.2. ↩ | {"url":"https://biologicalmodeling.org/chemotaxis/gillespie","timestamp":"2024-11-04T18:46:28Z","content_type":"text/html","content_length":"32972","record_id":"<urn:uuid:a5295f20-f7ab-47de-8d25-ac3c7a413d26>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00063.warc.gz"} |
Planck about inflation
The CMB spectrum measured by the Planck satellite points to a perfectly boring universe: the vanilla ΛCDM cosmological model, no hint of new light degrees of freedom beyond the standard model, no
hint of larger-than-expected neutrino masses, etc. However at the quantitative level things are a bit more interesting, as Planck has considerably narrowed down the parameter space of inflation. We
may not be far from selecting a small class out the huge zoo of inflationary models.
Simplest models of inflation involve a scalar field with a potential. During inflation, the value of the scalar field is such that the potential is large and positive, effectively acting as a
cosmological constant that supports a faster-than-light expansion of the universe. The potential should be almost but not exactly flat, so that the scalar field slowly creeps down the potential
slope; once it falls into the minimum inflation ends and the modern history begins. Clearly, that sounds like a spherical cow model rather than a fundamental picture. However, the single-field
slow-roll inflation works surprisingly well at the quantitative level. There is no sign of isocurvature perturbations that would point to a more complicated inflaton sector. There is no sign of
running of the spectral index that would point to departures from the slow-roll conditions. There is no sign of non-gaussianities, that would point to large self-interactions of the inflaton field.
There is no sign of wiggles in the CMB spectrum that would point to some violent events happening during inflation. One can say that the slow-roll inflation is like a spherical cow model that
correctly predicts not only the milk yield, but also the density, hue, creaminess, and even the timbre of moo the cow makes when it's being milked.
Let's look into more details of the slow-roll inflation. Assuming the standard kinetic term for the inflaton field φ, the model is completely characterized by the scalar potential V(φ). The important
parameters are the first and second derivatives of the potential at the time when the observable density fluctuations are generated. Up to normalization, these derivatives are the slow-roll
parameters ε and η (see the equation box for a precise definition). Both have to be much smaller than 1, otherwise the inflaton field evolves too quickly to support inflation. Several observables
measured by Planck depend primarily on ε and η. In particular, the spectral index, which measures the departure of the primordial density fluctuation spectrum from scale invariance, is given by ns -
1=2η-6ε. Since Planck measured ns=0.9603±0.0073, we know the order of magnitude of the slow-roll parameters: either ε or η or both have to be of order 0.01.
Another important observable that depends on the slow roll parameters is the tensor-to-scalar ratio
. The system of an inflaton coupled to gravity has 3 physical degrees of freedom: the scalar mode linked to curvature perturbations, and the tensor mode corresponding to gravitational waves. The
scalar mode was detected in a distant past by the COBE satellite and its amplitude
is of order 10^-10. The tensor mode has not been detected so far. From the box you see that the amplitude
of the tensor mode is directly sensitive to the value of the inflaton potential, and for the slow-roll inflation it is expected to be somewhat smaller than
. In fact, the relative amplitude of tensor and scalar fluctuations is a direct measure of the parameter ε:
r=At/As = 16ε.
Now, the latest limit from Planck is r≲0.11 at 95% confidence level and, given we expect ε∼0.01 to fit the spectral index, it is already a non-trivial constraint on the shape of the inflaton
That's why in the plot of the best-fit area in the
plane many inflationary models fall into the excluded region. Basically, power-law potentials V(φ)∼φ^n that are too steep, n≳2, are excluded. The quadratic potential V(φ) = m^2 φ^2, perhaps the most
popular one, is on the verge of being excluded. What survives are power-law potentials with n≲2, or hilltop models where inflation happens near a maximum of the potential. The latter is predicted
e.g. in the so-called
natural inflation
where the inflaton is a Goldstone boson with a periodic cosine potential.
So, the current situation is interesting but unsettled. However, the limit
≲0.11 may not be the last word, if the Planck collaboration manages to fix their polarization data. The tensor fluctuations can be better probed via the B-mode of the CMB polarization spectrum, with
the sensitivity of Planck often quoted around r∼0.05. If indeed the parameter ε is not much smaller than 0.01, as hinted by the spectral index, Planck may be able to pinpoint the B-mode and measure a
non-zero tensor-to-scalar ratio
That would be a huge achievement because we would learn the absolute scale of inflation, and get a glimpse into fundamental physics at 10^16 GeV!. Observing no signal and setting stronger limits
would also be interesting, as it would completely exclude power-law potentials. We'll see in 1 year.
See the original Planck paper for more details.
18 comments:
What do you think of "Inflationary paradigm in trouble after Planck2013"? It makes the weird criticism that yes, simple inflationary models match the data from Planck ... but these models are
"anything but simple", according to the authors' version of how inflation ought to be, if it were true. The paper could be subtitled "concern-trolling the inflationary paradigm".
Yes, i read it, it's a very silly paper. It basically says that inflation is in trouble because it does not fit some moronic multiverse ideas.
I think one does not even have to read the whole paper, just reading the title is enough to guess that it is rather some kind of trolling instead of giving real serious reasonable physics
reasoning ;-)
The title reminds me of trolling news and popular articles about HEP ...
question.You say that V(φ)∼φ^n that are too steep, n≳2, are excluded.Don't you need φ^4 term for spontaneous breaking of symmetry? Is this turned off after symmetry breaking? Thanks.
Kashyap Vasavada
During inflation the scalar field is not at a minimum, so you don't need any phi^4 terms to stabilize the potential (it's different than the Higgs field which has to sit in a minimum with a
non-zero vev today, to which end you need the negative |H|^2 and the positive |H|^4 term in the SM). Moreover, it is not said that the inflaton potential cannot have any phi^4 term, but that phi^
4 cannot be the dominant one during inflation.
None of the models described here are embedded within Particle physics, or as a matter of fact within visible sector.
Inflation dilutes all matter, so the last 50-60 efoldings of inflation must be driven by fields which can directly decay into the SM degrees of freedom.
In these respects none of these models t shed any light on how perturbations and matter are created in the Universe.
Thanks for answering my question. If you don't mind I have a related question.Ok.I sort of understand that phi^4 term is not as important for inflaton as for higgs. But I have problem in
understanding the whole concept. In the Lagrangian derivation of spontaneous breaking of symmetry (required for phase transition?), you change the sign of (mass)^2.This seems to me an
uncomfortable mathematical jugglery! Admittedly after getting a new lower vacuum, all the (mass)^2 are positive. Is there any physical understanding of what you are doing? Is there a better
derivation which does not do this trick? Thanks. Only recently I became aware of your blog. I plan to read it regularly.
I agree with Jester on "Inflationary paradigm in trouble after Planck2013", the authors are desperately seeking their heads high --- there are already number of flaws with respect to pre-Big Bang
scenarios originally suggested, unless one resolves the Big Bang singularity one cannot trust iota of the perturbation calculation. It is really sad to see some senior and well respected people
can still push this theory without any theoretical and observational backing. It is really sad and rotten science from Princeton and Harvard.
Jester, could this imply that the scalar field is massless to some degree?
I don't think one can put it this way. The data disfavor the inflaton mass term driving the expansion, but there can be a large mass term in the potential as long as some other terms provide
larger contributions to the vacuum energy.
Wow. A succinct and very clear summary, that post clears up several points I was a bit hazy on. Thanks for that, Cormac
given that Planck is selecting inflationary models, is there a plot translating this into constraints on the reheating temperature?
Reheating into what degrees of freedom?
if you just mean -- the equation of state parameter just to be radiation, then a very good estimate will be T \sim V^{1/4} ( where V = inflaton potential ) or T\sqrt{\Gamma M_p}, where \Gamma is
the decay width of the inflaton into say for instance light fermions. However this calculation does not take into account what are the real degrees of freedom.
However if you wish to ask you need just the Standard Model degrees of freedom, then you really have to know the micrphysics of the inflaton itself and how inflaton is embedded in a Beyond the
Standard Model theory. The latter is a very serious issue and only few papers really bother to go into such details.
The right question is to ask what should be the reheat temperature of the Universe where all the Standard Model degrees of freedom are in thermal equilibrium, this is what the Universe cares as
far as the BBN is concerned. Note that there is hardly any room for extra relativistic species.
I suspect the word 'not' is missing from the last sentence of the second paragraph. Tnx again for a great post
Jester, have you died? No new post since April. I'm missing your blog updates.
Two months in exile ... but now Woit's away it's time you reclaimed the HEP blogging throne!
Three months later... EPS conference and still nothing... Is particle physics officially over then? :)
Are you still alive? Yes I know I saw your name somewhere, but that might be a zombie. | {"url":"https://resonaances.blogspot.com/2013/04/planck-about-inflation.html","timestamp":"2024-11-08T08:18:20Z","content_type":"application/xhtml+xml","content_length":"109969","record_id":"<urn:uuid:8e19f71f-48d9-4e1a-9310-5b8a05841cc1>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00587.warc.gz"} |