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Discovering the Law of Cosines
A triangle with sides a, b and c is drawn in the left hand panel. A regular polygon of n sides is erected on each side of the triangle. Use the blue dot in the left hand panel to adjust the ratio of
b to a. Use the gold dot to vary the shape of the triangle. [You can animate the shape of the triangle by clicking on the icon in the lower left hand corner of either panel.] You can change the
polygon erected on each side of the triangle by using the polygon slider. | {"url":"https://www.geogebra.org/m/Ad6k5Tpt","timestamp":"2024-11-12T11:52:19Z","content_type":"text/html","content_length":"87744","record_id":"<urn:uuid:2cf43082-7814-4128-8ce9-2ae93d6eb572>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00305.warc.gz"} |
29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Cite as
Bret Benesh, Jamylle Carter, Deidra A. Coleman, Douglas G. Crabill, Jack H. Good, Michael A. Smith, Jennifer Travis, and Mark Daniel Ward. Periods in Subtraction Games (Keynote Speakers). In 29th
International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp.
8:1-8:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
Copy BibTex To Clipboard
author = {Benesh, Bret and Carter, Jamylle and Coleman, Deidra A. and Crabill, Douglas G. and Good, Jack H. and Smith, Michael A. and Travis, Jennifer and Ward, Mark Daniel},
title = {{Periods in Subtraction Games}},
booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)},
pages = {8:1--8:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-078-1},
ISSN = {1868-8969},
year = {2018},
volume = {110},
editor = {Fill, James Allen and Ward, Mark Daniel},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.8},
URN = {urn:nbn:de:0030-drops-89015},
doi = {10.4230/LIPIcs.AofA.2018.8},
annote = {Keywords: combinatorial games, subtraction games, periods, asymptotic structure} | {"url":"https://drops.dagstuhl.de/entities/volume/LIPIcs-volume-110","timestamp":"2024-11-05T00:34:58Z","content_type":"text/html","content_length":"402429","record_id":"<urn:uuid:d88c9824-583c-4694-9589-274f6b86b6c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00613.warc.gz"} |
Introduction to snapKrig
Getting started
snapKrig is for modeling spatial processes in 2-dimensions and working with associated grid data. There is an emphasis on computationally fast methods for kriging and likelihood but the package
offers much more, including its own (S3) grid object class sk.
This vignette introduces the sk class and basic snapKrig functionality before showing an example of how to do universal kriging with the Meuse soils data from the sp package.
We recommend using the more up-to-date sf package to work with geo-referenced vector data, and for geo-referenced raster data we recommend terra. Both are loaded below.
sk grid objects
Pass any matrix to sk to get a grid object. Start with a simple example, the identity matrix:
# define a matrix and pass it to sk to get a grid object
mat_id = diag(10)
g = sk(mat_id)
# report info about the object on console
#> 10 x 10 complete
#> [1] "sk"
An sk object stores the matrix values (if any) and the dimensions in a list, and it assigns default x and y coordinates to rows and columns. This makes it easy to visualize a matrix as a heatmap
sk has its own custom plot method. The result is similar to graphics::image(mat_id) except that image is not flipped and (when ij=TRUE) the axes use matrix row and column annotations instead of y and
snapKrig has many useful methods implemented for the sk class, including operators like + and ==
# make a grid of logical values
g0 = g == 0
#> 10 x 10 complete
#> (logical data)
# plot
plot(g0, ij=TRUE, col_grid='white', main='matrix off-diagonals')
col_grid). See sk_plot for more styling options.
Spatial statistics is full of large, structured matrices and I find these heatmaps helpful for getting some intuition about that structure. For example the next plot shows a covariance matrix for a
square grid of points (n=100)
# get a covariance matrix for 10 by 10 grid
vmat = sk_var(sk(10))
# plot the grid in matrix style
plot(sk(vmat), ij=TRUE, main='a covariance matrix')
Various symmetries stand out: the banding; the blocks; the Toeplitz structures - both within and among blocks; and the unit diagonal. Visualize any matrix this way. Just pass it to sk then plot.
snapKrig internally stores grid data as a matrix vectorization that uses the same column-major ordering as R’s default vectorization of matrices:
# extract vectorization
vec = g[]
# compare to R's vectorization
all.equal(vec, c(mat_id))
#> [1] TRUE
When you pass a matrix to c or as.vector, R turns it into a vector by stacking the columns (in order). sk vectorizes in the same order, and allows square-bracket indexing, g[i], to access elements of
this vector.
A good way to jump in and start exploring snapKrig modelling functionality is to simulate some data. This can be as simple as passing the size of the desired grid to sk_sim.
# simulate data on a rectangular grid
gdim = c(100, 200)
g_sim = sk_sim(gdim)
# plot the grid in raster style
plot(g_sim, main='an example snapKrig simulation', cex=1.5)
You can specify different covariance models and grid layouts in sk_sim . Here is another example with the same specifications except a smaller nugget effect (‘eps’), producing a smoother output.
# get the default covariance parameters and modify nugget
pars = sk_pars(gdim)
pars[['eps']] = 1e-6
# simulate data on a rectangular grid
g_sim = sk_sim(gdim, pars)
# plot the result
plot(g_sim, main='an example snapKrig simulation (near-zero nugget)', cex=1.5)
snapKrig is unusually fast at generating spatially auto-correlated data like this and it supports a number of different covariance models. In simple terms, this changes the general appearance, size,
and connectivity of the random blobs seen in the image above. See ?sk_corr for more on these models.
Covariance plots
Use sk_plot_pars to visualize a covariance parameter set by showing the footprint of covariances surrounding the central point in a grid. For our simulated data, that looks like this:
Exporting grids
The simulation plot calls above used ij=FALSE (the default), which displays the grid as a raster, much like a terra or raster layer plot call. sk grid objects are similar in content to terra’s
SpatRaster object
#> complete sk grid
#> 20000 points
#> range [-2.21, 2.16]
#> ..............................
#> dimensions : 100 x 200
#> resolution : 1 x 1
#> extent : [0, 99] x [0, 199]
However, snapKrig functionality is more focused on spatial modeling and kriging. Outside of that context we recommend managing raster data with other packages (terra and sf in particular). sk will
accept single and multi-layer rasters from the terra and raster packages, reshaping them as sk grid objects; and sk grids can be converted to SpatRaster or RasterLayer using sk_export.
snapKrig provides sk_rescale to change the size of a grid.
# upscale
g_sim_up = sk_rescale(g_sim, up=4)
# plot result
plot(g_sim_up, main='simulation data up-scaled by factor 4X', cex=1.5)
Setting up=4 requests every fourth grid point along each grid line, and the rest are discarded. This results in a grid with smaller dimensions and fewer points. Setting argument down instead of up
does the opposite, introducing down-1 grid lines in between each existing grid line and filling them with NAs.
# downscale
g_sim_down = sk_rescale(g_sim_up, down=4)
# plot result
plot(g_sim_down, main='up-scaled by factor 4X then down-scaled by factor 4X', cex=1.5)
This returns us to the dimensions of the original simulation grid, but we have an incomplete version now. A sparse sub-grid is observed and the rest is NA (having been discarded in the first
sk_rescale call).
Down-scaling usually refers to the process of increasing grid dimensions, then imputing (guessing) values for the empty spaces using nearby observed values. sk_rescale doesn’t do imputation, but its
result can be passed to sk_cmean to fill in the unobserved grid points.
# upscale
g_sim_down_pred = sk_cmean(g_sim_down, pars)
#> 25 x 50 complete sub-grid detected
# plot result
plot(g_sim_down_pred, main='down-scaled values imputed by snapKrig', cex=1.5)
sk_cmean uses conditional expectation to predict the 20,000 values in g_sim (at original resolution) based only on the 1250 observed points in g_sim_down (1/4 resolution). The function is optimized
for raster data of this form (NA except for a sub-grid), and extremely fast compared to most kriging packages, making snapKrig a powerful down-scaling tool.
These results look impressive - the predictions look almost identical to our earlier plot of the full dataset (g_sim). But we are cheating here. We knew exactly which model was best for imputation
(pars) because we used it to simulate the data in the first place. More often users will estimate pars from the data using maximum likelihood estimation (MLE).
Fitting models
We recommend using MLE to fit snapKrig models. This is the process of looking for the model parameters that maximize a statistic called the likelihood, which is a function of both the parameters and
the data. Roughly speaking, the likelihood scores how well the model parameters match the data.
To illustrate, consider the model (pars) that we used to generate the simulation data. Suppose the two range parameters in the model are unknown to us, but the other parameters are known. We could
make a list of plausible values for the ranges and check the likelihood for each one, given the data.
# pick two model parameters for illustration
p_nm = stats::setNames(c('y.rho', 'x.rho'), c('y range', 'x range'))
# set bounds for two parameters and define test parameters
n_test = 25
bds = sk_bds(pars, g_sim_up)[p_nm, c('lower', 'upper')]
bds_test = list(y=seq(bds['y.rho', 1], bds['y.rho', 2], length.out=n_test),
x=seq(bds['x.rho', 1], bds['x.rho', 2], length.out=n_test))
To organize the results, make a grid out of the test values (similar to expand.grid) then fill it with likelihood values in a loop.
# make a grid of test parameters
g_test = sk(gyx=bds_test)
p_all = sk_coords(g_test)
#> processing 625 grid points...
# fill in the grid with log-likelihood values
for(i in seq_along(g_test))
# modify the model parameters with test values
p_test = sk_pars_update(pars)
p_test[p_nm] = p_all[i,]
# compute likelihood and copy to grid
g_test[i] = sk_LL(sk_pars_update(pars, p_test), g_sim_up)
The resulting likelihood surface is plotted below, and its maximum is circled.
# plot the likelihood surface
plot(g_test, asp=2, main='log-likelihood surface', ylab=names(p_nm)[1], xlab=names(p_nm)[2], reset=FALSE)
# highlight the MLE
i_best = which.max(g_test[])
points(p_all[i_best,'x'], p_all[i_best,'y'], col='white', cex=1.5, lwd=1.5)
This should approximately match the true scale parameter values that were used to generate the data
# print the true values
print(c(x=pars[['x']][['kp']][['rho']], y=pars[['y']][['kp']][['rho']]))
#> x y
#> 14.14214 10.00000
So if we didn’t know pars ahead of time (and usually we don’t), we could instead apply this principle and simply churn through plausible parameter candidates until we find the best scoring one.
However this grid search approach is usually not a very efficient way of doing MLE, and there are many good alternatives (just have a look through the CRAN’s Optimization Task View). snapKrig
implements MLE for covariance models in sk_fit using stats::optim. The next section demonstrates it on a real life dataset.
Example data: Meuse soils
This section looks at real geo-referenced points in the Meuse soils dataset (Pebesma, 2009), which reports heavy metal concentrations in a river floodplain in the Netherlands. These points are used
in the kriging vignette for gstat, which we loosely follow in this vignette, and they are lazy-loaded with the sp package.
Users can access the Meuse data directly by calling data(meuse) and data(meuse.riv), which returns data frames containing coordinates. For this vignette, however, I use a helper function, get_meuse,
to represent the data in a more snapKrig-friendly sf class object. The function definition for get_meuse is hidden from this document for tidiness, but it can be found in the source code
(“meuse_vignette.Rmd”) just below this paragraph.
# load the Meuse data into a convenient format
meuse_sf = get_meuse()
# extract the logarithm of the zinc concentration as sf points
pts = meuse_sf[['soils']]['log_zinc']
pts is a geo-referenced sf-class points collection. This means that in addition to coordinates and data values, there is a CRS (coordinate reference system) attribute telling us how the coordinates
map to actual locations on earth. This can be important for properly aligning different layers. For example, in the plot below, we overlay a polygon representing the location of the river channel
with respect to the points. If this polygon had a different CRS (it doesn’t), we would have first needed to align it using sf::st_transform.
# set up a common color palette (this is the default in snapKrig)
.pal = function(n) { hcl.colors(n, 'Spectral', rev=TRUE) }
# plot source data using sf package
plot(pts, pch=16, reset=FALSE, pal=.pal, key.pos=1, main='Meuse log[zinc]')
plot(meuse_sf[['river_poly']], col='lightblue', border=NA, add=TRUE)
plot(st_geometry(pts), pch=1, add=TRUE)
Snapping point data
snapKrig works with a regular grid representation of the data, so the first step is to define such a grid and snap the Meuse points to it using sk_snap. The extent and resolution can be selected
automatically, as in…
# snap points with default settings
g = sk_snap(pts)
#> maximum snapping distance: 15.4262127060058
#> 155 x 155 incomplete
…or they can be set manually, for example by supplying a template grid with the same CRS as pts, or by specifying some of the grid properties expected by sk. Here we will request a smaller grid by
specifying a resolution of 50m by 50m
# snap again to 50m x 50m grid
g = sk_snap(pts, list(gres=c(50, 50)))
#> maximum snapping distance: 33.2640947569598
#> 78 x 56 incomplete
#> incomplete geo-referenced sk grid
#> 4368 points
#> 155 observed
#> range [4.73, 7.52]
#> ..............................
#> dimensions : 78 x 56
#> resolution : 50 x 50
#> extent : [329737.5, 333587.5] x [178622.5, 181372.5]
The units of argument ‘gres’, and of the snapping distance reported by sk_snap, are the same as the units of the CRS. This is often meters (as it is with Meuse), but if you aren’t sure you should
have a look at sf::st_crs(pts) for your pts.
Call plot on the output of sk_snap to see how these points look after snapping to the grid. As with sk object plots, you can overlay additional spatial vector layers using the add argument.
# plot gridded version using the snapKrig package
plot(g, zlab='log(ppb)', main='snapped Meuse log[zinc] data')
plot(meuse_sf[['river_poly']], col='lightblue', border=NA, add=TRUE)
Here we’ve set a fairly coarse grid resolution to keep the package build time short. The result is a somewhat pixelated-looking image and a high snapping error. This error can be controlled by
reducing ‘gres’ (the spacing between grid points). Users might want to try substituting gres=c(25, 25) or gres=c(5, 5) to get a sense of the speed of snapKrig on large problems.
Be warned that if the grid resolution is fine enough, individual pixels can become invisible in plot calls, giving the false impression that there is no data. When there really is no data, the output
of print(g) and summary(g) will say so. If you don’t believe them, call which(!is.na(g)) to locate the non-NAs in your grid.
The snapKrig model splits point values into two components: random spatial variation; and a non-random (but unknown) trend. This trend is assumed to be a linear combination of spatially varying
covariates, known throughout the area of interest. The process of fitting both components of the model and then generating predictions is called universal kriging.
In this example we use just one covariate, distance to the river, but users can also supply several, or none at all (simple and ordinary kriging are also supported). snapKrig will adjust for any
covariates, and fit the random spatial component to the remaining unexplained variation. This is similar to the way that we estimate variance from model residuals (observed minus fitted) in simple
linear regression.
To fit a model you only need to know your covariates at the observed point locations, but to do prediction with universal kriging you will need them at all prediction locations. In our case we can
create this layer directly by passing the data grid point locations and the river line geometry to sf::st_distance
# measure distances for every point in the grid
river_dist = sf::st_distance(sk_coords(g, out='sf'), meuse_sf[['river_line']])
#> processing 4368 grid points...
To create a new sk grid object containing these distances, simply copy g and replace its values with the numeric vector of distances from river_dist. We recommend also scaling all covariates for
numerical stability
# make a copy of g and insert the scaled distances as grid point values
X = g
X[] = scale( as.vector( units::drop_units(river_dist) ) )
#> complete geo-referenced sk grid
#> 1 layer
#> 4368 points
#> range [-1.44, 3.27]
#> ..............................
#> dimensions : 78 x 56
#> resolution : 50 x 50
#> extent : [329737.5, 333587.5] x [178622.5, 181372.5]
The result is plotted below, along with the center line of the river channel in black.
# plot the result
plot(X, zlab='distance\n(scaled)', main='distance to river covariate')
plot(meuse_sf[['river_line']], add=TRUE)
It is unusual to be able to generate covariates at arbitrary locations like this. More often users will have pre-existing covariates, and their layout will dictate the layout of the prediction grid.
A typical workflow therefore begins with an additional step:
1. consolidate all covariate layers into a common grid, g (possibly using terra::project)
2. snap the response data pts to this grid using sk_snap(pts, g)
3. fit the model and compute predictions
Model fitting
For the first part of step (3) we provide sk_fit, which fits a model to data by numerical maximum likelihood. Its default settings (isotropic Gaussian covariance) will work for many applications, and
they work well enough in this example. This makes model fitting very straightforward:
However, in order to get the best model fit (and the best predictions) we strongly recommend understanding and experimenting with the arguments to sk_fit. These control the covariance structure, the
parameter space, and other optimizer settings. We also encourage users to check diagnostics on the parameter list returned by sk_fit using functions like sk_plot_pars and sk_plot_semi.
sk_fit fit works by searching for the maximum of the (log) likelihood function for the model given the data, using R’s stats::optim. Finding the likelihood manually for a given parameter set is
simple. If the parameters are in the list form returned by sk_fit, simply pass it (along with the data and any covariates) to sk_LL.
For users with their own preferred optimization algorithms, snapKrig also provides the convenience function sk_nLL, which is a wrapper for sk_LL that negates its result (so the problem becomes
minimization), and accepts parameters in its first argument as a vector.
print and summary reported that g is an incomplete sk grid, and we saw from its mostly empty heatmap that the majority of the grid is unsampled (having NA grid point values). We are going to now fill
in these spatial gaps using kriging predictions from sk_cmean. This is the final step of universal kriging.
The call returns a complete version of the observed data grid g, where all values (including the observed ones) have been replaced by predictions using the model defined in fit_result_uk (returned
from sk_fit), and the covariates grid(s) in X.
plot(g_uk, zlab='log[zinc]', main='universal kriging predictions')
plot(meuse_sf[['river_line']], add=TRUE)
ie the trend) and the other is the random spatial component, which is interpolated from the observed points.
In ordinary and universal kriging these two components are interrelated - the trend estimate influences the spatial component estimate and vice versa. In some special cases however, the users may
wish to disentagle them (for example if the trend is known a priori, or a nonlinear trend is being modeled separately), in which case the response data (g) should be de-trended, and X should be set
to 0 (not NA) in the sk_fit and sk_cmean calls. This is called simple kriging
Of all linear unbiased predictors, the kriging predictor is by definition optimal at minimizing prediction uncertainty. This is a good reason to prefer kriging, but it doesn’t mean you shouldn’t
worry about uncertainty in your problem. In fact, one of the nice things about kriging theory is its explicit formula for prediction variance. We can compute it directly, rather than having to
To compute kriging variance, call sk_cmean with argument what='v'.
# compute conditional mean and variance
g_uk_var = sk_cmean(g, fit_result_uk, X, what='v', quiet=TRUE)
As before the function returns a complete grid, this time with kriging variance values. Taking square roots yields the standard error of prediction
plot(sqrt(g_uk_var), zlab='log[zinc]', main='universal kriging standard error')
plot(meuse_sf[['river_line']], add=TRUE)
plot(st_geometry(pts), pch=1, add=TRUE)
The observed point locations are outlined in this plot to emphasize how uncertainty increases with distance to the nearest observation. It also increases as values of the covariates veer into
extremes (locations far from the river channel), as these covariate values have no associated (zinc) observations.
Notice that even when a grid point coincides exactly with an observation, there is nonzero uncertainty. This reflects a spatially constant measurement error that is represented in the model by the
nugget effect. Find this parameter in list element ‘eps’ of the parameter list returned by sk_fit.
This nugget effect is important for realism, as virtually all real-life datasets have measurement error, but it is also important for numerical stability. While it is possible to set the nugget to
zero - producing an exact interpolator - this can have unpredictable results due to numerical precision issues.
So far we have been been working with the logarithms of the zinc concentrations. This produces something closer to a Gaussian random variable - a requirement of kriging theory. But when it comes to
predictions and applications, we are probably after the un-transformed values.
Taking exp(g_uk), while intuitive, would introduce a negative bias. The mistake is in assuming that E(f(X)) is the same as f(E(X)) (for expected value operator E and transformation f), which is only
true if f is linear.
In short, to get zinc concentration predictions on the original scale, we need a bias adjustment. We use a simplified version of the one given in Cressie (2015) - adding half the variance before
exponentiating. The two plots below shows the result on its own, and again with the original observed point data overlaid.
# prediction bias adjustment from log scale
g_uk_orig = exp(g_uk + g_uk_var/2)
# points on original scale
pts_orig = meuse_sf[['soils']]['zinc']
# prediction plot
zlim = range(exp(g), na.rm=TRUE)
plot(g_uk_orig, zlab='zinc (ppm)', main='[zinc] predictions and observations', cex=1.5, zlim=zlim)
plot(meuse_sf[['river_line']], add=TRUE)
# full plot
plot(g_uk_orig, zlab='zinc (ppm)', main='[zinc] predictions and observations', cex=1.5, zlim=zlim, reset=FALSE)
plot(meuse_sf[['river_line']], add=TRUE)
# overlay observation points
plot(pts_orig, add=TRUE, pch=16, pal=.pal)
plot(sf::st_geometry(pts_orig), add=TRUE)
The underlying heatmap is our final predictor and on top we have plotted the observed data. In order to make the color scales match, we have masked the heatmap in this plot to have the same range as
the observations. | {"url":"https://pbil.univ-lyon1.fr/CRAN/web/packages/snapKrig/vignettes/snapKrig_introduction.html","timestamp":"2024-11-03T22:13:39Z","content_type":"text/html","content_length":"712916","record_id":"<urn:uuid:16eb5483-99ae-42c6-a3b0-fa8e0260175a>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00193.warc.gz"} |
Van's Blog
In previous blogs, I described conversations with Andrew Bauman, an up-and-coming mathematician who is an undergrad at UALR and who spends spare time tinkering with problems in linear algebra and
quantum mechanics. Today, in one of our gym-facilitated meetings, Andrew brought three distinguished and enjoyable intellects besides himself. This led to a wide-ranging conversation that I will
attempt to recap here for posterity. We will start with a sidebar since the evening was full of them, some recursive, some eight layers deep.
Before tonight’s impromptu, mostly math and quantum computing discussion, I checked in with our pool lifeguard, a pole vaulter (not the back flipping fellow above, but certainly capable). A review of
video footage of his record pole vault revealed excellent form, with one specific moment that could benefit from an improvement that consisted of pressing to a handstand and walking on one’s hands.
This body position is attained for less than a second during the pole vault but is critical to obtaining greater heights. He showed me footage of his hand standing and walking, which was quite good.
It gave me ideas for a follow-up exercise involving an inverted shoulder shrug, an inverted pole grip change/grip walk, and an inverted kip-up that could benefit him further. I’m writing this here as
part of a stream-of-consciousness recap so I don’t forget it in the twists and turns of what follows.
Weight, there's more:
I strapped a 5 lb weight on each hip for my nightly mile walk/run tonight. However, due to the chance encounter with Andrew (and company), I did not make it to my walk, but I sported the weights like
a pair of revolvers from the Wild West. Anyone seeing our extended conversation would have to wonder, “Why a weight belt for talking about math?”. Answer, “Heavy Topic”!
Andrew paused his ping-pong game to introduce me to his ping-pong partner, Dr. Sudan Xing, a mathematician and professor at UALR who specializes in geometric projection and embedding problems.
A peek at Google Scholar introduces us to the central theme of Dr. Xing's work, which is focused on the Orlicz-Brunn-Minkowski theory and related Minkowski problems in convex geometry. This theory is
an extension of the classical Brunn-Minkowski theory, which deals with the relationship between the volumes of convex bodies and their Minkowski sums.
Now I’m a dolt, so I wanted to know what a simple Minkowski sum looked like, so I asked ChatGPT-4 to write me some code and produced the figure below. It was almost a 1-shot job; more recent work I
have done has taken nearly 40 shots/redos to get right. The code is here.
The mathematics involved in her work primarily comes from convex geometry, functional analysis, and measure theory. Key concepts include:
• Convex bodies and functions
• Minkowski addition and Orlicz addition
• L_p norms and Orlicz norms
• Surface area measures and the Minkowski problem
• Brunn-Minkowski and isoperimetric inequalities
• Log-concave functions and measures
Dr. Xing's work contributes to developing a more general theory of convex bodies and related geometric inequalities, with potential applications in mathematics and beyond.
Next, we met her bioinformatics colleague, Ju Ni. We discussed how exciting it was to live in the time of AI/ML and the OMIM database, where we can know the genes involved in almost any affliction of
human beings. We discussed the importance of visualizing gene and biochemical pathways for specific conditions. We discussed the particular example of Thyroid Cancer, one of the only truly “curable”
cancers, since it can be treated with Iodine-131, which is preferentially taken up by the thyroid, thus neutralizing the cancer, but alas, the thyroid as well, necessitating lifelong medication
We briefly referenced a certain Calculus book, a Facebook math site, and my AddSubMulDivia five-book series “that nobody reads.” We had a good laugh about how pathetic it is to care so much about
things no one else does.
Dr. Xing and I found out we shared appreciation for 2010 Fields medal laureate Cedric Villani, a mathematician and politician who is friends with the president of France. Quoting from my favorite
Villani revolutionized mathematical physics with contributions to optimal transport theory, kinetic theory, partial differential equations, and the study of Ricci curvature in metric spaces. His
work, notable for bridging pure mathematics with applied physics, includes groundbreaking analyses of the Boltzmann equation and its convergence to the Landau equation, shedding light on gas
behaviors in varied regimes. Villani's innovative use of optimal transport for exploring metric spaces with Ricci curvature bounds has influenced areas ranging from plasma physics to network
We talked about:
the advent of AI/ML is enabling the revisiting of unsolved math problems
symbolic algebra (now called CAS)
symbolic geometry, invented by my friend Phil Todd and his program Geometry Expressions™
Andrew mentioned he was working on a problem that starts with drawing a bisecting line on a piece of paper, and proving that the halfspaces generated by the line are distinct.
Dr. Xing responded with H+/H- halfspaces, then visualized the problem in 3D with her 'ping-pong paddle' analogy. She discussed the 'floating problem' – a sphere trimmed into equal-volume sections by
tangent planes. This process seems to give the sphere unique properties, but we switched topics too quickly.
We touched on projective geometry and how representation (explicit, implicit, parametric, iterated, chaotic) influences what we can understand about a problem. I mentioned the challenges of
intersecting closed tensor product surfaces made with B-splines and wondered if Dr. Xing's dualized projection approach could help.
I mentioned my curiosity about a set of planar ray tracing problems as a family of reachability problems that are quite interesting. These live under the heading “Visibility Polygon” and the Art
Gallery Problem.
We talked a lot about how people conceptualize various kinds of mathematical concepts.
At this time, Greg, a friend of Andrew’s, had arrived. He has a Master’s degree and was focusing on math education. Dr. Xing and Ju Ni had to go, so Greg, Andrew, and I started drilling down on
several topics.
The topic space exploded before we settled down and had a heart-to-heart on quantum computing and Bell’s Inequality.
Topic Explosion (Free Association Gone Wild)
• inner products and their connection to standard deviation and variance
• Hilbert spaces, norms
• distance metrics: Euclidean, Manhattan, Minkowski
• closure loss on the inclusion of zero
• maintaining state on chained binary operations of AddSubMulDiv-ia to enable reversibility and prevent the loss of structure of the path of a calculation that would otherwise be non-unique if
results were discarded at intermediate states. Undoability.
• skew and kurtosis being higher moments in statistics
• discrete and continuous distributions
• moments in structural mechanics and moments of inertia.I demonstrated how a phone tossed in space will land without a change in rotation in two of its three axes but not the third. The Veritasium
link below explains it better.
• Spaghetti Sort as an analogy to quantum computing simultaneous equation solving using entangled particles.
• The mystery of entanglement is the same as having tossed a coin that landed heads and automatically knowing that the other particle’s spin is “tails”.
• The Bloch sphere
• Bell’s inequality
• Alice and Bob’s experiment: A stream of entangled particles
• Quantum and Classical Interpretations of the experiment
• The interchangeability of streaming and fixed interval experiments
• Interval arithmetic
• The register operation of comparing A and B’s results,
• XNOR was the decision operator correctly identified by Greg
• Marvin Minsky’s XOR catastrophe started the AI winter, requiring two neurons. | {"url":"https://lvwarren.blogspot.com/","timestamp":"2024-11-13T02:50:31Z","content_type":"text/html","content_length":"85644","record_id":"<urn:uuid:2ff44eed-5baa-4fdc-bd33-fb36d323854b>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00527.warc.gz"} |
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All about Gear Effect - 5
Dave Tutelman -- February 22, 2009
The role of roll
After the previous pages were posted to my web site, Marcel Bal asked me about face roll. Specifically, he wanted to know if vertical face curvature was a good thing or a bad thing. He mentioned Tom
Wishon's GRT (graduated roll technology) drivers, which are specifically designed with minimal face roll except fairly high on the clubface -- and even then then a relatively small roll.
What a wonderful question! And why didn't I think to ask it before I posted the article? It would seem a logical culmination of all that work, because it requires almost everything to give a complete
answer. In fact, it probably calls for correcting for clubhead rotation; even though the rotation is small during impact, it can have a non-neglible effect on the answer to this question.
Let's see if we can come up with an answer.
General approach
We clubmakers tend to think of a single loft number as the ideal loft for a specific swing, where a swing is a combination of clubhead speed and angle of attack, and perhaps other effects like shaft
bend and wrist position at impact. I won't debate exactly what goes into it, but rather address the idea that the ideal loft is a single number at all heights on the clubface.
When we look for the ideal loft, we vary the loft while looking at launch conditions -- ball speed, launch angle, and backspin -- and how the combination affects the distance. Our simple model of
ideal loft therefore assumes that, for a given swing, the launch conditions (especially the backspin) is the same wherever on the clubface you strike the ball. Now we know that assumption is wrong!
Figure 5-1
Figure 5-1 shows the clubface angle (the loft) at various heights on the face, specifically at the center and a quarter-inch and half-inch above and below the center. The black line on the left is a
constant-loft face (in other words, no curvature, no face roll). Are the launch conditions going to be the same at each height? In a word, no!
• The backspin will vary quite a lot. The data we used to validate vertical gear effect showed a variation of 1300rpm over this same range of heights.
• The ball speed and launch angle may vary a small amount, because of face rotation caused by a high or low impact.
So we can't just assume that the same loft is the ideal, or optimum, loft at each height on the face. The red and blue segments are a more realistic picture. We can look at the launch conditions at
each face height and optimize the loft for that height, taking into account face rotation (and thus gear effect). We already know that the backspin will be less as we move up the face. A little work
with any trajectory program tells us emphatically that decreased backspin calls for increased launch angle, and vice versa. So we can expect that high on the clubface (low backspin) the best loft
should be higher than the center of the face. Conversely, the backspin is higher at the bottom of the face, so we can expect the optimum loft there to be lower.
If we take these ideal loft segments and string them together, as we do at the right on Figure 5-1, we see that the progression of lofts requires a curvature of the clubface. So, if we have a
collection of ideal lofts, that implies what the ideal face roll should be.
How can we go from a set of optimum lofts at various face heights to roll radius? Actually, that's the simple part. (The hard work was finding the optimum loft at each face height.)
Figure 5-2
Figure 5-2 shows a driver head, with the loft specified at two points. The two lofts are L[1] and L[2], and they are separated by a height difference of ΔH. What we will do is find a series of lofts
L[1], L[2], etc, at height increments of ΔH.
Those lofts constitute a curvature statement. But we are used to seeing the curvature in terms of radius -- as roll is usually specified. We will have to turn that series of lofts into a series of
radii of curvature. If they are all pretty similar, we can use the average as a fair description of the roll radius for the whole clubface. If there is a lot of variation, then the ideal clubface has
a graduated roll. This may be the same as Wishon's GRT graduated roll -- or different. We're going to find out which.
The figure hints at the computation, but let's be explicit. We have drawn two radii, perpendicular to the face at the two points of interest. Where they meet is the radius of curvature, at least for
the section of clubface between the two points. The angle at which they meet is the difference between the lofts
ΔL = L[2] - L[1]
The angle ΔL as a proportion of 360° is the same as ΔH as a proportion of the circumference 2πR, or:
Solving the proportion for R, we get:
R = 360 ΔH = 57.3 ΔH
2π ΔL ΔL
So the steps are:
1. Choose a set of heights on the clubface. We will choose -0.8" to +0.8" by 0.2" increments.
2. Optimize the loft at each point.
3. Look at the optimized lofts, and draw some conclusions about face roll.
Let's do it!
Equations we will need
If you're interested in the results but not the math, skip this sub-chapter.
We already have, or can easily find from what we've already done, the equations we need. They are summarized in the table below.
│ What it finds │ Formula │ Where or how │
│ │ │ we got it │
│ Ball speed V[b] (mph) │ │ Golf physics │
│ │ = 0.813 V[c] (1+e) cos (L) │ tutorial │
│ Backspin from loft s[o] (rpm) │ 160 V[c] sin (L) │ Golf physics │
│ │ │ tutorial │
│ Launch angle a (degrees) │ L (0.96 - .0071 L) │ Golf physics │
│ │ │ tutorial │
│ Torque moment arm y (inches) │ H - D sin a │ This article, │
│ │ │ vertical gear effect │
│ │ │ This is half the │
│ Correction to reduce ball speed │ │ backward velocity │
│ due to clubhead rotation (mph) │ │ of the face │
│ │ │ at release. │
│ │ │ See appendix │
│ │ │ This is half the │
│ Correction to increase launch angle │ │ rotation during │
│ due to clubhead rotation (degrees) │ │ impact. │
│ │ │ See appendix │
│ Gear effect topspin s (rpm) │ 25 V[b] y │ Basic vertical │
│ │ │ gear effect spin │
Now we have enough to create a spreadsheet to give us launch conditions for any loft and clubhead speed. We will build the spreadsheet using the above formulae, and then iteratively solve for the
optimum loft. That is:
• We will start with a constant clubhead speed (102.7mph, to give 150mph ball speed) and constant loft across the face (11°). Further assumptions about the driver:
□ We will assume a constant COR across the face at 0.83, the rules-allowed maximum. Designers have learned how to shape the face thickness to preserve COR across a lot of the face, so this
analysis assumes such a driver.
□ We continue (as in all the previous examples) with a zero angle of attack.
• For each height on the clubface (H= -0.8" through H= +0.8"), we will do the following:
1. Use the spreadsheet to compute a set of launch conditions (ball speed, launch angle, and backspin).
2. Plug the launch conditions into TrajectoWare Drive (TWD) and find the carry distance and angle of descent.
3. If this is the maximum distance we are likely to get, note it and the loft -- then move on to the next H. Otherwise, choose a new loft (higher or lower) and go back to step 1.
Optimized roll results
After we go through those calculations, we will have found the optimum loft for each height on the face. Then we will use that to picture the face roll curvature. And here is the result of those
│ Inputs │ Computed intermediate results │ Backspin │ Outputs │
│ H │ Loft │ Ball speed │ Launch angle │ y │ Correct'd ball │ Correct'd launch │ Due to │ Due to │ Net │ Carry Distance (TWD) │ Angle of │ Roll radius R │
│ │ │ │ │ │ speed │ angle │ loft │ gear effect │ │ │ Descent (TWD) │ │
│ -0.8 │ 4.7 │ 152.2 │ 4.4 │ -0.90 │ 148.3 │ 2.9 │ 1346 │ 3331 │ 4677 │ 213.9 │ 31.9 │ --- │
│ -0.6 │ 5.8 │ 152.0 │ 5.3 │ -0.72 │ 149.4 │ 4.2 │ 1660 │ 2692 │ 4353 │ 221.7 │ 31.8 │ 10.4 │
│ -0.4 │ 7.2 │ 151.6 │ 6.5 │ -0.55 │ 150.1 │ 5.7 │ 2059 │ 2057 │ 4117 │ 228.5 │ 34.0 │ 8.2 │
│ -0.2 │ 8.4 │ 151.2 │ 7.6 │ -0.37 │ 150.5 │ 7.0 │ 2400 │ 1396 │ 3797 │ 233.9 │ 34.2 │ 9.6 │
│ 0 │ 9.6 │ 150.7 │ 8.6 │ -0.19 │ 150.5 │ 8.3 │ 2740 │ 728 │ 3469 │ 237.9 │ 34.1 │ 9.6 │
│ 0.2 │ 11.4 │ 149.8 │ 10.0 │ -0.03 │ 149.8 │ 10.0 │ 3247 │ 99 │ 3347 │ 241.0 │ 35.4 │ 6.4 │
│ 0.4 │ 12.8 │ 149.0 │ 11.1 │ 0.15 │ 148.9 │ 11.4 │ 3640 │ -553 │ 3087 │ 242.6 │ 36.8 │ 8.8 │
│ 0.6 │ 14.3 │ 148.1 │ 12.3 │ 0.32 │ 147.6 │ 12.8 │ 4058 │ -1192 │ 2867 │ 242.8 │ 37.4 │ 7.6 │
│ 0.8 │ 16 │ 146.9 │ 13.6 │ 0.50 │ 145.7 │ 14.3 │ 4529 │ -1803 │ 2726 │ 242 │ 38.6 │ 6.7 │
Figure 5-3
When we look at the main result, the roll radius at each height on the face, we see neither a constant number nor a smooth curve. There is a trend, but it is a jagged one, as shown in the graph at
the right. It looks like the optimum face roll is in the 8"-10" range on the lower part of the face, and the 6"-8" range on the upper part. Why should it be this "noisy" a relationship?
• Take a look at the formula for roll radius. It is proportional to ΔH / ΔL, which is the slope of the height vs loft curve. But we are restricting our search to the best loft to a tenth of a
degree. That restricts ΔL to the nearest tenth of a degree, which limits the computed roll to a few discrete values, not a continuum. That is the major reason for "noise".
• By way of further explanation, the distance vs loft curve is fairly flat near its optimum -- and we are working near the optimum. That means that we could be off by a tenth of a degree of loft.
As we saw, numerical differentiation amplifies error. A tenth of a degree error in two successive readings could throw off the face roll by 2 inches.
• In fact, even the trend might be wrong; the correct answer might be a constant roll or, for that matter, a bigger difference between the top and bottom of the clubface. Why? The lower quarter of
the clubface (H≤ -0.4) shows very high net spin -- over 4000rpm -- because the loft spin is adding to a strong gear effect backspin. But TrajectoWare Drive starts to lose accuracy at spins over
4000rpm. So we cannot take very seriously any results from the bottom of the clubface.
• Finally, it is worth remembering that we assume a 0.83 COR across the entire height of the clubface. Any "sweet spot" effect is due entirely to roll radius and gear effect. If the face is hotter
in some spots than others, the COR profile has to be superimposed on our work, and may affect the optimum loft at some of the points.
Even with all these caveats, there is a reasonable conclusion to be drawn from the data. The face that the best possible loft progresses with height on the face tells us that the best possible face
must be curved. Thus... Face roll is useful in keeping the distance up for high and low mishits. How useful? Let's run our spreadsheet a few more times and see.
How sensitive is distance to the roll radius?
This time, instead of finding the loft (L) at each face height (H), we will assume a roll radius (R) and compute the distance at each face height. The loft at each height is easy enough to compute,
knowing the roll radius. Remember the equation we used earlier:
R = 57.3 ΔH / ΔL
If we choose some base H[o] and L[o] for the clubhead, we can find the loft at any face height by solving that equation.
We will continue to use a clubhead speed of 102.7mph, giving a center-hit ball speed of 150mph. From our previous work, we know that the maximum carry distance occurs at about H=½" or perhaps a
fraction higher, and the best loft at H=½" is 13.6°. Let is use these as H[o] and L[o]. That way, we have the same maximum distance no matter what the roll radius.
Figure 5-4: Ball speed = 150mph
Figure 5-4 is a graph showing how carry distance varies with H for different values of the roll radius R. The graph shows several things pretty clearly:
• A radius of 8" (the yellow curve) seems to give the best preservation of distance as the height of impact varies. It makes for the most forgiving clubhead -- at least forgiving of height errors.
• For R=8", the distance is within a yard of the maximum anywhere from ¼" to ¾" above the center of the face.
• Things are still quite good with a roll radius of 10" (the green curve). We still lose only a yard from ¼" to ¾" above the center. A lower strike, say on-center, is a yard shorter than the 8"
roll -- not bad, but probably measurable.
• We see a little more loss of distance at 6" and 12". Now we are losing three yards from a center hit.
• A nearly flat face without roll (R=40") drops off quite severely when struck either high or low of the optimum.
This tells us that roll matters! It is a good thing. It "makes the sweet spot bigger".
A flat face is way too sensitive to the height at which the ball is struck. Yes, it may preserve a nice-looking trajectory; a low strike will not produce a hot worm-burner. But, given the extra
backspin due to gear effect with a low strike, the worm-burner needs that spin and will give more distance -- and the "nice trajectory" from the flat face will balloon and fall short.
But is an 8" roll radius really best? It may well be. Then again, we don't see that much curvature on the clubheads that are being sold. Perhaps the gear effect model may be a little off. The physics
is definitely sound, but our estimate of things like vertical moment of inertia may be a little off. Certainly not by an order of magnitude. Most likely not even by a factor of two. But it might be
off enough so that the ideal radius is more like 10" or even 12". I would be surprised if it is off by more than that. And anything approaching a flat face has got to be wrong.
You might ask about another possible source of error in our conclusion. TrajectoWare Drive, the software that computed the distance, loses accuracy at higher spin rates. Is that an issue here? No, it
is not. Spin rates high enough to give inaccurate distances did show up, but only for distances under 230 yards, lower than the bottom of the graph. None of the data shown on the graph is affected by
TrajectoWare Drive error.
One roll fits all?
Is 8" -- or, for that matter, is any roll radius -- the best for all golfers? We haven't addressed that issue at all so far. Let's test the one-size-fits-all assertion by trying higher and lower
clubhead speeds and seeing how the ideal roll varies. So far, we have assumed in every case that the golfer has a 102.7mph clubhead speed, generating a ball speed of 150mph. Now we'll try ball speeds
of 180mph and 120mph.
Figure 5-5: Ball speed = 180mph
First we'll try a higher clubhead speed. Figure 5-5 is based on a clubhead speed of 122.5mph, which gives a maximum ball speed of 180mph. This would be a pretty big hitter on the PGA Tour, but not
the top ten and definitely not a long drive competitor. For this clubhead speed, the best distance occurs at H=0.6" and a loft at that height of 12.5°.
The 8" roll radius is still the best. Looking at the second best, this time it is the 6" roll. (At 150mph it was the 10" roll.) So perhaps the ideal roll is a little more curved for higher ball
speeds -- a bit above 8" at 150mph and a bit below at 180mph. But it is not a very big difference, considering that 20mph is a huge difference in clubhead speed, and 30mph a huge difference in ball
Figure 5-6: Ball speed = 120mph
How about lower ball speeds? Let's try a clubhead speed of 83mph, giving a ball speed of 120mph. All the drivers modeled had a 17.2° loft at a point 0.6" above the center of the face. We see the
result in Figure 5-6.
This time, the 8" roll shares top billing with the 10" roll, suggesting that perhaps a 9" roll might even be a little better. Still, that is not all that different from the higher clubhead speeds.
Face roll makes a significant difference in how forgiving a driver is to high or low misses. A flat face has a very small "sweet spot" in terms of height; the carry distance falls off sharply if
impact is above or below this height.
The optimum face roll varies a bit with clubhead speed and ball speed. But not much. I would estimate the best face roll for a 180mph ball speed is less than 2" different from that for 120mph ball
speed. This might be worth working the problem for a long drive competitor, but probably not for someone playing competitive golf. If the roll is optimized for 150mph, the lost distance is less than
a yard for any ball speed between 120mph and 180mph (compared with a roll optimized for that ball speed), for any strike at the middle of the face or higher.
The calculations show an optimum face roll in the vicinity of 8". This may or may not be correct, depending on the accuracy of the model and the assumptions involved.
This invites a discussion of sensitivities of the conclusion. The main sensitivity is the actual amount of ball spin due to vertical gear effect. For the range of interest, more gear effect spin
requires more face curvature. In particular:
• If the clubhead moment of inertia is higher than the model estimates, then the optimum roll radius may be more (a flatter face).
• If the CG is farther back in the head, then the optimum roll radius may be less (a more curved face).
• If the effect on launch angle and ball speed due to face rotation is higher than we used, then the optimum roll radius may be more (a flatter face).
• If the shaft tip is actually stiff enough to significantly limit vertical gear effect, then the optimum roll radius may be more (a flatter face).
We noted earlier that the model disagrees with two reports of the amount of gear effect spin, Wishon and Upshaw, which certainly do not agree with each other in any event. The model predicts spins
much higher than Wishon reports, and closer to Upshaw's. But I was unable to duplicate analytically Upshaw's reported dependency on shaft tip stiffness. Looking at the sensitivites above:
1. If Wishon's much smaller estimate of gear effect spin is correct, then the optimum roll radius would be much greater (a much flatter face). This explains his GRT driver heads, with a 15" roll for
the top third and a 20" roll for the bottom two thirds.
2. If Upshaw's shaft-sensitivity of gear effect is correct, then the optimum roll radius would become heavily dependent on the shaft. It might be anywhere from 8" to nearly flat, depending on the
shaft used.
Barring either of these extremes, the bullet-points above are unlikely to cause even a factor of two difference in the optimum roll. In real terms, the optimum roll is almost certainly between 8" and
12". For drivers! Hybrids or even fairway woods might be quite different, because their MOI and depth of CG are quite different.
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Biochemical Oxygen Demand (BOD) | BOD Equation & Solved Example
Biochemical Oxygen Demand, BOD is defined as the oxygen required by aerobic microorganisms to oxidise the biodegradable organic matter. In other words, the BOD indicates the amount of oxygen that
bacteria and other microorganisms consume in a water sample during a specific period at a constant standard temperature (20 degrees celsius) to degrade the water contents aerobically.
BOD is thus an indirect measure of the sum of all biodegradable organic substances in the water. This also indicates how much dissolved oxygen (milligram per litre) is needed in a given time for the
biological degradation of the organic wastewater constituents.
Some oxygen is always dissolved in water and it is known as dissolved oxygen (DO). DO is necessary for the survival of aquatic life and a minimum of 4 PPM of dissolved oxygen is required for the
survival of aquatic life. In the case of wastewater, organic matter present requires the DO to decompose them.
Microorganisms that decompose the organic matter in the presence of oxygen are known as aerobic bacteria. BOD is defined as the oxygen required by aerobic microorganisms to oxidise the biodegradable
organic matter.
It can be used for finding the quantity of oxygen required to stabilise the biodegradable organic matter. It helps us in deciding the size of the treatment units of the wastewater treatment plant. It
gives us an idea of the efficiency of the process i.e. how much BOD is being removed. Other than that we can find the strength of sewage.
We can further classify organic matters into two groups:
1. Carbonaceous matter: First Stage BOD
2. Nitrogenous matter: Second Stage BOD
Total BOD is the summation of the above two.
How to find the BOD?
The standard method used in the laboratory to find the BOD of a sample is the 'Dilution Method'. It is a time taking process, in 5 days 60% -70% of the organic matter is decomposed. In 20 days
95%-99% of the organic matter is oxidised. The water sample is diluted with aerated water and the initial DO is found, then it is incubated for 5 days at 20 Degrees Celsius. After these 5 days, we
again find the DO which is now known as the final DO.
BOD₅ at 20 Degrees Celsius = (Initial DO - Final DO) x Dilution Factor
The Dilution Factor is defined as the ratio of Volume of diluted sample to Volume of the undiluted sample.
Dilution Factor, DF = Diluted Sample Volume/Undiluted Sample Volume
Formulation of BOD Equation
To derive the BOD equation, Oxygen Equivalent of Organic Matter Present is plotted against Time. 'Oxygen Equivalent of Organic Matter Present' is nothing but the BOD remaining.
The rate of change of BOD directly varies with BOD remaining at that point in time (Lt)
Integrating both the sides from interval Lo to Lt in time interval 0 to t,
At any time, Yt = Lo - Lo x e^(-kt)
If we intend to find out the solution in base 10 instead of base e, then
Lt = Lo x 10^(-k't), Where k' = k/2.303
Here the K' is also known as the deoxygenation Constant, which depends upon the temperature. The formulas given here are in at a standard temperature of 20 Degrees Celsius. To convert to some other
K' (T Degree Cel.) = K' (20 Degree Cel.) x [1.047] ^(T-20)
To find the ultimate BOD, that happens when t tends to infinity.
To understand these formulas more properly, I have a solved example for you. Do check it out.
Solve this one!
BOD is a very important topic asked in GATE. Solve this previous year question to get an idea of the kind of question that gets asked.
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Cfd Convergence - Index CFD
Converge Cfd Software
Within this case, the particular problem is almost always a problem anywhere inside the simulation mesh. The particular equation of which diverged may provide clues the supply of your trouble. where
N will be the total number of grid points used for the main grid and D is the dimension from the flow domain. This specific effective grid improvement ratio can also be used with regard to
unstructured grids. One more check of typically the asymptotic range will certainly be discussed within the section within the grid convergence list. The order of accuracy is identified by the
purchase of the leading term of typically the truncation error plus is represented with respect to the scale from the discretization, h.
The examination regarding the spatial convergence of any simulation is usually a straight-forward approach for determining typically the ordered discretization error in a CFD simulation. The
technique involves performing typically the simulation on two or more successively finer grids. Typically the term grid convergence study is equivalent to typically the commonly used expression grid
refinement study. If the residuals are constantly poor, replace the order regarding interpolation. The commissions for any specific equation may simply refuse to converge, but they don’t in fact
Practical Cfd Modeling: Judging Convergence
Typically the pattern of the particular residuals provide hints to why a new simulation fails. Plus some solvers stabilize their residuals in different ways, making these focuses on completely
irrelevant. Now you see why CFD engineers rely about more than one tool to judge convergence.
• Further, typically the converged solution within the coarser grid after that can be utilized directly since the preliminary solution on the better grid.
• Equations may balance with both sensible and absurd solutions.
• After finishing the simulation, the CFD engineer need to check flow styles to ensure earning sense.
Usually, residuals below 1e-3 is an excellent beginning point to proceed to another check. The particular GCI is a measure of the proportion the computed value is away through the value of the
asymptotic numerical value. It indicates an error band on exactly how far the answer is from your asymptotic value. It indicates just how much the solution would change along with a further
improvement of the main grid. A small benefit of GCI shows that the calculation is within the asymptotic range. Typically the physical mechanism that leads to incompressible behavior is the
particular rapid propagation associated with pressure waves, which usually must move by means of a fluid very much faster compared to the materials speed of the fluid. Most often, typically the
numerical propagation associated with pressure waves is accomplished by several sort of time scheme that couples pressures to velocities.
Order Of Grid Concurrence
Yet , EFD is created to permit the remedy to be done when convergence continues to be achieved for those quantities that are of interest for the designer. For instance, when the application is
usually electronics and Now i’m interested in temperatures, exactly why should I have on the calculation after the temperature ranges appealing have ceased changing? Convergence is also often
assessed by the amount of residuals, the sum in which discretised equations are certainly not satisfied, and not by the error in the solution. The user should therefore be aware of this, in deciding
what concurrence criterion should end up being used to assess a new solution. Roache provides provided a methodology for that uniform credit reporting of grid processing studies. In a great iterative
solution, commissions are the solution imbalances. For statistical accuracy, you should expect residuals to be as small as feasible.
One usually looks for the residuals to achieve a certain stage and after that level-off because an indication regarding iterative convergence. For a time-marching, steady-state strategy, this
requires examining whether the particular residual has been lowered a certain quantity (usually 3-4) associated with orders of magnitude. Scored, its feasible to carry on until almost everything is
converged to be able to the round-off mistake of the equipment you’re using.
Example Grid Convergence Research
The area buy of accuracy is the order for your stencil representing typically the discretization of typically the equation at one location in the grid. Theglobal buy of accuracy views the propagation
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Article of the Month -
Article of the Month - April 2014
A Practical Deformation Monitoring Procedure and Software System for CORS Coordinate Monitoring
Meng Chan LIM and Halim SETAN, Malaysia
Meng Chan Lim Halim Setan, Malaysia
This article in .pdf-format (21 pages)
^1) This paper illustrates the combination of continuous GPS measurement with robust method for deformation detection to GPS station position change. A window-based software system for GPS
deformation detection and analysis via robust method, called Continuous Deformation Analysis System (ConDAS), has been developed at Universiti Teknologi Malaysia. This paper describes the design and
architecture of ConDAS and highlights the deformation analysis results from two assessments. The paper is a Malaysian Peer Review paper, which will be presented at FIG Congress 2014 16-21 June, in
Kuala Lumpur, Malaysia.
This paper illustrates the combination of continuous GPS measurement with robust method for deformation detection to GPS station position change. A software system named Continuous Deformation
Analysis System (ConDAS) has been developed at Universiti Teknologi Malaysia. It was specially designed to work with high precision GPS processing software (i.e. Bernese 5.0) for coordinate
monitoring. The main components of ConDAS are: parameter extraction (from Bernese output), deformation detection (via IWST and S-transformation) and graphical visualisation. Two assessments were
included in this paper. Test results show that the system performed satisfactorily, significant displacement can be detected and the stability information of all monitored stations can be obtained.
This paper highlights the architecture, the design of the software system and the results.
Continuous Global Positioning System (GPS) networks record station position changes with millimetre-level accuracy have revealed that GPS is capable of detecting the significant deformations on
various spatial and temporal scales (Ji and Herring 2011; Li and Kuhlmann 2010; Cai et al. 2008; Yu et al. 2006). However, a rigorous deformation analysis technique is still required for preparing a
versatile and comprehensive spatial displacement results. To date, several continuous deformation monitoring systems are operational, such as SCIGN (Hudnut et al. 2001), GOCA (Jager et al. 2006) and
DDS (Danisch et al. 2008). This study employs a robust method known as Iteratively Weighted Similarity Transformation (IWST) and final S-Transformation to the daily GPS position time series.
A window-based software system for GPS deformation detection and analysis via robust method, called Continuous Deformation Analysis System (ConDAS), has been developed at Universiti Teknologi
Malaysia. It is a software system that solely designed to work with high precision GPS processing software (i.e. Bernese 5.0) for coordinate monitoring. The main components of ConDAS are: parameter
extraction (from Bernese output), deformation detection and graphical visualisation. All these components are integrated in one environment using MATLAB.
This paper describes the design and architecture of ConDAS and highlights the deformation analysis results from two assessments. In fact, the robust IWST method that employed by ConDAS is typically
used for structural deformation monitoring such as dam, slope and etc. However, this study combines IWST and final S-Transformation techniques to Continuous Operating Reference Station (CORS)
coordinate monitoring. Larger monitoring area was analysed using robust method for the first time. Promising results have been obtained through the assessment.
This study is devoted to develop a software system that adapts to GPS deformation detection and analysis for GPS CORS network. Due to the extraordinary demands for displacement detection accuracy,
high precision GPS processing software, namely Bernese 5.0 is employed. Figure 1 outlines the process of entire study.
Figure 1: The outline of developed deformation detection system
ConDAS is designed to work with Bernese software for deformation detection and analysis, thus this study comprises of two parts: GPS data processing strategy (via Bernese) and deformation analysis
strategy (via ConDAS).
2.1 GPS Data Processing Strategy
There are numbers of GPS deformation monitoring study using Bernese to process GPS data (Haasdyk et al. 2010; Hu et al. 2005; Jia 2005; Janssen 2002; Vermeer 2002). By implementing the Bernese
software, data cleaning, cycle slip detection, ambiguity resolution and network adjustment of GPS data all can be achieved to meet the desired criteria. The processing procedure using Bernese
Processing Engine (BPE) with double difference is illustrated in Figure 2.
Basically, the entire GPS processing step is divided into three parts: preparation, pre-processing and processing. The preparation part deals with computing a priori coordinate file, preparing the
orbit and earth orientation files in Bernese formats, converting RINEX files to Bernese format, synchronising the receiver clocks to GPS time and producing an easy to read overview of available data.
Meanwhile, the pre-processing part handles the creation of single difference files, editing of the cycle slips and removal of suspect observation. The processing part is responsible to resolve the
ambiguity. After computing a solution with real valued ambiguities the Quasi Ionosphere Free (QIF) strategy is used to resolve ambiguities to their integer numbers. Subsequently, the processing part
computes and provides the fixed ambiguity solution. A summary results file is saved and dispensable output files are removed at the final stage of processing.
Figure 2: Double difference GPS processing method using BPE
The GPS data are post-processed using Bernese 5.0 software. In order to capitalise the deformation analysis context, some GPS data processing parameters and models were carefully configured as listed
in Table 1. In spite of the general script in Bernese GPS software 5.0, some processing scripts are slightly change to well fit the requirement of deformation analysis. For instance, the function of
script SNGDIF (in the pre-processing part) is used to generate single-difference files from two zero-difference files. In other word, program SNGDIF is employed to form baselines only from
zero-difference observation files. There are plenty of options to be selected for generating the single-difference observation files for network solution. In general the option OBS-MAX guarantees the
best performance for the processing of a network using correct correlations. However, there is another option that fit the demand of deformation analysis, called DEFINED option, in which only the
predefined baselines from a baseline definition file are created. The selection of DEFINED option has significant impact on the determination of total number of parameters involved for every epoch.
Table 1: Parameters and models used in GPS data processing
Generally, Bernese GPS Software allows user to control the processing strategies or even skip certain redundant scripts (highlighted in Figure 2) during processing. Thus, for deformation analysis
purpose, the following redundant scripts are skipped: HELMR1, R2S_SUM, R2S_SAV and R2S_DEL.
Besides, the command line of searching clock correction file (P1C1yymm.DCB and P1P2yymm.DCB) had been removed or disable from the R2S_COP script that stored in the directory C:\GPSUSER\SCRIPT.
Eventually, the GPS data processing was performed without the clock correction file and it has been verified that no significant influence on final Bernese output by incessant trial and error. At
last, three types of result files were generated for every 24-hour epoch, for instance: a priori coordinate file and adjusted coordinate file in STA folder (e.g.: APR110010.CRD & R1_110010.CRD),
along with covariance file in OUT folder (e.g.: R1_11001.COV).
2.2 Deformation Analysis Technique and Software Development
The determination of deformations is mainly formed from two parts. The first is the measurement of deformation and the second is the analysis of these measurements (Aguilera et al. 2007). However,
deformation analysis using the geodetic method mainly consists of a two-step analysis via independent adjustment of the network of each epoch, followed by deformation detection between the two epochs
(Setan and Singh 2001). In this case, network adjustment is handled by Bernese 5.0 software and ConDAS carries out the two-epoch deformation analysis.
Generally, conventional deformation analysis applied in geodesy (Caspary 1988) extracts the deformation vectors and the variance-covariance matrix. In this classification, Iteratively Weighted
Similarity Transformation (IWST) tends to compute the displacement vector and its variance-covariance matrix by iteratively changing weight matrix, W. In fact, IWST method belongs to the family of
“robust” methods. IWST method finds the best datum, with minimal distorting influence on the vector of displacement (Chrzanowski et al. 1994). However, for the deformation analysis here we strongly
recommended the final S-transformation (with respect to stable reference points) after the IWST is applied. A flow chart of IWST with final S-Transformation method is illustrated in Figure 3. For
further computation of IWST and S-Transformation, please refer to Lim (2012), Lim et al. (2010) and Chen et al. (1990).
Figure 3: Flow chart of IWST with final S-Transformation that deployed in ConDAS
Two-epoch deformation analysis was employed in this study. From Figure 3, when comparing the two epochs of data (Epoch i: Xi , Qx ; Epoch j: Xj , Qx ) , the vector of displacements (dc) and (Qdc) its
cofactor matrix are calculated as shown in Lim et al (2010):
(In Equations (1) and (2), Xi and Xj must be in the same datum. An S-transformation with respect to the same datum must be conducted before the calculation of d (Figure 3).
At the beginning of the deformation analysis, the first matrix which must be computed is the weight matrix, W. For the first iteration (k=1), the matrix W is equal to I (i.e. W=I), where all the
diagonal elements are 1 and all other elements are 0. In the second (k+1) and all subsequent iterations, the diagonal elements of the weight matrix are defined as:
For 1-D networks, there are some differences for the calculation of d’ and Qd’. Firstly, the displacements d are arranged in increasing order. The median is assigned unit weight 1 and zero weight is
assigned to the other displacements d. If the total number of d is an even number, the two middle (median) displacements d are assigned unit weight 1 and zero weight is assigned to the other
displacement d (Chen et al. 1990). Then, the new vector of displacements d’ and its cofactor matrix Qd’ are (Chen et al. 1990):
where tz = mean value of the middle displacements and
di = the displacement of point i.
S = similarity transformation matrix =
G = inner constraint matrix
For a 2-D network, the elements of the weight matrix W are computed as follow:
where k is the iteration number.
For a 3-D network, the elements of the weight matrix W are computed as follow:
It is possible that some
i. Setting a lower bound (e.g. 0.0001 m). If
ii. Replacing the weight matrix as
In this study, the first solution has been chosen and it is preferable to limit the weight matrix for avoiding the long computation.
After that,
The G matrix is an inner constraint matrix. The dimensions of the G matrix are different for 1-D, 2-D and 3-D networks. For a GPS network, the matrix G is illustrated as Equation 10.
Further details are given in Chen et al. (1990) and Chrzanowski et al. (1994). The iterative procedure continues until the absolute differences between the successive transformed displacements d are
smaller than a tolerance value
In the last iteration, a final S-transformation is performed to get the actual value of the displacement vector by using stable reference points (as verified by the previous IWST analysis) as datum.
Consequently, elements of weight matrix, W are assigned 1 for stable reference points and 0 for other points to achieve the final S-transformation. Hence, the principle of congruency testing (Setan
and Singh, 2001) is used for calculating the actual deformation displacement vector. In the final iteration, the displacement vector
When the vectors of the displacements and the variance-covariance matrix of each point are computed, the stability information of each point can be determined through a single point test. The
displacement values and the variance-covariance matrix are compared with a critical value. Assuming the point i is tested, then, the algorithms are as follow (Chen et al. 1990; Setan and Singh 2001)
If the above test passes
In principle, ConDAS has been developed using Matrix Laboratory (MATLAB). This software system is tentatively developed to detect the unstable stations in a deformation monitoring network by IWST
method and S-Transformation to analyse the GPS results in the deformation perspective. Figure 4 illustrates the overall workflow of ConDAS.
Figure 4: Architecture of ConDAS (Lim et al., 2011)
As an overall, ConDAS consists of three modules: parameters extraction module, deformation detection module and visualisation module. The architecture and function of modules are described
respectively as following.
2.2.1 Parameters Extraction Module
After high accuracy coordinates computation from Bernese GPS software, a posteriori variance factor, degree of freedom and variance-covariance matrices can be obtained from the result files. These
parameters are required in order to perform the two-epoch deformation analysis. In other words, these parameters are the inputs of deformation analysis. For this study, a Bernese parameter extraction
module has been created using MATLAB as illustrated in Figure 5(a). It was designed to suit with Bernese in order to extract the required parameters according to the format of Bernese results files.
A warning message will pop out if the specify parameters are unavailable in the Bernese output file. A deformation input file in text file (.txt) was generated after parameters extraction from
Bernese output as shown in Figure 5(b).
Figure 5(a): GUI of parameters extraction Figure 5(b): Example format of deformation input file.
2.2.2 Deformation Detection Module
The core of deformation analysis program is the implementation of IWST algorithm. However, initial checking of data and test on variance ratio are important to ensure that common points, similar
approximate coordinates and same points names are used in two epochs. Thus, there is a statistic test termed variance ratio test that need to be conducted in order to determine the compatible
weighting between two epochs, and any further analysis should be stopped at this stage if test is rejected. The test statistic is referred to as Equation 15 (Lim et al. 2010; Setan and Singh 2001).
with j and i represent the larger and smaller variance factors, F is the Fisher’s distribution, is the chosen significance level (typically = 0.05) and
In this module, two-epoch deformation analysis were performed in two stages: i) stability analysis of reference stations using IWST and single point test; and ii) deformation analysis of all stations
by final S-transformation and single point test. Indication of a set of stable control stations was crucial in order to compute the displacement vectors of all monitored stations respectively.
Deformation detection module of ConDAS as illustrated in Figure 6(a) currently utilises a single point test in detecting displacement that reject any point with its displacement extends beyond the
confidence region (Chrzanowski et al. 1994). It is flagged as unstable if a given point fails the test at the specified confidence level. At the final stage of program, a summarised deformation
output file could be generated as shown in Figure 6(b). It contains the summary of file used, statistical summary and station information whether the station is flagged as moved or stable.
Figure 6(a): GUI of deformation detection module Figure 6(b): Output file of deformation detection module
2.2.3 Visualisation Module
The function of the visualisation module is twofold: i) to view the stability results of every two-epoch analysis; ii) to generate the deformation trend over a selected period. In fact, the stability
results in numerical and graphical modes are provided and visualised together with error ellipse and displacement vector for every monitored station. Besides, fluctuation of displacement vector over
a period can be visualised via this module. Some data statistics (e.g.: maximum, minimum and standard deviation values over that particular period) also can be obtained from visualisation module.
Figure 7 presents the GUI of visualisation module
Figure 7: GUI of visualisation module.
3. TEST RESULTS
Two test results were included in this paper for assessment purpose. Due to the CORS coordinate monitoring is the aim of the study, two sets of GPS data were collected from Malaysia Real Time
Kinematic GNSS Network (MyRTKnet) and Iskandar Malaysia CORS Network (ISKANDARnet) (Shariff et al. 2009). The first set of data was used to validate the software system by Aceh earthquake incident,
the latter one was utilised to monitor the displacement trend of every GPS stations within the network.
3.1 Test Results 1: Validation of System using Aceh Earthquake
Lately, the validation of the entire system was conducted by using the existing GPS data set from MyRTKnet. The processed data set started from 4th Dec 2004 until 31st Dec 2004 (i.e. before and after
the Aceh earthquake incident on 26th Dec 2004). Total six of IGS stations (ALIC, DARW, DGAR, HYDE, KARR and KUNM) have been chosen as the control points and two stations from MyRTKnet: JHJY and LGKW
were selected as object points. Figure 8 illustrates network distribution of selected IGS and MyRTKnet stations.
Figure 8: Network distribution of six IGS stations and two MyRTKnet stations
However, only the stable control point (among the selected IGS station) that being verified by ConDAS can be used as datum to compute the displacement vectors of object points. Throughout the
analysis, all stations were stable before the earthquake happen. However, the results show the LGKW station was moved start from 26th Dec 2004 and onwards. These results are similar with findings
from Jhonny (2010). Figures 9 and 10 illustrate the fluctuation of displacement vectors for JHJY and LGKW.
Figure 9: Fluctuation of displacement vectors of station JHJY in Easting, Northing and Up. Figure 10: Fluctuation of displacement vectors of station LGKW in Easting, Northing and Up.
From Figure 9, there were no significant movements detected at station JHJY during the incident occurred and the days onwards. The maximum displacement vectors were varied from 0.003m to 0.023m.
However, significant movements were detected at station LGKW at the day of year 361 and onwards (Figure 10). The displacement vectors were diverged from 0.007m to 0.167m. Table 2 shows the stability
information in numerical results.
Table 2: The displacement vectors of station JHJY and LGKW
n/a = data not available
3.2 Test Results 2: Deformation Trend of ISKANDARnet
There were seven stations in the deformation monitoring network, four from the IGS stations were used as reference (i.e. COCO, NTUS, PIMO, XMIS) and three stations from ISKANDARnet (ISK1, ISK2 and
ISK3) were used as object points as illustrated in Figure 11.
Figure 11: The deformation monitoring network for ISKANDARnet
GPS data processing and two-epoch deformation analysis were performed using two years (1st Jan 2010 – 31st Dec 2011) GPS data. However, ISKANDARnet was undergone some rigorous on-site maintenance
during March, July and August of year 2010 and early of April until Jun of year 2011. Thus, no GPS data was available on that specified period. After the GPS data processing was carried out with
Bernese software, two-epoch deformation analysis (at 5% significance level) were performed in two stages: i) stability analysis of reference stations using IWST; and ii) deformation analysis of all
stations. The stability of reference stations was vital in order to select a set of stable reference stations to conduct the analysis for all stations in the monitoring network. The results of
stability analysis of two epoch’s data (4th and 5th Jan 2010) in Table 3 confirmed that all four reference stations were stable.
Table 3: Stability analysis of the four reference stations using IWST
Subsequently, deformation analysis of all seven stations was carried out via final S-transformations based on the stable reference points (Table 3). All seven stations were verified as stable (Table
4). Consequently, the results obtained illustrate that the movement experienced by the GPS CORS stations at cm level can be detected. However, there was no significant movement as shown in Table 4.
Table 4: Stability of all monitoring stations using final S-Transformation based on four stable reference points
Next, GPS data (1st Jan 2010 – 31st Dec 2012) have been processed and analysed continually using the devised technique. The epoch on 4th Jan 2010 and 1st Jan 2011 were selected as reference epoch for
year 2010 and year 2011 respectively that any epochs against it. The results of stability analysis show all the stations are stable. The fluctuation of CORS stations: ISK1, ISK2 and ISK3 can be
revealed through plotting in Northing, Easting and Up. Figure 12(a), 12(b), 12(c) show the variation of ISK1, ISK2 and ISK3 in Easting, Northing and Up for year 2010. Nevertheless, Figure 13(a), 13
(b), 13(c) show the variation of ISK1, ISK2 and ISK3 in Easting, Northing and Up for year 2011.
Figure 12(a): Variation of displacement vectors of ISK1 in Easting, Northing and Up for year 2010. Figure 12(b): Variation of displacement vectors of ISK2 in Easting, Northing and Up for year 2010.
Figure 12(c): Variation of displacement vectors of ISK3 in Easting, Northing and Up for year 2010. Figure 13(a): Variation of displacement vectors of ISK1 in Easting, Northing and Up for year 2011.
Figure 13(b): Variation of displacement vectors of ISK2 in Easting, Northing and Up for year 2011. Figure 13(c): Variation of displacement vectors of ISK3 in Easting, Northing and Up for year 2011.
With respect to Figure 12(a), Figure 12(b) and Figure 12(c), Easting component of ISK1, ISK2 and ISK3 were suspicious that undergo some position changes throughout 2010. However, the displacements
were considered still under the safe condition and this three monitored stations were deemed to be stable based on the computed deformation analysis results. As an overall, the largest standard
deviation of ISK1, ISK2 and ISK3 reached 1.3 centimetres. It illustrates the obtained results was promising enough in the context of consistency. Table 5 shows the data statistics of ISKANDARnet
stations in year 2010.
Table 5: Statistical analysis of ISK1, ISK2 and ISK3 for year 2010
Consequently, regarding to Figure 13(a), Figure 13(b) and Figure 13(c) deformation analysis of ISKANDARnet was interrupted due to some rigorous on-site maintenance and software up-grading throughout
the year 2011. GPS data was not available that caused gaps to occur all the way in the plotting. In particular, largest standard deviation of ISK1, ISK2 and ISK3 achieved 1.5 centimetres. From Figure
13(a), Figure 13(b) and Figure 13(c), Up component of three monitored stations was suddenly slumped from day of year 280 until day of year 365. Further site investigation was needed to ensure the
location was free from threats. Nevertheless, all monitored stations were deemed to be stable and no significant displacement was detected in year 2011. Table 6 illustrates the data statistics of
ISK1, ISK2, and ISK3 for year 2011.
Table 6: Statistical analysis of ISK1, ISK2 and ISK3 for year 2011
4. CONCLUSION
In this paper, the skeleton of continuous deformation analysis and visualisation of GPS CORS has been illustrated. A combination of strategy is devised to develop a compatible deformation detection
software system for CORS coordinate monitoring. To attain the millimeter accuracy, some special processing strategies had been applied in the Bernese GPS software. Three types of output files from
Bernese software were extracted for deformation detection and analysis. Consequently, a windows-based software system for GPS deformation detection via IWST and final S-transformation methods, called
ConDAS, has been described. It has been proven to have potential for providing high-quality stability information for CORS network. The test results show the suitability of this software system for
practical applications. Furthermore, the obtained results are very promising, indicating the suitability of combining IWST and final S-transformation techniques for CORS coordinate monitoring. The
future works tend to improve the flexibility of this software system in terms of data searching, loading and code embedding towards a fully automated deformation monitoring system.
The authors would like to thank Department of Survey and Mapping Malaysia (DSMM) for providing valuable MyRTKnet GPS data. The authors are grateful to the following agencies for research funds:
Ministry of Science, Technology and Innovation (MOSTI) for Science Fund (Vot. 79350), Ministry of Higher Education (MOHE) for RUG (Vot. Q.J130000.7127.02J69) and Land Surveyors Board (LJT) Malaysia.
The authors also grateful to GNSS & Geodynamics Research Group (FKSG, Universiti Teknologi Malaysia) which provide the research facility for data processing purposes.
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Setan, H. and Singh, R., 2001. Deformation Analysis of a Geodetic Monitoring Network. Geomatica. 55(3), 333-346.
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Lim Meng Chan was a M.Sc. student at the Dept of Geomatic Engineering, Faculty of Geoinformation and Real Estate, Universiti Teknologi Malaysia (UTM). She holds B. Eng. (Hons) in Geomatic (2008) and
M.Sc. in Satellite Surveying (2013). Her master project focuses in the area of GPS for continuous deformation monitoring under supervision of Prof. Dr. Halim Setan and Mr. Rusli Othman.
Dr. Halim Setan is a professor at the Faculty of Geoinformation and Real Estate, Universiti Teknologi Malaysia. He holds B.Sc. (Hons.) in Surveying and Mapping Sciences from North East London
Polytechnic (England), M.Sc. in Geodetic Science from Ohio State University (USA) and Ph.D from City University, London (England). His current research interests focus on precise 3D measurement,
deformation monitoring, least squares estimation, laser scanning and 3D modelling.
Prof. Dr. Halim Setan
Department of Geomatic Engineering,
Faculty of Geoinformation and Real Estate
Universiti Teknologi Malaysia (UTM)
81310 Johor Bharu
Tel. +607-5530908
Fax + 607-5566163
Email: halim@utm.my
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Feb 2004
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8:42AM 1 Stepwise regression and partial correlation for wildlife census time series
7:48AM 1 LME, where is the package?
7:22AM 2 logististic regression (GLM). How to get 95 pct. confidence limits?
3:36AM 2 matrix inverse in C
Friday February 27 2004
Time Replies Subject
11:15PM 2 importing S-Plus data files
10:39PM 2 browseURL question
9:09PM 2 a loop question
9:04PM 0 Johansen Procedure
7:16PM 3 How to recover t-statistics?
6:12PM 2 question about if else
5:18PM 1 question about setdiff()
4:26PM 3 load data for mypkg-Ex.R
4:05PM 5 How to save images?
3:56PM 4 question
1:10PM 2 Is there a way to deactivate partial matching in R?
11:28AM 4 Change the result data
10:34AM 0 Get R.lib , how to generate it
9:16AM 3 locator(n=0)
2:26AM 1 Outer with Three Vectors
Thursday February 26 2004
Time Replies Subject
11:25PM 3 Collapsing Categorical Variables
10:50PM 0 boot and sample question
10:42PM 1 unable to install dse in mac OS X 10.3
9:15PM 6 adding header info to write.table
8:47PM 3 saving plots as objects?
8:00PM 1 Loading SparseM on Win2K
6:11PM 1 writing polygons/segments to shapefiles (.shp) or other A rcGIS compatible file
6:08PM 0 Compiling third party C++ libraries
5:39PM 3 my own function given to lapply
4:13PM 3 Help with multicolored points in one plot
4:09PM 0 R on VMS
3:15PM 2 limit to complex plots?
3:08PM 2 RE: system.time(), sys.time() etc
2:06PM 1 Handling R Objects in C
1:47PM 1 variance estimator for the cumulative incidence function
1:05PM 2 Multidimensional scaling and distance matrices
11:58AM 2 Sweave and Xemacs on Windows2000?
11:33AM 2 Structural Equation Model
10:05AM 0 Machine Learning category
8:55AM 1 linking other C++ libraries
8:52AM 1 Memory limitation in GeoR - Windows or R?
5:50AM 1 Distance and Aggregate Data - Again...
3:57AM 1 Gnumeric - 1 Excel - ?
3:46AM 2 minimum value
3:02AM 2 return value in function
1:02AM 0 se.contrast ???????????
Wednesday February 25 2004
Time Replies Subject
11:11PM 1 distinct random number
10:34PM 0 books:
8:42PM 0 books: "Programming with Data: A Guide to the S Language" vs." S Programming"
8:06PM 2 LOOCV using R
6:16PM 4 Computing very large distance matrix
5:33PM 2 PWM Help
5:00PM 1 read.spss defaults
3:32PM 2 writing polygons/segments to shapefiles (.shp) or other ArCGIS compatible file
1:49PM 0 simtest for Dunnett
12:02PM 1 Pb with RODBC installation
10:58AM 0 k nearest neighbours between two matrix
9:58AM 0 RExcel and statistical tests
9:01AM 1 structure of mlm objects ?
3:03AM 1 lapack routine dgesdd, error code 1
2:47AM 2 circular filter
1:06AM 1 (no subject)
12:03AM 2 levelplot add line
Tuesday February 24 2004
Time Replies Subject
7:56PM 0 Course***Advanced R/Splus Programming @ 5 locations, March 2004
7:23PM 2 Statistical Quality Control
5:59PM 1 Inheriting from factors + co.
4:51PM 2 matrix() Help
4:45PM 2 Legends text
4:36PM 3 problem of install.packages in windows (R 1.81)
4:13PM 1 Accessing columns in data.frame using formula
4:01PM 0 Suggestions ?!?!
3:51PM 0 Blue book
3:32PM 6 be careful: using attach in R functions
3:30PM 2 convergence in polr
2:58PM 3 Calculate Distance and Aggregate Data?
2:21PM 5 Nonlinear Optimization
1:34PM 7 <no subject>
12:44PM 5 r: plots
11:37AM 1 rstandard does not produce standardized residuals
11:01AM 4 Computing the mode
8:52AM 4 would be nice ...
8:30AM 2 Filter out some levels?
6:05AM 0 se.contrast
6:04AM 0 (no subject)
3:24AM 0 Sweave and sep = "\t"
12:53AM 3 quesion on diag of matrix
Monday February 23 2004
Time Replies Subject
11:46PM 2 orthonormalization with weights
11:33PM 2 (2) Questions
10:18PM 1 intersection points of two functions
8:30PM 3 library nnet
8:13PM 1 HTTP Post connections in R
8:07PM 6 Need help on parsing dates
7:49PM 2 parameters' value
5:40PM 2 deleting elements from an array/object
5:36PM 1 Reference to use of MLR in industry and biology
4:16PM 0 Package "ref" implements references and referenceable data.frames for the S-language
3:57PM 2 Error in multiple xyplots
3:55PM 1 problem with unlist
2:25PM 0 Re: R for economists
1:38PM 0 Question concerning functions nlsList and nlme from nlme R library.
1:37PM 1 DLLs and the Floating Point Control Word.
1:01PM 0 fake
12:49PM 2 outputs of KNN prediction
12:43PM 1 border of a polygon in contour.kriging - geoR
10:40AM 2 plot(my.procrustes.model) from library {vegan}
9:55AM 0 Is there a /ddfm=satterth for R?
9:52AM 4 lme - problems with model
2:39AM 1 MiceR
2:37AM 1 dendrogram ultrametrics
2:03AM 0 (no subject)
12:45AM 3 Nearest Neighbour for prediction
Sunday February 22 2004
Time Replies Subject
10:55PM 1 a simple question about a matrix
10:46PM 0 countourplot background
10:28PM 0 contourplots
8:20PM 1 For loop help
7:58PM 2 how to plot multi- lines on one diagram
7:10PM 6 help for MLE
5:18PM 3 Simulation help
6:42AM 1 New Perl module Statistics::R
5:23AM 2 nested loop
4:54AM 0 butterworth filter code?
Saturday February 21 2004
Time Replies Subject
12:54PM 0 R/SigmaStat
10:15AM 2 RE:Including R plots in a Microsoft Word document
10:07AM 3 saving variables created by functions in the workspace
2:55AM 1 Stratified random sampling in R?
Friday February 20 2004
Time Replies Subject
11:54PM 0 Data Analyst Intern position in San Francisco
11:31PM 0 M-Plots in R
9:30PM 1 BATCH files
9:19PM 0 (no subject)
9:14PM 1 run R BATCH job in PHP
7:48PM 0 setting options when using eval
7:03PM 0 Installing OmegaHat OOP package
6:43PM 1 strptime() behaviour
5:22PM 1 A question on lme in R
4:58PM 2 passing object names in a vector to save?
3:54PM 9 R: Including R plots in a Microsoft Word document
3:30PM 0 How to make a plot to represent 500 repeated 95%CI
2:34PM 1 Stupid Limma question..
2:17PM 3 problem with abline for x.y
2:01PM 1 read.table with spaces
11:38AM 1 nlme and multiple comparisons
4:23AM 1 Confidence intervals for logistic regression
3:28AM 0 New Package: multinomRob
1:34AM 1 Sweave not found from MikTeX?
1:05AM 0 Using R remotely on a Mac OS X machine
12:49AM 1 Does pdf() not work on Trellis graphics?
Thursday February 19 2004
Time Replies Subject
11:45PM 1 piece wise application of functions
11:14PM 4 1024GB max memory on R for Windows XP?
9:57PM 0 lme problem
9:50PM 1 controlling nls errors
8:35PM 5 solving equations with several variables
7:52PM 2 R won't start
5:29PM 1 efficient matrix approx
4:40PM 1 Comparing two regression slopes
3:56PM 3 filling the area between two curves in a plot
3:53PM 2 read.socket - Strange strings. How to force sub to remove all occurences of a pattern?
3:46PM 0 More variables on pca
2:45PM 1 reshape direction=wide
2:22PM 0 polr warning message optim
1:38PM 1 Obtaining SE from the hessian matrix
1:26PM 6 R for economists (was: Almost Ideal Demand System)
1:20PM 3 suppressing non-integer labels for plot x-axis
1:03PM 3 F Dist
11:05AM 1 How to create a "nb" object?
8:30AM 1 Setting ylim while plotting an 'its' object
3:28AM 0 surprising revelation
12:19AM 1 latex problem with Sweave output file under Debian
Wednesday February 18 2004
Time Replies Subject
10:41PM 0 citation() doesn?t work
10:29PM 0 Conjugate function found disregard pervious posting...
8:59PM 2 boostrapping at R
8:44PM 2 using names() as a variable in a formula
7:38PM 1 Complex conjugate?
7:21PM 2 Area between CDFs
6:42PM 0 return a list of vectors from C?
6:39PM 1 NAs introduced by coercion warning?
6:30PM 0 (no subject)
6:03PM 5 overlay points on plot
5:29PM 3 Generalized Estimating Equations and log-likelihood calculation
4:45PM 6 interesting feature
4:34PM 2 building the development version
3:39PM 4 How to repeat a procedure
1:34PM 3 PNG Problem on Windows 98
1:10PM 0 Plotting a three parameter gamma distribution
10:04AM 0 Ang: How to write efficient R code
5:19AM 3 ANOVA procedure on the sufficient statistics
4:41AM 1 Printing values within a function
1:19AM 0 Discriminant Analysis Using Anova in R
12:05AM 3 persp and lines()
12:04AM 0 setMethod
Tuesday February 17 2004
Time Replies Subject
11:02PM 2 Test for pre-existing Win menu or item
10:38PM 4 Apply a function to each cell of a ragged matrix
9:54PM 1 RCMD SHLIB == Couldn't reserve space for cygwin's heap, Win32 ?
9:23PM 5 pass by reference -- how to do it
7:59PM 2 In praise of ->
7:28PM 3 parse error in GLMM function
7:13PM 1 Help with multiple graphs on one set of axis
6:28PM 0 New package -- mvpart
5:05PM 0 A log on Bayesian statistics, stochastic cost frontier, montecarl o markov chains, bayesian P-values
4:38PM 2 problem with fitdistr ?
3:55PM 0 ID mprxahov... thanks
3:24PM 2 interfacing C++ using .Call
3:14PM 0 ID tketcunbit... thanks
3:09PM 0 ID ikhltq... thanks
3:05PM 4 importing ascii grids (for gstat)
2:38PM 1 Comparison of % variance explained by each PC before AND after rotation
2:36PM 10 How to write efficient R code
2:23PM 2 Generating 2x2 contingency tables
1:26PM 2 Installing package on R
12:40PM 2 citation() doesn´t work
11:39AM 1 extracting standard error from lme output
11:20AM 1 Bug report for fracdiff
10:00AM 0 Bad Plotting subrange
9:51AM 2 Lattice graphics and strip function
8:32AM 4 normality test
3:14AM 1 varimax rotation in R
Monday February 16 2004
Time Replies Subject
10:25PM 1 xls2csv.pl: Script to translate Excel files into CSV
10:10PM 4 plot
10:05PM 0 How do we obtain Posterior Predictive (Bayesian) P-values in R (a sking a second time)
9:44PM 4 Matrix mulitplication
8:41PM 4 Questions about Matrix
8:04PM 2 question about matrix
7:50PM 1 Binary logistic model using lrm function
6:31PM 1 resizing a plot area when using mfrow
5:20PM 1 consensus trees/groups from clustering
4:37PM 1 2 bwplots - different colors
4:15PM 2 Data for use in maps()
4:14PM 1 understanding loops for "loop-plotting"
4:12PM 2 problem for installing package
3:50PM 0 specifying partial nesting in lme
3:44PM 0 How do we obtain Posterior Predictive (Bayesian) P-values in R
2:42PM 0 how to solve a linear equation system with polynomial factors?
2:01PM 0 error in nls, step factor reduced below minFactor
1:54PM 2 R Included with Open Infrastructure for Outcomes (OIO) system
12:17PM 1 aov and Error documentation
11:37AM 1 nlme_crossed AND nested random effects
10:46AM 1 Offset in GLMM
10:41AM 1 labRow/labCol options in heatmap()
10:25AM 1 repeated measures nonlinear regression
9:44AM 0 intercept row in anova()
2:18AM 0 unbalanced
Sunday February 15 2004
Time Replies Subject
8:48PM 5 Maximum likelihood estimation in R
8:40PM 1 panel data
7:27PM 4 father and son heights
6:56PM 1 manova() with a data frame
4:42PM 1 linear regression of data with standard deviation
4:22PM 0 linear regression and chi square test of data with standard deviation
5:18AM 2 help on compilation of R help file in LaTeX format.
4:49AM 1 source() function and crash of R!
1:16AM 1 Error Installing dse Package
Saturday February 14 2004
Time Replies Subject
11:10PM 1 Converting a number column to a factor in a data frame?
10:23PM 1 speed in batch mode versus interactive mode
6:07PM 0 Time Series?
4:51PM 2 ftp.stat.math.ethz.ch not accessible?
3:33PM 6 Beginner's question about t.test()
11:50AM 0 A course in using R for Epidemiology
11:03AM 2 converting data to date format
9:10AM 1 points inout a circle
1:41AM 1 PLEASE IGNORE PREVIOUS: How to configure ess-5.2.0beta3-1.i586.rpm, Xemacs and SuSE 9.0?
1:31AM 1 How to configure ess-5.2.0beta3-1.i586.rpm, Xemacs and SuSE 9.0?
12:56AM 1 Digital Image Processing
Friday February 13 2004
Time Replies Subject
11:48PM 0 Problems getting R to work from Java under Windows
10:16PM 1 parametric bootstrap and computing factor scores
9:12PM 0 profiling C code
8:35PM 3 Re: Re: Find Closest 5 Cases?
8:11PM 4 How to plot a blank plot
6:25PM 5 predict function
6:19PM 1 Problems with R CMD INSTALL on SUSE-LINUX 9.0
5:15PM 0 Frailty Model - parameter inferences
4:34PM 3 Calculate Closest 5 Cases?
4:33PM 0 generic/method .. find.package
3:06PM 1 Problems loading dataset in Rcmdr
1:17PM 0 R installation and reg-test
1:08PM 0 Windows dll compilation (mingw32): how to find R.h and ot her head er files when sketching short functions
12:37PM 1 Windows dll compilation (mingw32): how to find R.h and other head er files when sketching short functions
9:38AM 1 Parallel programming with R
7:11AM 0 Help in installing packages
3:42AM 1 IIA tests
2:51AM 2 Readbin and file position
2:07AM 1 How to get time differences in consistent units?
1:45AM 1 RES: AGREP
1:14AM 2 Puzzled by write.table function
12:12AM 0 how to get the cluster output as a text file but not a graphic one?
Thursday February 12 2004
Time Replies Subject
8:44PM 1 C-Code
8:07PM 1 Importing BSQ/BIP/BIL files into R
7:56PM 1 kriging prediction intervals
7:49PM 6 Basic Help
7:42PM 2 variances of values predicted using a lm object
4:39PM 1 suggestion "suggestion" and dataframe operations
4:32PM 0 How to predict ARMA models?
4:23PM 0 MARS in classification problem
4:06PM 1 How do you create a "MCMC" object?
3:41PM 1 left eigenvector
1:56PM 1 Porting let* from Common LISP to R
12:14PM 0 (no subject)
12:00PM 1 Almost Ideal Demand System
11:12AM 4 R-help
11:08AM 3 Debugging R Code
11:02AM 2 How to download
10:57AM 2 calling R from a shell script and have it display graphics
9:12AM 2 lattice: showing panels for factor levels with no values
6:25AM 1 Kernel Density Estimator for 2D Binned Data
3:22AM 3 How to detect whether a file exists or not?
Wednesday February 11 2004
Time Replies Subject
8:16PM 0 tobit Heteroscedasticity
7:38PM 0 gelman.diag question
6:51PM 1 how much memory? was: R does in memory analysis only?
6:00PM 1 64-bit Windows 2003 build of R
5:53PM 6 AGREP
5:33PM 1 About the macro defined in Rinternals.h
1:51PM 0 Re: Clinical Significance as a package
1:13PM 1 .Call setAttrib(ans,R_DimSymbol,dim); Crashes.
12:42PM 0 Error on imodwt function - package waveslim
12:31PM 1 Erro in loadNamespace
12:30PM 7 large fonts on plots
12:00PM 0 Bhat: installation problem(2)
11:52AM 1 Bhat: installation problem
11:48AM 1 Comment: R patterns
11:30AM 3 RGui (Windows) crashes after use of a Salford Fortran DLL
11:15AM 3 Any help with bootstrapping
9:48AM 0 The use of R for the elaboration of some index
7:42AM 1 MCD-Estimator in R
5:54AM 6 lapply and dynamically linked functions
4:07AM 0 Installation on Mac OS X 10.3.2 with Fink readline and headers
2:22AM 1 RW1081.exe installation
12:15AM 1 Clinical significance as a package?
12:07AM 1 levelplot colorkey
Tuesday February 10 2004
Time Replies Subject
10:09PM 0 Bug in concord package
9:44PM 1 generate random sample from ZINB
9:15PM 0 Permissions after install of R 1.8.1.
8:49PM 1 make check in 1.8.1.
8:23PM 2 Invoking R from PHP/Mysql environment
7:56PM 2 Constructing an environment from a data.frame
7:10PM 0 Course***R/S-plus Programming Techniques in Raleigh, February 26-27
5:55PM 2 Dotplot: y-labels from rownames
5:27PM 1 lattice: scales beginning at zero with relation="free"
4:34PM 0 name space conflict using RMySQL and ROracle
3:55PM 1 interfacing C code in Windows
3:52PM 3 how to get the GUI directory chooser on Windows?
2:48PM 6 R: lags
2:40PM 1 Diagnostic in multilevel models
1:59PM 3 coxph error
1:27PM 4 The ttest.c example in R under MS Windows
10:53AM 0 Evaluating R. I need to open "a dataset".
9:30AM 2 confidence-intervals in dotchart
9:00AM 3 confidence-intervals in barchart
2:42AM 2 How to compute the minimal distanct between a point and curve in N-dim space
2:16AM 0 GLMMpql: reporting on main effects
Monday February 9 2004
Time Replies Subject
9:38PM 1 RConsole
9:21PM 0 CART Data Mining 2004 Conference, San Francisco, SCHEDULE information
8:47PM 2 data.frame to matrix
8:22PM 5 simple question on picking out some rows of a matrix/data frame
7:09PM 0 STARMA model building
7:08PM 5 Printting 'for' and 'while' indices
7:05PM 0 Fit system of equations
6:51PM 0 Affy library: error on ReadAffy()
6:13PM 0 xtable table placement
5:00PM 1 Duncan's Multiple Range test
4:27PM 1 Importing a SAS file to R: Alas, STILL more problems--has anyone gotten this message before, and why
3:47PM 0 Another question, unfortunately. . . .(Installing "foreig n"/trying to import/export SAS files)
3:34PM 1 nedit syntax highlighting patterns for R?
3:24PM 0 Re: Another question, unfortunately.
2:38PM 1 Another question, unfortunately. . . .(Installing "foreign"/trying to import/export SAS files)
2:21PM 3 citing a package?
12:56PM 2 moments, skewness, kurtosis
12:48PM 1 Estimate Covariance Matrix of two vectors
11:27AM 1 Subset function of lm(); "rolling regressions"
10:24AM 2 Recursive partitioning with multicollinear variables
1:55AM 1 Can S-Plus packages be used in R without modification?
1:08AM 10 PhD student reading list, suggestions wanted
Sunday February 8 2004
Time Replies Subject
11:07PM 2 substitute, eval, quote and functions
9:07PM 5 iterating over files in a directory with R
8:39PM 2 parsing numbers from a string
6:01PM 0 bootstrap estimates for lme
5:38PM 1 APE: compar.gee( )
7:19AM 0 2D density contour plot
5:56AM 0 2D histogram
Saturday February 7 2004
Time Replies Subject
10:15PM 2 R does in memory analysis only?
7:43PM 1 Adding a color bar to image
4:45PM 1 Newbie help with calling R from C programs
5:17AM 1 Subset function of lm() does not seem to work
12:52AM 1 knn using custom distance metric
12:50AM 1 display functions in groupedData and lme
Friday February 6 2004
Time Replies Subject
10:48PM 1 Savitzky-Golay smoothing -- an R implementation
10:19PM 0 error message from regsubsets
10:18PM 0 problem with bagging
9:57PM 3 column names in matrix vs. data frame in R 1.8
8:10PM 2 vector of factors to POSIXlt
6:49PM 3 a grep/regexpr problem
5:19PM 0 erroneous additional weighting in plot.lm for glm objetcs?
4:50PM 1 problem to get coefficient from lm()
4:09PM 1 structured random effects
3:47PM 1 How to get the pseudo left inverse of a singular squarem atrix?
3:30PM 3 quantile function
3:07PM 4 more or less pager
2:36PM 1 nnet problem
12:55PM 1 0.1 + 0.2 != 0.3 revisited
12:16PM 0 How to remove method initialize./package methods
12:01PM 2 Converting a Dissimilarity Matrix
11:24AM 0 information on R
9:44AM 1 Any help
9:20AM 2 Normality Test on several groups
8:42AM 0 multiple plots in different windows
3:41AM 0 availability of heap or priority queue
Thursday February 5 2004
Time Replies Subject
11:08PM 2 Histograms by two factors
11:00PM 0 Plotting question
10:10PM 0 Thank you for your answers: Re: correction to the previously asked question (about merging factors)
10:02PM 1 for help about MLE in R
9:52PM 5 rgamma question
9:45PM 5 (Novice-) Problem with the plot-function
9:32PM 0 correction to the previously asked question (about mergin g factors)
8:22PM 0 What is the correct way to merge factors?
8:11PM 2 correction to the previously asked question (about merging factors)
8:03PM 0 Thanks for help
7:19PM 2 I am totally lost on how to install R . . .
6:15PM 1 What is the correct way of using function C() for factors:
5:17PM 2 Incomplete Factorial design
4:09PM 2 Available in S-plus, also in R1.8.1?
3:13PM 2 Savitzky-Golay smoothing for reflectance data
3:11PM 1 Multilevel in R
2:05PM 1 Installing odesolve under MacOSX
11:31AM 2 xyplot (lattice): colours of lines
10:29AM 2 (no subject)
10:29AM 0 Gamma Test package
2:08AM 2 Sweave problem
12:38AM 1 lines and dates
Wednesday February 4 2004
Time Replies Subject
11:45PM 1 RE: error (fwd)
9:59PM 0 Very Fast Multivariate Kernel Density Estimation
7:18PM 1 center or scale before analyzing using pls.pcr
4:13PM 0 Job opportunity
3:47PM 1 Fitting nonlinear (quantile) models to linear data.
2:48PM 0 Sweave and .Rd files
2:45PM 1 2 questions: batch file + R 1.8 on Red-Hat 9.0
2:31PM 5 Date Time Conversion problems...
1:51PM 0 help(Memory) [forwarded message]
1:51PM 3 Using huge datasets
1:50PM 0 New Discussion list about R in PORTUGUESE - Lista de discussão sobre a linguagem R em Portugues
1:37PM 5 nortest package
1:22PM 0 AlgDesign
1:19PM 3 number point under-flow
12:41PM 2 Latin 2 encoding + fonts
12:38PM 0 Implementing streams in R
11:41AM 5 Newbie question: histogram
11:38AM 1 implement a function
11:33AM 3 Various newbie questions
10:39AM 1 Returnin char back through the .Call interface
10:28AM 1 arima function
10:03AM 1 xypplot (lattice): colours of lines
9:44AM 0 (no subject)
2:39AM 1 Novice problems with write()
1:37AM 1 Clustering with 'agnes'
Tuesday February 3 2004
Time Replies Subject
10:51PM 1 Error in f(x, ...) : subscript out of bounds
9:58PM 1 S language
9:26PM 1 Locate a warning
8:50PM 4 how to change one of the axis to normal probability scale
8:42PM 0 S
7:48PM 0 Linux R installation problem, never mind....
7:27PM 1 Insightful acquires "S" language
4:39PM 2 problem with read.table
4:25PM 5 lm coefficients
3:29PM 1 Linux installation problem
3:17PM 1 Passing characters by .Call
2:23PM 2 Prompt / Console problem
1:47PM 0 GEE
11:57AM 4 R: lags and plots
10:40AM 3 Implementating streams in R
10:07AM 2 How to build a AR(q)-GARCH(q) process ?
9:43AM 5 creating a factor
9:22AM 1 Normal distribution
8:35AM 4 filled maps
7:59AM 3 R: plotting multiple functions
2:26AM 4 running R from PHP
1:46AM 1 output from multcomp and lm
Monday February 2 2004
Time Replies Subject
11:35PM 2 Re: packages
11:27PM 0 problems when compiling C code
11:25PM 1 how to label plots?
10:16PM 3 sorting by date
9:25PM 2 Nearest Neighbor Algorithm in R -- again.
8:13PM 3 ordering and plotting question
8:02PM 3 mvrnorm problem
7:46PM 1 Robust nonlinear regression - sin(x)/x?
6:29PM 0 problem building R on HPUX 11.23
5:18PM 1 filled contour + points
5:03PM 1 axes in boxplots
4:45PM 1 Order in barchart
3:13PM 1 glm.poisson.disp versus glm.nb
1:42PM 4 for loops?
9:51AM 2 ordering in dotplot
9:32AM 1 PLS discriminant analysis
1:40AM 2 print comment lines on `sink'ed files?
Sunday February 1 2004
Time Replies Subject
11:01PM 2 3 little questions
6:53PM 1 interactive 2-D plot interrogation
3:44PM 2 CART: rapart vs bagging
3:14PM 1 coxph
3:13PM 1 coxph in R
2:55PM 4 I can't make .C(...) web-page example.
2:05PM 5 Stepwise regression and PLS
8:39AM 4 Assistance with data import from Statistica | {"url":"https://thr3ads.net/r-help/2004/02","timestamp":"2024-11-06T14:37:13Z","content_type":"text/html","content_length":"107034","record_id":"<urn:uuid:90819d6c-64b4-4a31-b9c8-ca96bb7758cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00157.warc.gz"} |
Introduction to Computational Modeling: Hodgkin-Huxley Model
Computational modeling can be a tough nut to crack. I'm not just talking pistachio-shell dense; I'm talking walnut-shell dense. I'm talking a nut so tough that not even a nutcracker who's cracked
nearly every damn nut on the planet could crack this mother, even if this nutcracker is so badass that he wears a leather jacket, and that leather jacket owns a leather jacket, and that leather
jacket smokes meth.
That being said, the best approach to eat this whale is with small bites. That way, you can digest the blubber over a period of several weeks before you reach the waxy, delicious ambergris and
eventually the meaty whale guts of computational modeling and feel your consciousness expand a thousandfold. And the best way to begin is with a single neuron.
The Hodgkin-Huxley Model, and the Hunt for the Giant Squid
Way back in the 1950s - all the way back in the twentieth century - a team of notorious outlaws named Hodgkin and Huxley became obsessed and tormented by fevered dreams and hallucinations of the
Giant Squid Neuron. (The neurons of a giant squid are, compared to every other creature on the planet, giant. That is why it is called the giant squid. Pay attention.)
After a series of appeals to Holy Roman Emperor Charles V and Pope Stephen II, Hodgkin and Huxley finally secured a commission to hunt the elusive giant squid and sailed to the middle of the Pacific
Ocean in a skiff made out of the bones and fingernails and flayed skins of their enemies. Finally spotting the vast abhorrence of the giant squid, Hodgkin and Huxley gave chase over the fiercest seas
and most violent winds of the Pacific, and after a tense, exhausting three-day hunt, finally cornered the giant squid in the darkest netherregions of the Marianas Trench. The giant squid sued for
mercy, citing precedents and torts of bygone eras, quoting Blackstone and Coke, Anaxamander and Thales. But Huxley, his eyes shining with the cold light of purest hate, smashed his fist through the
forehead of the dread beast which erupted in a bloody Vesuvius of brains and bits of bone both sphenoidal and ethmoidal intermixed and Hodgkin screamed and vomited simultaneously. And there stood
Huxley triumphant, withdrawing his hand oversized with coagulate gore and clutching the prized Giant Squid Neuron. Hodgkin looked at him.
"Huxley, m'boy, that was cold-blooded!" he ejaculated.
"Yea, oy'm one mean cat, ain't I, guv?" said Huxley.
"'Dis here Pope Stephen II wanted this bloke alive, you twit!"
"Oy, not m'fault, guv," said Huxley, his grim smile twisting into a wicked sneer. "Things got outta hand."
Scene II
Drunk with victory, Hodgkin and Huxley took the Giant Squid Neuron back to their magical laboratory in the Ice Cream Forest and started sticking a bunch of wires and electrodes in it. To their
surprise, there was a difference in voltage between the inside of the neuron and the bath surrounding it, suggesting that there were different quantities of electrical charge on both sides of the
cell membrane. In fact, at a resting state the neuron appeared to stabilize around -70mV, suggesting that there was more of a negative electrical charge inside the membrane than outside.
Keep in mind that when our friends Hodgkin and Huxley began their quest, nobody knew exactly how the membrane of a neuron worked. Scientists had observed action potentials and understood that
electrical forces were involved somehow, but until the experiments of the 1940s and '50s the exact mechanisms were still unknown. However, through a series of carefully controlled studies, the
experimenters were able to measure how both current and voltage interacted in their model neuron. It turned out that three ions - sodium (Na+), potassium (K+), and chlorine (Cl-) - appeared to play
the most important role in depolarizing the cell membrane and generating an action potential. Different concentrations of the ions, along with the negative charge inside the membrane, led to
different pressures exerted on each of the ions.
For example, K+ was found to be much more concentrated inside of the neuron than outside, leading to a concentration gradient exerting pressure for the K+ ions to exit the cell; at the same time,
however, the attractive negative force inside the membrane exerting a countering electrostatic pressure, as positively charged potassium ions would be drawn toward the inside of the cell. Similar
characteristics of the sodium and chlorine ions were observed as well, as shown in the following figure:
Ned the Neuron, filled with Neuron Goo. Note that the gradient and electrostatic pressures, expressed in microvolts (mV) have arbitrary signs; the point is to show that for an ion like chlorine, the
pressures cancel out, while for an ion like potassium, there is slightly more pressure to exit the cell than enter it. Also, if you noticed that these values aren't 100% accurate, then
congratu-frickin-lations, you're smarter than I am, but there is no way in HECK that I am redoing this in Microsoft Paint.
In addition to these passive forces, Hodgkin and Huxley also observed an active, energy-consuming force in maintaining the resting potential - a mechanism which exchanged potassium for sodium ions,
by kicking out roughly three sodium ions for each potassium ion. Even with this pump though, there is still a whopping 120mV of pressure for sodium ions to enter. What prevents them from rushing in
there and trashing the place?
Hodgkin and Huxley hypothesized that certain channels in the neuron membrane were selectively permeable, meaning that only specific ions could pass through them. Furthermore, channels could be either
open or closed; for example, there may be sodium channels dotting the membrane, but at a resting potential they are usually closed. In addition, Hodgkin and Huxley thought that within these channels
were gates that regulated whether the channel was open or closed, and that these gates could be in either permissive or non-permissive states. The probability of a gate being in either state was
dependent on the voltage difference between the inside and the outside of the membrane.
Although this all may seem conceptually straightforward, keep in mind that Hodgkin and Huxley were among the first to combine all of these properties into one unified model - something which could
account for the conductances, voltage, and current, as well as how all of this affected the gates within each ion channel - and they were basically doing it from scratch. Also keep in mind that these
crazy mofos didn't have stuff like Matlab or R to help them out; they did this the old-fashioned way, by changing one thing at a time and measuring that shit by
. Insane. (Also think about how, in the good old days, people like Carthaginians and Romans and Greeks would march across entire continents for months, years sometimes, just to slaughter each other.
These days, my idea of a taxing cardiovascular workout is operating a stapler.) To show how they did this for quantifying the relationship between voltage and conductance in potassium, for example,
they simply applied a bunch of different currents, saw how it changed over time, and attempted to fit a mathematical function to it, which happens to fit quite nicely when you include n-gates and a
fourth-power polynomial.
After a series of painstaking experiments and measurements, Hodgkin and Huxley calculated values for the conductances and equilibrium voltages for different ions. Quite a feat, when you couple that
with the fact that they hunted down and killed their very own Giant Squid and then ripped a neuron out of its brain. Incredible. That is the very definition of alpha male behavior, and it's something
I want all of my readers to emulate.
Table 3 from Hodgkin & Huxley (1952) showing empirical values for voltages and conductances, as well as the capacitance of the membrane.
The same procedure was used for the n, m, and h gates, which were also found to be functions of the membrane voltage. Once these were calculated, then the conductances and voltage potential could be
found for any resting potential and any amount of injected current.
H & H's formulas for the n, m, and h gates as a function of voltage.
So where does that leave us? Since Hodgkin and Huxley have already done most of the heavy lifting for us, all we need to do is take their constants and equations they've already derived, and put it
into a script that we can then run through Matlab. At some point, just to get some additional exercise, we may also operate a stapler.
But stay focused here. Most of the formulas and constants can simply be transcribed from their papers into a Matlab script, but we also need to think about the final output that we want, and how we
are going to plot it. Note that the original Hodgkin and Huxley paper uses a differential formula for voltage to tie together the capacitance and conductance of the membrance, e.g.:
We can use a method like Euler first-order approximation to plot the voltages, in which each time step is based off of the previous one which is added to a function multiplied by a time step; in the
sample code below, the time step can be extremely small, thus giving a better approximation to the true shape of the voltage timecourse. (See the "calculate the derivatives" section below.)
The following code runs a simulation of the Hodgkin Huxley model over 100 milliseconds with 50mA of current, although you are encouraged to try your own and see what happens. The sample plots below
show the results of a typical simulation; namely, that the voltage depolarizes after receiving a large enough current and briefly becomes positive before returning to its previous resting potential.
The conductances of sodium and potassium show that the sodium channels are quickly opened and quickly closed, while the potassium channels take relatively longer to open and longer to close.The point
of the script is to show how equations from papers can be transcribed into code and then run to simulate what neural activity should look like under certain conditions. This can then be expanded into
more complex areas such as memory, cognition, and learning.
The actual neuron, of course, is nowhere to be seen; and thank God for that, else we would run out of Giant Squids before you could say Jack Robinson.
Book of GENESIS, Chapter 4 Original Hodgkin & Huxley paper
%===simulation time===
simulationTime = 100; %in milliseconds
%===specify the external current I===
changeTimes = [0]; %in milliseconds
currentLevels = [50]; %Change this to see effect of different currents on voltage (Suggested values: 3, 20, 50, 1000)
%Set externally applied current across time
%Here, first 500 timesteps are at current of 50, next 1500 timesteps at
%current of zero (resets resting potential of neuron), and the rest of
%timesteps are at constant current
I(1:500) = currentLevels; I(501:2000) = 0; I(2001:numel(t)) = currentLevels;
%Comment out the above line and uncomment the line below for constant current, and observe effects on voltage timecourse
%I(1:numel(t)) = currentLevels;
%===constant parameters===%
%All of these can be found in Table 3
gbar_K=36; gbar_Na=120; g_L=.3;
E_K = -12; E_Na=115; E_L=10.6;
%===set the initial states===%
V=0; %Baseline voltage
alpha_n = .01 * ( (10-V) / (exp((10-V)/10)-1) ); %Equation 12
beta_n = .125*exp(-V/80); %Equation 13
alpha_m = .1*( (25-V) / (exp((25-V)/10)-1) ); %Equation 20
beta_m = 4*exp(-V/18); %Equation 21
alpha_h = .07*exp(-V/20); %Equation 23
beta_h = 1/(exp((30-V)/10)+1); %Equation 24
n(1) = alpha_n/(alpha_n+beta_n); %Equation 9
m(1) = alpha_m/(alpha_m+beta_m); %Equation 18
h(1) = alpha_h/(alpha_h+beta_h); %Equation 18
for i=1:numel(t)-1 %Compute coefficients, currents, and derivates at each time step
%---calculate the coefficients---%
%Equations here are same as above, just calculating at each time step
alpha_n(i) = .01 * ( (10-V(i)) / (exp((10-V(i))/10)-1) );
beta_n(i) = .125*exp(-V(i)/80);
alpha_m(i) = .1*( (25-V(i)) / (exp((25-V(i))/10)-1) );
beta_m(i) = 4*exp(-V(i)/18);
alpha_h(i) = .07*exp(-V(i)/20);
beta_h(i) = 1/(exp((30-V(i))/10)+1);
%---calculate the currents---%
I_Na = (m(i)^3) * gbar_Na * h(i) * (V(i)-E_Na); %Equations 3 and 14
I_K = (n(i)^4) * gbar_K * (V(i)-E_K); %Equations 4 and 6
I_L = g_L *(V(i)-E_L); %Equation 5
I_ion = I(i) - I_K - I_Na - I_L;
%---calculate the derivatives using Euler first order approximation---%
V(i+1) = V(i) + deltaT*I_ion/C;
n(i+1) = n(i) + deltaT*(alpha_n(i) *(1-n(i)) - beta_n(i) * n(i)); %Equation 7
m(i+1) = m(i) + deltaT*(alpha_m(i) *(1-m(i)) - beta_m(i) * m(i)); %Equation 15
h(i+1) = h(i) + deltaT*(alpha_h(i) *(1-h(i)) - beta_h(i) * h(i)); %Equation 16
V = V-70; %Set resting potential to -70mv
%===plot Voltage===%
hold on
ylabel('Voltage (mv)')
xlabel('time (ms)')
title('Voltage over Time in Simulated Neuron')
%===plot Conductance===%
p1 = plot(t,gbar_K*n.^4,'LineWidth',2);
hold on
p2 = plot(t,gbar_Na*(m.^3).*h,'r','LineWidth',2);
legend([p1, p2], 'Conductance for Potassium', 'Conductance for Sodium')
xlabel('time (ms)')
title('Conductance for Potassium and Sodium Ions in Simulated Neuron')
20 comments:
1. Muy Buena la interpretacion
Desearia contactarme contigo para resolver algunos problemas de Hodgkin and Huxley, o me puedes brindar un tutorial, lo que ocurre tengo problemas en la codificacion, o explicar el codigo que has
insertado paso a paso en matlab
1. Al final, alguien te dio respuesta? Saludos
2. Hi,
I need some help in the HH model as i want to implement research paper in matlab and i m not getting any idea how to start with this. It will be very helpful if u suggest some approach for this.
This is the link of paper
1. Hi Priyanka,
Similar to what I did above, first specify all of the constants that they provide in the paper (e.g., the conductances, potentials, and capacitances that are given); then, set up your
equations just as they do in the paper. For equation 4, which involves a differential equation, I would also recommend calculating the value using the Euler method that I use in the post.
3. Hi
thank you dear ...
4. Hi,
Thanks for the beautiful demonstration and explanation. Could you let me know a few things:
1. what is the difference that you mean in the code with constant current and one with time steps? The ideal case in a neuron is one with constant current, is it? Correct me if I am wrong?
1. Hi Raunaq,
You can do either; the default in the script is to have current run for the first 500 milliseconds, and then shut off; you can alter this however you want, or make the current constant by
commenting out that line and uncommenting the one below it. It's just for demonstration purposes.
5. Great explanation Andrew. Could you briefly guide me how should I go about adding stochastic noise to this hh model?
1. Hey Jack,
Check out the normrnd function, and add that to the variable you're interested in (e.g., voltage, conductance, etc). For example, something like:
E_Na_mu = 115;
E_Na_std = 5;
E_Na = normrnd(E_Na_mu, E_Na_std);
2. Hi Jack,
Thanks a lot!
I will look into it. Also, would you have any suggestion where I could learn about coding a markov chain model for ion channel?
6. Andrew I saw your video and it was really good I was wondering if you can help me with an assignment?
7. Hi Andrew. I'm trying to ask a question, but I've not been able to comment. If this question comes up many times, I am sorry. I am assuming that the values in Table 3 from Hogkin Huxley that you
have above depend on the particular neuron type. Is that right? Also, the functions in the following table depend on the particular neuron type. Is that right?
8. Hi andrew.,How do I model the simulations to show excitory and inhibitory effects of neurotransmitters.
9. Hi Andrew! Thanks for the video, I have a quastion: How does the program know which is the first V(i)? Thanks you :)
10. Hey!!! why do you define symmetrical values for the constants from those on table 3??
E_K = -12; E_Na=115; E_L=10.6;
11. Hello Andrew! Thank you for this great blog! I´m working on similar actionpotential. My problem is, how to run my action potential on a one-dimentional ring? So that the actionpotential
Best greetings, Ewe
12. Thank you, Andrew!
13. wow
14. This is awesome, thanks Andy.
15. hello i have practically no idea why im here but i have to say you are one funny mf. I think i may have made the wrong degree choice >.< | {"url":"http://andysbrainblog.blogspot.com/2013/10/introduction-to-computational-modeling.html","timestamp":"2024-11-12T06:23:42Z","content_type":"application/xhtml+xml","content_length":"280386","record_id":"<urn:uuid:af75cd29-3c49-4152-8b0e-92b2c0077127>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00663.warc.gz"} |
The purpose of this series of assignments is to play with trigonometric functions. We use some formulas we already used in the topic Trigonometry Unit circle and simple formulas. There are also other
heavily used formulas: the so-called addition and subtraction formulas. Especially in this case we know: practice makes perfect.
1. Show:
2. Show:
3. Solve for :
4. Use the Simpson/Mollweide formulas for the following formula:
5. Show the following relation using some trigonometric formulas:
6. Show:
7. Solve for
8. A point moves in the
calculate the equation of the curve along which the point is moving.
9. Show:
10. Show: | {"url":"https://4mules.nl/en/addition-and-subtraction-formulas/assignments/","timestamp":"2024-11-06T20:56:36Z","content_type":"text/html","content_length":"43629","record_id":"<urn:uuid:7218b4e4-c33c-4417-a08c-f02ca6411628>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00240.warc.gz"} |
Anurupyena & Vinculum Division using Vedic Mathematics when Nikhilam and Paravartya is not possible to be applied BUT if divisor is either multiplied or divided by some factor then it is possible.
Anurupyena Sutra Division As seen earlier in Multiplication, Anurupyena means Proportion. Topics need to be known before starting: In this topic I will let you know the shortcut to divide numbers
using Anurupyena Sutra. But before checking Anurupyena Methos below concepts are required. Basic Requisites page. Anurupyena method from Multiplication Nikhilam Method of Division Paravartya Specific
Condition Required: As we know the meaning of Anurupyena (as proportion/ratio), we multiply/divide by factor to make divisor closer to larger number (To apply Nikhilam) OR to make closer to smaller
number (To apply Paravartya). Later we multiply/divide QUOTIENT with same factor. Also Read => More Division Sutras in Vedic Mathematics Anurupyena Division Tricks: Its always good to use factors for
multiplication instead of division because if division is used then on dividing Quotient by that factor might create non-integer i.e. a decimal quotient. To avoid overheads its better to use
multiplication. It might follow or followed by Vinculum for simplicity purpose. Vinculum As seen earlier with bigger digits, calculation/process gets little bulky. So using Vinculum we need to
convert bigger digits to smaller. Prerequisites: Vinculum and How to Play with Quotients and Remainders. Nikhilam and Paravartya Methods of Division As discussed earlier when we have larger divisor
we apply Nikhilam and when we have smaller divisor we apply Paravartya. But when we have 1 or more larger digits (6,7,8,9) in divisor then calculating answer becomes little lengthy/time consuming
also big multiplications are to be done. So we can convert such divisors in Vinculum Number. Example: # 2621/828 For already seen this example in … [Read more...] | {"url":"http://mathlearners.com/tag/paravartya/","timestamp":"2024-11-02T18:13:02Z","content_type":"text/html","content_length":"49026","record_id":"<urn:uuid:2241e5b5-9588-44aa-af76-4bc7e6fee340>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00164.warc.gz"} |
Fuzzy PID control of a two-link flexible manipulator
For a flexible manipulator system, the unwanted vibrations deteriorate usually the performance of the system due to the coupling of large overall motion and elastic vibration. This paper focuses on
the active vibration control of a two-link flexible manipulator with piezoelectric materials. The multi flexible body dynamics (MFBD) model of the two-link flexible manipulator attached with
piezoelectric sensors and actuators is established firstly. Based on the absolute nodal coordinate formulation (ANCF), the motion equations of the manipulator system are derived and motion process
and dynamic responses of the system are simulated. According to the time varying feature of system, a fuzzy PID controller is developed to depress the vibration. This controller can tune control
gains online accommodating to the variation of the system. Control results obtained by the fuzzy PID control and the conventional PID control indicate that the fuzzy PID controller can effectively
suppress the elastic vibration of the manipulator system and performs better than the conventional PID controller.
1. Introduction
Manipulators are widely used in industrial and research environments which are too hazardous or unpleasant for a human worker. High performance robotic systems are quantified by a high speed of
operation, high end-position accuracy, and lower energy consumption [1, 2]. However, the accuracy is deteriorated by excessive vibration of the flexible links. Flexural vibration in flexible
manipulators has been the main research challenge in the modeling and control of such systems [3]. There exists the coupling of large overall motion and elastic vibration during the movement of
flexible manipulator system. It is a typical nonlinear time-varying system, yet to be adequately resolved.
During the past 30 years, considerable researches on the development of dynamic models of flexible manipulators have been carried out. In the most previous research works, the kinematic description
of flexible manipulators that undergo large displacements is the floating frame of reference formulation. The mechanical model of flexible link robot, being described by differential equations and
thus possessing an infinite number of dimensions, is difficult to be used directly in system analysis or control design. Usually the motion equations are truncated to some finite dimensional models
with either the assumed modes method (AMM) or the finite element method (FEM) [4]. In [5-8], flexible manipulators using Lagrange’s equation and the assumed mode method are studied. Many authors used
the finite element method to derive the equations of motion [9-12]. Theodore and Ghosal [10] provided a detailed comparison between the assumed mode method and the finite element method used for
flexible manipulators.
The main drawback of the floating frame of reference formulation is that the flexible bodies are simplified to linear models. These linear models simplified for the convenience of simulation are
unreasonable in many cases. In these researches, nonlinear material and nonlinear elastic deflections introduced by large amplitude vibration of the manipulator are not permissible [3]. However, the
absolute nodal coordinate formulation (ANCF) proposed by Shabana [13], supports both nonlinear geometric deformation as well as nonlinear material formulations and can be used dynamic analysis and
simulation of multibody systems with large deformations. In this paper, a more accurate model for the dynamic analysis of a two-link flexible manipulator system is obtained using this formulation,
and motion process and dynamic responses of the system are simulated.
Besides the effort on dynamic modeling and analysis of flexible manipulator systems, researchers carried out a lot of work on vibration control of the manipulators. Wang [14], Meirovitch [15] and
Shaheed et al. [16] studied vibration control of one-link manipulator with PD control, optimal control and adaptive control, respectively. Moreover, there has been a lot of interest in the area of
active vibration control of manipulators by using piezoelectric materials as sensors and actuators. H. C. Shin and S. B. Choi [1, 17], Z. C. Qiu et al. [18], V. Bottega et al. [19] controlled the
end-point position of a two-link flexible manipulator employing servomotors mounted at the hub and piezoceramic actuators attached to the surfaces of each flexible link. Despite intense research work
have been devoted to the vibration control of manipulator system, most of the previous studies are control experiments or the control simulations based on AMM multi flexible body models, there is no
study available on the vibration control simulation using the nonlinear finite element model in ANCF. This paper reports vibration control of the two-link flexible manipulator system based on a more
accurate model.
In addition, it is well known that the fuzzy PID control technique combining the traditional PID control with the fuzzy control algorithm can adaptively adjust the PID parameters online by using the
fuzzy logic. For this reason, the fuzzy PID control technique can effectively improve the control accuracy and is extensively used in processes where systems dynamics is either very complex or
exhibit a highly nonlinear characteristic [20, 21]. In [22], experiments are carried out to evaluate the effectiveness of the fuzzy PID control method applied for hydraulic systems. In [23-25], fuzzy
PID controllers which can adaptively adjust controller parameters on-line are designed to control SRM (Switched Reluctance Motor) system, HVAC (heating, ventilating and air-conditioning) system, and
APS (atmospheric pressure simulator) system. These references listed here are merely part of applications of fuzzy PID controllers. However, little research has been done to explore how this control
method applied into the vibration control of flexible manipulator systems. In this paper, we focus on the vibration control of a two-link flexible manipulator system using fuzzy PID controllers.
According to the time varying feature of the system, a fuzzy PID controller, which can tune control gains online accommodating to the variation of the system is developed to depress the vibration of
flexible links.
The organization of this paper is as follows. In Section 2, the rigid flexible coupling motion equations of the system are derived based on absolute nodal coordinate formulation. The simulation multi
flexible body dynamics (MFBD) model of the manipulator system with piezoelectric sensors and actuators is established using multibody system analysis software RecurDyn. In Section 3, a fuzzy PID
controller suitable for the time varying system is developed and the coupled vibration control process for the two-link manipulator is simulated. Then, control effects are discussed for different
cases and the performance of fuzzy PID controller is compared with that of conventional PID controller in Section 4. Several conclusions are summarized in the last section.
2. Dynamic modeling
2.1. Description of the system
Fig. 1 shows the schematic of a two-link flexible manipulator system. The system consists of two flexible links with bonded piezoelectric actuators and sensors, two revolute joints, and two motors.
The first flexible link is clamped on the hub of the shoulder joint. The elbow joint attached at the tip of the first link connects two flexible links together. Under the effect of torques from drive
motors, the motion of flexible links is rotational motion about $Y$-axes of shoulder joint and elbow joint. Two PZT actuators are perfectly attached to the upper surfaces of the flexible links near
the location of each root, respectively. Moreover, two piezofilm sensors are bonded to lower surfaces of the flexible links.
A flexible link of the manipulator system featuring by surface-bonded piezoelectric actuator and sensor is shown in Fig. 2. Subscript $i$ denotes the $i$th flexible link, and subscripts $a$, $b$ and
$s$ denote actuator, beam, and sensor, respectively. ${d}_{i}$ represents the location of actuator and sensor, measured from the root of the $i$th link, ${d}_{1}=$0.06 m and ${d}_{2}=$0.04 m. The
dimensions and mechanical properties of flexible links, piezoelectric actuators and sensors are given in Table 1.
Fig. 1A two-link flexible manipulator system featuring piezoelectric actuators and sensors
Fig. 2The ith flexible link of the manipulator system
Table 1Dimensional and mechanical properties of the flexible links and piezoelectric actuators and sensors
Dimension ($m$×$m$×$m$) Young’s modulus, $E$ (GPa) Density, $\rho$ (kg/m^3) Piezoelectric constant, ${d}^{31}$ (m/V)
Link 1 0.55×0.08×0.002 69 2700 –
Link 2 0.45×0.04×0.0012 69 2700 –
Actuator 1 0.1×0.08×0.0008 71.4 7350 200×10^-12
Actuator 2 0.08×0.04×0.0008 71.4 7350 200×10^-12
Sensor 1 0.02×0.02×2.5e-4 2.5 1800 1.5×10^-12
Sensor 2 0.02×0.02×2.5e-4 2.5 1800 1.5×10^-12
2.2. Sensors and actuators
Fig. 3 is a schematic of the vibration control. It is assumed that the piezofilm sensors are thin with respect to the flexible links, hence the strain in each of them is regarded as uniform over
thickness. The piezoelectric sensor can be treated as a parallel plate capacitor, and the charge $Q\left(t\right)$ stored across the electrodes of the capacitor can be expressed as:
$Q\left(t\right)={E}_{s}{w}_{s}{d}_{s}^{31}{\int }_{{l}_{s}}\epsilon \left(t\right)dx,$
where ${w}_{s}$ and ${l}_{s}$ are the width and length of the sensor, ${E}_{s}$ and ${d}_{s}^{31}$ are elastic modulus and piezoelectric constant of the sensor. $\epsilon \left(t\right)$ is the
strain component along the sensor length direction.
Assuming the value of $\epsilon \left(t\right)$ to be average over the piezofilm sensor length, the voltage generated by the sensor can be rewritten as:
$Q\left(t\right)={E}_{s}{w}_{s}{l}_{s}{d}_{s}^{31}\epsilon \left(t\right).$
Because the piezoelectric sensor has very high output impedance, the output of the sensor is passed through a charge amplifier. Then, the output voltage ${V}_{s}\left(t\right)$ of the charge
amplifier is defined as:
${V}_{s}\left(t\right)=\frac{Q\left(t\right)}{{C}_{f}}=\frac{{E}_{s}{w}_{s}{l}_{s}{d}_{s}^{31}}{{C}_{f}}\epsilon \left(t\right),$
where ${C}_{f}$ is the capacitance of the amplifier.
Fig. 3Schematic of the vibration control
According to Eq. (3), it can be found that the output voltage of the charge amplifier is proportional to the amount of electric charge generated by the sensor. The output signal of the charge
amplifier is then fed to the controller designed for the vibration control of the flexible links. The output of controller $u\left(t\right)$ is amplified by a power amplifier. The final voltage
applied to the actuator is the product of ${K}_{a}$ and $u\left(t\right)$:
where ${V}_{a}\left(t\right)$ is the voltage applied to the piezoceramic actuator, and ${K}_{a}$ is the power amplification factor.
Finally, the rectangular piezoelectric actuator is equivalent to a pair of torques $M\left(t\right)$ with opposite signs and proportional to ${V}_{a}\left(t\right)$:
where ${w}_{b}$ is the width of the actuator, ${h}_{b}$ and ${h}_{a}$ are the thicknesses of the beam and PZT actuator, respectively. ${E}_{a}$ and ${d}_{a}^{31}$ are the Young’s modulus and
piezoelectric constant of the PZT actuator.
2.3. Absolute nodal coordinate formulation for the flexible links
Absolute nodal coordinate formulation (ANCF) can be used in the large rotation and deformation analysis of flexible bodies that undergo arbitrary displacements [13]. In this subsection, the ANCF is
briefly introduced and the motion equations of flexible links are derived by employing ANCF.
In the absolute nodal frame formulation, the element nodal coordinates are defined in the inertial frame. Here, we define $\mathbf{e}$ as the vector of element nodal coordinates, and then these nodal
coordinates are used with a global shape function $\mathbf{S}$. The global shape function $\mathbf{S}$ has a complete set of rigid body modes that can describe arbitrary rigid body translational and
rotational displacements. Therefore, the global position vector of an arbitrary point on the element can be described as:
By differentiating Eq. (6) with respect to time, the absolute velocity vector $\stackrel{˙}{\mathbf{e}}$ can be defined. Using the element nodal coordinate and velocity vectors, the kinetic energy
$T$ and potential energy $U$ of the element are given as:
where ${\mathbf{M}}_{e}$ and ${\mathbf{K}}_{e}$ are the mass and stiffness matrices of the element, respectively. It is worth noting that, in the absolute coordinate formulation, ${\mathbf{M}}_{e}$
is a constant matrix and ${\mathbf{K}}_{e}$ is a highly nonlinear function of the element coordinates and changes over time even in the case of linear elastic problems.
Employing the Lagrange’s equation, one can obtain the equations of motion of the finite element as follows:
where ${\mathbf{Q}}_{e}$ is the vector of generalized nodal force described in the absolute coordinate system. Using the equation of the finite element, connectivity conditions between the finite
elements can be imposed and the equations of the elements can be assembled to obtain the equations of motion of flexible links in the multibody system.
Employing the Lagrange multiplier theorem, one can obtain the equations of motion for the constrained manipulator system as follows:
$\mathbf{A}\stackrel{¨}{\mathbf{q}}+\mathbf{B}\mathbf{q}-\mathbf{Q}+{\mathrm{\Phi }}_{q}^{T}\lambda =0,\mathbf{}\mathbf{}\mathbf{}\mathrm{\Phi }\left(\mathbf{q},t\right)=0,$
where $\mathbf{q}$ is the global vector of element nodal coordinates. $\mathbf{A}$ and $\mathbf{B}$ are the global mass matrix and stiffness matrix assembled by element mass matrices and element
stiffness matrices. $\mathbf{Q}$ is the global force vector described in the absolute coordinate system. ${\mathrm{\Phi }}_{q}^{T}$ and $\lambda$ are Jacobian and Lagrange multiplier, respectively.
2.4. Modeling in RecurDyn
The electromechanical model of the two-link manipulator is established and the kinematic process and vibration of the system can be simulated based on the model. RecurDyn, a powerful multibody
dynamic analysis software from Korea, is employed here. It combines MBD (Multi Body Dynamics) to analyze the motion of rigid bodies and a nonlinear FEM (Finite Element Method) to analyze the motion,
stress, and deformation of flexible bodies. RecurDyn’s solver combines these two components into a single solver. This makes RecurDyn a fast, robust, and reliable solver [26].
The MFBD modeling of the manipulator system can be described as three basic steps. First of all, the finite element models of flexible links are constructed in ANSYS. After that, the finite element
models are imported into RecurDyn. Finally, constraints and drivers are added to the model of the flexible manipulator system for kinematics and dynamics simulation.
The finite element models of shoulder link and elbow link are established in ANSYS as shown in Fig. 4. Because the thickness of the piezofilm sensors are much thinner and softer than the flexible
links, the influences of the sensors on the system inertia and stiffness properties are neglected in the simulation.
The finite element models are imported into the dynamic simulation software RecurDyn. Two rotational joints and two drivers are included to represent the shoulder and elbow joints, respectively. The
flexible links are driven by the drivers, and rotate around axes of the rotational joints. Finally, the dynamic model of the two-links manipulator system is finished with RecurDyn as shown in Fig. 5.
Fig. 4Finite element models of flexible links
Fig. 5The multi flexible body dynamical model of the manipulator
3. Fuzzy PID control
The driving torques applied on the joints and subsequent rigid body motion are treated as disturbances to the system which exciting vibrations of the flexible links. To suppress the vibration of the
manipulator system, a fuzzy PID controller is developed. The fuzzy PID controller is a hybrid controller combining a conventional PID controller with an adaptive fuzzy controller, and can tune
control gains online accommodating to the variation of the system.
3.1. Conventional PID control
Since PID controller has the advantages of simple structure, high stability and easy design, it has been most widely used in industrial process control. Firstly, a pure PID control is applied to the
two-link flexible robot manipulator. The block diagram of active vibration control with two simplified PID controllers is shown in Fig. 6. In the PID controller, the difference between the reference
signal and the sensor signal is used as input. The strain, amplified by the charge amplifier, is taken as the sensor signal. The reference input, denoted by $r\left(t\right)$, is assigned a zero
$e\left(t\right)=r\left(t\right)-{K}_{s}\epsilon \left(t\right),$
where ${K}_{s}$ is the sensor amplification factor.
Fig. 6Block diagram of active vibration control with a traditional PID controllers
In the PID controller the proportional, integral, and derivative terms are summed to calculate the output of the controller. Defining $u\left(t\right)$ as the ideal continuous controller output, the
final output of a PID controller is given by Eq. (12):
$u\left(t\right)={k}_{P}e\left(t\right)+{k}_{I}{\int }_{0}^{t}e\left(t\right)dt+{k}_{D}\frac{de\left(t\right)}{dt},$
where $e\left(t\right)$ is error. ${k}_{P}$, ${k}_{I}$ and ${k}_{D}$ are proportional gain, integral gain, and derivative gain, respectively.
We are concerned with digital control, and for small sampling periods, Eq. (12) may be approximated by a discrete approximation:
${u}_{n}={k}_{P}{e}_{n}+{k}_{I}\sum _{j=1}^{n}{e}_{j}+{k}_{D}\left({e}_{n}-{e}_{n-1}\right),$
where index $n$ refers to the time step.
The output of the PID controller is amplified by power amplification, i.e., the applied voltage to the actuator is multiplied by ${K}_{a}$. As mentioned before, the control torques $M$ applied to
manipulator system is calculated via Eq. (5).
The performance of the PID controller directly depends on an appropriate choice of the PID gains. Table 2 shows the effect of PID parameters on system response. In practice, the values of PID
parameters should be settled within a reasonable range.
Table 2Effect of PID parameters on system response
Rise time Overshoot Settling time Steady state error
${k}_{P}$ Reduce Increase Small change Reduce
${k}_{I}$ Reduce Increase Increase Eliminate
${k}_{D}$ Small change Reduce Reduce Small change
3.2. Fuzzy PID controller design
Conventional PID controllers have proven to be very effective for systems that can be modeled relatively precisely by mathematical equations. However, they have been found to be inefficient in
handling systems that are either too complex or too vague to be described by accurate mathematical models. What’s more, the success of the PID controller depends on an appropriate choice of the PID
gains and the selection of PID parameters in most cases is not an easy task. It takes a great deal of experience to transform design requirements and objectives to the performance index that will
produce the desired performance. To replace experience with analytical tools is very important for complex systems or those without precise descriptions. To determine proper control gains
analytically, a fuzzy inference module is designed especially for the conventional PID controller to adjust the PID gains online, according to the error and its change rate. The active vibration
control system with fuzzy PID controllers is shown in Fig. 7.
In this subsection, we briefly describe the standard procedure for designing a fuzzy controller, which includes of fuzzification, control rule base establishment, and defuzzification [22, 23].
3.2.1. Fuzzification of input and output variables
The structure of the fuzzy logic system based on the Mamdani inference method includes two inputs and three outputs as shown in Fig. 8. The inputs to the fuzzy inference system are error $e$ and the
rate of change of the error $ec$, and the outputs are increments of PID gains $\mathrm{\Delta }{k}_{P}$, $\mathrm{\Delta }{k}_{I}$, and $\mathrm{\Delta }{k}_{D}$, respectively. Through fuzzy logic
knowledge, the fuzzy PID tuners which tune PID parameters (${k}_{P}$, ${k}_{I}$, ${k}_{D}$) can be established by using the following equation:
${k}_{P}={k}_{P0}+\mathrm{\Delta }{k}_{P},{k}_{I}={k}_{I0}+\mathrm{\Delta }{k}_{I},{k}_{D}={k}_{D0}+\mathrm{\Delta }{k}_{D},$
where ${k}_{P0}$, ${k}_{I0}$, ${k}_{D0}$ are initial values of fuzzy PID controller gains.
Fig. 7Block diagram of Fuzzy PID control
Fig. 8The structure of the fuzzy logic system based on the Mamdani inference method
In order to transform the input and output data into proper semantic value, it is necessary to carry out fuzzification of the input and output variables. In this research, the fuzzy range of inputs
and outputs is separated into 7 semantic variables, and corresponding fuzzy subsets are [NB, NM, NS, ZO, PS, PM, PB]. Where NB is negative big; NM is negative middle; NS is negative small; ZO is
zero; PS is positive small; PM is positive middle; PB is positive big.
The membership functions of two inputs are implemented with seven Gaussian membership functions and scale within the range of [–6, 6]. The membership functions of three outputs are seven triangular
membership functions and scale within the range of [–3, 3]. The membership functions are illustrated in Figs. 9 and 10.
3.2.2. Fuzzy control rules
The key to realize self-tuning fuzzy PID control is to find out fuzzy relation between inputs and outputs by using the experience of experts or input-output data. The fuzzy inference rules between
inputs and outputs are given as in the form of.
Rule: If $e$ is ${A}_{i}$ and $ec$ is ${B}_{j}$, Then $\mathrm{\Delta }{k}_{p}$ is ${C}_{ij}$, $\mathrm{\Delta }{k}_{i}$ is ${D}_{ij}$ and $\mathrm{\Delta }{k}_{d}$ is ${E}_{ij}$. Where ${A}_{i}$, $
{B}_{j}$, ${C}_{ij}$, ${D}_{ij}$, ${E}_{ij}$ (${A}_{i}$, ${B}_{j}$, ${C}_{ij}$, ${D}_{ij}$, ${E}_{ij}\in \mathrm{}$[NB, NM, NS, ZO, PS, PM, PB]) are linguistic values of inputs and outputs. Table 3
gives out all 49 possible rules.
Fig. 9Membership functions of e and ec
Fig. 10Membership functions of kP, kI and kD
Table 3The fuzzy rule base for PID gains
${k}_{P}$ / ${k}_{I}$ / ${k}_{D}$
NB NM NS ZO PS PM PB
NB PB/NB/NB PB/NB/NB PM/NB/NM PM/NB/NM PS/NM/NS ZO/NM/ZO ZO/NS/ZO
NM PB/NB/NB PB/NB/NB PM/NB/NM PS/NM/NS PS/NM/NS ZO/NS/ZO NS/ZO/ZO
NS PM/ZO/NB PM/NS/NM PM/NM/NS PS/NM/NS ZO/NS/ZO NS/NS/PS NS/ZO/PS
$e$ ZO PM/ZO/NM PM/NS/NM PS/NS/NS ZO/NS/ZO NS/NS/PS NM/NS/PM NM/ZO/PM
PS PS/ZO/NM PS/ZO/NS ZO/ZO/ZO NS/ZO/PS NS/ZO/PS NM/ZO/PM NM/ZO/PB
PM PS/PB/ZO ZO/PS/ZO NS/PS/PS NM/PS/PS NM/PS/PM NM/PS/PB NB/PB/PB
PB ZO/PB/ZO ZO/PM/ZO NM/PM/PS NM/PM/PM NM/PS/PM NB/PS/PB NB/PB/PB
3.2.3. Defuzzification
Product-inference rule and center average defuzzzifier are adopted to accomplish fuzzy implication and synthesis calculation respectively. The output of $\mathrm{\Delta }{k}_{P}$ ($\mathrm{\Delta }
{k}_{I}$ and $\mathrm{\Delta }{k}_{D}$ are similar) from the fuzzy inference system is:
$\mathrm{\Delta }{k}_{p}\left(e,ec\right)=\frac{{\sum }_{n}^{N}{\omega }^{n}\mathrm{\Delta }{k}_{P}^{n}}{{\sum }_{n}^{N}{\omega }^{n}},$
${\omega }^{n}={\mu }_{{A}_{i}^{n}}\left(e\right){\mu }_{{B}_{j}^{n}}\left(ec\right),$
where $n$ denotes the $n$th fuzzy rule and $N$ is the number of rules in the rule base. $\mathrm{\Delta }{k}_{P}^{n}\in R$ is any point at which ${\mu }_{{C}_{ij}^{n}}\left(\mathrm{\Delta }{k}_{P}\
right)$ achieves its maximum value, ${\mu }_{{C}_{ij}^{n}}\left(\mathrm{\Delta }{k}_{P}\right)=\text{1}$. ${\mu }_{{A}_{i}}\left(e\right)$ and ${\mu }_{{B}_{j}}\left(ec\right)$ are the Gaussian
membership function of the input $e$ and $ec$, respectively:
${\mu }_{{A}_{i}^{n}}\left(e\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{e-{c}_{i}}{{\sigma }_{i}}\right)}^{2}\right],$
${\mu }_{{B}_{j}^{n}}\left(ec\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{ec-{c}_{j}}{{\sigma }_{j}}\right)}^{2}\right],$
where ${c}_{i}$ and ${\sigma }_{i}$ are the corresponding centers and standard deviations of Gauss membership functions, respectively.
4. Simulations and results
To simulate vibration control of the manipulator system, Matlab/Simulink and RecurDyn are utilized jointly. The procedure of the active vibration control simulation is illustrated in Fig. 11.
Firstly, the dynamic model is achieved from RecurDyn as described previously, and the inputs and outputs of model are defined, respectively. The inputs are control torques, and the outputs are strain
values at the sensor location. Next, the controllers which calculate control forces are designed in Simulink. Finally, solution is set up and the co-simulation is accomplished step by step.
Fig. 11Flow chart of co-simulation
In this part, the motion processes and dynamic responses of the system are simulated under two different cases: (1) Only the motor of joint 2 drives, joint 1 is locked all the way. Flexible link 2
swings within [0, $\pi /$3] rad and the variation of angles for joint 1 and joint 2 are shown in Fig. 12(a). (2) Joint 1 uniformly rotates from 0 rad to 2$\pi /$3 rad within two seconds and then
stops spanning. Joint 2 is locked before joint 1 arrives at desired position. During the span of [2, 3] s, joint 2 uniformly rotates from 0 rad to $\pi /$3 rad, and then stops. Fig. 12(b) describes
the rotating angles of joint 1 and joint 2 for the latter case.
If we take no account of elastic deformation, the motion of the manipulator system becomes multi rigid body movement. The displacement responses of any point for this rigid system are only depending
on the large overall motion.
The curves in Figs. 13 and 14 represent the displacements of endpoints for two links. By comparison of the tip displacement curves of flexible system (dotted lines) and rigid system (solid lines), it
is found that the disturbances of the driving torques and subsequent rigid body motion can excite vibrations of the flexible links. As pointed out before, there exists the coupling between the large
overall motion and elastic vibration during the movement of flexible manipulator system. The stationarity of the flexible manipulator system is deteriorated by unfavorable vibrations. The elastic
vibration of each link decays very slowly under only the influence of inherent material damping property. It also suggests that adding active vibration control to manipulator system is necessary.
Fig. 12Rotation angles of joint 1 and joint 2
Fig. 13The motion curves of endpoints for case 1
Fig. 14The motion curves of endpoints for case 2
To verify the effectiveness of the fuzzy PID controller, active vibration control simulations are carried out. In the following simulation, conventional PID control and fuzzy PID control strategies
are used for controlling vibrations of the flexible links respectively, for comparison. The sensor and power amplification factors, ${K}_{a}$ and ${K}_{s}$ are, respectively, taken as 50. The gains
of conventional PID controllers are fixed, while the fuzzy PID controllers tune the PID gains online within given ranges. Initial values and adjustable ranges of PID gains for fuzzy PID controllers
are listed in Table 4. The saturation blocks are assembled into the controllers to impose upper and lower bounds on the control voltages. On the one hand, the given initial values and adjustable
ranges ensure that adjusted gains are within in reasonable ranges. On the other hand, the control voltages are restricted within the limit of actuator voltage by the saturation blocks. Thus, the
stability of fuzzy PID controllers can be guaranteed.
Table 4Initial values and adjustable ranges of the gains for the controllers
PID controller 1 PID controller 2 Fuzzy PID controller 1 Fuzzy PID controller 2
Initial value:2.4 Initial value:2.0
Proportional gain ${k}_{P}$ 2.4 2.0
Range: [0.8, 4.0] Range: [0.8, 3.2]
Initial value:0.6 Initial value:0.4
Integral gain ${k}_{I}$ 0.6 0.4
Range: [0, 1.2] Range: [0, 0.8]
Initial value:0.04 Initial value:0.04
Derivative gain ${k}_{D}$ 0.04 0.04
Range: [0, 0.08] Range: [0, 0.08]
The tip deflection curves of two links under motion process 1 are shown in Fig. 15. In the absence of control, the elastic vibration of each link is really obvious. Although the lowest frequency
vibration caused by reciprocating motion is not be effectively suppressed, the higher vibration of each link is distinctly suppressed by active vibration control. This is because the frequency of
reciprocating motion is only 0.5 Hz, hence the energy dissipated by the damping effect of piezoelectric actuators is rarely. It is also observed that, the fuzzy PID control method achieves better
results than the conventional PID control.
Fig. 15Tip deflection responses of the flexible links for case 1
For motion process 2, the vibrations of each link are shown in Fig. 16. The rms (root-mean-square) displacements are calculated and utilized to evaluate the control efficiency of the fuzzy
controller. The rms displacement values of link 1 under no control, PID control and fuzzy PID control are 8 mm, 5.1 mm and 4.1 mm, respectively. The rms displacement values of link 2 under no
control, PID control and fuzzy PID control are 10.5 mm, 5 mm and 3.9 mm, respectively. Referring at values of the uncontrolled system response, rms displacement values of link 1 and link 2 with PID
control are decreased approximately 36 % and 52 %, respectively. Further, the control efficiency can be increased over 10 % through the use of fuzzy controller.
Fig. 16Tip deflection responses of the flexible links for case 2
Fig. 17Applied voltages to the PZT actuators for case 1
Fig. 18Applied voltages to the PZT actuators for case 2
Figs. 17 and 18 show voltages applied to the PZT actuators. When elastic vibration of flexible link is violent, the control voltages calculated by the fuzzy PID controller are larger compared with
the pure PID controller. On the contrary, the control voltages calculated by the fuzzy PID controller are smaller than the pure PID controller while the simulated responses of the system are small.
It proves that the fuzzy PID controller tunes control gains online accommodating to the variation of the system. This makes the fuzzy PID controller become a faster controller, and obtain smaller
steady error and better control quality.
Through these comparisons, a conclusion can be drawn that the proposed the fuzzy PID controller can improve the performance yielded by a PID controller. By the fuzzy PID control, the unwanted
vibration of flexible links is effectively suppressed. The active control significantly decreases the time it will take to an assigned position and improve work efficiency. Hence, the simulation
results provide some guidance for studying active vibration control of manipulator systems.
5. Conclusions
This paper focuses on multi flexible body dynamics analysis and active vibration control of a two-link flexible manipulator system. The main contributions of the present paper can be summarized as
1) By using absolute nodal coordinate formulation, the motion equations of the manipulator system are derived. The MFBD model of the system is established, and the electromechanical coupling
relations of piezoelectric actuators and sensors are analyzed.
2) The MFBD simulation analysis of the manipulator system has been completed. Tip displacement of each link is compared with those for a rigid manipulator system. It is found that the disturbances of
the driving torque and subsequent rigid body motion can excite vibrations of the flexible links. There exists the coupling between large overall motion and elastic vibration during the movement of
flexible manipulator system.
3) A fuzzy PID controller, which can tune the PID gains online, is developed and applied to the manipulator system. The elastic vibration of the flexible links is efficiently suppressed using fuzzy
PID control. The performances of fuzzy PID controllers and conventional PID controllers are compared and discussed. Simulation results indicate that the fuzzy PID controller achieves better results
than the conventional PID controller. In addition, the control methods described in this paper can be extended to control the vibration of other space manipulator system.
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About this article
13 December 2015
15 February 2016
Vibration generation and control
flexible manipulator
multibody dynamics
absolute nodal formulation
vibration control
fuzzy PID controller
This work was supported by the National Natural Science Foundation of China (No. 11302160) of the second author.
Copyright © 2016 JVE International Ltd.
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Powers by Multiplication
Powers by Multiplication: Learn
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Python SymPy Tutorial - Symbolic Computation Library - CodersLegacy
Python SymPy Tutorial – Symbolic Computation Library
SymPy is a free open source library in Python, used for Symbolic and Mathematical Computation. In this tutorial, we will cover how to effectively use the Python SymPy Library to perform mathematical
operations on expressions, Symbolic computations and other various algebraic properties.
Symbolic Computation with SymPy
SymPy allows us represent data and values in a different manner, that is 100% accurate. Let’s take a look at the following example.
import math
What this does, is returns an approximation of the sqaure root of 7. It’s not a 100% exact answer. The more decimal places you include, the more accurate the answer gets. So technically this will
never be an exact representation.
Now let’s try the same thing, but with SymPy instead.
import sympy
What SymPy returned here, was not a mathematical value, but rather a symbolic representation of the sqrt of 7. Thus, it can never be inaccurate. We can use this symbolic representation in other
calculations, with a higher degree of accuracy.
Here’s an another interesting snippet.
import sympy
import math
print(math.pow(math.sqrt(7), 2))
print(sympy.sqrt(7) ** 2)
We took the sqrt root of 7 with both libraries, then applied a power of 2 on both. SymPy returned the correct value, whereas the math library didn’t (because it works off approximations).
Other interesting things SymPy can do, is return simplified representations.
The next thing to cover in this Tutorial, is how to create and manipulate expressions in Python SymPy.
Mathematical Expressions with SymPy
SymPy gives us the ability to create actual expressions using variables (called symbols in SymPy). For example, if you want to represent the equation 2x^2 + 4x + 5 in Python, how would you do so? You
could represent such an equation in a string, but it would be of little use without SymPy.
SymPy gives us the ability to create and handle such expressions that can actually be used in computation. Python SymPy has various functions which could perform actual operations such as
differentiation and integration, and return the resultant expression! (We’ll take a look at these later in the tutorial)
So how do we create such an expression? Well it’s simple. First we define a symbol, which represents an unknown/variable like “x” or “y”.
from sympy import symbols
x, y = symbols("x y")
We can now use x and y to construct expressions. No need to make a string or anything, just do it normally. Let’s try out a few examples.
from sympy import symbols
x, y = symbols("x y")
expr1 = 2*x + 4*y # 2x + 2y
expr2 = 2*(x**2) + 5 # 2(x^2) + 5
expr3 = x**2 + y**2 # x^2 + y^2
Modifying SymPy Expressions
We can even do cool stuff like modify these expressions, by adding, subtracting or multiplying constants and other symbols from it.
expr1 += 5
2*x + 4*y
2*x + 4*y + 5
SymPy will even automatically adjust the expressions as you modify it. For example, if you have an expression "2x", and you add "x" to it, you might expect it to just concatenate and become 2x + x.
But nope, SymPy will automatically simplify it to 3x.
Here’s a short code snippet showing this.
expr = 2*y + x + 5
expr -= x
expr += 3
x + 2*y + 5
2*y + 5
2*y + 8
Substituting values into Expressions
Now that we know how to create and modify expressions in SymPy, let’s take a look at how to evaluate them. What we need to do here, is use the subs() function to substitute the symbols with numerical
expr1 = 2*(x**2) + 5 # 2(x^2) + 5
print("Expr 1: (x=2) ", expr1.subs(x, 2))
Expr 1: (x=2) 13
Now let’s try this for an expression which has multiple unknowns.
expr2 = 2*x + 4*y # 2x + 2y
print("Expr 2: (x=2, y=4) ", expr2.subs( {x: 2, y: 4} ))
Expr 2: (x=2, y=4) 20
We can also substitute them with other symbols, if that’s what we want.
x, y, a, b = symbols("x y a b")
expr3 = x**2 + y**2 # x^2 + y^2
print(expr3.subs({x: a, y: b}))
a**2 + b**2
Solving Equations with SymPy (root finding)
What’s even cooler, is that SymPy can literally solve entire equations for you, and return the root(s). No need to code the entire thing yourself, just use a single function along with the SymPy
expression and a list of root(s) will be returned.
Let’s try to use the SymPy solve() on the expression x^2 - x - 6.
import sympy
from sympy import symbols
x, y = symbols("x y")
expr = x**2 - x - 6
[-2, 3]
Let’s try this out on another expression.
expr = (x + 1)*(x - 1)*(x + 5)
x**3 + 5*x**2 - x - 5
[-5, -1, 1]
In the next section, we will cover several such operations, and explain how you can use them in your SymPy code.
Trigonometry with SymPy
Trigonometry is a pretty deal in most of Mathematica, so you might be wondering how you can include trigonometric functions and identities within your mathematical expressions. Let’s take a look!
Here’s a simple expression, sin(x). Let’s plug in a few values, just to verify the output. (These input values of x are in radians)
from sympy import symbols, expand, solve, trigsimp
from sympy import sin, cos, tan, acos, asin, atan, sinh, cosh, tanh, sec, cot, csc
x, y = symbols("x y")
expr = sin(x)
print(expr.subs(x, 0))
print(expr.subs(x, (90/57.3)))
As you have probably already noticed from the imports in the previous code example, SymPy gives us access to all the different variants of the trigonometric functions. “asin” represents inverse sin
(arc sin), whereas “sinh” represents hyperbolic sin. The same pattern applies to cos and tan as well.
Another cool thing we can do SymPy, is to simplify trigonometric identities. (No need to memorize any of them anymore!) SymPy will automatically attempt to simplify any expression you pass into the
trigsimp() function. Let’s take a look at a few examples.
x, y = symbols("x y")
print(trigsimp(sin(x)**2 + cos(x)**2))
print(trigsimp(sin(x)**4 - 2*cos(x)**2*sin(x)**2 + cos(x)**4))
cos(4*x)/2 + 1/2
Differentiation and Integration in SymPy
The last main topic we will discuss in this Tutorial is how to differentiate and integrate expressions in Python SymPy.
In order to differentiate expressions using SymPy, we can use the diff() method on any expressions Depending on the type of parameters passed to diff(), it will return the differential of that
The first parameter for diff() is the expression that you want to differentiate. The second parameter is what you wish to derivative with respect to. e.g: “Derivate with respect to x”.
Let’s take a look at an example.
expr = x**2 - x - 6
print(diff(expr, x))
2*x - 1
If you wish to differentiate an expression multiple times, there are two ways of doing so. The first method is by simply including the symbol you wish to derivate with respect to, multiple times.
expr = x**4
print(diff(expr, x))
print(diff(expr, x, x))
print(diff(expr, x, x, x))
Alternatively, you can include an integer “n” as a parameter after the symbol, and it will differentiate it “n” times.
expr = x**4
print(diff(expr, x, 1))
print(diff(expr, x, 2))
print(diff(expr, x, 3))
Furthermore, you can also differentiate with respect to multiple symbols within a single diff() function.
expr = y*x**2 + x*y**2
print(diff(expr, x, y))
2*(x + y)
Integration with SymPy
It’s time for Integration with SymPy. Let’s take a look at how we can integrate various mathematical expressions and obtain their integral forms.
Similarly to how differentiation works, we have a function for integration in SymPy called integrate(). It also takes the same parameters, which is the symbol by which we wish to integrate the
expression by.
Let’s take a look at some examples.
from sympy import symbols, diff, integrate
x, y = symbols("x y")
expr = 2*x
print(integrate(expr, x))
A slightly more complex expression being integrated.
expr = x**2 - x - 6
print(integrate(expr, x))
x**3/3 - x**2/2 - 6*x
Here’s what happens when you integrate an expression with multiple symbols, with respect to just one of those symbols.
expr = x + y + 2
print(integrate(expr, x))
x**2/2 + x*(y + 2)
For more information, check out our dedicated tutorial on Differentiation and Integration with SymPy.
Python SymPy Tutorial Series
Is that all there is to SymPy though? Of course not! There are many more advanced features and functions yet to be covered.
Here is a complete breakdown of all the individual concepts that we have covered in Python SymPy on this website, along with links to their dedicated tutorials.
This marks the end of the Python SymPy Tutorial. Any suggestions or contributions for CodersLegacy are more than welcome. Questions regarding the tutorial content can be asked in the comments section
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Service Life Prediction of Solid Rocket Propellants Considering Random Thermal Enviro
Okan Yilmaz Bayindir Kuran Gokhan Ozgen
Solid propellant rocket motor is the primary propulsion technology used for tactical missiles. Its widespread usage gives rise to diversity of the environments which it is handled and stored. These
uncontrolled thermal environments induce random stresses and strains in the propellant of a rocket motor which provoke mechanical damage along with chemical degradation. In this study, a service life
prediction technique which is based on response surface method is used and explained. A time dependent random function is used for the temperature model. Solid rocket propellant is modeled using
linear viscoelastic material model. Mechanical properties of the propellant corresponding different temperatures and loading rates are found from the mechanical tests performed on test samples. A
three dimensional finite element model is used to predict stresses and strains induced on the propellant. A cumulative damage model is used since during the storage and the deployment stresses in the
propellant accumulate with time. Aging behavior of the propellant is taken into consideration as well and Layton model is used for this purpose. Response surface method is used to construct surrogate
models in terms of parameters associated with the material properties and the propellant temperature. Latin Hypercube Sampling (LHS) is used for the generation of multivariate samples. Limit state
functions are used for failure modes of the propellant. The instantaneous reliability indexes and the probability of failure for thermal loading are predicted by means of the First Order Second
Moment (FOSM) Method. The progressive reliability of the propellant is illustrated on a rocket motor.
Fulltext (PDF) | {"url":"http://aiac.ae.metu.edu.tr/paper.php?No=AIAC-2011-030","timestamp":"2024-11-12T02:37:19Z","content_type":"text/html","content_length":"2855","record_id":"<urn:uuid:b0352c4a-ea8b-40d7-a4a4-4502b796821b>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00437.warc.gz"} |
MURAL - Maynooth University Research Archive Library
Number of items: 6.
Kilbane, D. and Cummings, A. and Heffernan, Daniel and O'Sullivan, G. (2006) Characterization of the structure and eigenvalue spectra of the compound states of Sm IX. Physica Scripta, 73 (2). pp.
Kilbane, D. and Cummings, A. and O'Sullivan, G. and Heffernan, Daniel (2006) Quantum statistics of a kicked particle in an infinite potential well. Chaos, Solitons and Fractals, 30 (2). pp. 412-423.
Kilbane, D. and Cummings, A. and O'Sullivan, G. and Heffernan, Daniel (2006) The classical-quantum correspondence of a kicked particle in an infinite potential well. Chaos, Solitons and Fractals, 30
(2). pp. 424-440.
Cummings, A. and O'Sullivan, G. and Hanan, W.G. and Heffernan, Daniel (2001) Multifractal analysis of selected rare-earth elements. Journal of Physics B: Atomic, Molecular and Optical Physics, 34
(13). pp. 2547-2573.
Cummings, A. and O'Sullivan, G. and Heffernan, Daniel (2001) Signatures of quantum chaos in rare-earth elements: I. Characterization of the Hamiltonian matrices and coupling matrices of Ce I and Pr I
using the statistical predictions of Random Matrix Theory. Journal of Physics B: Atomic, Molecular and Optical Physics, 34 (17). pp. 3407-3446.
Cummings, A. and O'Sullivan, G. and Heffernan, Daniel (2001) Signatures of quantum chaos in rare-earth elements: II. Characterization of the energy eigenvalues and dipole moments of Ce I and Pr I.
Journal of Physics B: Atomic, Molecular and Optical Physics, 34 (17). pp. 3447-3477. | {"url":"https://mural.maynoothuniversity.ie/view/creators/Cummings=3AA=2E=3A=3A.html","timestamp":"2024-11-11T04:33:40Z","content_type":"application/xhtml+xml","content_length":"15446","record_id":"<urn:uuid:63122976-4509-42e0-93f0-87403a49454c>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00149.warc.gz"} |
Python Calculator Code
Oops, something went wrong. Please try again in a few moments.
class Calculator:
Class to handle basic calculator operations.
- num1, num2: float
The two numbers on which the calculator operations will be performed.
def __init__(self, num1: float, num2: float):
Constructor to instantiate the Calculator class.
- num1: float
The first number for the calculator operations.
- num2: float
The second number for the calculator operations.
self.num1 = num1
self.num2 = num2
def add(self) -> float:
Adds the two numbers.
- float:
The sum of the two numbers.
return self.num1 + self.num2
def subtract(self) -> float:
Subtracts the second number from the first number.
- float:
The difference between the two numbers.
return self.num1 - self.num2
def multiply(self) -> float:
Multiplies the two numbers.
- float:
The product of the two numbers.
return self.num1 * self.num2
def divide(self) -> float:
Divides the first number by the second number.
- float:
The quotient of the division.
if self.num2 == 0:
raise ValueError("Cannot divide by zero.")
return self.num1 / self.num2
# Example usage of the Calculator class:
# Creating an instance of the Calculator class
calculator = Calculator(10, 5)
# Performing addition
result_add = calculator.add()
print(f"The sum of {calculator.num1} and {calculator.num2} is {result_add}.")
# Performing subtraction
result_subtract = calculator.subtract()
print(f"The difference between {calculator.num1} and {calculator.num2} is {result_subtract}.")
# Performing multiplication
result_multiply = calculator.multiply()
print(f"The product of {calculator.num1} and {calculator.num2} is {result_multiply}.")
# Performing division
result_divide = calculator.divide()
print(f"The division of {calculator.num1} by {calculator.num2} is {result_divide}.")
except ValueError as e:
print(f"Error while performing division: {e}") | {"url":"https://codepal.ai/code-generator/query/BQ0fR9Tg/python-calculator-code","timestamp":"2024-11-07T13:54:26Z","content_type":"text/html","content_length":"114406","record_id":"<urn:uuid:7597d3fa-b035-426b-b12b-4c9e67f43652>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00323.warc.gz"} |
Convert digital filter state-space parameters to second-order sections form
[sos,g] = ss2sos(A,B,C,D) returns second-order section form sos with gain g that is equivalent to the state-space system represented by input arguments A, B, C, and D. The input state-space system
must be single-output and real.
[sos,g] = ss2sos(A,B,C,D,iu) specifies index iu that indicates which input of the state-space system A, B, C, D the function uses in the conversion.
[sos,g] = ss2sos(A,B,C,D,order) specifies the order of the rows in sos with order.
[sos,g] = ss2sos(A,B,C,D,iu,order) specifies both the index ui and the order of the rows order.
[sos,g] = ss2sos(A,B,C,D,iu,order,scale) specifies the desired scaling of the gain and the numerator coefficients of all second-order sections.
sos = ss2sos(___) embeds the overall system gain g in the first section. You can specify an input combination from any of the previous syntaxes.
Second-Order Section Form of Filter
Design a fifth-order Butterworth lowpass filter, specifying a cutoff frequency of $0.2\pi$ rad/sample and expressing the output in state-space form. Convert the state-space result to second-order
sections. Visualize the frequency response of the filter.
[A,B,C,D] = butter(5,0.2);
sos = ss2sos(A,B,C,D)
sos = 3×6
0.0013 0.0013 0 1.0000 -0.5095 0
1.0000 1.9996 0.9996 1.0000 -1.0966 0.3554
1.0000 2.0000 1.0000 1.0000 -1.3693 0.6926
Mass-Spring System
A one-dimensional discrete-time oscillating system consists of a unit mass, $m$, attached to a wall by a spring of unit elastic constant. A sensor measures the acceleration, $a$, of the mass.
The system is sampled at ${F}_{s}=5$ Hz. Generate 50 time samples. Define the sampling interval $\Delta t=1/{F}_{s}$.
Fs = 5;
dt = 1/Fs;
N = 50;
t = dt*(0:N-1);
The oscillator can be described by the state-space equations
$\begin{array}{c}x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right),\\ y\left(k\right)=Cx\left(k\right)+Du\left(k\right),\end{array}$
where $x={\left(\begin{array}{cc}r& v\end{array}\right)}^{T}$ is the state vector, $r$ and $v$ are respectively the position and velocity of the mass, and the matrices
$A=\left(\begin{array}{cc}\mathrm{cos}\Delta t& \mathrm{sin}\Delta t\\ -\mathrm{sin}\Delta t& \mathrm{cos}\Delta t\end{array}\right),\phantom{\rule{1em}{0ex}}B=\left(\begin{array}{c}1-\mathrm{cos}\
Delta t\\ \mathrm{sin}\Delta t\end{array}\right),\phantom{\rule{1em}{0ex}}C=\left(\begin{array}{cc}-1& 0\end{array}\right),\phantom{\rule{1em}{0ex}}D=\left(\begin{array}{c}1\end{array}\right).$
A = [cos(dt) sin(dt);-sin(dt) cos(dt)];
B = [1-cos(dt);sin(dt)];
C = [-1 0];
D = 1;
The system is excited with a unit impulse in the positive direction. Use the state-space model to compute the time evolution of the system starting from an all-zero initial state.
u = [1 zeros(1,N-1)];
x = [0;0];
for k = 1:N
y(k) = C*x + D*u(k);
x = A*x + B*u(k);
Plot the acceleration of the mass as a function of time.
Compute the time-dependent acceleration using the transfer function to filter the input. Express the transfer function as second-order sections. Plot the result. The result is the same in both cases.
sos = ss2sos(A,B,C,D);
yt = sosfilt(sos,u);
Input Arguments
A — State matrix
State matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then A is of size n-by-n.
B — Input-to-state matrix
Input-to-state matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then B is of size n-by-p.
C — Output-to-state matrix
Output-to-state matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then C is of size q-by-n.
D — Feedthrough matrix
Feedthrough matrix, specified as a matrix. If the system has p inputs and q outputs and is described by n state variables, then D is of size q-by-p.
iu — Index
1 (default) | integer
Index, specified as an integer.
order — Row order
'up' (default) | 'down'
Row order in sos, specified as one of these values:
• 'down' — Order the sections so that the first row of sos contains the poles that are closest to the unit circle.
• 'up' — Order the sections so that the first row of sos contains the poles that are farthest from the unit circle.
The zeros are paired with the poles that are closest to them.
scale — Scaling of gain and numerator coefficients
'none' (default) | 'inf'
Scaling of the gain and numerator coefficients, specified as one of these values:
• 'none' — Apply no scaling.
• 'inf' — Apply infinity-norm scaling.
• 'two' — Apply 2-norm scaling.
Using infinity-norm scaling in conjunction with up-ordering minimizes the probability of overflow in the realization. Using 2-norm scaling in conjunction with down-ordering minimizes the peak
round-off noise.
Infinity-norm and 2-norm scaling are appropriate for only direct-form II implementations.
Output Arguments
sos — Second-order section representation
Second-order section representation, returned as a matrix. sos is an L-by-6 matrix of the form
$\text{sos}=\left[\begin{array}{cccccc}{b}_{01}& {b}_{11}& {b}_{21}& 1& {a}_{11}& {a}_{21}\\ {b}_{02}& {b}_{12}& {b}_{22}& 1& {a}_{12}& {a}_{22}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ {b}_{0L}& {b}_{1L}& {b}_{2L}& 1&
{a}_{1L}& {a}_{2L}\end{array}\right]$
whose rows contain the numerator and denominator coefficients b[ik] and a[ik] of the second-order sections of H(z), which is given by
$H\left(z\right)=g\prod _{k=1}^{L}{H}_{k}\left(z\right)=g\prod _{k=1}^{L}\frac{{b}_{0k}+{b}_{1k}{z}^{-1}+{b}_{2k}{z}^{-2}}{1+{a}_{1k}{z}^{-1}+{a}_{2k}{z}^{-2}}$
g — Overall system gain
real-valued scalar
Overall system gain, returned as a real-valued scalar.
If you call the function with one output argument, the function embeds the gain in the first section, H[1](z), so that
$H\left(z\right)=\prod _{l=1}^{L}{H}_{l}\left(z\right)={H}_{1}\left(z\right)×{H}_{2}\left(z\right)×\cdots ×{H}_{L}\left(z\right).$
Embedding the gain in the first section when scaling a direct-form II structure is not recommended and can result in erratic scaling. To avoid embedding the gain, use the function with two outputs:
sos and g.
The ss2sos function uses this four-step algorithm to determine the second-order section representation for an input state-space system.
1. Find the poles and zeros of the system given by A, B, C, and D.
2. Use the function zp2sos, which first groups the zeros and poles into complex conjugate pairs using the cplxpair function. zp2sos then forms the second-order sections by matching the pole and zero
pairs according to these rules:
1. Match the poles that are closest to the unit circle with the zeros that are closest to those poles.
2. Match the poles that are next closest to the unit circle with the zeros that are closest to those poles.
3. Continue this process until all of the poles and zeros are matched.
The ss2sos function groups real poles into sections with the real poles that are closest to them in absolute value. The same rule holds for real zeros.
3. Order the sections according to the proximity of the pole pairs to the unit circle. The ss2sos function normally orders the sections with poles that are closest to the unit circle last in the
cascade. You can specify for ss2sos to order the sections in the reverse order by setting the order input to 'down'.
4. Scale the sections by the norm specified by the scale input. For arbitrary H(ω), the scaling is defined by
${‖H‖}_{p}={\left[\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{|H\left(\omega \right)|}^{p}d\omega \right]}^{1/p}$
where p can be either ∞ or 2. For details, see the references. This scaling is an attempt to minimize overflow or peak round-off noise in fixed-point filter implementations.
[1] Jackson, Leland B. Digital Filters and Signal Processing. Boston: Kluwer Academic Publishers, 1996.
[2] Mitra, Sanjit Kumar. Digital Signal Processing: A Computer-Based Approach. New York: McGraw-Hill, 1998.
[3] Vaidyanathan, P. P. “Robust Digital Filter Structures.” Handbook for Digital Signal Processing (S. K. Mitra and J. F. Kaiser, eds.). New York: John Wiley & Sons, 1993.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• Any character or string input must be a constant at compile time.
Version History
Introduced before R2006a | {"url":"https://it.mathworks.com/help/signal/ref/ss2sos.html","timestamp":"2024-11-04T17:58:15Z","content_type":"text/html","content_length":"117307","record_id":"<urn:uuid:616be2e8-9adf-485d-b9ed-cdd5c52f7675>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00222.warc.gz"} |
How to Calculate PPM for Fertilizer | ehow.com
Things You'll Need
• Calculator
• Water-soluble fertilizer
• Balance or scale capable of measuring to 0.1 oz. or 0.01 g
Many gardening guides and plant-care instructions specify fertilizer concentrations in parts per million (ppm). The authors of such guides do this to avoid ambiguity. Instructions such as "fertilize
with 200 ppm nitrogen solution" are much more precise than "fertilize with house-plant fertilizer at one-tenth strength." Two situations arise in which ppm calculations become necessary. The first
involves preparing a fertilizer solution with a specified ppm concentration of a nutrient (nitrogen, phosphorous or potassium). In this case, the amount of fertilizer that needs to be added to a
given volume of water must be calculated. The second situation involves calculating the ppm of a specific nutrient in an already-prepared fertilizer solution.
Calculating the Amount of Fertilizer to Achieve a Given PPM
Step 1
Obtain the amount of nitrogen (N), phosphorous (P) and potassium (K) in the fertilizer from the fertilizer's label. These are the so-called N-P-K values, such as 10-20-10. The numbers refer to
percentages. A 10-20-10 fertilizer contains 10 percent nitrogen, 20 percent phosphorous and 10 percent potassium by weight.
Step 2
Determine the mass (in grams) of fertilizer to dissolve per liter of water to achieve the desired ppm value by dividing by the percent (in decimal form) of the desired nutrient, then divide by 1,000.
If a fertilizer of 200 ppm nitrogen is desired, then:
200 ppm / 0.10 / 1,000 = 2 g of fertilizer per liter (L)
Step 3
Determine the number of liters of fertilizer to be prepared. For sake of convenience, 1 gallon = 3.8 liters. Thus, if the quantity is known in gallons, multiply by this quantity by 3.8 to convert to
Step 4
Multiply the grams of fertilizer from Step 2 by the desired number of liters from Step 3 to determine the quantity of fertilizer to use. For example, to prepare 1 gallon:
(2 g) x (3.8 L) = 7.6 g fertilizer
Step 5
Weigh the quantity of fertilizer on a scale or balance. If necessary, this quantity can be converted to ounces by multiplying by 0.0353:
(7.6 g) x (0.0353 oz./g) = 0.268 oz. fertilizer
Step 6
Combine the weighed fertilizer and the measured quantity of water and mix well.
Calculating the PPM of an Already-Prepared Fertilizer Solution
Step 1
Convert the quantity of fertilizer added to a solution to grams if the gram value is not already known by dividing by 0.0353 g/oz.; thus, if 0.25 oz. was added to 2 gallons of water, then:
(0.25 oz.) / (0.0353 g/oz.) = 7.1 g fertilizer
Step 2
Determine grams of the individual nutrient by multiplying the result from Step 1 by the percent of the nutrient (marked on the fertilizer's container). Thus, for a 10-0-0 fertilizer, the nitrogen
content would be
(7.1 g fertilizer) x (0.1 nitrogen) = 0.71 g nitrogen
Step 3
Determine the volume of the solution in liters by multiplying gallons by 3.8 L/gallon. Thus, for the example in Step 1,
(2 gallons) x (3.8 L/gallon) = 7.6 L
Step 4
Determine ppm by dividing grams of nutrient from Step 2 by liters of solution from Step 3 and multiplying by 1,000:
(0.71 g) / (7.6 L) x 1,000 = 92 ppm nitrogen | {"url":"https://www.ehow.com/how_6390455_calculate-ppm-fertilizer.html","timestamp":"2024-11-03T22:28:49Z","content_type":"text/html","content_length":"351552","record_id":"<urn:uuid:7499b1c7-1db5-4b0e-9186-cbb1ea7c6052>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00708.warc.gz"} |
Generically finite morphisms
Lemma 29.51.1. Let $X$, $Y$ be schemes. Let $f : X \to Y$ be locally of finite type. Let $\eta \in Y$ be a generic point of an irreducible component of $Y$. The following are equivalent:
1. the set $f^{-1}(\{ \eta \} )$ is finite,
2. there exist affine opens $U_ i \subset X$, $i = 1, \ldots , n$ and $V \subset Y$ with $f(U_ i) \subset V$, $\eta \in V$ and $f^{-1}(\{ \eta \} ) \subset \bigcup U_ i$ such that each $f|_{U_ i} :
U_ i \to V$ is finite.
Comments (2)
Comment #5084 by GGTTTTLL on
In the proof of lemma 02NW, third paragraph, it says "Since each $\xi_i$ maps to a generic point of an irreducible component of $Y$, we see that each $\xi_i$ is a generic point of an irreducible
component of $X$. " does not seem correct in general, take any morphism to $Spec k$ for example. Maybe say: Since by (2), $\xi_i\in U_i$ maps to $\eta$ by finite morphisms, $\xi_i$ is a generic
point of $X$.
Comment #5295 by Johan on
What happened here probably happens all over the place: I used that if you have a scheme $S$ (locally) of finite type over a field with finitely many points, then those points are all generic
points. Moreover, if this happens for a generic fibre $S = X_\eta$, then those points are all generic points of irreducible components of $X$. This is just part of my arsenal of things that are
true and it is almost impossible to see (for me) that I am using something like this. So thanks to GGTTTTLL for catching this. I fixed it in a slightly different way, see this commit.
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Properties of number 746496
746496 has 77 divisors (see below), whose sum is σ = 2237371. Its totient is φ = 248832.
The previous prime is 746483. The next prime is 746497. The reversal of 746496 is 694647.
The square root of 746496 is 864.
It is a perfect power (a square), and thus also a powerful number.
It is a Jordan-Polya number, since it can be written as (4!)^2 ⋅ (3!)^4.
746496 is a `hidden beast` number, since 7 + 4 + 649 + 6 = 666.
It is a Harshad number since it is a multiple of its sum of digits (36).
It is a Duffinian number.
It is a nialpdrome in base 12 and base 16.
It is a self number, because there is not a number n which added to its sum of digits gives 746496.
It is not an unprimeable number, because it can be changed into a prime (746497) by changing a digit.
746496 is an untouchable number, because it is not equal to the sum of proper divisors of any number.
It is a polite number, since it can be written in 6 ways as a sum of consecutive naturals, for example, 248831 + 248832 + 248833.
746496 is a Friedman number, since it can be written as (9-7)*6^6*(4+4), using all its digits and the basic arithmetic operations.
2^746496 is an apocalyptic number.
746496 is the 864-th square number.
It is an amenable number.
It is a practical number, because each smaller number is the sum of distinct divisors of 746496
746496 is an abundant number, since it is smaller than the sum of its proper divisors (1490875).
It is a pseudoperfect number, because it is the sum of a subset of its proper divisors.
746496 is an frugal number, since it uses more digits than its factorization.
746496 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 38 (or 5 counting only the distinct ones).
The product of its digits is 36288, while the sum is 36.
The cubic root of 746496 is about 90.7143155924.
Multiplying 746496 by its sum of digits (36), we get a 4-th power (26873856 = 72^4).
746496 divided by its sum of digits (36) gives a 4-th power (20736 = 12^4).
The spelling of 746496 in words is "seven hundred forty-six thousand, four hundred ninety-six". | {"url":"https://www.numbersaplenty.com/746496","timestamp":"2024-11-14T14:54:09Z","content_type":"text/html","content_length":"11122","record_id":"<urn:uuid:77f3b997-250c-42a0-b77e-67e49a296165>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00052.warc.gz"} |
How do you write an equation in standard form given (-2, 7) m=-4? | HIX Tutor
How do you write an equation in standard form given (-2, 7) m=-4?
Answer 1
The Standard form of the equation is
#y=mx+c# In the problem #x=-2; y=7 and m = -4#
We shall find #c#
#mx+c=y# #-4(-2)+c=7# #8+c=7# #c=7-8=-1#
The equation is - [Substitute #m=-4 and c=-1# in #y=mx+c#]
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Answer 2
To write an equation in standard form given the point (-2, 7) and slope m = -4, you can use the point-slope form of a linear equation, which is y - y1 = m(x - x1).
Substitute the given point (-2, 7) and slope m = -4 into the formula:
y - 7 = -4(x - (-2))
y - 7 = -4(x + 2)
Expand and rewrite in standard form (Ax + By = C):
y - 7 = -4x - 8
y = -4x - 8 + 7
y = -4x - 1
Rearrange to standard form:
4x + y = -1
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-write-an-equation-in-standard-form-given-2-7-m-4-8f9af92675","timestamp":"2024-11-14T11:12:37Z","content_type":"text/html","content_length":"578253","record_id":"<urn:uuid:e0ba48e7-1500-4765-97bb-2532b4aed99d>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00117.warc.gz"} |
Improving Lattice based cryptosystems using the Hermite Normal
Improving Lattice based cryptosystems using the Hermite Normal Form
Authors: Daniele Micciancio
Cryptography and Lattices Conference - CaLC 2001. March 29-30, 2001, Providence, Rhode Island. Lecture Notes in Computer Science 2146. Springer-Verlag, pp. 126-145
[BibTeX] [PostScript] [PDF]
Abstract: We describe a simple technique that can be used to substantially reduce the key and ciphertext size of various lattice based cryptosystems and trapdoor functions of the kind proposed by
Goldreich, Goldwasser and Halevi (GGH). The improvement is significant both from the theoretical and practical point of view, reducing the size of both key and ciphertext by a factor $n$ equal to the
dimension of the lattice (i.e., several hundreds for typical values of the security parameter.) The efficiency improvement is obtained without decreasing the security of the functions: we formally
prove that the new functions are at least as secure as the original ones, and possibly even better as the adversary gets less information in a strong information theoretical sense. The increased
efficiency of the new cryptosystems allows the use of bigger values for the security parameter, making the functions secure against the best cryptanalytic attacks, while keeping the size of the key
even below the smallest key size for which lattice cryptosystems were ever conjectured to be hard to break. | {"url":"https://cseweb.ucsd.edu/~daniele/papers/HNFcrypt.html","timestamp":"2024-11-07T16:58:11Z","content_type":"application/xhtml+xml","content_length":"2257","record_id":"<urn:uuid:1ffebca1-7f5c-4452-9d0d-52325c51df6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00421.warc.gz"} |
SU-CS224N APR042024
See stochastic gradient descent
see word2vec
Or, we can even use a simpler approach, window-based co-occurance
• goal: we want to capture linear meaning components in a word vector space correct
• insight: the ratio of co-occurrence probabilities are linear meaning components
Therefore, GloVe vectors comes from a log-bilinear:
$$w_{i} \cdot w_{j} = \log P(i|j)$$
such that:
$$w_{x} \cdot (w_{a} - w_{b}) = \log \frac{P(x|a)}{P(x|b)}$$
Evaluating a NLP System
• evaluate on the specific target task the system is trained on
• evaluate speed
• evaluate understandability
• real task + attempt to replace older system with new system
• maybe expensive to compute
Word Sense Ambiguity
Each word may have multiple different meanings; each of those separate word sense should live in a different place. However, words with polysemy have related senses, so we usually average:
$$v = a_1 v_1 + a_2v_2 + a_3v_3$$
where \(a_{j}\) is the frequency for the \(j\) th sense of the word, and \(v_1 … v_{3}\) are separate word senses.
sparse coding
if each sense is relatively common, at high enough dimensions, sparse coding allows you to recover the component sense vectors from the average word vectors because of the general sparsity of the
vector space.
Word-Vector NER
Create a window +- n words next to each word to classify; feed the entire sequence’s embeddings concatenated into a neural classifier, and use a target to say whether the center word is a entity/
person/no label, etc.
These simplest classifiers usually use softmax, which, without other activations, gives a linear decision boundary.
With a neural classifier, we can add enough nonlinearity in the middle to make the entire representation, but the final output layer will still contain a linear classifier. | {"url":"https://www.jemoka.com/posts/kbhsu_cs224n_apr042024/","timestamp":"2024-11-03T13:54:00Z","content_type":"text/html","content_length":"8105","record_id":"<urn:uuid:4b3e36e1-3dfb-4269-afcd-7d66ee77976b>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00495.warc.gz"} |
HELP!!!!!!! Functions F(x) And G(x) Are Shown: F(x) = X2 G(x) = X2 + 8x + 16 In Which Direction And By
A.) Left By 4 Units
Step-by-step explanation:
Functions f(x) and g(x) are shown below: f(x) = x2 g(x) = x2 + 8x + 16
The direction in which and by how many units the f(x) should be shifted to obtain the g(x) is given by as follows.
f(x) = x²
g(x) = x² + 8x + 16
we have to consider, the dependent term while selecting the direction.
Therefore, except x², only x term is dependent term.
In g(x), the x term is given by 8.
Therefore, we need to shift the f(x) left by 4 units.
Therefore, option A is correct.
Option A, Left by 4 units
Step-by-step explanation:
Step 1: Convert g(x) to a function square
We currently have g(x) in this order: [tex]ax^2 + bx + c[/tex]
However, we want g(x) to be in this order: [tex](ax + c)^{2}[/tex]
The first thing we have to do is to factor it out:
[tex]g(x) = (x + 4)(x + 4)[/tex]
[tex]g(x) = (x+4)^{2}[/tex]
Step 2: Now we can see which way we need to move it
The original form is: [tex]f(x) = (ax - b)^{2}[/tex]
Since the - has changed to a +, that means that we moved -4 spaces down the x-axis. This means that we move left by 4 units.
Answer: Option A, Left by 4 units
Look at the graphs below to make sure:
[tex]P(32) = \frac{1}{90}[/tex]
[tex]P(Odd) = \frac{1}{2}[/tex]
[tex]P(Multiples\ 5) = \frac{1}{5}[/tex]
Step-by-step explanation:
Sample Space = 10 to 99
First, we calculate the sample size (n)
[tex]n = 99 - 10 + 1[/tex]
[tex]n = 90[/tex]
Solving (a): P(32)
In 10 to 99, there is only 1 32.
[tex]P(32) = \frac{n(32)}{n}[/tex]
[tex]P(32) = \frac{1}{90}[/tex]
Solving (b): P(Odd)
There are 45 odd numbers between 10 and 99
[tex]n(Odd) = 45[/tex]
[tex]P(Odd) = \frac{n(Odd)}{n}[/tex]
[tex]P(Odd) = \frac{45}{90}[/tex]
[tex]P(Odd) = \frac{1}{2}[/tex]
Solving (c): P(Multiple of 5)
There are 18 multiples of 5 between 10 and 99
[tex]P(Multiples\ 5) = \frac{18}{90}[/tex]
[tex]P(Multiples\ 5) = \frac{1}{5}[/tex] | {"url":"https://diemso.unix.edu.vn/question/help-br-functions-fx-and-gx-are-shown-br-br-fx-x2-gx-x2-8x-1-aky5","timestamp":"2024-11-15T01:24:05Z","content_type":"text/html","content_length":"71072","record_id":"<urn:uuid:55ee370b-3dd6-4407-8035-82d616ee4bde>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00194.warc.gz"} |
Introduction to Isotropic Invariant
What is/are Isotropic Invariant?
Isotropic Invariant
An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in
terms of derivatives of position. ^[1] The turbulent states for each B R has been compared using an anisotropic invariant map in the horseshoe vortex regime, top surface regime and in the wake
regime. ^[2] We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy–Green strain tensor. ^[3] The anisotropic
invariant maps show the near bed anisotropy inclining to be a two-component isotropy subjected to no seepage and seepage flow. ^[4] The approach helps us successfully in determining the fiber
strains, for a family of symmetrically and asymmetrically oriented fibers, with the aid of a single anisotropic invariant. ^[5] Anisotropic invariant map (AIM) has been plotted; nature of turbulence
is found to be non-homogeneous and anisotropic even at low Re (= 3200) and non-homogeneity increases as Re increases. ^[6] The ground substance is the common so-called compressible neo-Hooke model
and the standard reinforcement part is augmented by second order anisotropic invariants. ^[7] The turbulent states for each B R has been compared using an anisotropic invariant map in the horseshoe
vortex regime, top surface regime and in the wake regime. ^[1] The anisotropic invariant maps show the near bed anisotropy inclining to be a two-component isotropy subjected to no seepage and seepage
flow. ^[2] Anisotropic invariant map (AIM) has been plotted; nature of turbulence is found to be non-homogeneous and anisotropic even at low Re (= 3200) and non-homogeneity increases as Re increases. | {"url":"https://academic-accelerator.com/Manuscript-Generator/Isotropic-Invariant","timestamp":"2024-11-07T02:53:44Z","content_type":"text/html","content_length":"248243","record_id":"<urn:uuid:e8dca28d-4d0a-42c9-b93f-1cf96cbf6036>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00581.warc.gz"} |
Measuring Acceleration due to Gravity using a simple Pendulum. - GCSE Science - Marked by Teachers.com
Measuring Acceleration due to Gravity using a simple Pendulum.
Measuring Acceleration due to Gravity using a simple Pendulum
Simple Harmonic Motion is very similar to the motion of a bob of a simple pendulum. A simple pendulum is made of a metallic bob suspended by fine string. Using these apparatus the acceleration due to
gravity had to be measured. The acceleration due to gravity on earth is 9.81ms-2, therefore every second the particle will increase it’s speed by 9.81ms-1. Although, this may not be the case, due to
air resistance.
Acceleration is directly proportional to force and acts towards a fixed position.
To derive a value for g, the S.H.M. equation was used:
For calculating ‘g’ the formula had to be rearranged:
T2 = 4π2 l
g = 4π2 l
Where: g is the acceleration to gravity; T is the period & l is the length.
Out of the above, the length of the pendulum had to be measured and then the period with that information. Instead of using the formula, a graph could be drawn of length against T2, calculate the
gradient, invert it then multiply by 4π…
g = 4π2 l
“”””“””””.. all the weight is at the centre of the bob… - measuring to middle of bob. Keep bob small and dense is important….”””””
5b in analysis.. line of best fit – reason for lack of accuracy – strings got a weight therefore centre of gravity of whole pendulum will not be centre of bob.. Length of pendulum will be shorter
than what measuring…”””””””””””
This is a preview of the whole essay | {"url":"https://www.markedbyteachers.com/gcse/science/measuring-acceleration-due-to-gravity-using-a-simple-pendulum.html","timestamp":"2024-11-02T05:17:43Z","content_type":"text/html","content_length":"68088","record_id":"<urn:uuid:efcfe2fd-2fae-4f86-9fa4-26544b9debf5>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00609.warc.gz"} |
Limits by Direct Substitution
Lesson Video: Limits by Direct Substitution Mathematics • Second Year of Secondary School
In this video, we will learn how to use the direct substitution method to evaluate limits.
Video Transcript
In this video, we will learn how to use the direct substitution method to evaluate limits. There are certain conditions which need to be met in order to use direct substitution. Weβ ll be covering
these conditions and looking at some examples.
Letβ s start by recalling the definition of a limit. We say that, for some function π of π ₯, which is defined near π , the limit of π of π ₯ as π ₯ approaches π is π Ώ. What this
means is that the closer π ₯ gets to π , the closer the value of π of π ₯ gets to π Ώ. Formally, we write this like this. The limit as π ₯ goes to π of π of π ₯ is equal to π Ώ.
In order to find a limit using direct substitution, we simply substitute π ₯ equals π into π of π ₯. And so the result we get for direct substitution is that the limit as π ₯ goes to π of
π of π ₯ is equal to π of π . This is what weβ ll be using in order to find limits using direct substitution.
However, there are only certain cases when weβ re allowed to use direct substitution in order to find a limit. The functions which weβ re taking the limit of β so thatβ s π of π ₯ β must
satisfy at least one of the following conditions.
The first condition which will allow us to use direct substitution is that if π of π ₯ is a polynomial, that is, if π of π ₯ is of the form given here, where π naught to π π are all
constants. Now remember that this also includes constant functions.
The second condition which could enable us to use direct substitution is if π of π ₯ is a rational function. This means that π of π ₯ is equal to some π of π ₯ over π of π ₯, where π
of π ₯ and π of π ₯ are both polynomials. And we also require that π of π is nonzero, since if π of π is equal to zero, then the denominator of π of π will be equal to zero.
And so π of π will be undefined.
The third condition is that π of π ₯ is either a trigonometric, exponential, or logarithmic function. For the fourth condition, we have that π of π ₯ is a power function. So this means that π
of π ₯ is equal to π ₯ to the power of π , where π is a real number. Letβ s note that this also includes negative and fractional powers, such as π ₯ to the power of negative one-half,
which is also equal to one over the square root of π ₯.
For the fifth condition, we have that if π of π ₯ is a sum, difference, product, or quotient of functions for which direct substitution works. So that could be a combination of any of the types
of functions weβ ve already covered in these conditions.
The final condition is if π of π ₯ is a composition of functions for which direct substitution works. What this means is that if π of π ₯ is equal to π of β of π ₯, where β of π ₯
allows substitution at π and π of π ₯ allows substitution at β of π . Now we have covered all the conditions under which we can use direct substitution in order to solve a limit. Weβ re
now ready to look at an example.
Determine the limit as π ₯ goes to negative five of negative nine π ₯ squared minus six π ₯ minus nine.
Here weβ re asked to find the limit of the function negative nine π ₯ squared minus six π ₯ minus nine. And so we can write this down as π of π ₯. We can see that π of π ₯ is simply a
polynomial. Therefore, weβ re able to use direct substitution in order to find this limit.
Direct substitution tells us that the limit of π of π ₯ as π ₯ goes to π is equal to π of π . Applying this to our question, we can say that the limit of π of π ₯ as π ₯ goes to
negative five is equal to π of negative five. In order to reach our solution, we simply substitute negative five into π of π ₯. This gives us negative nine multiplied by negative five squared
minus six multiplied by negative five minus nine. You may find it helpful to put brackets around the negative numbers so that when we multiply the negative numbers, we do not forget about the
negative sign.
Firstly, letβ s expand the negative five squared. If we remembered that a negative number timesed by a negative number gives a positive number, then we will get that negative five squared is equal
to 25. So we can write negative nine multiplied by 25. Next, we can multiply the negative six by the negative five, to get positive 30. And then we simply have to subtract the nine on the end.
Multiplying the negative nine by the 25 gives us negative 225. And we subtract the nine from the 30 to give us plus 21. Adding 21 to negative 225 gives us a solution of negative 204.
In this example, we saw how to apply direct substitution when finding the limit of a polynomial function. Next, weβ ll be considering some functions and trying to work out whether or not they
satisfy the conditions to use direct substitution.
Which of the following functions satisfies the conditions for direct substitution of the limit, the limit of π of π ₯ as π ₯ goes to zero? A) π of π ₯ is equal to π ₯ squared plus five π ₯
over π ₯ squared minus two π ₯. B) π of π ₯ is equal to π ₯ squared minus five π ₯ plus six over π ₯ minus two sin π ₯. C) π of π ₯ is equal to π ₯ if π ₯ is greater than three and π
₯ minus three if π ₯ is less than or equal to three. D) π of π ₯ is equal to π ₯ plus one over π ₯. And E) π of π ₯ is equal to two π ₯ if π ₯ is greater than zero and two π ₯ minus
one if π ₯ is less than or equal to zero.
In order to find which of these functions we can use direct substitution in order to find the limit, we must test the conditions for which direct substitution works on each of the functions. For
function A, we have π of π ₯ is equal to π ₯ squared plus five π ₯ over π ₯ squared minus two π ₯. This is a rational function. And if we write π of π ₯ as π of π ₯ over π of π ₯,
then we obtain that π of π ₯ is equal to π ₯ squared plus five π ₯ and π of π ₯ is equal to π ₯ squared minus two π ₯.
Since both π of π ₯ and π of π ₯ are polynomials, the only condition which we need to check is that the denominator of the fraction is nonzero at π ₯ equals zero. The denominator of our
function is π of π ₯. So letβ s work out what π of zero is. We simply substitute zero into π ₯ squared minus two π ₯, which gives us zero squared minus two times zero. And zero squared and
two times zero are both zero. So therefore, π of zero is equal to zero. This means that the denominator of our function is equal to zero at π ₯ equals zero. And so weβ re not able to use direct
substitution in order to find this limit.
Moving on to function B, we have π of π ₯ is equal to π ₯ squared minus five π ₯ plus six over π ₯ minus two sin π ₯. And this here is a quotient of two functions. So we can write π of π
₯ as π of π ₯ over π of π ₯, where π of π ₯ is equal to π ₯ squared minus five π ₯ plus six and π of π ₯ is equal to π ₯ minus two sin π ₯.
In order to use direct substitution to find the limit of this function, we again need to check what the denominator equals at π ₯ equals zero. We find π of zero is equal to zero minus two
multiplied by sin of zero. And this is also equal to negative two sin of zero. However, sin of zero is equal to zero. And this gives us that π of zero must also be equal to zero.
So similarly to function A, we have found that the denominator of function B is also zero at π ₯ equals zero. And therefore, again, we cannot use direct substitution in order to find the limit of
this function.
Moving on to function C, we have that π of π ₯ is equal to π ₯ if π ₯ is greater than three and π ₯ minus three if π ₯ is less than or equal to three. In order to find whether we can use
direct substitution to find the limit of this function, we first need to consider whether the limit actually exists for this function. In order for the limit to exist, we require that the one-sided
limits on either side of zero are equal.
So this means that the limit as π ₯ approaches zero from above of π of π ₯ must be equal to the limit as π ₯ approaches zero from below of π of π ₯. As π ₯ approaches zero from above, we
have that π of π ₯ will be equal to π ₯ minus three. And this is because when π ₯ is just larger than zero, π ₯ will still be less than or equal to three. And so, therefore, our function π
of π ₯ will be equal to π ₯ minus three. And this is just a polynomial function.
And so we can use direct substitution in order to find the limit from above. And so we find the limit as π ₯ approaches zero from above of π of π ₯ is equal to π of zero, which is also equal
to zero minus three, or just negative three.
Next, if we consider the limit as π ₯ approaches zero from below, we know that π of π ₯ is again equal to π ₯ minus three, since zero is less than or equal to three, meaning that our π ₯-value
is less than or equal to three. So therefore, π of π ₯ must be equal to π ₯ minus three. Again, this is a polynomial. So we will use direct substitution in order to find the limit as π ₯
approaches zero from below, giving us that the limit is equal to π of zero, which is again equal to zero minus three, or negative three.
And so we have found that the limit as π ₯ approaches zero from above and the limit as π ₯ approaches zero from below are equal. And therefore, it satisfies our condition for the limit to exist. We
can add on to the end of our condition that these two one-sided limits will also be equal to the limit as π ₯ approaches zero of π of π ₯.
Now we know that the limit exists, we just need to check that we can use direct substitution to solve it. At π ₯ equals zero, π of π ₯ is equal to π ₯ minus three, which is a polynomial
function, meaning that we can use direct substitution in order to find the limit of this function.
Function D is π of π ₯ is equal to π ₯ plus one over π ₯. This is again a rational function. And we can again say that itβ s equal to π of π ₯ over π of π ₯, where π of π ₯ is
equal to π ₯ plus one and π of π ₯ is just equal to π ₯. We substitute π ₯ equals zero into the denominator of the fraction, which is π of π ₯. We find that π of zero is equal to zero.
Therefore, the denominator of π of π ₯ is equal to zero at π ₯ equals zero. Direct substitution can therefore not be used in order to find this limit.
For function E, we have π of π ₯ is equal to two π ₯ if π ₯ is greater than zero and two π ₯ minus one if π ₯ is less than or equal to zero. For this piecewise function, we again need to
consider the one-sided limits to the left and right of zero in order to check whether the limit itself exists. Remembering that, in order for the limit to exist, the limit as π ₯ approaches zero
from above must be equal to the limit as π ₯ approaches zero from below of π of π ₯.
When π ₯ is approaching zero from above β so this means π ₯ is just a bit larger than zero β we have that π of π ₯ is equal to two π ₯, since π ₯ is just larger than zero. π of π ₯ is
equal to two π ₯ is just a polynomial. And so we can find the limit as π ₯ approaches zero from above using direct substitution. We find that itβ s equal to π of zero, which is equal to two
times zero, or just zero.
Next, weβ ll consider the limit as π ₯ approaches zero from below. When π ₯ is smaller than or equal to zero, we have that π of π ₯ is equal to two π ₯ minus one, which is again just a
polynomial. We can therefore find the limit using direct substitution. This gives us that the limit as π ₯ approaches zero from below of π of π ₯ is equal to π of zero, or two times zero
minus one. And since two times zero is just zero, we get that this is equal to negative one.
Now if we compare these one-sided limits, we can see that theyβ re not equal to one another. This tells us that the limit does not exist here. And so we cannot use direct substitution in order to
find this limit. We find that the solution to this question is C.
Given π of π ₯ is equal to the modulus of π ₯ plus 11 minus the modulus of π ₯ minus 18, find the limit as π ₯ goes to four of π of π ₯.
Now our list of conditions under which direct substitution works does not include the modulus function. However, we can think of the modulus function in another way. We can write the modulus of π ₯
as the square root of π ₯ squared. Since finding the modulus of a number is simply taking the absolute value of that number and taking the square and then the root of a number will also give us the
absolute value of that number. The square root of π ₯ squared can also be written as π ₯ squared to the power of one-half.
π ₯ squared and π ₯ to the power of a half are both power functions. We know that we can apply direct substitution to power functions. And π ₯ squared to the power of a half is simply a
composition of two power functions. Therefore, we can also apply direct substitution to this. From this, we can see that we can apply direct substitution to the function mod π ₯.
Now weβ re ready to consider the function π of π ₯. We will consider the modulus of π ₯ plus 11 and the modulus of π ₯ minus 18. Here weβ re simply taking the moduluses of two polynomial
functions, which are π ₯ plus 11 and π ₯ minus 18. This is simply a composition of a polynomial function and a modulus function. And since we know we can apply direct substitution to both
polynomial functions and modulus functions, we know that we can also apply direct substitution to these composite functions.
Now π of π ₯ is simply the difference of these two composite functions. Therefore, we can also apply direct substitution to π of π ₯. So the limit as π ₯ approaches four of π of π ₯ is
equal to π of four, which is also equal to the modulus of four plus 11 minus the modulus of four minus 18, or mod 15 minus mod of negative 14, which is also equal to 15 minus 14. From this, we
find that the solution to this question is simply one.
Now we have seen a variety of limits which can be found using direct substitution and some which cannot. Let us recap some of the key points of this video. We have that the formula for using a direct
substitution is that the limit of π of π ₯ as π ₯ approaches π is equal to π of π . In order to use direct substitution, the limit must exist and the function must be defined at the
point at which weβ re taking the limit. And finally, the function weβ re taking the limit of must be a polynomial, rational, trigonometric, exponential, logarithmic, or power function or a sum,
difference, product, quotient, or composition of any of these types of functions. | {"url":"https://www.nagwa.com/en/videos/676173213586/","timestamp":"2024-11-14T20:22:57Z","content_type":"text/html","content_length":"277836","record_id":"<urn:uuid:2232c688-645b-469e-be07-1a70fc06966b>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00746.warc.gz"} |
probability of failure on demand tables
H. requirements, architectural constraints per Route 2. REFERENCES Bento J.-P., S. Bjore, G. Ericsson, A. Hasler, C.-D. Lyden, L. Wallin, K. Porn, O. Table 2 Failure rates - Primary Element
incremental, Route 2. For low demand mode, the failure measure is based on average Probability of dangerous Failure on Demand (PFDavg), whereas for high demand mode it is based on average Frequency
of Dangerous failure per hour. We describe the philosophies that are standing behind the PFD and the THR. Table 1 - Failure Rates These failure rates reflect currently-used industry data such as in
[i]. attention to each device’s Safety Failure Fraction (SFF) and Probability of Failure on Demand (PFDavg). The Probability of Failure on Demand (PFD) is a measure of the effectiveness of a safety
function. Failure Category . For example, the reactor system has an emergency quench water system piped to the reactor in the event of a runaway. The probability of failure and spurious trip rate are
functions of the reliability of the specific piece of equipment. 1) Where PFDavg is the average probability of failure The design of safety systems are often such that to work in the background,
monitoring a process, but not doing anything until a safety limit is overpassed when they … PFD - probability of failure upon demand Failure on demand occurs when a safety system is called upon to
react following an initiating event but fails to react. General Definition of Risk Reduction Factor The term Risk Reduction Factor (RRF) is very commonly used in discussions related to functional
safety and safety instrumented systems. -EN61508, PFD, Probability of Failure of the Markov Model is quite simple in this case because on Demand, Heterogeneous Structure, Homogenous theformulaof 1001
- Structure is well understood and In this case, the SIL value is derived from the PFD value (probability of failure on demand). For low demand service, the check valve probability of failure should
be used as the PFD for the backflow prevention IPL. In a 1oo1 voting arrangement there is no failure tolerance to either dangerous failures or safe failures. Low demand mode is typical in the process
industry. Probability of Failure on Demand Like dependability, this is also a probability value ranging from 0 to 1, inclusive. guaranteed to fail when activated). § Failure rates / Probability of
failure on demands etc § Types of data: Technical data, Operational data, ... 1 is the occurrence of the first failure, etc. http://www.SafeGuardProfiler.com Contents: SIL Verification Probability of
Failure on Demand (PFD) Equation A. Okubanjo, et al Nigerian Journal of Technology, Vol. Vico 46 21100 Varese Italy b Politecnico di Milano Dip. (tables B.2 to B.5 and B.10 to B.13 assume β = 2 × βD)
... 5.0 × 10-6 25 × 10-6 PFD G Average probability of failure on demand for the group of voted Channels (If the sensor, logic or final element subsystem comprises of only one voted group, then PFDG
is equivalent to PFDS, PFDL or PFDFE respectively) PFD S The failure rate “λ” is a variable determining the reliability of products. For each device in the SIF, both of these numbers have to be
compared to the rules outlined in the safety standards to ensure that they are sufficient for use in the required SIL of the SIS. Failure rate is the frequency with which an engineered system or
component fails, expressed in failures per unit of time. There are four discrete integrity levels: SIL 1, 2, 3 and 4. RRF = 1/PFDavg (Eq. The control valve is continuously modulated by the control
branch of the PLC systems and therefore a limited degree of diagnostic coverage can be assumed. AVG) requirements. A comparison shows, how the philosophies are connected and which connections between
PFH and PFD are implied. Low demand mode For low demand mode, it can be assumed that the safety system is not required more than once per year. In the paper, we will study the PFD and its connection
with the probability of failure per hour and failure rates of equipment using very simple models. AVERAGE PROBABILITY OF FAILURE ON DEMAND ESTIMATION FOR BURNER MANAGEMENT SYSTEMS A. This could be
determined using an FMEA (failure mode and effects analysis) or FTA (fault tree analysis). 3.1.15. unavailability as per 3.1.12 in the functional safety standard terminology (e.g. Probability of
failure on demand (PFD) PFD is probability of failure on demand. For comparison purposes, the failure probability of a steel pipe (mean values and distributions of tensile strength, modulus of
elasticity, and thickness listed in Table 5.6) is also evaluated using Monte Carlo simulation. k-out-of-n: G) systems subject to partial and full tests. Table 5 – Safety Integrity Level with
Architecture for Type B Subsystems 14 Table 6 – Low demand mode and continuos probabilities of failure 15 Table 7 – Performance Levels classification according to PFH D 16 Table 8 – Mean time to
dangerous failure of each channel (MTTF D) 16 Table 9 – Diagnostic coverage (DC) 17 Following 30 iterations, an instantaneous average failure probability of 2.85% is determined. Some typical
protection layer Probability of Failure on Demand (PFD) • BPCS control loop = 0.10 • Operator response to alarm = 0.10 • Relief safety valve = 0.001 • Vessel failure at maximum design pressure = 10-4
or better (lower) Source: A. Frederickson, Layer of Protection Analysis, www.safetyusersgroup.com, May 2006 2.3. Failure Rate and Event Data for use within Risk Assessments (06/11/17) Introduction 1.
PFD can be determined as an average probability or maximum probability over a time period. Reading the tables if you have a SIL 3 high demand safety function then the PFH needs to be < 1e-7/h (100
FIT). Abstract: For the assessment of the "safety integrity level" (SIL) in accordance with the standard EN 61508 it is among other things also necessary to calculate the "probability of failure on
demand" (PFD) of a safety related function. Probability of Failure on Demand PFD. It expresses the likelihood that the safety function does not work when required to. The calculated PFD value should
be verified as better than the minimum required PFD value as shown in the Table 1 by a factor of 25%. Non-approximate equations are introduced for probability of failure on demand (PFD) assessment of
a MooN architecture (i.e. These target failure measures are tabulated in Table 3. Probability of Failure on Demand (PFD) References IEC 61508-1 Functional safety of electrical / electronic /
programmable electronic safety-related systems - Part 1: General requirements, 1st edn. 6. 4, October 2017 1219 whenever the equipment under control (EUC) goes to a hazardous situation causing a real
… Recognising High Demand Mode IEC 61508[2]) Note 1 to entry: “Failure on demand” means here “failure likely to be observed when a demand occurs”. As this data meets Route 2. It indicates how many
instruments on average fail within a certain time span, indicated in “failure in time” unit. H. may be used. For low demand a SIL 3 safety function needs to have an average probability of failure on
demand of less than 0.001. Thereto a set of equations is given in the standard mentioned above. See Tables 1 and 2 for additional information. The PFD for a loop depends on the failure rates of all
the components in the loop. The check valve can be considered to be in low demand service if the demand rate on the check valve is less than once per year. For purposes of comparison, we have set a
value of PFD (average probability of failure on demand) and STR It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering.. “PF”, is the probability of a
malfunction or failure of the system. Probability of Failure on Demand average- This is the probability that a system will fail dangerously, and not be able to perform its safety function when
required. Failure Rate (FIT) Flowmeter ... average Probability of Failure on Demand (PFD. It is a measure of safety system performance, in terms of the probability of failure on demand. Identifying
the required amount of risk reduction is extremely important especially when evaluating existing legacy Burner Management Systems. The probability of failure, abbr. di Scienza e Alta Tecnologia, Via
G.B. The higher the SIL level, the higher the associated safety level and the lower the probability that a system will fail to perform properly. IEC 61508/61511 and ISA 84.01 use PFDavg as the system
metric upon which the SIL is defined. IEC 61508: Effect of Test Policy on the Probability of Failure on Demand of Safety Instrumented Systems Sergio Contini, Sabrina Copelli*, Massimo Raboni ,
Vincenzo Torretta , Carlo Sala Cattaneo , Renato Rota b a Università degli Studi dell’Insubria Dip. A PFD value of zero (0) means there is no probability of failure (i.e. The Chemicals, Explosives
and Microbiological Hazardous Division 5, CEMHD5, has an established set of failure rates that have been in use for several years. Moreover, we present a reasoning, why a probability of failure on
demand (PFD) might be misleading. Operating modes: Low demand and high demand Table 2.1 Control valve failure rates per million hours Fail shut 7 Fail open 3 Leak to atmosphere 2 Slow to move 2 Limit
switch fails to operate 1 The PFD of the complete SIS loop including the initiator, logic solver and final element shall be calculated. 36, No. Partial tests may occur at different time instants
(periodic or not) until the full test. The aspect to be verified is the Probability of Failure on Demand (PFD). CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): This paper
will discuss how quantitative methods can be utilized to select the appropriate Safety Integrity Level associated with Burner Management Systems. H. compliant . When asked “what does RRF mean?” most
functional safety practitioners will simply provide a mathematical equation in response, specifically. 3.5. it is 100% dependable – guaranteed to properly perform when needed), while a PFD value of
one (1) means it is completely undependable (i.e. This document details those items and their failure rates. The failure rate of a system usually depends on … , inclusive C.-D. Lyden, L. Wallin, K.
Porn, O Nigerian Journal of Technology, Vol has! Okubanjo, et al Nigerian Journal of Technology, Vol are four discrete levels! Incremental, Route 2 from the PFD of the system metric upon which the
value. Equations is given probability of failure on demand tables the process industry PFD ) is a variable determining the reliability of.! Are standing behind the PFD of the system metric upon which
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mode and effects analysis ) specifically! Management systems of the effectiveness of a MooN architecture ( i.e how the philosophies that are standing behind the for... Initiator, logic solver and
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Temperature Explained
Unit: K
Otherunits: °C, °F, °R, °Rø, °Ré, °N, °D, °L, °W
Dimension: wikidata
Intensive: Yes
Derivations: ,
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measured with a thermometer. It reflects the average kinetic energy of the
vibrating and colliding atoms making up a substance.
Thermometers are calibrated in various temperature scales that historically have relied on various reference points and thermometric substances for definition. The most common scales are the Celsius
scale with the unit symbol °C (formerly called centigrade), the Fahrenheit scale (°F), and the Kelvin scale (K), the latter being used predominantly for scientific purposes. The kelvin is one of the
seven base units in the International System of Units (SI).
Absolute zero, i.e., zero kelvin or −273.15 °C, is the lowest point in the thermodynamic temperature scale. Experimentally, it can be approached very closely but not actually reached, as recognized
in the third law of thermodynamics. It would be impossible to extract energy as heat from a body at that temperature.
Temperature is important in all fields of natural science, including physics, chemistry, Earth science, astronomy, medicine, biology, ecology, material science, metallurgy, mechanical engineering and
geography as well as most aspects of daily life.
Many physical processes are related to temperature; some of them are given below:
See main article: Scale of temperature. Temperature scales need two values for definition: the point chosen as zero degrees and the magnitudes of the incremental unit of temperature.
The Celsius scale (°C) is used for common temperature measurements in most of the world. It is an empirical scale that developed historically, which led to its zero point being defined as the
freezing point of water, and as the boiling point of water, both at atmospheric pressure at sea level. It was called a centigrade scale because of the 100-degree interval.^[3] Since the
standardization of the kelvin in the International System of Units, it has subsequently been redefined in terms of the equivalent fixing points on the Kelvin scale, so that a temperature increment of
one degree Celsius is the same as an increment of one kelvin, though numerically the scales differ by an exact offset of 273.15.
The Fahrenheit scale is in common use in the United States. Water freezes at and boils at at sea-level atmospheric pressure.
Absolute zero
At the absolute zero of temperature, no energy can be removed from matter as heat, a fact expressed in the third law of thermodynamics. At this temperature, matter contains no macroscopic thermal
energy, but still has quantum-mechanical zero-point energy as predicted by the uncertainty principle, although this does not enter into the definition of absolute temperature. Experimentally,
absolute zero can be approached only very closely; it can never be reached (the lowest temperature attained by experiment is 38 pK).^[4] Theoretically, in a body at a temperature of absolute zero,
all classical motion of its particles has ceased and they are at complete rest in this classical sense. Absolute zero, defined as, is exactly equal to, or .
Absolute scales
Referring to the Boltzmann constant, to the Maxwell–Boltzmann distribution, and to the Boltzmann statistical mechanical definition of entropy, as distinct from the Gibbs definition,^[5] for
independently moving microscopic particles, disregarding interparticle potential energy, by international agreement, a temperature scale is defined and said to be absolute because it is independent
of the characteristics of particular thermometric substances and thermometer mechanisms. Apart from absolute zero, it does not have a reference temperature. It is known as the Kelvin scale, widely
used in science and technology. The kelvin (the unit name is spelled with a lower-case 'k') is the unit of temperature in the International System of Units (SI). The temperature of a body in a state
of thermodynamic equilibrium is always positive relative to absolute zero.
Besides the internationally agreed Kelvin scale, there is also a thermodynamic temperature scale, invented by Lord Kelvin, also with its numerical zero at the absolute zero of temperature, but
directly relating to purely macroscopic thermodynamic concepts, including the macroscopic entropy, though microscopically referable to the Gibbs statistical mechanical definition of entropy for the
canonical ensemble, that takes interparticle potential energy into account, as well as independent particle motion so that it can account for measurements of temperatures near absolute zero.^[5] This
scale has a reference temperature at the triple point of water, the numerical value of which is defined by measurements using the aforementioned internationally agreed Kelvin scale.
Kelvin scale
Many scientific measurements use the Kelvin temperature scale (unit symbol: K), named in honor of the physicist who first defined it. It is an absolute scale. Its numerical zero point,, is at the
absolute zero of temperature. Since May, 2019, the kelvin has been defined through particle kinetic theory, and statistical mechanics. In the International System of Units (SI), the magnitude of the
kelvin is defined in terms of the Boltzmann constant, the value of which is defined as fixed by international convention.^[6]
Statistical mechanical versus thermodynamic temperature scales
Since May 2019, the magnitude of the kelvin is defined in relation to microscopic phenomena, characterized in terms of statistical mechanics. Previously, but since 1954, the International System of
Units defined a scale and unit for the kelvin as a thermodynamic temperature, by using the reliably reproducible temperature of the triple point of water as a second reference point, the first
reference point being at absolute zero.
Historically, the temperature of the triple point of water was defined as exactly 273.16 K. Today it is an empirically measured quantity. The freezing point of water at sea-level atmospheric pressure
occurs at very close to .
Classification of scales
There are various kinds of temperature scale. It may be convenient to classify them as empirically and theoretically based. Empirical temperature scales are historically older, while theoretically
based scales arose in the middle of the nineteenth century.^[7] ^[8]
Empirical scales
Empirically based temperature scales rely directly on measurements of simple macroscopic physical properties of materials. For example, the length of a column of mercury, confined in a glass-walled
capillary tube, is dependent largely on temperature and is the basis of the very useful mercury-in-glass thermometer. Such scales are valid only within convenient ranges of temperature. For example,
above the boiling point of mercury, a mercury-in-glass thermometer is impracticable. Most materials expand with temperature increase, but some materials, such as water, contract with temperature
increase over some specific range, and then they are hardly useful as thermometric materials. A material is of no use as a thermometer near one of its phase-change temperatures, for example, its
In spite of these limitations, most generally used practical thermometers are of the empirically based kind. Especially, it was used for calorimetry, which contributed greatly to the discovery of
thermodynamics. Nevertheless, empirical thermometry has serious drawbacks when judged as a basis for theoretical physics. Empirically based thermometers, beyond their base as simple direct
measurements of ordinary physical properties of thermometric materials, can be re-calibrated, by use of theoretical physical reasoning, and this can extend their range of adequacy.
Theoretical scales
Theoretically based temperature scales are based directly on theoretical arguments, especially those of kinetic theory and thermodynamics. They are more or less ideally realized in practically
feasible physical devices and materials. Theoretically based temperature scales are used to provide calibrating standards for practical empirically based thermometers.
Microscopic statistical mechanical scale
In physics, the internationally agreed conventional temperature scale is called the Kelvin scale. It is calibrated through the internationally agreed and prescribed value of the Boltzmann constant,^
[6] referring to motions of microscopic particles, such as atoms, molecules, and electrons, constituent in the body whose temperature is to be measured. In contrast with the thermodynamic temperature
scale invented by Kelvin, the presently conventional Kelvin temperature is not defined through comparison with the temperature of a reference state of a standard body, nor in terms of macroscopic
Apart from the absolute zero of temperature, the Kelvin temperature of a body in a state of internal thermodynamic equilibrium is defined by measurements of suitably chosen of its physical
properties, such as have precisely known theoretical explanations in terms of the Boltzmann constant. That constant refers to chosen kinds of motion of microscopic particles in the constitution of
the body. In those kinds of motion, the particles move individually, without mutual interaction. Such motions are typically interrupted by inter-particle collisions, but for temperature measurement,
the motions are chosen so that, between collisions, the non-interactive segments of their trajectories are known to be accessible to accurate measurement. For this purpose, interparticle potential
energy is disregarded.
In an ideal gas, and in other theoretically understood bodies, the Kelvin temperature is defined to be proportional to the average kinetic energy of non-interactively moving microscopic particles,
which can be measured by suitable techniques. The proportionality constant is a simple multiple of the Boltzmann constant. If molecules, atoms, or electrons^[9] ^[10] are emitted from material and
their velocities are measured, the spectrum of their velocities often nearly obeys a theoretical law called the Maxwell–Boltzmann distribution, which gives a well-founded measurement of temperatures
for which the law holds.^[11] There have not yet been successful experiments of this same kind that directly use the Fermi–Dirac distribution for thermometry, but perhaps that will be achieved in the
The speed of sound in a gas can be calculated theoretically from the gas's molecular character, temperature, pressure, and the Boltzmann constant. For a gas of known molecular character and pressure,
this provides a relation between temperature and the Boltzmann constant. Those quantities can be known or measured more precisely than can the thermodynamic variables that define the state of a
sample of water at its triple point. Consequently, taking the value of the Boltzmann constant as a primarily defined reference of exactly defined value, a measurement of the speed of sound can
provide a more precise measurement of the temperature of the gas.^[13]
Measurement of the spectrum of electromagnetic radiation from an ideal three-dimensional black body can provide an accurate temperature measurement because the frequency of maximum spectral radiance
of black-body radiation is directly proportional to the temperature of the black body; this is known as Wien's displacement law and has a theoretical explanation in Planck's law and the Bose–Einstein
Measurement of the spectrum of noise-power produced by an electrical resistor can also provide accurate temperature measurement. The resistor has two terminals and is in effect a one-dimensional
body. The Bose-Einstein law for this case indicates that the noise-power is directly proportional to the temperature of the resistor and to the value of its resistance and to the noise bandwidth. In
a given frequency band, the noise-power has equal contributions from every frequency and is called Johnson noise. If the value of the resistance is known then the temperature can be found.^[14] ^[15]
Macroscopic thermodynamic scale
Historically, till May 2019, the definition of the Kelvin scale was that invented by Kelvin, based on a ratio of quantities of energy in processes in an ideal Carnot engine, entirely in terms of
macroscopic thermodynamics. That Carnot engine was to work between two temperatures, that of the body whose temperature was to be measured, and a reference, that of a body at the temperature of the
triple point of water. Then the reference temperature, that of the triple point, was defined to be exactly . Since May 2019, that value has not been fixed by definition but is to be measured through
microscopic phenomena, involving the Boltzmann constant, as described above. The microscopic statistical mechanical definition does not have a reference temperature.
Ideal gas
A material on which a macroscopically defined temperature scale may be based is the ideal gas. The pressure exerted by a fixed volume and mass of an ideal gas is directly proportional to its
temperature. Some natural gases show so nearly ideal properties over suitable temperature range that they can be used for thermometry; this was important during the development of thermodynamics and
is still of practical importance today.^[16] ^[17] The ideal gas thermometer is, however, not theoretically perfect for thermodynamics. This is because the entropy of an ideal gas at its absolute
zero of temperature is not a positive semi-definite quantity, which puts the gas in violation of the third law of thermodynamics. In contrast to real materials, the ideal gas does not liquefy or
solidify, no matter how cold it is. Alternatively thinking, the ideal gas law, refers to the limit of infinitely high temperature and zero pressure; these conditions guarantee non-interactive motions
of the constituent molecules.^[18] ^[19] ^[20]
Kinetic theory approach
The magnitude of the kelvin is now defined in terms of kinetic theory, derived from the value of the Boltzmann constant.
Kinetic theory provides a microscopic account of temperature for some bodies of material, especially gases, based on macroscopic systems' being composed of many microscopic particles, such as
molecules and ions of various species, the particles of a species being all alike. It explains macroscopic phenomena through the classical mechanics of the microscopic particles. The equipartition
theorem of kinetic theory asserts that each classical degree of freedom of a freely moving particle has an average kinetic energy of where denotes the Boltzmann constant. The translational motion of
the particle has three degrees of freedom, so that, except at very low temperatures where quantum effects predominate, the average translational kinetic energy of a freely moving particle in a system
with temperature will be .
Molecules, such as oxygen (O[2]), have more degrees of freedom than single spherical atoms: they undergo rotational and vibrational motions as well as translations. Heating results in an increase of
temperature due to an increase in the average translational kinetic energy of the molecules. Heating will also cause, through equipartitioning, the energy associated with vibrational and rotational
modes to increase. Thus a diatomic gas will require more energy input to increase its temperature by a certain amount, i.e. it will have a greater heat capacity than a monatomic gas.
As noted above, the speed of sound in a gas can be calculated from the gas's molecular character, temperature, pressure, and the Boltzmann constant. Taking the value of the Boltzmann constant as a
primarily defined reference of exactly defined value, a measurement of the speed of sound can provide a more precise measurement of the temperature of the gas.^[13]
It is possible to measure the average kinetic energy of constituent microscopic particles if they are allowed to escape from the bulk of the system, through a small hole in the containing wall. The
spectrum of velocities has to be measured, and the average calculated from that. It is not necessarily the case that the particles that escape and are measured have the same velocity distribution as
the particles that remain in the bulk of the system, but sometimes a good sample is possible.
Thermodynamic approach
Temperature is one of the principal quantities in the study of thermodynamics. Formerly, the magnitude of the kelvin was defined in thermodynamic terms, but nowadays, as mentioned above, it is
defined in terms of kinetic theory.
The thermodynamic temperature is said to be absolute for two reasons. One is that its formal character is independent of the properties of particular materials. The other reason is that its zero is,
in a sense, absolute, in that it indicates absence of microscopic classical motion of the constituent particles of matter, so that they have a limiting specific heat of zero for zero temperature,
according to the third law of thermodynamics. Nevertheless, a thermodynamic temperature does in fact have a definite numerical value that has been arbitrarily chosen by tradition and is dependent on
the property of particular materials; it is simply less arbitrary than relative "degrees" scales such as Celsius and Fahrenheit. Being an absolute scale with one fixed point (zero), there is only one
degree of freedom left to arbitrary choice, rather than two as in relative scales. For the Kelvin scale since May 2019, by international convention, the choice has been made to use knowledge of modes
of operation of various thermometric devices, relying on microscopic kinetic theories about molecular motion. The numerical scale is settled by a conventional definition of the value of the Boltzmann
constant, which relates macroscopic temperature to average microscopic kinetic energy of particles such as molecules. Its numerical value is arbitrary, and an alternate, less widely used absolute
temperature scale exists called the Rankine scale, made to be aligned with the Fahrenheit scale as Kelvin is with Celsius.
The thermodynamic definition of temperature is due to Kelvin. It is framed in terms of an idealized device called a Carnot engine, imagined to run in a fictive continuous cycle of successive
processes that traverse a cycle of states of its working body. The engine takes in a quantity of heat from a hot reservoir and passes out a lesser quantity of waste heat to a cold reservoir. The net
heat energy absorbed by the working body is passed, as thermodynamic work, to a work reservoir, and is considered to be the output of the engine. The cycle is imagined to run so slowly that at each
point of the cycle the working body is in a state of thermodynamic equilibrium. The successive processes of the cycle are thus imagined to run reversibly with no entropy production. Then the quantity
of entropy taken in from the hot reservoir when the working body is heated is equal to that passed to the cold reservoir when the working body is cooled. Then the absolute or thermodynamic
temperatures, and, of the reservoirs are defined such that^[21]
The zeroth law of thermodynamics allows this definition to be used to measure the absolute or thermodynamic temperature of an arbitrary body of interest, by making the other heat reservoir have the
same temperature as the body of interest.
Kelvin's original work postulating absolute temperature was published in 1848. It was based on the work of Carnot, before the formulation of the first law of thermodynamics. Carnot had no sound
understanding of heat and no specific concept of entropy. He wrote of 'caloric' and said that all the caloric that passed from the hot reservoir was passed into the cold reservoir. Kelvin wrote in
his 1848 paper that his scale was absolute in the sense that it was defined "independently of the properties of any particular kind of matter". His definitive publication, which sets out the
definition just stated, was printed in 1853, a paper read in 1851.^[22] ^[23] ^[24] ^[25]
Numerical details were formerly settled by making one of the heat reservoirs a cell at the triple point of water, which was defined to have an absolute temperature of 273.16 K.^[26] Nowadays, the
numerical value is instead obtained from measurement through the microscopic statistical mechanical international definition, as above.
Intensive variability
In thermodynamic terms, temperature is an intensive variable because it is equal to a differential coefficient of one extensive variable with respect to another, for a given body. It thus has the
dimensions of a ratio of two extensive variables. In thermodynamics, two bodies are often considered as connected by contact with a common wall, which has some specific permeability properties. Such
specific permeability can be referred to a specific intensive variable. An example is a diathermic wall that is permeable only to heat; the intensive variable for this case is temperature. When the
two bodies have been connected through the specifically permeable wall for a very long time, and have settled to a permanent steady state, the relevant intensive variables are equal in the two
bodies; for a diathermal wall, this statement is sometimes called the zeroth law of thermodynamics.^[27] ^[28] ^[29]
In particular, when the body is described by stating its internal energy, an extensive variable, as a function of its entropy, also an extensive variable, and other state variables, with), then the
temperature is equal to the partial derivative of the internal energy with respect to the entropy:^[28] ^[29] ^[30]
Likewise, when the body is described by stating its entropy as a function of its internal energy, and other state variables, with, then the reciprocal of the temperature is equal to the partial
derivative of the entropy with respect to the internal energy:^[28] ^[30] ^[31]
The above definition, equation (1), of the absolute temperature, is due to Kelvin. It refers to systems closed to the transfer of matter and has a special emphasis on directly experimental
procedures. A presentation of thermodynamics by Gibbs starts at a more abstract level and deals with systems open to the transfer of matter; in this development of thermodynamics, the equations (2)
and (3) above are actually alternative definitions of temperature.^[32]
Local thermodynamic equilibrium
Real-world bodies are often not in thermodynamic equilibrium and not homogeneous. For the study by methods of classical irreversible thermodynamics, a body is usually spatially and temporally divided
conceptually into 'cells' of small size. If classical thermodynamic equilibrium conditions for matter are fulfilled to good approximation in such a 'cell', then it is homogeneous and a temperature
exists for it. If this is so for every 'cell' of the body, then local thermodynamic equilibrium is said to prevail throughout the body.^[33] ^[34] ^[35] ^[36] ^[37]
It makes good sense, for example, to say of the extensive variable, or of the extensive variable, that it has a density per unit volume or a quantity per unit mass of the system, but it makes no
sense to speak of the density of temperature per unit volume or quantity of temperature per unit mass of the system. On the other hand, it makes no sense to speak of the internal energy at a point,
while when local thermodynamic equilibrium prevails, it makes good sense to speak of the temperature at a point. Consequently, the temperature can vary from point to point in a medium that is not in
global thermodynamic equilibrium, but in which there is local thermodynamic equilibrium.
Thus, when local thermodynamic equilibrium prevails in a body, the temperature can be regarded as a spatially varying local property in that body, and this is because the temperature is an intensive
Basic theory
Temperature is a measure of a quality of a state of a material.^[38] The quality may be regarded as a more abstract entity than any particular temperature scale that measures it, and is called
hotness by some writers.^[39] ^[40] ^[41] The quality of hotness refers to the state of material only in a particular locality, and in general, apart from bodies held in a steady state of
thermodynamic equilibrium, hotness varies from place to place. It is not necessarily the case that a material in a particular place is in a state that is steady and nearly homogeneous enough to allow
it to have a well-defined hotness or temperature. Hotness may be represented abstractly as a one-dimensional manifold. Every valid temperature scale has its own one-to-one map into the hotness
manifold.^[42] ^[43]
When two systems in thermal contact are at the same temperature no heat transfers between them. When a temperature difference does exist heat flows spontaneously from the warmer system to the colder
system until they are in thermal equilibrium. Such heat transfer occurs by conduction or by thermal radiation.^[44] ^[45] ^[46] ^[47] ^[48] ^[49] ^[50] ^[51]
Experimental physicists, for example Galileo and Newton,^[52] found that there are indefinitely many empirical temperature scales. Nevertheless, the zeroth law of thermodynamics says that they all
measure the same quality. This means that for a body in its own state of internal thermodynamic equilibrium, every correctly calibrated thermometer, of whatever kind, that measures the temperature of
the body, records one and the same temperature. For a body that is not in its own state of internal thermodynamic equilibrium, different thermometers can record different temperatures, depending
respectively on the mechanisms of operation of the thermometers.
Bodies in thermodynamic equilibrium
For experimental physics, hotness means that, when comparing any two given bodies in their respective separate thermodynamic equilibria, any two suitably given empirical thermometers with numerical
scale readings will agree as to which is the hotter of the two given bodies, or that they have the same temperature.^[53] This does not require the two thermometers to have a linear relation between
their numerical scale readings, but it does require that the relation between their numerical readings shall be strictly monotonic.^[54] ^[55] A definite sense of greater hotness can be had,
independently of calorimetry, of thermodynamics, and of properties of particular materials, from Wien's displacement law of thermal radiation: the temperature of a bath of thermal radiation is
proportional, by a universal constant, to the frequency of the maximum of its frequency spectrum; this frequency is always positive, but can have values that tend to zero. Thermal radiation is
initially defined for a cavity in thermodynamic equilibrium. These physical facts justify a mathematical statement that hotness exists on an ordered one-dimensional manifold. This is a fundamental
character of temperature and thermometers for bodies in their own thermodynamic equilibrium.^[7] ^[42] ^[43] ^[56] ^[57]
Except for a system undergoing a first-order phase change such as the melting of ice, as a closed system receives heat, without a change in its volume and without a change in external force fields
acting on it, its temperature rises. For a system undergoing such a phase change so slowly that departure from thermodynamic equilibrium can be neglected, its temperature remains constant as the
system is supplied with latent heat. Conversely, a loss of heat from a closed system, without phase change, without change of volume, and without a change in external force fields acting on it,
decreases its temperature.^[58]
Bodies in a steady state but not in thermodynamic equilibrium
While for bodies in their own thermodynamic equilibrium states, the notion of temperature requires that all empirical thermometers must agree as to which of two bodies is the hotter or that they are
at the same temperature, this requirement is not safe for bodies that are in steady states though not in thermodynamic equilibrium. It can then well be that different empirical thermometers disagree
about which is hotter, and if this is so, then at least one of the bodies does not have a well-defined absolute thermodynamic temperature. Nevertheless, any one given body and any one suitable
empirical thermometer can still support notions of empirical, non-absolute, hotness, and temperature, for a suitable range of processes. This is a matter for study in non-equilibrium thermodynamics.
Bodies not in a steady state
When a body is not in a steady-state, then the notion of temperature becomes even less safe than for a body in a steady state not in thermodynamic equilibrium. This is also a matter for study in
non-equilibrium thermodynamics.
Thermodynamic equilibrium axiomatics
For the axiomatic treatment of thermodynamic equilibrium, since the 1930s, it has become customary to refer to a zeroth law of thermodynamics. The customarily stated minimalist version of such a law
postulates only that all bodies, which when thermally connected would be in thermal equilibrium, should be said to have the same temperature by definition, but by itself does not establish
temperature as a quantity expressed as a real number on a scale. A more physically informative version of such a law views empirical temperature as a chart on a hotness manifold.^[42] ^[57] ^[59]
While the zeroth law permits the definitions of many different empirical scales of temperature, the second law of thermodynamics selects the definition of a single preferred, absolute temperature,
unique up to an arbitrary scale factor, whence called the thermodynamic temperature.^[7] ^[42] ^[60] ^[61] ^[62] ^[63] If internal energy is considered as a function of the volume and entropy of a
homogeneous system in thermodynamic equilibrium, thermodynamic absolute temperature appears as the partial derivative of internal energy with respect the entropy at constant volume. Its natural,
intrinsic origin or null point is absolute zero at which the entropy of any system is at a minimum. Although this is the lowest absolute temperature described by the model, the third law of
thermodynamics postulates that absolute zero cannot be attained by any physical system.
Heat capacity
See also: Heat capacity and Calorimetry. When an energy transfer to or from a body is only as heat, the state of the body changes. Depending on the surroundings and the walls separating them from the
body, various changes are possible in the body. They include chemical reactions, increase of pressure, increase of temperature and phase change. For each kind of change under specified conditions,
the heat capacity is the ratio of the quantity of heat transferred to the magnitude of the change.^[64]
For example, if the change is an increase in temperature at constant volume, with no phase change and no chemical change, then the temperature of the body rises and its pressure increases. The
quantity of heat transferred,, divided by the observed temperature change,, is the body's heat capacity at constant volume:
$C_V = \frac.$If heat capacity is measured for a well-defined amount of substance, the specific heat is the measure of the heat required to increase the temperature of such a unit quantity by one
unit of temperature. For example, raising the temperature of water by one kelvin (equal to one degree Celsius) requires 4186 joules per kilogram (J/kg).
See also: Timeline of temperature and pressure measurement technology and International Temperature Scale of 1990. Temperature measurement using modern scientific thermometers and temperature scales
goes back at least as far as the early 18th century, when Daniel Gabriel Fahrenheit adapted a thermometer (switching to mercury) and a scale both developed by Ole Christensen Rømer. Fahrenheit's
scale is still in use in the United States for non-scientific applications.
Temperature is measured with thermometers that may be calibrated to a variety of temperature scales. In most of the world (except for Belize, Myanmar, Liberia and the United States), the Celsius
scale is used for most temperature measuring purposes. Most scientists measure temperature using the Celsius scale and thermodynamic temperature using the Kelvin scale, which is the Celsius scale
offset so that its null point is =, or absolute zero. Many engineering fields in the US, notably high-tech and US federal specifications (civil and military), also use the Kelvin and Celsius scales.
Other engineering fields in the US also rely upon the Rankine scale (a shifted Fahrenheit scale) when working in thermodynamic-related disciplines such as combustion.
The basic unit of temperature in the International System of Units (SI) is the kelvin. It has the symbol K.
For everyday applications, it is often convenient to use the Celsius scale, in which corresponds very closely to the freezing point of water and is its boiling point at sea level. Because liquid
droplets commonly exist in clouds at sub-zero temperatures, is better defined as the melting point of ice. In this scale, a temperature difference of 1 degree Celsius is the same as a increment, but
the scale is offset by the temperature at which ice melts .
By international agreement,^[65] until May 2019, the Kelvin and Celsius scales were defined by two fixing points: absolute zero and the triple point of Vienna Standard Mean Ocean Water, which is
water specially prepared with a specified blend of hydrogen and oxygen isotopes. Absolute zero was defined as precisely and . It is the temperature at which all classical translational motion of the
particles comprising matter ceases and they are at complete rest in the classical model. Quantum-mechanically, however, zero-point motion remains and has an associated energy, the zero-point energy.
Matter is in its ground state,^[66] and contains no thermal energy. The temperatures and were defined as those of the triple point of water. This definition served the following purposes: it fixed
the magnitude of the kelvin as being precisely 1 part in 273.16 parts of the difference between absolute zero and the triple point of water; it established that one kelvin has precisely the same
magnitude as one degree on the Celsius scale; and it established the difference between the null points of these scales as being (= and =). Since 2019, there has been a new definition based on the
Boltzmann constant,^[67] but the scales are scarcely changed.
In the United States, the Fahrenheit scale is the most widely used. On this scale the freezing point of water corresponds to and the boiling point to . The Rankine scale, still used in fields of
chemical engineering in the US, is an absolute scale based on the Fahrenheit increment.
Historical scales
See also: Conversion of scales of temperature. The following temperature scales are in use or have historically been used for measuring temperature:
Plasma physics
The field of plasma physics deals with phenomena of electromagnetic nature that involve very high temperatures. It is customary to express temperature as energy in a unit related to the electronvolt
or kiloelectronvolt (eV/k[B] or keV/k[B]). The corresponding energy, which is dimensionally distinct from temperature, is then calculated as the product of the Boltzmann constant and temperature,
. Then, 1eV/
is . In the study of
QCD matter
one routinely encounters temperatures of the order of a few hundred MeV/
, equivalent to about .
Continuous or discrete
When one measures the variation of temperature across a region of space or time, do the temperature measurements turn out to be continuous or discrete? There is a widely held misconception that such
temperature measurements must always be continuous.^[68] This misconception partly originates from the historical view associated with the continuity of classical physical quantities, which states
that physical quantities must assume every intermediate value between a starting value and a final value.^[69] However, the classical picture is only true in the cases where temperature is measured
in a system that is in equilibrium, that is, temperature may not be continuous outside these conditions. For systems outside equilibrium, such as at interfaces between materials (e.g., a metal/
non-metal interface or a liquid-vapour interface) temperature measurements may show steep discontinuities in time and space. For instance, Fang and Ward were some of the first authors to successfully
report temperature discontinuities of as much as 7.8 K at the surface of evaporating water droplets.^[70] This was reported at inter-molecular scales, or at the scale of the mean free path of
molecules which is typically of the order of a few micrometers in gases^[71] at room temperature. Generally speaking, temperature discontinuities are considered to be norms rather than exceptions in
cases of interfacial heat transfer.^[72] This is due to the abrupt change in the vibrational or thermal properties of the materials across such interfaces which prevent instantaneous transfer of heat
and the establishment of thermal equilibrium (a prerequisite for having a uniform equilibrium temperature across the interface).^[73] ^[74] Further, temperature measurements at the macro-scale
(typical observational scale) may be too coarse-grained as they average out the microscopic thermal information based on the scale of the representative sample volume of the control system, and thus
it is likely that temperature discontinuities at the micro-scale may be overlooked in such averages. Such an averaging may even produce incorrect or misleading results in many cases of temperature
measurements, even at macro-scales, and thus it is prudent that one examines the micro-physical information carefully before averaging out or smoothing out any potential temperature discontinuities
in a system as such discontinuities cannot always be averaged or smoothed out.^[75] Temperature discontiuities, rather than merely being anomalies, have actually substantially improved our
understanding and predictive abilities pertaining to heat transfer at small scales.
Theoretical foundation
See also: Thermodynamic temperature. Historically, there are several scientific approaches to the explanation of temperature: the classical thermodynamic description based on macroscopic empirical
variables that can be measured in a laboratory; the kinetic theory of gases which relates the macroscopic description to the probability distribution of the energy of motion of gas particles; and a
microscopic explanation based on statistical physics and quantum mechanics. In addition, rigorous and purely mathematical treatments have provided an axiomatic approach to classical thermodynamics
and temperature.^[76] Statistical physics provides a deeper understanding by describing the atomic behavior of matter and derives macroscopic properties from statistical averages of microscopic
states, including both classical and quantum states. In the fundamental physical description, the temperature may be measured directly in units of energy. However, in the practical systems of
measurement for science, technology, and commerce, such as the modern metric system of units, the macroscopic and the microscopic descriptions are interrelated by the Boltzmann constant, a
proportionality factor that scales temperature to the microscopic mean kinetic energy.
The microscopic description in statistical mechanics is based on a model that analyzes a system into its fundamental particles of matter or into a set of classical or quantum-mechanical oscillators
and considers the system as a statistical ensemble of microstates. As a collection of classical material particles, the temperature is a measure of the mean energy of motion, called translational
kinetic energy, of the particles, whether in solids, liquids, gases, or plasmas. The kinetic energy, a concept of classical mechanics, is half the mass of a particle times its speed squared. In this
mechanical interpretation of thermal motion, the kinetic energies of material particles may reside in the velocity of the particles of their translational or vibrational motion or in the inertia of
their rotational modes. In monatomic perfect gases and, approximately, in most gas and in simple metals, the temperature is a measure of the mean particle translational kinetic energy, 3/2 k[B]T. It
also determines the probability distribution function of energy. In condensed matter, and particularly in solids, this purely mechanical description is often less useful and the oscillator model
provides a better description to account for quantum mechanical phenomena. Temperature determines the statistical occupation of the microstates of the ensemble. The microscopic definition of
temperature is only meaningful in the thermodynamic limit, meaning for large ensembles of states or particles, to fulfill the requirements of the statistical model.
Kinetic energy is also considered as a component of thermal energy. The thermal energy may be partitioned into independent components attributed to the degrees of freedom of the particles or to the
modes of oscillators in a thermodynamic system. In general, the number of these degrees of freedom that are available for the equipartitioning of energy depends on the temperature, i.e. the energy
region of the interactions under consideration. For solids, the thermal energy is associated primarily with the vibrations of its atoms or molecules about their equilibrium position. In an ideal
monatomic gas, the kinetic energy is found exclusively in the purely translational motions of the particles. In other systems, vibrational and rotational motions also contribute degrees of freedom.
Kinetic theory of gases
Maxwell and Boltzmann developed a kinetic theory that yields a fundamental understanding of temperature in gases.^[77] This theory also explains the ideal gas law and the observed heat capacity of
monatomic (or 'noble') gases.^[78] ^[79] ^[80]
The ideal gas law is based on observed empirical relationships between pressure (p), volume (V), and temperature (T), and was recognized long before the kinetic theory of gases was developed (see
Boyle's and Charles's laws). The ideal gas law states:^[81]
is the number of
of gas and is the
gas constant
This relationship gives us our first hint that there is an absolute zero on the temperature scale, because it only holds if the temperature is measured on an absolute scale such as Kelvin's. The
ideal gas law allows one to measure temperature on this absolute scale using the gas thermometer. The temperature in kelvins can be defined as the pressure in pascals of one mole of gas in a
container of one cubic meter, divided by the gas constant.
Although it is not a particularly convenient device, the gas thermometer provides an essential theoretical basis by which all thermometers can be calibrated. As a practical matter, it is not possible
to use a gas thermometer to measure absolute zero temperature since the gases condense into a liquid long before the temperature reaches zero. It is possible, however, to extrapolate to absolute zero
by using the ideal gas law, as shown in the figure.
The kinetic theory assumes that pressure is caused by the force associated with individual atoms striking the walls, and that all energy is translational kinetic energy. Using a sophisticated
symmetry argument,^[82] Boltzmann deduced what is now called the Maxwell–Boltzmann probability distribution function for the velocity of particles in an ideal gas. From that probability distribution
function, the average kinetic energy (per particle) of a monatomic ideal gas is^[79] ^[83]
where the Boltzmann constant is the ideal gas constant divided by the Avogadro number, and $v_\text = \sqrt = \sqrt$ is the root-mean-square speed.^[84] This direct proportionality between
temperature and mean molecular kinetic energy is a special case of the equipartition theorem, and holds only in the classical limit of a perfect gas. It does not hold exactly for most substances.
Zeroth law of thermodynamics
See main article: Zeroth law of thermodynamics.
When two otherwise isolated bodies are connected together by a rigid physical path impermeable to matter, there is the spontaneous transfer of energy as heat from the hotter to the colder of them.
Eventually, they reach a state of mutual thermal equilibrium, in which heat transfer has ceased, and the bodies' respective state variables have settled to become unchanging.^[85] ^[86] ^[87]
One statement of the zeroth law of thermodynamics is that if two systems are each in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other.^[88] ^[89] ^
This statement helps to define temperature but it does not, by itself, complete the definition. An empirical temperature is a numerical scale for the hotness of a thermodynamic system. Such hotness
may be defined as existing on a one-dimensional manifold, stretching between hot and cold. Sometimes the zeroth law is stated to include the existence of a unique universal hotness manifold, and of
numerical scales on it, so as to provide a complete definition of empirical temperature.^[59] To be suitable for empirical thermometry, a material must have a monotonic relation between hotness and
some easily measured state variable, such as pressure or volume, when all other relevant coordinates are fixed. An exceptionally suitable system is the ideal gas, which can provide a temperature
scale that matches the absolute Kelvin scale. The Kelvin scale is defined on the basis of the second law of thermodynamics.
Second law of thermodynamics
See main article: Second law of thermodynamics. As an alternative to considering or defining the zeroth law of thermodynamics, it was the historical development in thermodynamics to define
temperature in terms of the second law of thermodynamics which deals with entropy. The second law states that any process will result in either no change or a net increase in the entropy of the
universe. This can be understood in terms of probability.
For example, in a series of coin tosses, a perfectly ordered system would be one in which either every toss comes up heads or every toss comes up tails. This means the outcome is always 100% the same
result. In contrast, many mixed (disordered) outcomes are possible, and their number increases with each toss. Eventually, the combinations of ~50% heads and ~50% tails dominate, and obtaining an
outcome significantly different from 50/50 becomes increasingly unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.
As temperature governs the transfer of heat between two systems and the universe tends to progress toward a maximum of entropy, it is expected that there is some relationship between temperature and
entropy. A heat engine is a device for converting thermal energy into mechanical energy, resulting in the performance of work. An analysis of the Carnot heat engine provides the necessary
relationships. According to energy conservation and energy being a state function that does not change over a full cycle, the work from a heat engine over a full cycle is equal to the net heat, i.e.
the sum of the heat put into the system at high temperature, q[H] > 0, and the waste heat given off at the low temperature, q[C] < 0.^[91]
The efficiency is the work divided by the heat input:
where w[cy] is the work done per cycle. The efficiency depends only on |q[C]|/q[H]. Because q[C] and q[H] correspond to heat transfer at the temperatures T[C] and T[H], respectively, |q[C]|/q[H]
should be some function of these temperatures:
Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, a heat engine operating between T[1] and T[3] must have the same efficiency
as one consisting of two cycles, one between T[1] and T[2], and the second between T[2] and T[3]. This can only be the case if
which implies
Since the first function is independent of T[2], this temperature must cancel on the right side, meaning f(T[1], T[3]) is of the form g(T[1])/g(T[3]) (i.e. = = =, where g is a function of a single
temperature. A temperature scale can now be chosen with the property that
Substituting (6) back into (4) gives a relationship for the efficiency in terms of temperature:
For T[C] = 0K the efficiency is 100% and that efficiency becomes greater than 100% below 0K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0K is
the minimum possible temperature. In fact, the lowest temperature ever obtained in a macroscopic system was 20nK, which was achieved in 1995 at NIST. Subtracting the right hand side of (5) from the
middle portion and rearranging gives^[21] ^[91]
where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, whose change characteristically vanishes for a complete cycle if it is
defined by
where the subscript indicates a reversible process. This function corresponds to the entropy of the system, which was described previously. Rearranging (8) gives a formula for temperature in terms of
fictive infinitesimal quasi-reversible elements of entropy and heat:
For a constant-volume system where entropy S(E) is a function of its energy E, dE = dq[rev] and (9) gives
i.e. the reciprocal of the temperature is the rate of increase of entropy with respect to energy at constant volume.
Definition from statistical mechanics
Statistical mechanics defines temperature based on a system's fundamental degrees of freedom. Eq.(10) is the defining relation of temperature, where the entropy
is defined (up to a constant) by the logarithm of the number of
of the system in the given macrostate (as specified in the
microcanonical ensemble
is the Boltzmann constant and
is the number of microstates with the energy
of the system (degeneracy).
When two systems with different temperatures are put into purely thermal connection, heat will flow from the higher temperature system to the lower temperature one; thermodynamically this is
understood by the second law of thermodynamics: The total change in entropy following a transfer of energy
from system 1 to system 2 is:
\DeltaS=-(dS/dE)[1] ⋅ \DeltaE+(dS/dE)[2] ⋅ \DeltaE=\left(
and is thus positive if
From the point of view of statistical mechanics, the total number of microstates in the combined system 1 + system 2 is
, the logarithm of which (times the Boltzmann constant) is the sum of their entropies; thus a flow of heat from high to low temperature, which brings an increase in total entropy, is more likely than
any other scenario (normally it is much more likely), as there are more microstates in the resulting macrostate.
Generalized temperature from single-particle statistics
It is possible to extend the definition of temperature even to systems of few particles, like in a quantum dot. The generalized temperature is obtained by considering time ensembles instead of
configuration-space ensembles given in statistical mechanics in the case of thermal and particle exchange between a small system of fermions (N even less than 10) with a single/double-occupancy
system. The finite quantum grand canonical ensemble,^[92] obtained under the hypothesis of ergodicity and orthodicity,^[93] allows expressing the generalized temperature from the ratio of the average
time of occupation
of the single/double-occupancy system:
is the
Fermi energy
. This generalized temperature tends to the ordinary temperature when
goes to infinity.
Negative temperature
See main article: Negative temperature. On the empirical temperature scales that are not referenced to absolute zero, a negative temperature is one below the zero-point of the scale used. For
example, dry ice has a sublimation temperature of which is equivalent to .^[95] On the absolute Kelvin scale this temperature is . No body can be brought to exactly (the temperature of the ideally
coldest possible body) by any finite practicable process; this is a consequence of the third law of thermodynamics.^[96] ^[97] ^[98]
The internal kinetic theory temperature of a body cannot take negative values. The thermodynamic temperature scale, however, is not so constrained.
For a body of matter, there can sometimes be conceptually defined, in terms of microscopic degrees of freedom, namely particle spins, a subsystem, with a temperature other than that of the whole
body. When the body is in its own state of internal thermodynamic equilibrium, the temperatures of the whole body and of the subsystem must be the same. The two temperatures can differ when, by work
through externally imposed force fields, energy can be transferred to and from the subsystem, separately from the rest of the body; then the whole body is not in its own state of internal
thermodynamic equilibrium. There is an upper limit of energy such a spin subsystem can attain.
Considering the subsystem to be in a temporary state of virtual thermodynamic equilibrium, it is possible to obtain a negative temperature on the thermodynamic scale. Thermodynamic temperature is the
inverse of the derivative of the subsystem's entropy with respect to its internal energy. As the subsystem's internal energy increases, the entropy increases for some range, but eventually attains a
maximum value and then begins to decrease as the highest energy states begin to fill. At the point of maximum entropy, the temperature function shows the behavior of a singularity, because the slope
of the entropy as a function of energy decreases to zero and then turns negative. As the subsystem's entropy reaches its maximum, its thermodynamic temperature goes to positive infinity, switching to
negative infinity as the slope turns negative. Such negative temperatures are hotter than any positive temperature. Over time, when the subsystem is exposed to the rest of the body, which has a
positive temperature, energy is transferred as heat from the negative temperature subsystem to the positive temperature system.^[99] The kinetic theory temperature is not defined for such subsystems.
See main article: Orders of magnitude (temperature).
• For Vienna Standard Mean Ocean Water at one standard atmosphere when calibrated strictly per the two-point definition of thermodynamic temperature.
• The value is approximate. The difference between K and °C is rounded to to avoid false precision in the Celsius value.
• For a true black-body (which tungsten filaments are not). Tungsten filament emissivity is greater at shorter wavelengths, which makes them appear whiter.
• Effective photosphere temperature. The difference between K and °C is rounded to to avoid false precision in the Celsius value.
• The difference between K and °C is within the precision of these values.
• For a true black-body (which the plasma was not). The Z machine's dominant emission originated from electrons (soft x-ray emissions) within the plasma.
See also
Notes and references
Bibliography of cited references
• Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, (1st edition 1968), third edition 1983, Cambridge University Press, Cambridge UK, .
• Buchdahl, H.A. (1966). The Concepts of Classical Thermodynamics, Cambridge University Press, Cambridge.
• Jaynes, E.T. (1965). Gibbs vs Boltzmann entropies, American Journal of Physics, 33(5), 391–398.
• Middleton, W.E.K. (1966). A History of the Thermometer and its Use in Metrology, Johns Hopkins Press, Baltimore.
• Miller . J . 2013 . Cooling molecules the optoelectric way . Physics Today . 66 . 1 . 12–14 . 10.1063/pt.3.1840 . 2013PhT....66a..12M . free .
• Partington, J.R. (1949). An Advanced Treatise on Physical Chemistry, volume 1, Fundamental Principles. The Properties of Gases, Longmans, Green & Co., London, pp. 175–177.
• Pippard, A.B. (1957/1966). Elements of Classical Thermodynamics for Advanced Students of Physics, original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK.
• Quinn, T.J. (1983). Temperature, Academic Press, London, .
• Schooley, J.F. (1986). Thermometry, CRC Press, Boca Raton, .
• Roberts, J.K., Miller, A.R. (1928/1960). Heat and Thermodynamics, (first edition 1928), fifth edition, Blackie & Son Limited, Glasgow.
• Thomson, W. (Lord Kelvin) (1848). On an absolute thermometric scale founded on Carnot's theory of the motive power of heat, and calculated from Regnault's observations, Proc. Camb. Phil. Soc.
(1843/1863) 1, No. 5: 66–71.
• Thomson. W. (Lord Kelvin). William Thomson, 1st Baron Kelvin. On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's
Observations on Steam. Transactions of the Royal Society of Edinburgh. March 1851. XX. part II. 261–268, 289–298.
• Truesdell, C.A. (1980). The Tragicomical History of Thermodynamics, 1822–1854, Springer, New York, .
• Tschoegl, N.W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam, .
• Zeppenfeld . M. . Englert . B.G.U. . Glöckner . R. . Prehn . A. . Mielenz . M. . Sommer . C. . van Buuren . L.D. . Motsch . M. . Rempe . G. . 2012 . Sysiphus cooling of electrically trapped
polyatomic molecules . Nature . 491 . 7425. 570–573 . 10.1038/nature11595. 1208.0046 . 2012Natur.491..570Z . 23151480. 4367940 .
Further reading
• Chang, Hasok (2004). Inventing Temperature: Measurement and Scientific Progress. Oxford: Oxford University Press. .
• Zemansky, Mark Waldo (1964). Temperatures Very Low and Very High. Princeton, NJ: Van Nostrand.
External links
Notes and References
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59. Web site: Water Science School . Frozen carbon dioxide (dry ice) sublimates directly into a vapor. . USGS.
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62. Book: Kittel, Charles . Thermal Physics. Charles Kittel. Kroemer, Herbert . Herbert Kroemer. 1980. 2nd. W.H. Freeman Company. 978-0-7167-1088-2. Appendix E.
63. This the Hawking Radiation for a Schwarzschild black hole of mass M = . It is too faint to be observed.
64. Web site: World record in low temperatures . 2009-05-05 . live . https://web.archive.org/web/20090618075820/http://ltl.tkk.fi/wiki/LTL/World_record_in_low_temperatures . 2009-06-18 .
65. Results of research by Stefan Bathe using the PHENIX detector on the Relativistic Heavy Ion Collider at Brookhaven National Laboratory in Upton, New York. Bathe has studied gold-gold,
deuteron-gold, and proton-proton collisions to test the theory of quantum chromodynamics, the theory of the strong force that holds atomic nuclei together. Link to news release.
66. http://public.web.cern.ch/public/Content/Chapters/AboutCERN/HowStudyPrtcles/HowSeePrtcles/HowSeePrtcles-en.html How do physicists study particles? | {"url":"https://everything.explained.today/Temperature/","timestamp":"2024-11-06T11:53:23Z","content_type":"text/html","content_length":"144660","record_id":"<urn:uuid:db03e5b8-dd10-4cbb-8979-80a2934c3675>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00780.warc.gz"} |
In a metal fabrication process, metal rods are produced to a… | Wiki Cram
In a metal fabrication process, metal rods are produced to a…
In а metаl fаbricatiоn prоcess, metal rоds are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control
specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. Which of the following statements is true?
In а metаl fаbricatiоn prоcess, metal rоds are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control
specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. Which of the following statements is true?
Disclаimer: This is а chаllenge questiоn and intended tо be mоre difficult! Without using don't cares, how many rows with a valid input (v=1) would a 10x4 priority encoder
have in its truth table?
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Weak topological insulating phases of h
SciPost Submission Page
Weak topological insulating phases of hard-core-bosons on the honeycomb lattice
by Amrita Ghosh, Eytan Grosfeld
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Amrita Ghosh · Eytan Grosfeld
Submission information
Preprint Link: https://arxiv.org/abs/2010.16126v2 (pdf)
Date submitted: 2020-11-03 15:20
Submitted by: Grosfeld, Eytan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Condensed Matter Physics - Theory
Approaches: Theoretical, Computational
We study the phases of hard-core-bosons on a two-dimensional periodic honeycomb lattice in the presence of an on-site potential with alternating sign along the different y-layers of the lattice.
Using quantum Monte Carlo simulations supported by analytical calculations, we identify a weak topological insulator, characterized by a zero Chern number but non-zero Berry phase, which is
manifested at either density 1/4 or 3/4, as determined by the potential pattern. Additionally, a charge-density-wave insulator is observed at 1/2-filling, whereas the phase diagram at intermediate
densities is occupied by a superfluid phase. The weak topological insulator is further shown to be robust against any amount of nearest-neighbor repulsion, as well as weak next-nearest-neighbor
repulsion. The experimental realization of our model is feasible in an optical lattice setup.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2020-12-9 (Invited Report)
• Cite as: Anonymous, Report on arXiv:2010.16126v2, delivered 2020-12-09, doi: 10.21468/SciPost.Report.2269
This manuscript studies a bosonic analogue of weak topological insulating phases on the honeycomb lattice. Specifically the model studied are on a two-dimensional periodic honeycomb lattice in the
presence of an on-site potential with alternating sign along the different y-direction of the lattice. Using quantum Monte Carlo simulations and analytical calculations, the authors identify a
bosonic weak topological insulator, characterized by a zero Chern number but non-zero Berry phase, which is manifested at either density 1/4 or 3/4, as determined by the potential pattern. They also
map out the full phase diagram, including a charge-density-wave insulator at 1/2-filling and superfluid at intermediate densities. Supprisely the weak topological insulator is further shown to be
robust against any amount of nearest-neighbor repulsion, as well as weak next-nearest-neighbor repulsion. The proposed model may be experimentally realized using cold atoms in an optical lattice. I
find the results are interesting from both theoretical and experimental aspects, so I recommend its publication.
I have the following comments:
1: a main character of weak topological insualtor is the existence of edge states on the edges along specific directions. Here the edge state is quasi-1D superfluid. One may calculate the
single-particle correlator b^{dagger}_i b_j. The decaying behavior may reflect such information: it is insulating if the decay is exponential with the distance, and is gapless superfluid if the decay
follows a power law.
2: Since the authors are studying a bosonic model, the Chern number and Berry phase for bosons should be calculated to characterize the bosonic weak topological insulator. as well as weak
next-nearest-neighbor repulsion. The experimental realization of our model is feasible in an optical lattice setup.
Report #1 by Anonymous (Referee 1) on 2020-11-30 (Invited Report)
• Cite as: Anonymous, Report on arXiv:2010.16126v2, delivered 2020-11-30, doi: 10.21468/SciPost.Report.2245
1 - The paper is very well written and easy to follow.
2 - The results are presented in an intuitive, pedagogical manner.
3 - The authors perform a detailed study of their model, using multiple order parameters, topological invariants, as well as varying boundary conditions.
1 - The symmetry classification and topological protection of the model is insufficiently discussed (see report below).
2 - Is it not clear to what extent the work meets the acceptance criteria of Scipost Physics (specifically, the list of "Expectations"), as opposed to Scipost Physics Core.1 - The symmetry
classification and topological protection of the model is insufficiently discussed (see report below).
2 - Is it not clear to what extent the work meets the acceptance criteria of Scipost Physics (specifically, the list of "Expectations"), as opposed to Scipost Physics Core.
The authors study the topological phases of hard-core bosons on a hexagonal lattice in which the onsite potential is modulated. They find that WTI phases appear once the onsite potential is larger
than the nearest neighbor hopping strength, and that these phases are robust against NN repulsion as well as against weak NNN repulsion.
The paper is very well written. I enjoyed reading it. Results are presented in an intuitive, pedagogical way, making them easy to follow. There are however two points that I think the authors should
address. These points are listed above, and I detail them here:
1) The authors discuss WTI phases appearing in symmetry class BDI and use a Hamiltonian that is non-interacting (I'm referring to Eq. 1, before the NN and NNN repulsion are added). However, the
single-particle Hamiltonian of Eq. 1 does not belong to symmetry class BDI. It does have time-reversal symmetry T=K, meaning it is real, but there is no chiral symmetry. There is no unitary that
anti-commutes with H because of the non-zero onsite modulation and chemical potential.
Consistent with this lack of chiral symmetry, the edge states discussed by the authors do not appear at E=0. In class BDI, it is not just translation symmetry, but also chiral symmetry which protects
the WTI. Because of chiral symmetry all states come in +E, -E pairs (as can be seen from the bandstructures of Fig. 6). The edge states of a WTI should be pinned to the middle of the E=0 gap, such
that they cannot be removed from this gap without breaking symmetries. In the authors' model however, edge states appear in the gap between bands 1 and 2 (or 3 and 4), away from E=0. What symmetry is
responsible for their topological protection? Why can't they, in principle, be shifted up or down in energy such that they hybridize with the bulk states and dissapear?
2) While the work is novel and well presented, the authors should spend more time discussing if/how their paper meets the expectations of Scipost Physics (https://scipost.org/SciPostPhys/about#
criteria). From my reading of the paper as it is now, it seems to me that it instead meets the acceptance criteria of Scipost Physics Core (https://scipost.org/SciPostPhysCore/about), provided that
the point (1) above is addressed.
Requested changes
1 - Show explicitly what are the symmetries of their model and its symmetry class.
2 - Prove that their phases are topologically protected. This means to prove that there does not exist a symmetry-preserving perturbation which removes the edge states, for instance by shifting their
energies away from their respective gaps. | {"url":"https://scipost.org/submissions/2010.16126v2/","timestamp":"2024-11-13T09:23:35Z","content_type":"text/html","content_length":"39756","record_id":"<urn:uuid:4b4ff033-039f-43dd-8f73-65e5dee13212>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00348.warc.gz"} |
11Apr, 2021
Loan is a liability for business hence f has to pay . It is accounting entry where loan is credited a sits liability to show in books of accounts and when loan is paid loan account is debited to
adjust the loan paid
Example Amar Traders took a loan of Rs 100000 .
Journal entry for the same will be\
When Loan is taken from Bank
Particular Amount
Bank A/c Dr 100000
To loan A/c 100000
Being loan taken from bank
Next when its installment will be due and paid
Now Amar pay the installment of loan is 12000 which include interest on loan Rs 7000 and Loan Rs 5000. Every month like in first month he pay Rs 5000 loan amount and Rs 7000 interest on loan
therefore he has to first make due entry and then paid the interest and loan amount.
Loan and its interest payable entry
Particular Amount
Loan A/c Dr. 5000
Interest on loan A/c Dr. 7000
To loan Payable A/c 5000
To Interest on Loan payable A/c 7000
(being on interest and loan due
After making a due entry when loan and interest is paid then in the books of accounts payble interest on loan and loan is debited so they are adjusted and payment is made.
Particular Amount
loan Payable A/c 5000
Interest on loan Payable A/c Dr. 7000
To bank a/c 12000
(being on interest and loan due paid
Another example of understanding of loan .
Another example of understanding of loan .
Suppose A ltd took loan from bank on 1 Dec 2020 Rs 100000 and interest payable Rs. 12,000 financial year closed on 31 March 2021.
Case 1
If interest paid Rs 2000
Interest on Loan A/c Dr. 4000
To Bank A/c 2000.
To Interest on Loan payable A/c 2000
(Being interest on Loan paid)
Case 2
If Interest paid Rs 8000
Interest paid Rs 8000
Under such condition our liability is to paid Rs 4000 but we
paid Rs 8000 means we paid loan in advance under such condition journal entry
for the same will be.
│Particular │Amount │
│Interest on loan A/c Dr. │8000 │
│ To bank a/c │ 8000 │
│ │ │
│ │ │
│(being on interest and loan paid) │
│Particular │Amount│
│Prepaid Interest on loan A/c │4000 │
│Dr. │ │
│ To Interest on loan │ 4000│
│ (being on interest and │ │
│loan paid) │ │
Case 3
If No Interest paid
Under such condition we have to make a entry of Rs 4000 in our books as interest payable
│Particular │Amount│
│ Interest on loan A/c Dr. │8000 │
│ To Interest on loan payble │ 8000│
│ (being on interest on loan payable) │ │ | {"url":"http://www.thevistaacademy.com/how-to-make-journal-entry-of-loan-in-accounting/","timestamp":"2024-11-05T16:59:25Z","content_type":"text/html","content_length":"127859","record_id":"<urn:uuid:4163134f-ce26-4cbd-ba34-ec66269e29b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00792.warc.gz"} |
Circle Area Calculator
Use the circle area calculator to calculate the area, diameter, circumference or radius of a circle.
Circle Area
The length `C` of the circumference (perimeter) of a circle is calculated using the formula:
`C = 2pir`, where `r` is the radius of the circle.
The radius may also be expressed using the diameter so that `2r = d`:
The area `A` left inside the circle is calculated using the formula:
`A=pir^2`, where `r` is the radius of the circle or correspondingly:
`A=pi/4d^2`, where `d` is the diameter of the circle
If the diameter of the circle `d` and the length of the circumference `C` are known, the area can be calculated (without the figure `pi`) using the formula:
`A = Cd/4`
Information About the Circle Area Calculator
Use the circle area calculator to calculate the area, diameter, circumference or radius of a circle.
Sources and more information | {"url":"https://www.mycalculators.net/mathematics/circle-area","timestamp":"2024-11-05T23:21:11Z","content_type":"text/html","content_length":"49948","record_id":"<urn:uuid:9e33fc7b-573f-4545-9e99-130543fb847f>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00379.warc.gz"} |
iQ Abacus Math School
Algebra (II)
Please note that the price was determined according to the available videos and tests so far. If the lessons that you are interested in are not yet available, please email to info@iQabacus.com before
you make a payment.
Elementary Algebra by Harold R. Jacobs is an atypical algebra text that has been around for almost forty years in essentially the same edition because it has been so popular. The Algebra (II) course
covers the 2nd half of the book from Chapters 9 through 17.
Elementary Algebra covers all concepts typical of a 2-year algebra course. It invites students to explore algebra concepts in a friendlier environment than other texts. Cartoons, comic strips (e.g.,
Broom Hilda, B.C., Wizard of Id, and Doonesbury), interesting and creative applications, puzzles, and even poetry capture the interest of students who struggle with abstract mathematics. For example,
a lesson on mixture problems opens with the story of Archimedes and the King of Syracuse’s golden crown which the king suspected was not really solid gold. Jacobs then sets up a volume and weight
equation based on the problem.
In addition to stories and practical applications, Jacobs uses the rectangle-building (or area-model) concept throughout the text to demonstrate how concepts work. This is the same rectangle building
idea used by Math-U-See and some other manipulative systems. While Jacobs’ book shows pictures, not requiring the use of manipulatives, students can still use manipulatives if they are helpful. I
think most students should really benefit from this approach when they are learning to multiply and factor polynomials. (This last feature makes this text a particularly good one to use after
Math-U-See’s Pre-Algebra level. If you don’t already have manipulatives, you might want to check out the Algebra Tiles from Nasco.)
The hardcover textbook is divided into seventeen chapters, with each chapter subdivided into a number of lessons. Students should be able to read through the instructional material and examples on
their own, working independently most of the time. If your student benefits from a teacher's presentation, you might want to get the optional DVD Instruction set. These six DVDs have about six hours
of lesson presentation by Dr. Callahan.
In the original book, there were four exercise sets at the end of each lesson. The first set reviews concepts from earlier lessons along with some from the current lesson. The second set concentrates
primarily on topics taught in the current lesson. The third set of problems is generally fairly similar to the second set. These three sets present problems ranging from simple computation through
word and application problems. The most challenging problems are in the fourth set. These are often complex problems requiring critical thinking skills. Since the third set is similar to the second
set in the level of difficulty, Master Books has removed the third set of problems from the textbook. They have put those sets in the teacher guide, so they are still available for those who might
want to use them. Generally, you will have students work through at least the first two sets of problems. By assigning appropriate problems, you can use the text with students of varying
capabilities. Answers to the second set from each lesson are in the back of the student text so students can see if they are getting the correct answers. (If students are tempted to look at answers
ahead of time, you might assign problems from the third set instead.)
Each chapter concludes with a "Summary and Review" section that briefly summarizes key concepts. This is followed by two sets of problems that review concepts from the entire chapter. The text
includes answers for only one of these sets of chapter review problems. The teacher’s guide is the source for the rest of the answers for chapter reviews as well as the answers for the first, third,
and fourth problem sets from the lessons. The teacher guide also has tests and test answer keys. A complete solutions manual is also available for students who need help figuring out what steps they
have missed when they can't solve a problem.
The student text and the teacher guide are the most critical components, but if you want the solutions manual or the DVDs, you will save purchasing either the curriculum pack with the textbook,
teacher guide, and solutions manual or the curriculum pack that also includes the set of DVDs.
Jacob's Elementary Algebra provides solid coverage of first-year algebra and remains one of the easiest courses for students to use and understand.
Source: https://cathyduffyreviews.com/homeschool-reviews-core-curricula/math/math-grades-9-12/jacobs-elementary-algebra#
Note: You have not purchased this course yet,
or login to continue the course. | {"url":"https://www.iqabacus.com/course_list/29","timestamp":"2024-11-02T07:46:59Z","content_type":"text/html","content_length":"44922","record_id":"<urn:uuid:7e76e226-dbc9-45cf-ad46-3ba859ee8798>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00468.warc.gz"} |
[CA] Canada
Institut Périmètre de physique théorique / Perimeter Institute [PI]
Name Perimeter Institute
Name (original vn) Institut Périmètre de physique théorique
Acronym PI
Country Canada
Type Research Institute
Status Active
ROR id (link) https://ror.org/013m0ej23
Crossref Org ID (link) No Crossref Org ID found
ROR information
ROR ID (link) https://ror.org/013m0ej23
Primary Name Perimeter Institute
Perimeter Institute for Theoretical Physics
Alternative Names Institut Périmètre de Physique Théorique
Country Canada
Website(s) http://www.perimeterinstitute.ca/
Funder Registry instances associated to this Organization:
• Institut Périmètre de physique théorique
51 Publications associated to this Organization :
10 publications
11 publications
14 publications
7 publications
1 publication
6 publications
2 publications
Fellows affiliated to this Organization
Support history
List of the subsidies (in one form or another) which
SciPost has received from this Organization. Click on a row
to see more details.
Type Amount Date
Donation €3000 2024-01-01 until 2024-12-31
Total support obtained: €3000
Balance of SciPost expenditures versus support received
The following Expenditures (Perimeter Institute) table compiles the expenditures by SciPost to publish all papers which are associated to Perimeter Institute , and weighed by this Organization's
Help! What do these terms mean?
Concept Acronym Definition
Associated An Organization's Associated Publications is the set of papers in which the Organization (or any of its children) is mentioned in author affiliations, or in the
Publications acknowledgements as grant-giver or funder.
Number of Associated NAP Number of Associated Publications, compiled (depending on context) for a given year or over many years, for a specific Journal or for many, etc.
A fraction of a unit representing an Organization's "weight" for a given Publication.
The weight is given by the following simple algorithm:
Publication Fraction PubFrac • First, the unit is split equally among each of the authors.
• Then, for each author, their part is split equally among their affiliations.
• The author parts are then binned per Organization.
By construction, any individual paper's PubFracs sum up to 1.
Expenditures We use the term Expenditures to represent the sum of all outflows of money required by our initiative to achieve a certain output (depending on context).
Average Publication APEX For a given Journal for a given year, the average expenditures per Publication which our initiative has faced. All our APEX are listed on our APEX page.
Total Associated Total expenditures ascribed to an Organization's Associated Publications (given for one or many years, Journals etc depending on context).
PubFrac share The fraction of expenditures which can be associated to an Organization, based on PubFracs. This is defined as APEX times PubFrac, summed over the set of Publications
defined by the context (e.g. all Associated Publications of a given Organization for a given Journal in a given year).
Subsidy support Sum of the values of all Subsidies relevant to a given context (for example: from a given Organization in a given year).
Difference between incoming and outgoing financial resources for the activities under consideration (again defined depending on context).
Impact on reserves
• A positive impact on reserves means that our initiative is sustainable (and perhaps even able to grow).
• A negative impact on reserves means that these activities are effectively depleting our available resources and threatening our sustainability.
Year (click to toggle details) NAP Total associated expenditures PubFrac Subsidy Impact on
share support reserves
Cumulative 51 €29973 €14220 €3000 €-11220
Per year:
2024 (ongoing) 10 €8000 €3494 €3000 €-494
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2024.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share
SciPostPhys €800 7 €5600 2.801 €2240
SciPostPhysCore €800 3 €2400 1.568 €1254
2023 11 €5445 €2601 €0 €-2601
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2023.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share
SciPostPhys €495 10 €4950 5.221 €2584
SciPostPhysCore €495 1 €495 0.035 €17
2022 14 €6658 €3624 €0 €-3624
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2022.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share
SciPostPhys €444 10 €4440 5.718 €2538
SciPostPhysCodeb €444 2 €888 0.2 €88
SciPostPhysLectNotes €665 2 €1330 1.501 €998
2021 7 €4710 €2441 €0 €-2441
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2021.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share
SciPostPhys €642 5 €3210 2.167 €1391
SciPostPhysCore €600 1 €600 1.0 €600
SciPostPhysLectNotes €900 1 €900 0.5 €450
2020 1 €620 €310 €0 €-310
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2020.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share
SciPostPhys €620 1 €620 0.5 €310
2019 6 €3080 €1385 €0 €-1385
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2019.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share
SciPostPhys €440 4 €1760 1.401 €616
SciPostPhysLectNotes €660 2 €1320 1.166 €769
2018 2 €1460 €365 €0 €-365
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2018.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share
SciPostPhys €730 2 €1460 0.501 €365
2017 0 €0 €0 €0 €0
The following table give an overview of expenditures , compiled for all Publications which are associated to Perimeter Institute for 2017.
You can see the list of associated publications under the Publications tab.
Expenditures (Perimeter Institute)
Journal APEX NAP Total associated expenditures PubFracs PubFracs expenditures share | {"url":"https://scipost.org/organizations/27/","timestamp":"2024-11-10T17:50:15Z","content_type":"text/html","content_length":"96173","record_id":"<urn:uuid:c3cb7387-1c49-4133-9856-2e9bb3ec87ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00027.warc.gz"} |
XSL,XPath: How to use xpath expression written in the source xml
For example my xml source has some attribute node whose value is an xpath expression. How would I use that value as an xpath expression in my xsl?e.g.<tag attr=/tag2/somechild/><tag2><somechild>value
</somechild></tag2><tag3><somechild2>value2</somechild2></tag3>In my case, the value of the attribute "attr" is dynamic (changing), and I want my xsl file to access the value according to the value
of "attr" (an xpath expression). How can I do this? That is, how can I write the xpath expression that would select the value that the value of the "attr" points to? In essence, I would like to have
something like: <xsl:for-each select="the xpath expression in the value of attr">, and<xsl:value-of select="the xpath expression in the value of attr">but how can I do this? Thank you very much!
The only standard way I could think of is that you'll first generate an XSLT stylesheet containing those XPath expressions and then perform the transformation with the generated stylesheet. You'll
need to declare the XSLT namespace with two prefixes, the second of which should be mapped with namespace alias.Something like this:
<xsl:stylesheet version="1.0"xmlns:xsl="http://www.w3.org/1999/XSL/Transform"xmlns:oxsl="http://example.com/"><!--The namespace URI of this doesn't really matter<xsl:namespace-alias stylesheet-prefix="oxsl" result-prefix="xsl"/><xsl:template match="/"> <oxsl:stylesheet version="1.0"> <oxsl:template match="/"> <oxsl:value-of select="{//tag/@attr}"/> </oxsl:template> </oxsl:stylesheet></xsl:template></xsl:stylesheet>
But you'll probably need to use the server to generate the stylesheet, so that you could perform another transformation.The other way (if you can't use the server) would be to generate both the
output and the stylesheet into one output, so that the output looks like:
<?xml-stylesheet type="text/xsl" href="#stylesheet"?><tag><xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform" xml:id="stylesheet>...</xsl:stylesheet></tag>
Thanks! I'll try this!
This topic is now archived and is closed to further replies. | {"url":"https://w3schools.invisionzone.com/topic/14829-xslxpath-how-to-use-xpath-expression-written-in-the-source-xml/","timestamp":"2024-11-03T06:30:35Z","content_type":"text/html","content_length":"77603","record_id":"<urn:uuid:4b965e9d-479d-4eb7-8289-6ce994edbbe9>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00810.warc.gz"} |
Jack Lee - Primary School Science Tuition - The Smart Student
In this article, we’ll be discussing how to apply the Laws of Indices to solve this Secondary 3 A-Math Exponentials question from Mayflower Secondary School.
Read More
Estimation & Approximation: How To Estimate Using Given Information
Posted by Jack Lee | Oct 21, 2024 | Secondary 1 Math, Estimation & Approximation |
In this article, we’ll be discussing how to use given information in the question to solve this Secondary 1 Math Estimation & Approximation question without using a calculator.
Read More | {"url":"https://thepiquelab.com/blog/author/jacklee/","timestamp":"2024-11-09T11:17:25Z","content_type":"text/html","content_length":"140806","record_id":"<urn:uuid:c41b4a75-1b55-42bb-a231-a86f16d7152e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00091.warc.gz"} |
Analyzing Experiments Using the marginaleffects Package for R | Code Horizons
Analyzing Experiments Using the marginaleffects Package for R
Vincent Arel-Bundock will provide hands-on exercises and more insights for analyzing experiments using the marginaleffects package for R during the Interpreting and Communicating Statistical Results
with R short course on March 27-29.
2×2 Experiments
A 2×2 factorial design is a type of experimental design that allows researchers to understand the effects of two independent variables (each with two levels) on a single dependent variable. The
design is popular among academic researchers as well as in industry when running A/B tests.
In this blog, we illustrate how to analyze these designs with the the marginaleffects package for R. As we will see, marginaleffects includes many convenient functions for analyzing both experimental
and observational data, and for plotting our results.
Fitting a Model
We will use the mtcars dataset. We’ll analyze fuel efficiency, mpg (miles per gallon), as a function of am (transmission type) and vs (engine shape).
vs is an indicator variable for if the car has a straight engine (1 = straight engine, 0 = V-shaped). am is an indicator variable for if the car has manual transmission (1 = manual transmission, 0=
automatic transmission). There are then four types of cars (1 type for each of the four combinations of binary indicators).
Let’s start by creating a model for fuel efficiency. For simplicity, we’ll use linear regression and model the interaction between vs and am.
library(modelsummary) ## See ?mtcars for variable definitions
fit <- lm(mpg ~ vs + am + vs:am, data=mtcars) # equivalent to ~ vs*am
We can plot the predictions from the model using the plot_predictions function. From the plot below, we can see a few things:
• Straight engines (vs=1) are estimated to have better expected fuel efficiency than V-shaped engines (vs=0).
• Manual transmissions (am=1) are estimated to have better fuel efficiency for both V-shaped and straight engines.
• For straight engines, the effect of manual transmissions on fuel efficiency seems to increase.
plot_predictions(fit, by = c("vs", "am"))
Evaluating Effects From The Model Summary
Since this model is fairly simple, the estimated differences between any of the four possible combinations of vs and am can be read from the regression table, which we create using the modelsummary
modelsummary(fit, gof_map = c("r.squared", "nobs"))
We can express the same results in the form of a linear equation:
With a little arithmetic, we can compute estimated differences in fuel efficiency between different groups:
• 4.700 mpg between am=1 and am=0, when vs=0.
• 5.693 mpg between vs=1 and vs=0, when am=0.
• 7.629 mpg between am=1 and am=0, when vs=1.
• 8.621 mpg between vs=1 and vs=0, when am=1.
• 13.322 mpg between a car with am=1 and vs=1, and a car with am=0 and vs=0.
Reading off these differences from the model summary is relatively straightforward in very simple cases like this one. However, it becomes more difficult as more variables are added to the model, not
to mention obtaining estimated standard errors becomes nightmarish. To make the process easier, we can leverage the avg_comparisons() function from the marginaleffects package to compute the
appropriate quantities and standard errors.
Using avg_comparisons To Estimate All Differences
The grey rectangle in the graph below is the estimated fuel efficiency when vs=0 and am=0, that is, for an automatic transmission car with a V-shaped engine.
Let’s use avg_comparisons to get the difference between straight engines and V-shaped engines when the car has automatic transmission. In this call, the variables argument indicates that we want to
estimate the effect of a change of 1 unit in the vs variable. The newdata=datagrid(am=0) determines the values of the covariates at which we want to evaluate the contrast.
variables = "vs",
newdata = datagrid(am = 0))
#> Term Contrast Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> vs 1 - 0 5.69 1.65 3.45 <0.001 10.8 2.46 8.93
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo
As expected, the results produced by avg_comparisons() are exactly the same as those which we read from the model summary table. The contrast that we just computed corresponds to the change
illustrasted by the arrow in this plot:
The next difference that we compute is between manual transmissions and automatic transmissions when the car has a V-shaped engine. Again, the call to avg_comparisons is shown below, and the
corresponding contrast is indicated in the plot below using an arrow.
variables = "am",
newdata = datagrid(vs = 0))
#> Term Contrast Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> am 1 - 0 4.7 1.74 2.71 0.00678 7.2 1.3 8.1
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo
The third difference we estimated was between manual transmissions and automatic transmissions when the car has a straight engine. The model call and contrast are:
variables = "am",
newdata = datagrid(vs = 1))
#> Term Contrast Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> am 1 - 0 7.63 1.86 4.11 <0.001 14.6 3.99 11.3
#> Columns: rowid, term, contrast, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high, predicted, predicted_hi, predicted_lo
The last difference and contrast between manual transmissions with straight engines and automatic transmissions with V-shaped engines. We call this a “cross-contrast” because we are measuring the
difference between two groups that differ on two explanatory variables at the same time. To compute this contrast, we use the cross argument of avg_comparisons:
variables = c("am", "vs"),
cross = TRUE)
#> C: am C: vs Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> 1 - 0 1 - 0 13.3 1.65 8.07 <0.001 50.3 10.1 16.6
#> Columns: term, contrast_am, contrast_vs, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
The 2×2 design is a very popular design, and when using a linear model, the estimated differences between groups can be directly read off from the model summary, if not with a little arithmetic.
However, when using models with a non-identity link function, or when seeking to obtain the standard errors for estimated differences, things become considerably more difficult. This vignette showed
how to use avg_comparisons to specify contrasts of interests and obtain standard errors for those differences. The approach used applies to all generalized linear models and effects can be further
stratified using the by argument (although this is not shown in this vignette.)
https://codehorizons.com/wp-content/uploads/2023/09/graph-6-1.png 691 1728 Kaity Turner https://codehorizons.com/wp-content/uploads/2020/03/Code-Horizons-Coding-Seminars-Logo-Aqua.png Kaity Turner
2023-09-06 14:54:572024-10-11 15:11:26Analyzing Experiments Using the marginaleffects Package for R | {"url":"https://codehorizons.com/analyzing-experiments-using-the-marginaleffects-package-for-r/","timestamp":"2024-11-02T20:06:27Z","content_type":"text/html","content_length":"91061","record_id":"<urn:uuid:a227e377-a32b-4e45-b505-b732f1ca68b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00623.warc.gz"} |
Studentized residual
Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |
Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory
In statistics, a Studentized residual, named in honor of William Sealey Gosset, who wrote under the pseudonym Student, is a residual adjusted by dividing it by an estimate of its standard deviation.
Studentization of residuals is an important technique in the detection of outliers.
Errors versus residuals[ ]
It is very important to understand the difference between errors and residuals in statistics. Consider the simple linear regression model
${\displaystyle Y_i=\alpha_0+\alpha_1 x_i+\varepsilon_i,}$
where the errors ε[i], i = 1, ..., n, are independent and all have the same variance σ^2. The residuals are not the true, and unobservable, errors, but rather are estimates, based on the observable
data, of the errors. When the method of least squares is used to estimate α[0] and α[1], then the residuals, unlike the errors, cannot be independent since they satisfy the two constraints
${\displaystyle \sum_{i=1}^n \widehat{\varepsilon}_i=0}$
${\displaystyle \sum_{i=1}^n \widehat{\varepsilon}_i x_i=0.}$
${\displaystyle \varepsilon_i}$ is the ith error, and
${\displaystyle \widehat{\varepsilon}_i}$ is the ith residual.) Moreover, the residuals, unlike the errors, do not all have the same variance: the variance (counter-intuitively) decreases as the
corresponding x-value gets farther from the average x-value. The fact that the variances of the residuals differ, even though the variances of the true errors are all equal to each other, is the
principal reason for the need for Studentization.
How to Studentize[ ]
For this simple model, the design matrix is
${\displaystyle X=\left[\begin{matrix}1 & x_1 \\ \vdots & \vdots \\ 1 & x_n \end{matrix}\right]}$
and the "hat matrix" H is the matrix of the orthogonal projection onto the column space of the design matrix:
${\displaystyle H=X(X^T X)^{-1}X^T.}$
The "leverage" h[ii] is the ith diagonal entry in the hat matrix. The variance of the ith residual is
${\displaystyle \mbox{var}(\widehat{\varepsilon}_i)=\sigma^2(1-h_{ii}).}$
The corresponding Studentized residual is then
${\displaystyle {\widehat{\varepsilon}_i\over \widehat{\sigma} \sqrt{1-h_{ii}\ }}}$
where ${\displaystyle \widehat\sigma}$ is an appropriate estimate of σ.
Internal and external Studentization[ ]
The estimate of σ^2 is
${\displaystyle \widehat{\sigma}^2={1 \over n-m}\sum_{j=1}^n \widehat{\varepsilon}_j^2.}$
where m is the number of parameters in the model (2 in our example). But it is desirable to exclude the ith observation from the process of estimating the variance when one is considering whether the
ith case may be an outlier. Consequently one may use the estimate
${\displaystyle \widehat{\sigma}_{(i)}^2={1 \over n-m-1}\sum_{j=1}^n \widehat{\varepsilon}_j^2,}$
based on all but the ith case. If the latter estimate is used, excluding the ith case, then the residual is said to be externally Studentized; if the former is used, including the ith case, then it
is internally Studentized.
If the errors are independent and normally distributed with expected value 0 and variance σ^2, then the probability distribution of the ith externally Studentized residual is a Student's
t-distribution with n − m − 1 degrees of freedom, and can range from ${\displaystyle -\infty }$ to ${\displaystyle +\infty }$.
On the other hand, the internally Studentized residuals are in the range ${\displaystyle 0 \pm \sqrt{\mathrm{r.d.f.}}}$, where r.d.f. is the number of residual degrees of freedom, namely n − m. If
"i.s.r." represents the internally Studentized residual, and again assuming that the errors are independent identically distributed Gaussian variables, then
${\displaystyle \mathrm{i.s.r.}^2 = \mathrm{r.d.f.}{t^2 \over t^2+\mathrm{r.d.f.}-1}}$
where t is distributed as Student's t-distribution with r.d.f. − 1 degrees of freedom. In fact, this implies that i.s.r.^2/r.d.f. follows the beta distribution B(1/2,(r.d.f. − 1)/2). When r.d.f. = 3,
the internally Studentized residuals are uniformly distributed between ${\displaystyle -\sqrt{3}}$ and ${\displaystyle +\sqrt{3}}$.
If there is only one residual degree of freedom, the above formula for the distribution of internally Studentized residuals doesn't apply. In this case, the i.s.r.'s are all either +1 or -1, with 50%
chance for each.
The standard deviation of the distribution of internally Studentized residuals is always 1, but this does not imply that the standard deviation of all the i.s.r.'s of a particular experiment is 1.
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Grade 7 McGraw Hill Glencoe - Answer Keys Chapter 8: Measure Figures; Lesson 1: Circumference
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Grade 7 McGraw Hill Glencoe - Answer Keys
Chapter 8: Measure Figures; Lesson 1: Circumference
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Find the radius or diameter of each circle with the given dimension.
• Question 1
d = 3 m
• Question 2
r = 14 ft
• Question 3
d = 20 in
Find the circumference of each circle. Use 3.14 or\(\frac{22}{7}\) for \(\pi\). Round to the nearest tenth if necessary.
• Question 4
\(C\approx\) \(\text{m}\)
• Question 5
\(C\approx\) \(\text{yd}\)
• Question 6
Building on the Essential Question A circle has a circumference of about 16.3 meters and a diameter of about 5.2 meters. What is the relationship between the circumference and diameter of this
• Type below:
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School Math
Fullerton District Jr. High/ Middle School Math Tutoring Services
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Our Algebra 2 program builds on the concepts learned in Algebra 1, introducing students to more advanced algebra topics. Students will explore complex equations, polynomials, and functions. Our
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At Optimus Learning School, our Jr. High Math Tutoring Services for the Fullerton School District are designed to support students at every level of their academic journey. Whether your child needs
foundational support in Math 7 or is preparing for advanced courses like Algebra 2 or Trigonometry, we provide the personalized instruction they need to excel. | {"url":"https://www.optimuslearningschool.com/junior-high/math/fullerton/","timestamp":"2024-11-03T06:26:17Z","content_type":"text/html","content_length":"170677","record_id":"<urn:uuid:76f115d8-296a-4c7c-a642-7e697cff028b>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00722.warc.gz"} |
S2DW code documentation
By Jason McEwen and Yves Wiaux
The S2DW code provides functionality to perform the scale discretised wavelet transform on the sphere developed in our paper: Exact reconstruction with directional wavelets on the sphere (ArXiv|DOI).
Routines are provided to compute wavelet and scaling coefficients from the spherical harmonic coefficients of a signal on the sphere and to synthesise the spherical harmonic coefficients of the
original signal from its wavelet and scaling coefficients. The reconstruction of the spherical harmonic coefficients of the original signal is exact to numerical precision. Typically, maximum
reconstruction errors are of the order 10^(-12) or smaller. Please see our paper for further details of the wavelet transform and a discussion of typical reconstruction errors and execution times of
this implementation.
It is considerably more accurate and efficient to perform our wavelet transform on the sphere in harmonic space, hence this is the approach adopted in the S2DW code. The S2DW library itself considers
only the spherical harmonic representation of data defined on the sphere and not real space representations. Many different pixelisations schemes for the sphere exist, with corresponding algorithms
to perform forward and inverse spherical harmonic transforms. These algorithms are not always exact, hence the core functionality of the S2DW code operates on the spherical harmonic coefficients of
signals only. Users are then free to use any pixelisation of the sphere and the computation of spherical harmonic coefficients is the users' concern.
A number of optional utility programs are also provided in the S2DW code. These enable users to perform the scale discretised wavelet transform on data defined on the sphere in real space. The
HEALPix pixelisation of the sphere is adopted for this purpose. The spherical harmonic transform on a HEALPix pixelisation is not exact, hence the reconstruction accuracy of our S2DW code for real
space data is limited by the accuracy of the forward and inverse spherical harmonic transforms provided by HEALPix.
For further details see the README.txt file.
The S2DW library requires only the FFTW and CFITSIO packages. If one wishes to also use the utility programs then the HEALPix and S2 packages are also required. Please note that all S2DW code is
written in Fortran 90 and a suitable Fortran compiler will be required.
Compiling library
Before compiling you may need to edit the makefile to specify your compiler and to link with the appropriate libraries. Once you have set the makefile up for your system then the S2DW library may be
compilied by running:
>> make lib
For details on how to use the S2DW library code see the documentation below.
A test program is provided to test the installation of the S2DW library. To make this program simply run:
>> make test
The test may then be performed by running:
>> make runtest
If you see the message 'Tests passed!' printed then the S2DW library has been installed correctly and is reconstructing the spherical harmonic coefficients of a random test signal exactly from its
wavelet and scaling coefficients. By default the test is performed at a band limit of B=64. To run the test for other band limits run:
>> ./bin/s2dw_test xx
where xx is the band limit of the test.
Compiling utility programs
The utility programs require the HEALPix and S2 packages to handle real space representations of data on the sphere (these are not required for the S2DW library). Before attempting to compile the
utility programs please ensure that these libraries are linked correctly in the makefile. Once these libraries are linked you may compile the S2DW utility programs by running:
>> make prog
For details on how to use the S2DW utility programs see the documentation below.
By default, the S2DW code ships with only this single documentation page. The remainder of the documentation is automatically generated from the source code using f90doc. Please ensure that you have
f90doc installed in order to generate this documentation. Once f90doc is installed the documentation may be generated by running:
>> make docs
Cleaning up
To clean up your version of the S2DW code and return all code to its original state run:
>> make clean
To remove all documentation, except the front page (i.e. this file), run:
>> make cleandocs
Library documentation
The S2DW library contains all functionality to compute the scale discretised wavelet transform on the sphere from the spherical harmonic coefficients of a signal. Real space representations of
signals on the sphere are not considered. An overview of the modules that comprise the S2DW library is given here. Please click on the link for each module for detailed documentation.
s2dw_core_mod: Provides core functionality to perform a scale discretised wavelet transform on the sphere.
s2dw_dl_mod: Functionality to compute specified plane of the Wigner dl matrix.
s2dw_stat_mod: Functionality to compute statistics of wavelet coefficients.
s2dw_error_mod: Functionality to handle errors that may occur in the S2DW library. Public S2DW error codes are defined, with corresponding private error comments and default halt execution status.
s2dw_fileio_mod: Functionality to read and write S2DW formatted fits files containing wavelet and scaling coefficients.
s2dw_types_mod: Definition of intrinsic types and constants used in the S2DW library.
Utility program documentation
The S2DW utility programs provide functionality for dealing with data on the sphere in real space, where the HEALPix pixelisation of the sphere is adopted. Once spherical harmonic coefficients are
computed, the utility programs make use to the S2DW library to perform the scale discretised wavelet transform on the sphere. Please click on the link for each program for detailed documentation.
s2dw_analysis: Computes the S2DW wavelet and scaling coefficients of a Healpix sky map.
s2dw_synthesis: Reconstructs a Healpix sky map from S2DW wavelet and scaling coefficients.
s2dw_test: Performs S2DW transform analysis and synthesis and check that the original signal is reconstructed exactly (to numerical precision). Note that this utility program does not deal with real
space representations of data on the sphere and hence does not require the HEALPix or S2 packages.
s2dw_wav2sky: Converts wavelet coefficients read from a S2DW formatted fits/matlab file to a sky Healpix fits file.
s2dw_wavplot: Computes a Healpix sky map of wavelet for a given j for subsequent plotting.
s2dw_fits2mat: Converts a fits S2DW file containing wavelet and scaling coefficients to a matlab S2DW file.
s2dw_mat2fits: Converts a matlab S2DW file containing wavelet and scaling coefficients to a fits S2DW file.
MATLAB usage
The data associated with a scale discretised wavelet tranform of a signal (including wavelet and scaling coefficients and other parameters) may be written to either a FITS or Matlab m file. The
binary FITS files are considerably smaller and faster to read/write. However, for users who wish to analyse wavelet coefficients in Matlab it is also possible to write matlab readable m files.
Utility routines also exist to convert between these two file types. A Matlab routine to write the data back to an m file from within Matlab is provided in the ./matlab subdirectory:
s2dw_matlab_wav_write.m: Write S2DW data manipulated in Matlab to a S2DW m file that can be read by Matlab and the S2DW Fortran code.
For support or to report any bugs please contact Jason McEwen (jason.mcewen AT ucl.ac.uk).
Authors: J. D. McEwen & Y. Wiaux
Version: 1.1 - August 2013
Version History
• Version 1.1
Parallelised code and implemented some optimisations.
• Version 1.0
Initial public release of S2DW code. Development functionality removed from library and obselete matlab interface functions removed.
• Version 0.3
Dynamic memory allocation of wavelet coefficients added to reduce memory requirements. Functionality to read/write matlab files added. Some further test/development functions removed.
• Version 0.2
Synthesis reordered to provide memory and execution time performance improvements. Dynamic temporary memory allocation added to reduce memory requirements. Some test/development functions
• Version 0.1
Original S2DW version.
S2DW package to compute the scale discretised wavelet transform on the sphere
Copyright (C) 2008 Yves Wiaux & Jason McEwen
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
General Public License for more details (LICENSE.txt).
You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301, USA.
We thank Gilles Puy very much for testing the S2DW code.
The code to compute Wigner dl functions was provided by Daniel Mortlock.
This documentation has been generated largely by f90doc, with some minor modifications.
Last modified: December 2008 | {"url":"http://astro-informatics.github.io/s2dw/","timestamp":"2024-11-11T10:14:18Z","content_type":"text/html","content_length":"13060","record_id":"<urn:uuid:16a9f4f5-e33a-4f57-aa55-897df2caa42c>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00011.warc.gz"} |
Teaching Assistantship | College of Science
2022 Teaching Assistantship
Seungsu Lee Awarded Teaching Assistantship from the University of Utah
Graduate student Seungsu Lee has received a Teaching Assistantship Award from the University of Utah. The award is designed to bolster undergraduate education while providing graduate students with
experience teaching in undergraduate environments. The opportunity is for full-time graduate teaching assistants.
“Receiving the award means a lot to me in different ways,” said Lee. “It tells me that my proposal is effective and will help many people who study math. Also, the award ensures support from the
department and my mentor in implementing my proposal into an actual class. In terms of my career, the award confirms my teaching skills. I learned English as a second language, and I have a strong
Korean accent, so receiving the award proves that one can develop communication and teaching skills to teach mathematics efficiently regardless one’s background.”
"When I teach, I love to communicate with students, tell them what they’re doing correctly, and teaching them how to do mathematical reasoning. In particular, I like the moment when students
understand what I’m teaching about a mathematical concept, and I can see the “aha” moment in their faces."
Lee will be teaching an asynchronous online class for Math 2270—Linear Algebra--and will have responsibility for creating lecture videos for the department website. Asynchronous learning allows an
instructor flexibility in creating a learning environment that will allow for different kinds of learners and learning styles. Lee’s academic advisor is Professor Karl Schwede, and his mentor for the
project is Assistant Professor (Lecturer) Matt Cecil.
“I like to chat about mathematics with other people,” said Lee. “When I teach, I love to communicate with students, tell them what they’re doing correctly, and teaching them how to do mathematical
reasoning. In particular, I like the moment when students understand what I’m teaching about a mathematical concept, and I can see the “aha” moment in their faces.”
When Lee was a child, his father showed him the magic square. The magic square is a square array of numbers in which all the rows, columns, and diagonals add up to the same sum, which is called the
magic constant. This is the fun part in working through the square—you get the same number when you add numbers for each row, column, or even diagonals. “As far as I can remember, the magic square
marked the first time that I ever saw a mathematical puzzle,” said Lee. He was very interested in the algorithm to solve the magic square. As he got older, he started to do more and more math. When
he was in high school, he had a great math teacher, who showed him rigorous ways to think about calculus by using epsilon and delta. This was a turning point for Lee that made him decide to forge a
career in math.
He completed his undergraduate degree at Yonsei University in South Korea. “I got interested in algebraic geometry when I was an undergraduate,” he said. “Unfortunately, my university’s graduate
school didn’t focus on this area of math, so I searched online and was excited to see that the U’s Math Department has a huge research group in algebraic geometry. I was so happy to be accepted to
the department’s program.” After he earns a Ph.D., he plans to seek a research position. | {"url":"https://science.utah.edu/news/seungsu-lee/","timestamp":"2024-11-02T22:09:36Z","content_type":"text/html","content_length":"64115","record_id":"<urn:uuid:74223ab9-bc3f-43cf-b12c-982c6aeb70c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00611.warc.gz"} |
Proof Search on Bilateralist Judgments over Non-deterministic Semantics
TABLEAUX 21's proceedings, Lecture Notes in Computer Science, Springer
The bilateralist approach to logical consequence maintains that judgments of different qualities should be taken into account in determining what-follows-from-what. We argue that such an approach may
be actualized by a two-dimensional notion of entailment induced by semantic structures that also accommodate non-deterministic and partial interpretations, and propose a proof-theoretical apparatus
to reason over bilateralist judgments using symmetrical two-dimensional analytical Hilbert-style calculi. We also provide a proof-search algorithm for finite analytic calculi that runs in at most
exponential time, in general, and in polynomial time when only rules having at most one formula in the succedent are present in the concerned calculus. | {"url":"https://vitorgreati.me/publications/2021-09-02-tableaux-21-psbilatndsem.html","timestamp":"2024-11-03T07:44:12Z","content_type":"text/html","content_length":"8396","record_id":"<urn:uuid:6c7533dd-0566-4576-a266-6f4875a170a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00046.warc.gz"} |
ML Was Hard Until I Learned These 5 Secrets! - TechNewsIndexML Was Hard Until I Learned These 5 Secrets! | TechNewsIndex
Look at all this math and code you need to understand to learn machine learning. It can be very hard, for me at least; it was as well, until I learned these 5 secrets, which honestly aren’t even
secrets, but no one really teaches you them, although everyone should know them. I mean, I spent the last 3 and a half years studying machine learning, and it took me way too long to learn these
secrets on my own.
So let me reveal them to you so that you don’t have to struggle for that long. You’re thinking of math the wrong way around, which already sets you up for failure at the very beginning, but isn’t it
your fault? Back when I started learning machine learning, some of my professors would simply throw a formula on screen and tell us, This is the loss function for a decision tree. And that was it. I
and most of my peers were confused and simply stared at the formula, waiting for a magical aha moment where the formula made sense. I was always asking myself how those smart scientists could
understand math so well that they could develop new algorithms and just think in the language of math. Until I realized I was thinking the wrong way around, I was focusing too much on the actual
mathematical formulas in the realm of math instead of taking a step back and thinking like a scientist. I mean, I was literally looking at a formula and trying to understand the formula as a whole,
which for me now, after learning the secret, just doesn’t make any sense anymore. Don’t think of math as something abstract; make it human-interpretable. You need to realize that you need to think
the other way around. Think of the idea a human had, understand it, and then think of how to translate it into the language of math.
This may sound very confusing, but math is not a standalone language in which people think. Scientists think just like you and me, in natural language. They just know how to translate their ideas
into their formalisms of math, which then allows them to be implemented and then further developed using the rules of math. As mentioned, I was always looking at a formula as a whole, but each
component of a formula is just a component of this human idea. For example, a sum or a product are literally just a for loop that can have some conditions that are literally equivalent to an if-else
statement in code. Of course, this is easier said than done, and to understand a mathematical concept using human ideas requires someone to actually properly teach you these human ideas and how to
translate them step by step. But in my experience, there are two scenarios. One, the teacher does that already, but you don’t understand why he does it because you were never explicitly taught to
think that way, or two, the teacher really only looks at the formulas and derivations. In that case, you need to try to figure out the original human idea yourself by, for example, looking it up
online, but the good thing is that you now know it is not your fault and there is an intuitive understanding of the math that you can find. Math is just the formalization of a human idea. Very few
people actually think in the language of math; it’s just a tool, but when it comes to every intermediate derivation step, you often actually think in the language of math, which is very difficult
unless you know the next secret. This secret literally changed the way I look at scary math derivations like this one. Again, jumping back in time to when I watched a lecture at college, my professor
would explain the intuitive idea of an ML algorithm, show the translation into the language of math, and then show us where we want the formula to end up to make it more efficient or simply actually
work as an algorithm that can be implemented.
But at some point he would go off on a derivation spree, writing out one step after the other and expecting us to understand why he did what he did. Everyone was confused and, of course, scared and
annoyed. But these derivations are simpler than you might think. Not easy, but much simpler to execute after learning this one secret. I realized each step was simply applying one specific rule or
definition. I realized that, up to a certain degree, these mathematical derivations or transformations just require you to have a list of rules and tricks you need to collect that you can then simply
apply. During the lectures, for each step I saw, I would explicitly look for the rule and definition they used and write that down on my list. When solving or reading math derivations on my own, for
homework assignments, or during an exam, in most cases, I would literally just do some sort of pattern matching. I would look at where I currently am, go down my list of rules and definitions, and
apply what fits the pattern. Of course, some patterns, rules, and definitions are harder to spot than others, but after doing this for long enough, you just start to memorize certain patterns. But in
general, for most ML math, this secret technique does work wonders. You need to collect your mathematical toolkit and learn to recognize when you can apply a rule. Which means practice, practice,
practice. But math is, of course, far from all there is to ML. Coding is also a very challenging skill. Learning the basics of Python and then an ML library like PyTorch is really cool and fun. You
simply follow a tutorial online and have a really steep learning curve. You follow a recipe of steps and implement a lot of code. You really see and feel the progress you are making. But then, when
you want to go further and learn to implement actual algorithms or ML pipelines on your own, you hit a wall. All of a sudden, you sit on one annoying problem for several hours, have written perhaps
five lines of code, and you think you are not making any progress. This is very, very frustrating and is the point where a lot of people determine that coding is really hard and that they will never
be able to really learn to code. Writing five lines of code in three hours is pathetic. I mean, I always thought I was so stupid for writing code that never worked until I debugged it for hours. This
can get so bad that you don’t even want to start coding because you know it will fail. But that is simply the wrong way to think. Actually, writing code for one hour will very likely mean, let’s say,
three hours of debugging. Once I learned that this is what it really means to be coding, I suddenly felt so relieved and not stupid anymore.
Coding ML models didn’t feel impossible or hard anymore because I was doing exactly what was normal and expected. And nowadays, there are amazing tools that I can’t live without, like github copilot,
that can generate code for you and explain code for you. But there is so much that you simply learn through your own experience or the experience of others. That’s why I have a completely free weekly
newsletter where I share my experience as a machine learning researcher, including actionable tips, AI news, and more. I’ll just pin a comment below with the link to sign up. But anyway, you have to
realize that writing code is not actually coding; debugging is coding. This realization really helps you with implementing things on your own, step by step. But when you have to work with an existing
code base where you have everything at once, you will probably still be overwhelmed. So let’s look at the next secret tip. There are two cases where you will need to understand complex code. The
first one is when building on top of an existing repository. I remember when I started working on my first larger project, where I built on top of an existing code base. It was so much code. I
literally had no idea where to start. Just like in the previous secret, I again felt like learning to read code was as impossible as writing code. All those tutorials and smaller personal projects
didn’t prepare me for this much code. I tried reading each source file and started to write code as soon as possible into places I thought made sense, but that unsurprisingly led to a lot of
headaches and wasting time writing code that was destined to fail. I had to learn the hard way that there is a very simple strategy for understanding large code bases. I wish someone would have
simply told me once how to approach a challenge like this. With most large ML repositories, you have a train.py and an eval.py file. Those should always be the starting points. I find those files,
set a breakpoint in the beginning, and start stepping through the code with the debugger. I cannot emphasize enough how simply yet insanely effective this technique is. It’s literally like cheating.
You can step through the data preprocessing, the training loop, the actual model, the evaluation metrics, and every other detail. Depending on the codebase and your experience, this takes just a few
hours, and you will have an amazing overview of the codebase and a much better feeling for where to add the new code for your own idea. That said, you might not always want to build on top of an
existing highly optimized codebase but simply want to understand an algorithm better. For example, when you want to understand PPO, a famous reinforcement learning algorithm, I would not recommend
looking at the optimized implementation; that’s way too overkill and complex. Luckily, for many important models, there are minimal educational implementations that just implement the main idea so
that people can understand the model. And here, yet again, the best way is to set a breakpoint at the beginning of the main function and then just start debugging.
Finally, there is one fundamental secret to mastering machine learning that you need to know. This final secret ties everything we just discussed together and is the one reason that will determine
your success or failure in mastering ML. 34% of organizations consider poor AI skills, expertise, or knowledge as the top reason blocking successful AI adoption, according to an IBM study from 2022.
Why do you think people fail to learn machine learning and fall into the category of people with poor AI skills? Is it because it is hard? Yes, but it was also hard for every person who has now
mastered it. People fail to master ML because they stop learning machine learning too early and give up. And why do people give up? They have false expectations and don’t enjoy the process of
learning. They think mastering ML is hard because they didn’t learn it in a few weeks. Or, because they didn’t understand a video explaining an ML concept the first time, they will never understand
it. I took my first introductory AI college course about 3 and a half years ago. After that semester, I took my first real ML course, along with working on my first ML projects. After that semester,
I continued with my first deep learning courses and continued working on projects and reading a lot of papers. I definitely didn’t understand everything the first time, but I knew that was normal.
Over time, I learned the secrets mentioned before and that somewhat mastering ML takes time. I failed my first ML interviews for internships at Amazon, Neuro, and Google DeepMind, but now I am
working with an ex-meta professor and collaborating with a Google DeepMind researcher. It takes time. Period. This is not a skill you learn over a weekend. The 10,000-hour rule applies here as well;
if you spend 10,000 hours on a specific skill, you will master it. I’m absolutely not trying to discourage you, but rather the opposite. I don’t want to be some weird influencer who wants to sell you
a dream of mastering ML in a few weeks. I want to encourage you to really learn machine learning, to really learn the theory, and to really gather practical experience. And the mastery of machine
learning comes after learning fundamentals by really working on projects, encountering real-world problems, and reading real-state-of-the art papers or blog posts.
By having this expectation that it will take time, you relax way more, and the learning process becomes much easier, more enjoyable, and more successful in the end. All these secrets are universally
true, no matter how you decide to learn machine learning. And I say that because there are mainly three ways to do so. | {"url":"https://technewsindex.com/artificial-intelligence/ml-was-hard-until-i-learned-these-5-secrets/","timestamp":"2024-11-09T22:16:23Z","content_type":"text/html","content_length":"81556","record_id":"<urn:uuid:c79d0008-af49-4c16-85b4-5406de62ef57>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00719.warc.gz"} |
Proposal for clustering functions
geometry[] ST_ClusterIntersecting(geometry geom)
Aggregate function returning an array of GeometryCollections representing the connected components of a set of geometries.
• accepts [Multi]Point, [Multi]LineString, [Multi]Polygon geometries of any type that can be converted into GEOS (I can't think of a situation where [Multi]Point would be useful, but that doesn't
mean there isn't one…)
• return a geometry array (my current implementation returns a GeometryCollection, but the recursive semantics of ST_Dump then undo all of the hard work)
Example: if run on a table containing all of the LineStrings in the image below, would return an array with two MultiLineString geometries (red and blue)
geometry[] ST_ClusterWithin(geometry geom, double precision distance)
Aggregate function returning an array of GeometryCollections?/MultiPoints?, where any component is reachable from any other component with jump of no more than the specified distance.
• like ST_ClusterIntersecting, but uses a distance threshold rather than intersection when determining if two geometries should be included in the same component. Could have an implementation very
similar to ST_ClusterIntersecting, or could be restricted to points and maybe have a more efficient implementation.
• differs from k-means in that a distance is provided, not a number of clusters
Example: In the picture below, an array of five MultiPoints would be returned (color-coded). The threshold distance in this case was more than the orange line but less than the pink line.
for help on using the wiki. | {"url":"https://trac.osgeo.org/postgis/wiki/DevClusteringFunctions","timestamp":"2024-11-09T10:55:44Z","content_type":"text/html","content_length":"11244","record_id":"<urn:uuid:24ac63d2-d1a9-4775-9fdc-aec7663bebf6>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00753.warc.gz"} |
Faculty Directory | VCU Department of Mathematics and Applied Mathematics
Joy Whitenack
Emeritus Professor
• Ed.D. in Mathematics Education, 1995, Peabody College of Vanderbilt University
• M.S. in Mathematics, 1991, Middle Tennessee State University
• B.A. in Mathematics, 1987, Christopher Newport University
• B.S. in Elementary Education, 1978, James Madison University
Research Interests
Whitenack's research interests include supporting elementary mathematics specialists’ transition into leadership roles, teacher learning, instructional design theory and children’s understanding of
Select Publications
• Whitenack, J. W., Cavey, L. O. & Henney, C. (in press). Empowering young mathematical learners: A guide for parents and teachers. Reston, VA: National Council of Teachers of Mathematics.
• Whitenack, J. W., Cavey, L. O. & Ellington, A. J. (2014). The role of framing in productive classroom discussions: A case for teacher learning. Journal of Mathematical Behavior, 33, 42-55.
• Moreau, D., & Whitenack, J. W. (2013). Coaching individual teachers. In Campbell, P., A. Ellington, W. Haver, & (Eds.), The elementary mathematics specialist’s handbook (pp. 31–49). Reston, VA:
National Council of Teachers of Mathematics.
• Minervino, S., Robertson, P, & Whitenack, J. W. (2013). Turning challenges into opportunities. In Campbell, P., A. Ellington, V. Inge, & W. Haver (Eds.), The elementary mathematics specialist’s
handbook (pp. 177–189). Reston, VA: National Council of Teachers of Mathematics.
• Whitenack, J. W. & Ellington, A. J. (2013). Supporting middle school mathematics specialists’ work: A case for learning and changing teachers’ perspectives. The Mathematics Enthusiast, 10(3),
• Ellington, A. J. & Whitenack, J. W. (2010, May). Fractions and the funky cookie. Teaching Children Mathematics, 532-539.
• Cavey, L. O., Whitenack, J. W., & Lovin, L. (2007). Investigating teachers’ mathematics teaching understanding: A case for coordinating perspectives, Educational Studies in Mathematics, 64,
• Whitenack, J. W., & Yackel, E. (2002, May). Making mathematical arguments in the primary grades: The importance of explaining and justifying one’s ideas. Teaching Children Mathematics,8(9),
• Whitenack, J. W., Knipping, N. & Novinger, S. (2001). Coordinating theories of learning to account for second-grade children’s arithmetical understandings. Mathematical Thinking and Learning,
• Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analysis of classroom video recordings and transcripts. Educational Studies in Mathematics, 30, 213-228.
• National Council of Teachers of Mathematics
• The North American Group for the Psychology of Mathematics Education
• Virginia Council of Teachers of Mathematics
• Greater Richmond Council of Teachers of Mathematics
Professional Activities
• Virginia Council of Teachers of Mathematics, Board member and Chair of the Scholarship Committee
• Greater Richmond Council of Teachers of Mathematics, Board member | {"url":"https://math.vcu.edu/directory/faculty/whitenack.html","timestamp":"2024-11-05T12:49:06Z","content_type":"text/html","content_length":"26803","record_id":"<urn:uuid:8898cef0-07a4-4ff7-8548-6cc3fc6cf66a>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00371.warc.gz"} |
Understanding Machine Learning
Today Georgia Tech had the launch event for our new
Machine Learning Center
. A panel discussion talked about different challenges in machine learning across the whole university but one common theme emerged: Many machine learning algorithms seem to work very well but we
don't know why. If you look at a neural net (basically a weighted circuit of threshold gates) trained for say voice recognition, it's very hard to understand why it makes the choices it makes.
Obfuscation at its finest.
Why should we care? A few reasons:
• Trust: How do we know that the neural net is acting correctly? Beyond checking input/output pairs we can't do any other analysis. Different applications have a different level of trust. It's okay
if Netflix makes a bad movie recommendation, but if a self-driving car makes a mistake...
• Fairness: Many examples abound of algorithms trained on data will learn intended or unintended biases in that data. If you don't understand the program how do figure out the biases?
• Security: If you use machine learning to monitor systems for security, you won't know what exploits still might exist, especially if your adversary is being adaptive. If you can understand the
code you could spot and fix security leaks. Of course if the adversary had the code, they might find exploits.
• Cause and Effect: Right now at best you can check that a machine learning algorithm only correlates with the kind of output you desire. Understanding the code might help us understan the
causality in the data, leading to better science and medicine.
What if P = NP? Would that help. Actually it would makes things worse. If you had a quick algorithm for NP-complete problems, you could use it to find the smallest possible circuit for say matching
or traveling salesman but you would have no clue why that circuit works.
Sometimes I feel we put to much pressure on the machines. When we deal with humans, for example when we hire people, we have to trust them, assume they are fair, play by the rules without at all
understanding their internal thinking mechanisms. And we're a long way from figuring out cause and effect in people.
10 comments:
1. If P=NP you could also find the shortest proof in your favorite formal system that the smallest possible circuit does what you wanted it to do, as well as any other claim you are wondering that
may be true about the circuit. That proof might not be comprehensible to you, but it could be written in a format where proof assistant software such as HOL or Coq could parse it and convince you
it is correct. So if P=NP (with feasible low constants) I think that would definitely help.
1. no, correctness proofs can still be unfeasibily long.
2. Obviously correctness proofs can be unfeasibly long. Those would also be among those explanations of program behavior that we cannot understand. I was addressing Lance's claim that:
"What if P = NP? Would that help. Actually it would makes things worse. If you had a quick algorithm for NP-complete problems, you could use it to find the smallest possible circuit for say
matching or traveling salesman but you would have no clue why that circuit works."
My point was simple: if short understandable explanations of a circuit's behavior exist at all, then we could find those explanations under P=NP (with the usual caveats of low-order
constants). So P=NP would arguably help, not make things worse.
3. How are both statements together consistent? 1. "correctness proofs can still be unfeasibily long". 2. "proof assistant software such as HOL or Coq could parse it and convince you it is
correct". Sorry do not understand implications of P=NP
2. Lance concludes "… we're a long way from figuring out cause and effect in people."
The NIH web site "ClinicalTrials.gov" provides a concrete overview of our present understanding of "cause and effect in people".
For example, a search of clinical trials concerning "personality" and "therapy", restricted to clinical trials presently recruiting volunteers, yields a peer-reviewed listing — a listing that
even can be conveniently alphabetized — of conditions in respect to which an understanding of "cause and effect in people" is objectively lacking.
Q Are we to infer that cognitive scientists, complexity theorists, and psychiatric physicians soon will be reading each other's literature?
A They already are! Not often in STEAM-history, does cross-pollination of trans-disciplinary ideas occur as vigorously, fertilely, dangerously, even distressingly, and none-the-less excitingly as
at present.
Q Does this mean that programmers increasingly are consciously acting as (informatic) therapists to their neural networks, and concomitantly, psychiatrists increasingly are consciously acting as
(epigenetic/connectomic) programmers to their human clients?
A For more-and-more STEAM-workers — young ones especially — the plain-and-simple answer is "yes (obviously)".
Whether welcome or not (opinions differ, obviously), isn't this transformative unification empirically evident (nowadays) to pretty much everyone?
3. To your list of reasons to care, I would add another: that the problem of understanding which learning tasks are feasible for these methods is a fascinating and important one from a purely
intellectual point of view.
4. Given a number x and a set S of n positive integers, MINIMUM is the problem of deciding whether x is the minimum of S. We can easily obtain an upper bound of n comparisons: find the minimum in
the set and check whether the result is equal to x. Is this the best we can do? Yes, since we can obtain a lower bound of (n - 1) comparisons for the problem of determining the minimum and
another obligatory comparison for checking whether that minimum is equal to x. A representation of a set S with n positive integers is a Boolean circuit C, such that C accepts the binary
representation of a bit integer i if and only if i is in S. Given a positive integer x and a Boolean circuit C, we define SUCCINCT-MINIMUM as the problem of deciding whether x is the minimum bit
integer which accepts C as input. For certain kind of SUCCINCT-MINIMUM instances, the input (x, C) is exponentially more succinct than the cardinality of the set S that represents C. Since we
prove that SUCCINCT-MINIMUM is at least as hard as MINIMUM in order to the cardinality of S, then we could not decide every instance of SUCCINCT-MINIMUM in polynomial time. If some instance (x,
C) is not in SUCCINCT-MINIMUM, then it would exist a positive integer y such that y < x and C accepts the bit integer y. Since we can evaluate whether C accepts the bit integer y in polynomial
time and we have that y is polynomially bounded by x, then we can confirm SUCCINCT-MINIMUM is in coNP. If any single coNP problem cannot be solved in polynomial time, then P is not equal to coNP.
Certainly, P = NP implies P = coNP because P is closed under complement, and therefore, we can conclude P is not equal to NP.
You could read the details in:
5. I will remove the link http://vixra.org/pdf/1704.0335v1.pdf and use instead:
6. A video in youtube with the explanation of my P versus NP solution!!!
7. https://www.youtube.com/watch?v=W5-Xb9fd4JM | {"url":"https://blog.computationalcomplexity.org/2017/04/understanding-machine-learning.html?m=1","timestamp":"2024-11-08T12:26:08Z","content_type":"application/xhtml+xml","content_length":"78264","record_id":"<urn:uuid:6e13e5b7-0e4b-4af2-a639-b6aac7f556ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00645.warc.gz"} |
Math Expression: Learn about improper fractions and mixed fractions
Improper Fractions and Mixed Fractions
Lesson Objective
In this lesson, we will learn about improper fractions and mixed fractions. Also, we will see how we can convert from improper to mixed fraction and vice versa.
About This Lesson
In understanding fractions, we had seen some ideas behind improper fractions and mixed fractions.
This lesson will explain these two types of fractions in detail and show how they are related.
Next, we will learn a method to quickly convert between these two types of fractions.
You can proceed by reading the study tips first or watch the math video. You can try out the practice questions after that.
Study Tips
Tip #1 - Understand the difference
An improper fraction can be converted into a mixed fraction. Note that, these two fractions are equivalent. The only difference is at the way they are written. See the picture below:
The math video below will explain more about it.
Tip #2 - Improper to Mixed Fractions
To quickly convert an improper fraction to a mixed fraction, we can use the 'long division' method. The picture below shows an example on converting ^11/[5].
The math video below and the practice questions will explain this in detail.
Tip #3 - Mixed to Improper Fractions
To quickly convert a mixed fraction to an improper fraction, we use the steps shown in the picture below.
Below is an example on converting 2 ^1/[5].
The math video below and the practice questions will explain this in detail.
Math Video
Click play to watch video
Please support us by downloading our Fraction Basics app and subscribe to get all 12 video lessons and All Access pass to 8 Zapzapmath Home apps with 180 math games from as low as US$1.67/month:
Apple App Store (iOS) | Google Play (Android)
Math Video Transcript
In this lesson, we will learn about improper fractions, and mixed fractions.
Also, we will see how we can convert improper to mixed fraction, and vice versa.
Consider this fraction, 3 over 5.
Now, we can visually represent this fraction, with this long piece of bar.
Since the denominator is 5, we can divide this bar into 5 equal parts.
Next, with the numerator as 3, 3 out of 5 parts can be colored green.
Now, since the numerator is smaller than the denominator, this fraction is a proper fraction.
Alright, let's increase the numerator from 3, 4, 5.
Note that, from 5 onwards, this fraction is now considered as an I.F, because the numerator, is equals or greater than the denominator.
Let's further increase the numerator of this improper fraction until 11.
Now, if we observe carefully, we can actually use these bars to convert this I.F, to M.F.
Here's how.
Since, all the parts in this 2 bars are green, these bars can be considered as 2 whole green bars.
As for the remaining bar, we have 1 out of 5 part colored as green.
So here, is the mixed fraction 2, 1 over 5, converted from I.F, 11 over 5.
As you can see, using these bars to convert I.F to M.F, is quite tedious.
Therefore, we need to learn a quicker way of doing this.
Here's how we can quickly convert the I.F, 11 over 5, to a M.F.
First, we know that 11 over 5 is the same as 11 divides 5.
So, by doing the division, we get the quotient as 2, which is actually the whole number for the mixed fraction.
Next, 2 multiply by 5 gives 10. 11 minus 10 gives the remainder as 1.
This remainder, 1, becomes the mixed fraction numerator, and it is actually the green part here.
Here, we can see that, we had successfully converted this improper fraction to mixed fraction.
Next, let's convert this M.F back to I.F.
First, we multiply 5 with 2. This gives 10.
This multiplication is actually the same as, finding the 10 green colored parts here.
Next, notice that, there is 1 more part to include.
We can include it by adding, 10 with 1. This gives 11, where it is actually the I.F's numerator.
Here, we had successfully done the conversion from M.F to I.F.
That is all for this lesson. Try out the practice question to test your understanding.
Practice Questions & More
Multiple Choice Questions (MCQ)
Now, let's try some MCQ questions to understand this lesson better.
You can start by going through the series of questions on improper fractions and mixed fractions or pick your choice of question below.
• Question 1 on converting improper to mixed fractions
• Question 2 on converting mixed to improper fractions
Site-Search and Q&A Library
Please feel free to visit the Q&A Library. You can read the Q&As listed in any of the available categories such as Algebra, Graphs, Exponents and more. Also, you can submit math question, share or
give comments there. | {"url":"https://www.mathexpression.com/improper-fractions.html","timestamp":"2024-11-08T21:13:50Z","content_type":"text/html","content_length":"34945","record_id":"<urn:uuid:bec8f468-4195-4e9b-ba0f-0e570e350781>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00480.warc.gz"} |
Motion Vectors for hardware particles
I am trying to write an expression which will mimic the mv2DToxik motion vectors, but for hardware particles. Therefore removing the need to instance some geo to get the motion vectors to render.
I have not got very far before I came across some vector maths...
Here is the setup which works for a camera pointing exactly down the z-axis.
Here is the render loaded into Nuke. Notice the RGB values.
I now need to find a way to convert World Velocity to Screen Space Velocity.
In a MEL expression. Hmmm, time to ask the
After some great advice from Zoharl, I grabbed some code which uses the camera's worldInverseMatrix to transform the velocity vector.
Here is the expression:
float $mult=0.5;
//get the particle's World Space velocity
vector $vel=particleShape1.worldVelocity;
float $xVel=$vel.x;
float $yVel=$vel.y;
float $zVel=$vel.z;
// create particle's velocity matrix which is in World Space
matrix $WSvel[1][4]=<<$xVel,$yVel,$zVel,1>>;
// get the camera's World Inverse Matrix
float $v[]=`getAttr camera1.worldInverseMatrix`;
matrix $camWIM[4][4]=<< $v[ 0], $v[ 1], $v[ 2], $v[ 3]; $v[ 4], $v[ 5], $v[ 6], $v[ 7]; $v[ 8], $v[ 9], $v[10], $v[11]; $v[12], $v[13], $v[14], $v[15] >>;
//multiply particle's velocity matrix by the camera's World Inverse Matrix to get the velocity in Screen Space
matrix $SSvel[1][4]=$WSvel * $camWIM;
vector $result = <<$SSvel[0][0],$SSvel[0][1],$SSvel[0][2]>>;
float $xResult = $mult * $result.x;
float $yResult = $mult * $result.y;
float $zResult = $mult * $result.z;
So far it seems to be working, but I will try to test it and see if it breaks down.
Thanks to Zoharl on the CGTalk forum and to
who came up with the original matrix manipulation code. | {"url":"http://www.particle-effects.com/2011/11/motion-vectors-for-hardware-particles.html","timestamp":"2024-11-09T12:57:45Z","content_type":"text/html","content_length":"47265","record_id":"<urn:uuid:0ef035ac-cf02-4327-bde4-61b2b3579e9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00012.warc.gz"} |
How much power is produced if a voltage of 12 V is applied to a circuit with a resistance of 24 Omega? | Socratic
How much power is produced if a voltage of #12 V# is applied to a circuit with a resistance of #24 Omega#?
2 Answers
There are 3 variations of the power formula:
$P = V \cdot I , \text{ "P = I^2*R, " and } P = {V}^{2} / R$
If you know any one of them, the others are available thru the use of Ohm"s Law and a bit of algebra.
For this problem, use $P = {V}^{2} / R$
$P = {\left(12 V\right)}^{2} / \left(24 \Omega\right) = 6 {V}^{2} / \Omega = 6 W$
I hope this helps,
Power in this case is given by the equation:
$P = {V}^{2} / R$
□ $P$ is the power in watts
□ $V$ is the voltage in volts
□ $R$ is the resistance in ohms
So, we get:
$P = {\left(12 \setminus \text{V}\right)}^{2} / \left(24 \setminus \Omega\right)$
$= \frac{144 \setminus {\text{V}}^{2}}{24 \setminus \Omega}$
$= 6 \setminus \text{W}$
Impact of this question
1976 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/how-much-power-is-produced-if-a-voltage-of-12-v-is-applied-to-a-circuit-with-a-r-1#637345","timestamp":"2024-11-13T11:52:42Z","content_type":"text/html","content_length":"35276","record_id":"<urn:uuid:43b7a96f-b4da-41a9-8136-6fa517ac2233>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00758.warc.gz"} |
A student sits on a rotating stool holding two 2.9-kg objects.
When his arms are extended...
Answer #1
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before J
after J
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Hacking the slot machine
Wired reports that a Russian group designed a brilliant slot machine cheat that they’ve used to bilk casinos out of millions of dollars. The article is sketchy on technical details, but there’s
enough information there for me to speculate how it was done.
Understand, I don’t know anything about the internals of the software running on these machines, but I know enough about pseudorandom number generators, their use and misuse, to offer a plausible
explanation of the vulnerability and how it’s exploited. I also know a few people in the industry. What I describe below is possible, and from my experience quite likely to have happened. Whether
it’s exactly what happened, or if it’s even close to the mark, I have no way of knowing.
First, a little background.
In modern computer controlled slot machines (say, anything built in the last 30 years), the machine uses random numbers to determine the results of a spin. In concept, this is like rolling dice, but
the number of possibilities is huge: on the order of about four billion. In theory, every one of those four billion outcomes is equally likely every time you roll the dice. That would be true in
practice if the computer were using truly random numbers.
But computers are deterministic; they don’t do random. Instead, they use algorithms that simulate randomness. As a group, these algorithms are called pseudorandom number generators, or PRNGs. You can
probably guess that PRNGs differ in how well they simulate true randomness. They also differ in ease of implementation, speed, and something called “period.” You see, a PRNG is just a mathematical
way to define a deterministic, finite sequence. Given a starting state (called the seed), the PRNG will generate a finite set of values before it “wraps around” to the beginning and starts generating
the same sequence all over again. Period is the number of values generated before wrapping. If you know what PRNG is being used and you know the initial state (seed), then you know the sequence of
numbers that will be generated.
The machines in question were purchased on the secondary market sometime before 2009. It’s probably safe to say that the machines were manufactured sometime between 1995 and 2005. During that era,
the machines were almost certainly running 32 bit processors, and likely were generating 32 bit random numbers using PRNGs that maintained 32 bits of state. That means that there are 2^32 (four
billion and change) possible starting states, each of which has a maximum period of 2^32. Overall, there are 2^64 possible states for the machine to be in. That’s a huge number, but it’s possible to
compute and store every possible sequence so that, if somebody gave you a few dozen random numbers in sequence, you could find out where they came from and predict the next number. It’d take a few
days and a few terabytes of disk storage to pre-compute all that, but you’d only have to do it once.
It’s likely that the PRNG used in these machines is a linear congruential generator which, although potentially good if implemented well, is easy to reverse-engineer. That is, given a relatively
small sequence of generated numbers, it’s possible to compute the seed value and predict the next values in the sequence. All this can be done without knowing exactly which specific LCG algorithm is
being used.
The hackers did have the source code of the program (or they disassembled the ROM), but they didn’t have access to the raw numbers as they were generated. Instead, they had to deduce the random
number based on the outcome of the spin. But again, that just takes a little (okay, more than just a little, but not too much) computation time to create a table that maps reel positions to the
random sequence.
My understanding is that slot machines continually generate random numbers on a schedule, even when the machine is idle. Every few milliseconds, a new number is generated. If it’s not used, then it’s
discarded and the next number is generated. So if you know where you are in the sequence at a given time, then you can predict the number that will be generated at any time in the future. Assuming,
of course, that your clock is synchronized with the slot machine’s clock.
If you refer back to the article, you’ll see that the agent who was working the machine would record the results of several spins, then walk away and consult his phone for a while before coming back
to play. That phone consultation was almost certainly uploading the recorded information to the central site, which would crunch the numbers to determine where the machine was in the random sequence.
The system knows which numbers in the sequence correspond to high payouts, so it can tell the phone app when to expect them. The agent then goes back to the machine and watches his phone while
hovering his finger over the spin button. When the phone says spin, he hits the button.
The system isn’t perfect. With perhaps up to 200 random numbers being generated every second, and human reaction time being somewhat variable, no player will hit the big payout every time. But he’s
increased his odds tremendously. Imagine somebody throwing one gold coin into a bucket of a million other coins, and another gold coin into a bucket of 200 other coins. You’re given the choice to
blindly choose from one of the two buckets. Which would you choose from?
That might all sound complicated, but it’s really pretty simple in concept. All they did was create a map of the possibilities and devise a way to locate themselves on the map. Once you know where
you are on the map, then the rest is a simple matter of counting your steps. Creating the map and the location algorithm likely took some doing, but conceptually it’s very simple.
The above explanation is overly broad, I’ll admit, and I wave my hand over a number of technical details, but people at work with whom I’ve discussed this generally agree that this is, at least in
broad strokes, how the hackers did it. Understand, I work at a company that develops slot machine games for mobile devices, and several of the programmers here used to work for companies that make
real slot machines. They know how these machines work.
When I originally saw the article, I assumed that some programmer had made a mistake in coding or using the PRNG. But after thinking about it more critically, I believe that these machines are
representative of the state of the art in that era (1995-2005). I don’t think there was a design or implementation failure here. The only failure would be that of not imagining that in a few years it
would be possible for somebody who didn’t have a supercomputer to compute the current machine state in a few minutes and exploit that knowledge. This isn’t a story about incompetence on the part of
the game programmers, but rather a story about the cleverness of the crooks who pulled it off. I can admire the technical prowess it took to achieve the hack while still condemning the act itself and
the people who perpetrated it. | {"url":"https://blog.mischel.com/2017/02/07/hacking-the-slot-machine/","timestamp":"2024-11-07T07:01:39Z","content_type":"text/html","content_length":"30802","record_id":"<urn:uuid:70f0cd4b-d293-4613-abaa-7047ebf56af4>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00150.warc.gz"} |
Tegmark's mathematical universe hypothesis
I am sorry for the misunderstanding, I did not mean to refuse discussing the idea here. I stated that it is inefficient to list all the information here again. you said you don't want to look at
my website, I said ok see the FQXI ones and I could have elaborated on any question/comment that you might have had.
The reality is no one will take your work seriously, if you have a website but have never published in a reputable journal.
Is there no one in academia you can reach out to?
The reality is no one will take your work seriously, if you have a website but have never published in a reputable journal.
Is there no one in academia you can reach out to?
Good question. I know what you are saying and it is true. Yes I am planning on a blitz to contact people in the academia and find some sympathetic ones. I have tried half heartedly to contact some
professors at my country's university but I found them not suitable. Still I have one in mind that I am saving him when I feel that I have reached a good convincing stage. Actually he is well known,
I remember he co-authored a paper that made a stir in the physics community
Good question. I know what you are saying and it is true. Yes I am planning on a blitz to contact people in the academia and find some sympathetic ones. I have tried half heartedly to contact
some professors at my country's university but I found them not suitable. Still I have one in mind that I am saving him when I feel that I have reached a good convincing stage. Actually he is
well known, I remember he co-authored a paper that made a stir in the physics community
What about your thesis supervisor? Or a lecturer at uni you got on with?
What about your thesis supervisor? Or a lecturer at uni you got on with?
I did my masters in 1987, my Advisors are long Dead (Peter Unsworth RIP, best advisor and a friend). he was so funny , I still remember his jokes.
Last edited:
I did my masters in 1987, my Advisors are long Dead (Peter Unsworth RIP, best advisor and a friend). he was so funny , I still remember his jokes.
Ok so you are about 60, never published. This is going to be difficult for you unless you find that ally.
Perhaps there is something you can patent?
Ok so you are about 60, never published. This is going to be difficult for you unless you find that ally.
Perhaps there is something you can patent?
I appreciate your kind help. I am more push to 70 with a start of a brain meltdown
I will touch on the results that I have obtained which can pickup Fine Structure Constant (FSC) automatically. Even in my system which I believe it to be fundamental the FSC is a maddening
number, its like a water in an ocean it is everywhere but when you try to catch it it seeps through your hand
FSC alpha=.007297352568 , 1/alpha= 137.0359991 almost 137.036
so the trick that I could use and came naturally is to simulate and obtain two curves that crossed each other and when solved it gave an approximate value for FSC using wolfram alpha. of course I
have simulation so the curves are constructed by curve fitting resultant points. So the data can be fit to multiple appropriate curves but non of them can give high accuracy because the
simulation is based on PRNG and
no matter how much iteration or number of point you make still you get errors
due to simulation and curve fitting accuracy.
https://www.wolframalpha.com/input?i=solve y=x^2, y=397.1161/x+549*exp(-19.57724/x) for x,y https://www.wolframalpha.com/input?i=solve y=12.0163*x-3.18925, y=1.00*x^2 for x,y
and its final result
https://www.wolframalpha.com/input?i=(13801296569 + 120163 sqrt(13163446569))/200000000&assumption="ClashPrefs" -> {"Math"}
Note I am using the pro version if you have a problem but I think it is not a problem to use it with the free version. please tell me if you have a problem it can be overcome.
I just want to clarify a point in this post. The errors quickly diminish as the number of iterations and simulated point are increased, however as accuracy increases but after certain increase in
iterations and points it will become harder and harder to increase the accuracy (i.e. more accurate significant figures) as anybody who has done PRNG simulation would have noticed. | {"url":"https://sciforums.com/threads/tegmarks-mathematical-universe-hypothesis.166226/page-6","timestamp":"2024-11-07T06:34:58Z","content_type":"text/html","content_length":"86686","record_id":"<urn:uuid:97fea2ab-98f0-4b99-a197-85dddaeffe39>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00397.warc.gz"} |
Resources :: Algorithmic Combinatorics on Words REU
F. Blanchet-Sadri, Algorithmic Combinatorics on Partial Words, Chapman & Hall/CRC Press, Boca Raton, FL, 2008.
Book Chapters
F. Blanchet-Sadri, “Open Problems on Partial Words,” In G. Bel-Enguix, M.D. Jimenez-Lopez and C. Martin-Vide (Eds.), New Developments in Formal Languages and Applications, Ch. 2, Vol. 3,
Springer-Verlag, Berlin, Heidelberg, 2008, pp 11-58.
F. Blanchet-Sadri, B. Blakeley, J. Gunter, S. Simmons and E. Weissenstein, “Classifying All Avoidable Sets of Partial Words of Size Two,” In C. Martin-Vide (Ed.), Scientific Applications of
Language Methods, Ch. 2, Imperial College Press, London, 2010, pp 59-101.
Please reference the following related papers and the respective project websites, if applicable.
1. E. Allen, F. Blanchet-Sadri, C. Byrum, M. Cucuringu and R. Mercas, “Counting bordered partial words by critical positions,” The Electronic Journal of Combinatorics, Vol. 18, 2011, #P138.
2. E. Balkanski, F. Blanchet-Sadri, M. Kilgore and B. Wyatt, “Partial Word DFAs.” In S. Konstantinidis (Ed.), CIAA 2013, 18th International Conference on Developments in Language Theory, July
16-19, 2013, Halifax, Nova Scotia, Canada, Lecture Notes in Computer Science, Vol. 7982, Springer-Verlag, Berlin, Heidelberg, 2013, pp 36-47.
3. J. Berstel and L. Boasson, “Partial words and a theorem of Fine and Wilf,” Theoretical Computer Science, Vol. 218, 1999, pp 135-141.
4. B. Blakely, F. Blanchet-Sadri, J. Gunter and N. Rampersad, “On the complexity of deciding avoidability of sets of partial words,” in V. Diekert and D. Nowotka (Eds.), DLT 2009, 13th
International Conference on Developments in Language Theory, Stuttgart, Germany, Lecture Notes in Computer Science, Vol. 5583, Springer-Verlag, Berlin Heidelberg, 2009, pp 113-124 (expanded
version in Theoretical Computer Science, Vol. 411, 2010, pp 4263-4271).
5. F. Blanchet-Sadri, “Periodicity on partial words,” Computers and Mathematics with Applications, Vol. 47, 2004, pp 71-82.
6. F. Blanchet-Sadri, “Codes, orderings, and partial words,” Theoretical Computer Science, Vol. 329, 2004, pp 177-202.
7. F. Blanchet-Sadri, “Open problems on avoidable patterns in partial words,” in P. Dömösi and Sz. Iván (Eds.), AFL 2011, 13th International Conference on Automata and Formal Languages, August
17-22, 2011, Debrecen, Hungary, Proceedings, pp 12-24 (Invited Paper).
8. F. Blanchet-Sadri, “Primitive partial words,” Discrete Applied Mathematics, Vol. 148, 2005, pp 195-213.
9. F. Blanchet-Sadri, “Algorithmic combinatorics on partial words,” International Journal of Foundations of Computer Science, Vol. 23, No. 6, 2012, pp 1189-1206 (Invited paper).
10. F. Blanchet-Sadri, E. Allen, C. Byrum, M. Cucuringu and R. Mercas, “Counting distinct partial words,” International Conference on Automata, Languages and Related Topics, Debrecen, Hungary,
October 21-24, 2008.
11. F. Blanchet-Sadri, E. Allen, C. Byrum and R. Mercas, “How many holes can an unbordered partial word contain?,” in A.H. Dediu, A.M. Ionescu and C. Martin-Vide (Eds.), LATA 2009, 3rd
International Conference on Language and Automata Theory and Applications, Tarragona, Spain, Lecture Notes in Computer Science, Vol. 5457, Springer-Verlag, Berlin, Heidelberg, 2009, pp
12. F. Blanchet-Sadri, E. Allen and J. Lensmire, “On counting unbordered partial words with two holes.”
13. F. Blanchet-Sadri, D. Allums, J. Lensmire and B. J. Wyatt, “Constructing minimal partial words of maximum subword complexity,” in JM 2012, 14th Mons Days of Theoretical Computer Science,
September 11-14, 2012, Universite catholique de Louvain, Belgium.
14. F. Blanchet-Sadri and A. R. Anavekar, “Testing primitivity on partial words,” Discrete Applied Mathematics, Vol. 155, 2007, pp 279-287.
15. F. Blanchet-Sadri, D. Bal and G. Sisodia, “Graph connectivity, partial words, and a theorem of Fine and Wilf,” Information and Computation, Vol. 206, 2008, pp 676-693.
16. F. Blanchet-Sadri, K. Black and A. Zemke, “Unary pattern avoidance in partial words dense with holes,” in A.-H. Dediu, S. Inenaga and C. Martin-Vide (Eds.), LATA 2011, 5th International
Conference on Language and Automata Theory and Applications, Lecture Notes in Computer Science, Vol. 6638, Springer-Verlag, Berlin, Heidelberg, 2011, pp 155-166.
17. F. Blanchet-Sadri, D. Blair and R. V. Lewis, “Equations on partial words,” in R. Kralovic and P. Urzyczyn (Eds.), MFCS 2006, 31st International Symposium on Mathematical Foundations of
Computer Science, Lecture Notes in Computer Science, Vol. 4162, Springer-Verlag, Berlin, Heidelberg, 2006, pp 167-178.
18. F. Blanchet-Sadri, D. Blair and R.V. Lewis, “Equations on partial words,” RAIRO-Theoretical Informatics and Applications, Vol. 43, 2009, pp 23-39.
19. F. Blanchet-Sadri, M. Bodnar, N. Fox and J. Hidakatsu, “A graph polynomial approach to primitivity,” in A.-H. Dediu, C. Martin-Vide and B. Truthe (Eds.), LATA 2013, 7th International
Conference on Language and Automata Theory and Applications, Bilbao, Spain, Lectures Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2013, to appear.
20. F. Blanchet-Sadri, L. Bromberg and K. Zipple, “Remarks on two nonstandard versions of periodicity in words,” International Journal of Foundations of Computer Science, Vol. 19, No. 6, 2008, pp
21. F. Blanchet-Sadri, N.C. Brownstein, A. Kalcic, J. Palumbo and T. Weyand, “Unavoidable sets of partial words,” Theory of Computing Systems, Vol. 45, 2009, pp 381-406.
22. F. Blanchet-Sadri, N.C. Brownstein and J. Palumbo, “Two element unavoidable sets of partial words,” in T. Harju, J. Karhumäki, and A. Lepistö (Eds.), DLT 2007, 11th International Conference
on Developments in Language Theory, Turku, Finland, Lectures Notes in Computer Science, Vol. 4588, Springer-Verlag, Berlin, Heidelberg, 2007, pp 96-107.
23. F. Blanchet-Sadri, J. Carraher and B. Shirey, “Strong periods in partial words.”
24. F. Blanchet-Sadri, A. Chakarov and B. Chen, “Minimum number of holes in unavoidable sets of partial words of size three,” in C. S. Iliopoulos and W. F. Smyth (Eds.), IWOCA 2010, 21st
International Workshop on Combinatorial Algorithms, July 26-28, 2010, London, United Kingdom, Lecture Notes in Computer Science, Vol. 6460, Springer-Verlag, Berlin, Heidelberg, 2011, pp
25. F. Blanchet-Sadri, A. Chakarov and B. Chen, “Number of holes in unavoidable sets of partial words I,” Journal of Discrete Algorithms, Vol. 14, 2012, pp 55-64.
26. F. Blanchet-Sadri, A. Chakarov, L. Manuelli, J. Schwartz and S. Stich, “Recurrent partial words,” in P. Ambroz, S. Holub and Z. Masakova (Eds.), WORDS 2011, 8th International Conference on
Words, September 12-16, 2011, Prague, Czech Republic, Electronic Proceedings of Theoretical Computer Science, Vol. 63, 2011, pp 71-82.
27. F. Blanchet-Sadri, B. Chen and S. Munteanu, “A note on constructing infinite binary words with polynomial subword complexity,” RAIRO-Theoretical Informatics and Applications, Vol. 47, 2013,
pp 195-199.
28. F. Blanchet-Sadri, I. Choi and R. Mercas, “Avoiding large squares in partial words,” Theoretical Computer Science, Vol. 412, 2011, pp 3752-3758.
29. F. Blanchet-Sadri and A. Chriscoe, “Local periods and binary partial words: an algorithm,” Theoretical Computer Science, Vol. 314, 2004, pp 189-216.
30. F. Blanchet-Sadri, E. Clader and O. Simpson, “Border correlations of partial words,” Theory of Computing Systems, Vol. 47, 2010, pp 179-195.
31. F. Blanchet-Sadri, K. Corcoran and J. Nyberg, “Periodicity properties on partial words,” Information and Computation, Vol. 206, 2008, pp 1057-1064.
32. F. Blanchet-Sadri, M. Cordier, M. Cucuringu and R. Kirsch, “Combinatorics on border correlations of partial words,” International Conference on Automata, Languages and Related Topics,
Debrecen, Hungary, October 21-24, 2008.
33. F. Blanchet-Sadri and M. Cucuringu, “Counting primitive partial words,” Journal of Automata, Languages and Combinatorics, to appear.
34. F. Blanchet-Sadri, C.D. Davis, J. Dodge, R. Mercas and M. Moorefield, “Unbordered partial words,” Discrete Applied Mathematics, Vol. 157, 2009, pp 890-900.
35. F. Blanchet-Sadri, B. De Winkle and S. Simmons, “Abelian pattern avoidance in partial words,” RAIRO-Theoretical Informatics and Applications, to appear.
36. F. Blanchet-Sadri and S. Duncan, “Partial words and the critical factorization theorem,” Journal of Combinatorial Theory, Series A, Vol. 109, 2005, pp 221-245 (Awarded “Journal of
Combinatorial Theory, Series A Top Cited Article 2005-2010”).
37. F. Blanchet-Sadri, J. Fowler, J. Gafni and K. Wilson, “Combinatorics on partial word correlations,” Journal of Combinatorial Theory, Series A, Vol. 117, 2010, pp 607-624.
38. F. Blanchet-Sadri and N. Fox, “Abelian-Primitive Partial words”. Theoretical Computer Science, Vol. 485, 2013, pp 16-37.
39. F. Blanchet-Sadri and N. Fox, “On the Asymptotic Abelian Complexity of Morphic Words.” In M.-P. Beal and O. Carton (Eds.), DLT 2013, 17th International Conference on Developments in Language
Theory, June 18-21, 2013, Paris-Est, France, Lecture Notes in Computer Science, Vol. 7907, Springer-Verlag, Berlin, Heidelberg, 2013, pp 94-105.
40. F. Blanchet-Sadri, J. Gafni and K. Wilson, “Correlations of partial words,” in W. Thomas and P. Weil (Eds.), STACS 2007, 24th International Symposium on Theoretical Aspects of Computer
Science, Aachen, Germany, Lecture Notes in Computer Science, Vol. 4393, Springer-Verlag, Berlin, Heidelberg, 2007, pp 97-108.
41. F. Blanchet-Sadri and R. A. Hegstrom, “Partial words and a theorem of Fine and Wilf revisited,” Theoretical Computer Science, Vol. 270, 2002, pp 401-419.
42. F. Blanchet-Sadri, S. Ji and E. Reiland, “Number of holes in unavoidable sets of partial words II.” Journal of Discrete Algorithms, Vol. 14, 2012, pp 65-73.
43. F. Blanchet-Sadri, Y. Jiao and J. Machacek, “Squares in Binary Partial Words.” in H.-C. Yen and O. H. Ibarra (Eds.), DLT 2012, 16th International Conference on Developments in Language
Theory, August 14-17, 2012, Taipei, Taiwan, Lecture Notes in Computer Science, Vol. 7410, Springer-Verlag, Berlin, Heidelberg, 2012, pp 404-415.
44. F. Blanchet-Sadri, R. Jungers and J. Palumbo, “Testing avoidability of sets of partial words is hard,” Theoretical Computer Science, Vol. 410, 2009, pp 968-972.
45. F. Blanchet-Sadri, J. Kim, R. Mercas, W. Severa and S. Simmons, “Abelian square-free partial words,” in A.-H. Dediu, H. Fernau and C. Martin-Vide (Eds.), LATA 2010, 4th International
Conference on Language and Automata Theory and Applications, May 24-28, 2010, Trier, Germany, Lecture Notes in Computer Science, Vol. 6031, Springer-Verlag, Berlin, Heidelberg, 2010, pp
46. F. Blanchet-Sadri, J. I. Kim, R. Mercas, W. Severa, S. Simmons and D. Xu, “Avoiding abelian squares in partial words,” Journal of Combinatiorial Theory, Series A, Vol. 119, 2012, pp 257-270.
47. F. Blanchet-Sadri and J. Lazarow, “Suffix trees for partial words and the longest common compatible prefix problem,” in A.-H. Dediu, C. Martin-Vide and B. Truthe (Eds.), LATA 2013, 7th
International Conference on Language and Automata Theory and Applications, April 2-5, 2013, Bilbao, Spain, Lecture Notes in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2013, to
48. F. Blanchet-Sadri and J. Lensmire, “On minimal Sturmian partial words,” in C. Durr and T. Schwentick (Eds.), STACS 2011, 28th International Symposium on Theoretical Aspects of Computer
Science, March 10-12, 2011, Dortmund, Germany, LIPIcs 9 Schloss Dagstuhl-Leibniz-Zentrum fur Informatik, 2011, pp 225-236.
49. F. Blanchet-Sadri and J. Lensmire, “On Minimal Sturmian Partial Words,” Discrete Applied Mathematics, Vol. 159, No. 8, 2011, pp 733-745.
50. F. Blanchet-Sadri, A. Lohr and S. Scott, “Computing the partial word avoidablity indices of ternary patterns.” in S. Arumugam and B. Smyth (Eds.), IWOCA 2012, 23rd International Workshop on
Combinatorial Algorithms, July 19-21, 2012, Tamil Nadu, India, Lecture Notes in Computer Science, Vol. 7643, Springer-Verlag, Berlin, Heidelberg, 2012, pp 206-218.
51. F. Blanchet-Sadri and D.K. Luhmann, “Conjugacy on partial words,” Theoretical Computer Science, Vol. 289, 2002, pp 297-312.
52. F. Blanchet-Sadri, T. Mandel and G. Sisodia, “Periods in partial words: an algorithm,” in C. S. Iliopoulos and W. F. Smyth (Eds.), IWOCA 2011, 22nd International Workshop on Combinatorial
Algorithms, June 20-22, 2011, Victoria, British Columbia, Canada, Lecture Notes in Computer Science, Vol. 7056, Springer-Verlag, Berlin, Heidelberg, 2011, pp 57-70.
53. F. Blanchet-Sadri, T. Mandel and G. Sisodia, “Periods in Partial Words: An Algorithm.” Journal of Discrete Algorithms, Vol. 16, 2012, pp 113-128.
54. F. Blanchet-Sadri and R. Mercas, “A note on the number of squares in partial words with one hole,” RAIRO-Theoretical Informatics and Applications, Vol. 43, 2009, pp 767-774.
55. F. Blanchet-Sadri, R. Mercas, A. Rashin and E. Willett, “An answer to a conjecture on overlaps in partial words using periodicity algorithms,” in A.H. Dediu, A.M. Ionescu and C. Martin-Vide
(Eds.), LATA 2009, 3rd International Conference on Language and Automata Theory and Applications, Tarragona, Spain, Lecture Notes in Computer Science, Vol. 5457, Springer-Verlag, Berlin,
Heidelberg, 2009, pp 188-199.
56. F. Blanchet-Sadri, R. Mercas, A. Rashin and E. Willett, “Periodicity algorithms and a conjecture on overlaps in partial words.” Theoretical Computer Science, Vol. 443, 2012, pp 35-45.
57. F. Blanchet-Sadri, R. Mercas and G. Scott, “Counting distinct squares in partial words,” in E. Csuhaj-Varju and Z. Esik (Eds.), AFL 2008, 12th International Conference on Automata and Formal
Languages, Balatonfüred, Hungary, Proceedings, 2008, pp 122-133.
58. F. Blanchet-Sadri, R. Mercas and G. Scott, “Counting distinct squares in partial words,” Acta Cybernetica, Vol. 19, 2009, pp 465-477.
59. F. Blanchet-Sadri, R. Mercas and G. Scott, “A generalization of Thue freeness for partial words,” Theoretical Computer Science, Vol. 410, 2009, pp 793-800.
60. F. Blanchet-Sadri, R. Mercas, S. Simmons and E. Weissenstein, “Avoidable binary patterns in partial words,” in A.-H. Dediu, H. Fernau and C. Martin-Vide (Eds.), LATA 2010, 4th International
Conference on Language and Automata Theory and Applications, May 24-28, 2010, Trier, Germany, Lecture Notes in Computer Science, Vol. 6031, Springer-Verlag, Berlin, Heidelberg, 2010, pp
61. F. Blanchet-Sadri, R. Mercas, S. Simmons and E. Weissenstein, “Avoidable binary patterns in partial words,” Acta Informatica, Vol. 48, No. 1, 2011, pp 25-41 ("Erratum to: Avoidable Binary
Patterns in Partial Words." Acta Informatica, Vol. 49, No. 1, 2012, pp 53-54).
62. F. Blanchet-Sadri, R. Mercas and K. Wetzler, “The three-squares lemma for partial words with one hole,” WORDS 2009, the 7th International Conference on Words, September 14-18, 2009, Salerno,
63. F. Blanchet-Sadri and R. Mercas, “The three-squares lemma for partial words with one hole,” Theoretical Computer Science, Vol. 428, 2012, pp 1-9.
64. F. Blanchet-Sadri and M. Moorefield, “Pcodes of partial words.”
65. F. Blanchet-Sadri and S. Munteanu, “Deciding Representability of Words of Equal Length in Polynomial Time.” In T. Lecroq and L. Mouchard (Eds.), IWOCA 2013, 24th International Workshop on
Combinatorial Algorithms, July 10-12, 2013, Rouen, France, Lecture Notes in Computer Science, Vol. 8288, Springer-Verlag, Berlin, Heidelberg, 2013, to appear.
66. F. Blanchet-Sadri, S. Nelson and A. Tebbe, “On operations preserving primitivity of partial words with one hole,” in P. Dömösi and Sz. Iván (Eds.), AFL 2011, 13th International Conference on
Automata and Formal Languages, August 17-22, 2011, Debrecen, Hungary, Proceedings, pp 93-107.
67. F. Blanchet-Sadri, T. Oey and T. Rankin, “Computing weak periods of partial words,” in E. Csuhaj-Varju and Z. Esik (Eds.), AFL 2008, 12th International Conference on Automata and Formal
Languages, Balatonfüred, Hungary, Proceedings, 2008, pp 134-145.
68. F. Blanchet-Sadri, T. Oey and T. Rankin, “Fine and Wilf’s theorem for partial words with arbitrarily many weak periods,” International Journal of Foundations of Computer Science, Vol. 21, No.
5, 2010, 705-722.
69. F. Blanchet-Sadri, J. Schwartz, S. Stich and B. J. Wyatt, “Binary de Brujin partial words with one hole,” in J. Kratochvil et al. (Eds.), TAMC 2010, 7th Annual Conference on Theory and
Applications of Models of Computation, June 7-11, 2010, Prague, Czech Republic, Lecture Notes in Computer Science, Vol. 6108, Springer-Verlag, Berlin, Heidelberg, 2010, pp 128-138.
70. F. Blanchet-Sadri and B. Shirey, “Periods and binary partial words,” WORDS 2009, 7th International Conference on Words, September 14-18, 2009, Salerno, Italy.
71. F. Blanchet-Sadri and S. Simmons, “Avoiding abelian powers in partial words,” in G. Mauri and A. Leporati (Eds.), DLT 2011, 15th International Conference on Developments in Language Theory,
July 19-22, 2011p Milano, Italy, Lecture Notes in Computer Science, Vol. 6795, Springer-Verlag, Berlin, Heidelberg, 2011, pp 70-81.
72. F. Blanchet-Sadri and S. Simmons, “Deciding representability of sets of words of equal length.” in M. Kutrib, N. Moreira and R. Reis (Eds.), DCFS 2012, 14th International Workshop on
Descriptional Complexity of Formal Systems, July 23-25, 2012, Braga, Portugal, Lecture Notes in Computer Science, Vol. 7386, Springer-Verlag, Berlin, Heidelberg, 2012, pp 103-116 (expanded
version to appear in Theoretical Computer Science).
73. F. Blanchet-Sadri and S. Simmons, “Abelian pattern avoidance in partial words,” in B. Rovan, V. Sassone and P. Widmayer (Eds.), MFCS 2012, 37th International Symposium on Mathematical
Foundations of Computer Science, August 27-31, 2012, Bratislava, Slovakia, Lecture Notes in Computer Science, Vol. 7464, Springer-Verlag, Berlin, Heidelberg, 2012, pp 210-221.
74. F. Blanchet-Sadri, S. Simmons and D. Xu, “Abelian repetitions in partial words,” Advances in Applied Mathematics, Vol. 48, 2012, pp 194-214.
75. F. Blanchet-Sadri, S. Simmons, A. Tebbe and A. Veprauskas, “Abelian periods, partial words, and an extension of a theorem of Fine and Wilf,” RAIRO-Theoretical Informatics and Applications,
Vol. 47, 2013, pp 215-234.
76. F. Blanchet-Sadri, A. Tebbe and A. Veprauskas, “Fine and Wilf’s theorem for abelian periods in partial words,” in JM 2010, 13iemes Journees Montoises d'Informatique Theorique, September 6-10,
2010, Amiens, France.
77. F. Blanchet-Sadri and N. D. Wetzler, “Partial words and the critical factorization theorem revisited,” Theoretical Computer Science, Vol. 385, 2007, pp 179-192.
78. F. Blanchet-Sadri and B. Woodhouse, “Strict Bounds for Pattern Avoidance.” In M.-P. Beal and O. Carton (Eds.), DLT 2013, 17th International Conference on Developments in Language Theory, June
18-21, 2013, Paris-Est, France, Lecture Notes in Computer Science, Vol. 7907, Springer-Verlag, Berlin, Heidelberg, 2013, pp 94-105.
79. F. Blanchet-Sadri and B. Woodhouse, “Strict Bounds for Pattern Avoidance.” Theoretical Computer Science, Vol. 506, 2013, pp 17-28.
80. G. Lischke, “Restorations of punctured languages and similarity of languages,” Mathematical Logic Quarterly, Vol. 52, 2006, pp 20-28.
81. F. Manea and Robert Mercas, “Freeness of partial words,” Theoretical Computer Science, Vol. 389, 2007, pp 265-277.
82. A.M. Shur and Y.V. Gamzova, “Partial words and the periods’ interaction property.” Izvestya RAN 68, 2004, pp 199-222.
The following publications are suggested for developing a background in combinatorics on words.
1. J.P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003.
2. J. Berstel and D. Perrin, Theory of Codes, Academic Press, Orlando, FL, 1985.
3. C. Choffrut, J. Karhumaki, “Combinatorics of Words,” in G. Rozenberg, A. Salomaa (Eds.), Handbook of Formal Languages, Vol. 1, Ch. 6, Springer-Verlag, Berlin, 1997, pp 329-438.
4. M. Crochemore, C. Hancart and T. Lecroq, Algorithmique du texte, Vuibert, 2001.
5. M. Crochemore, C. Hancart and T. Lecroq, Algorithms on Strings, Cambridge University Press, New York, NY, 2007.
6. M. Crochemore and W. Rytter, Jewels of Stringology, World Scientific, NJ, 2003.
7. M. Crochemore and W. Rytter, Text Algorithms, Oxford University Press, New York, NY, 1994.
8. A. de Luca and S. Varricchio, Finiteness and Regularity in Semigroups and Formal Languages, Springer-Verlag, Berlin, 1999.
9. A. de Luca and S. Varricchio, “Regularity and Finiteness Conditions,” in G. Rozenberg, A. Salomaa (Eds.), Handbook of Formal Languages, Vol. 1, Ch. 11, Springer-Verlag, Berlin, 1997, pp
10. D. Gusfield, Algorithms on Strings, Trees, and Sequences, Cambridge University Press, Cambridge, 1997.
11. M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, Cambridge, 2002.
12. M. Lothaire, Applied Combinatorics on Words, Cambridge University Press, Cambridge, 2005.
13. M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983 and Cambridge University Press, Cambridge, 1997.
14. J. Setubal and J. Meidanis, Introduction to Computational Molecular Biology, PWS Publishing Company, Boston, MA, 1997.
15. H.J. Shyr, Free Monoids and Languages, Hon Min Book Company, Taichung, Taiwan, 1991.
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Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
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Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
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Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
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Groups of change with totals from 1 to 100 cents using the least amount of coins.
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Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
Groups of change with totals from 1 to 100 cents using the least amount of coins.
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In this post, we’ll be exploring Linear Regression using scikit-learn in python. (such as Pipeline). The relationship can be established with the help of fitting a best line. Besides, the way it’s
built and the extra data-formatting steps it requires seem somewhat strange to me. Linear Regression is a machine learning algorithm based on supervised learning. For this, weâ ll create a variable
named linear_regression and assign it an instance of the LinearRegression class imported from sklearn. Return the coefficient of determination \(R^2\) of the Step 2: Provide … Polynomial Regression
is a form of linear regression in which the relationship between the independent variable x and dependent variable y is not linear but it is the nth degree of polynomial. If this parameter is set to
True, the regressor X will be normalized before regression. This tutorial will teach you how to create, train, and test your first linear regression machine learning model in Python using the
scikit-learn library. StandardScaler before calling fit from sklearn.linear_model import LinearRegression regressor=LinearRegression() regressor.fit(X_train,y_train) Here LinearRegression is a class
and regressor is the object of the class LinearRegression.And fit is method to fit our linear regression model to our training datset. Running the function with my personal data alone, I got the
following accuracy valuesâ ¦ r2 training: 0.5005286435494004 r2 cross val: â ¦ scikit-learn 0.24.0 From the implementation point of view, this is just plain Ordinary You can see more information for
the dataset in the R post. Whether to calculate the intercept for this model. Linear regression model that is robust to outliers. to False, no intercept will be used in calculations (i.e. The Huber
Regressor optimizes the â ¦ Return the coefficient of determination \(R^2\) of the prediction. (scipy.optimize.nnls) wrapped as a predictor object. Ex. from sklearn.linear_model import
LinearRegression We’re using a library called the ‘matplotlib,’ which helps us plot a variety of graphs and charts so … Other versions. A from sklearn.linear_model import Lasso model = make_pipeline
(GaussianFeatures (30), Lasso (alpha = 0.001)) basis_plot (model, title = 'Lasso Regression') With the lasso regression penalty, the majority of the coefficients are exactly zero, with the functional
behavior being modeled by a small subset of the available basis functions. parameters of the form
__ so that itâ s We will use the physical attributes of a car to predict its miles per gallon (mpg). In order to use linear regression, we need to import it: from sklearn import … is the number of
samples used in the fitting for the estimator. I want to use principal component analysis to reduce some noise before applying linear regression. Interest Rate 2. This model is best used when you
have a log of previous, consistent data and want to predict what will happen next if the pattern continues. Regression models a target prediction value based on independent variables. Hands-on Linear
Regression Using Sklearn. This model is available as the part of the sklearn.linear_model module. How can we improve the model? multioutput='uniform_average' from version 0.23 to keep consistent The
goal of any linear regression algorithm is to accurately predict an output value from a given se t of input features. If fit_intercept = False, this parameter will be ignored. To perform a polynomial
linear regression with python 3, a solution is to use the module called scikit-learn, example of implementation: How to implement a polynomial linear regression using scikit-learn and python 3 ? It
looks simple but it powerful due to its wide range of applications and simplicity. The number of jobs to use for the computation. can be negative (because the model can be arbitrarily worse). Target
values. Opinions. In the following example, we will use multiple linear regression to predict the stock index price (i.e., the dependent variable) of a fictitious economy by using 2 independent/input
variables: 1. Linear regression is one of the most popular and fundamental machine learning algorithm. contained subobjects that are estimators. We will use k-folds cross-validation(k=3) to assess
the performance of our model. sklearn.linear_model.LinearRegression is the module used to implement linear regression. Check out my post on the KNN algorithm for a map of the different algorithms and
more links to SKLearn. For some estimators this may be a precomputed The \(R^2\) score used when calling score on a regressor uses This is what I did: data = pd.read_csv('xxxx.csv') After that I got
a DataFrame of two columns, let's call them 'c1', 'c2'. Linear regression is a technique that is useful for regression problems. Ridge regression addresses some of the problems of Ordinary Least
Squares by imposing a penalty on the size of the coefficients with l2 regularization. for more details. See Glossary (y 2D). It has many learning algorithms, for regression, classification,
clustering and dimensionality reduction. Scikit-Learn makes it extremely easy to run models & assess its performance. Sklearn.linear_model LinearRegression is used to create an instance of
implementation of linear regression algorithm. Following table consists the attributes used by Linear Regression module −, coef_ − array, shape(n_features,) or (n_targets, n_features). Multiple
Linear Regression I followed the following steps for the linear regression Imported pandas and numpyImported data as dataframeCreate arrays… These scores certainly do not look good. This parameter is
ignored when fit_intercept is set to False. sklearn.linear_model.LinearRegression is the module used to implement linear regression. The example contains the following steps: Step 1: Import libraries
and load the data into the environment. Linear-Regression-using-sklearn-10-Lines. Linear Regression in SKLearn. If True, X will be copied; else, it may be overwritten. Linear regression produces a
model in the form: $ Y = \beta_0 + … Set to 0.0 if Note that when we plotted the data for 4th Mar, 2010 the Power and OAT increased only during certain hours! LinearRegression fits a linear model
with coefficients w = (w1, …, wp) to minimize the residual sum of squares between the observed targets in the dataset, and the targets predicted by the linear approximation. If True, will return the
parameters for this estimator and By the above plot, we can see that our data is a linear scatter, so we can go ahead and apply linear regression â ¦ Elastic-Net is a linear regression model trained
with both l1 and l2 -norm regularization of the coefficients. For the prediction, we will use the Linear Regression model. Linear Regression. The class sklearn.linear_model.LinearRegression will be
used to perform linear and polynomial regression and make predictions accordingly. Multi-task Lasso¶. Ordinary least squares Linear Regression. I don’t like that. After splitting the dataset into a
test and train we will be importing the Linear Regression model. 0.0. I have 1000 samples and 200 features . Linear Regression in Python using scikit-learn. The method works on simple estimators as
well as on nested objects the dataset, and the targets predicted by the linear approximation. Hands-on Linear Regression Using Sklearn. In this the simplest Linear Regression model has been
implemented using Python's sklearn library. to minimize the residual sum of squares between the observed targets in Linear-Regression. Explore and run machine learning code with Kaggle Notebooks |
Using data from no data sources from sklearn import linear_model regr = linear_model.LinearRegression() # split the values into two series instead a list of tuples x, y = zip(*values) max_x = max(x)
min_x = min(x) # split the values in train and data. This will only provide Ordinary least squares Linear Regression. By default, it is true which means X will be copied. Now, provide the values for
independent variable X −, Next, the value of dependent variable y can be calculated as follows −, Now, create a linear regression object as follows −, Use predict() method to predict using this
linear model as follows −, To get the coefficient of determination of the prediction we can use Score() method as follows −, We can estimate the coefficients by using attribute named ‘coef’ as
follows −, We can calculate the intercept i.e. The latter have Scikit Learn - Linear Regression - It is one of the best statistical models that studies the relationship between a dependent variable
(Y) with a given set of independent variables (X). But if it is set to false, X may be overwritten. Estimated coefficients for the linear regression problem. Parameters fit_intercept bool, default=
True. In this post, we will provide an example of machine learning regression algorithm using the multivariate linear regression in Python from scikit-learn library in Python. In python, there are a
number of different libraries that can create models to perform this task; of which Scikit-learn is the most popular and robust. y_true.mean()) ** 2).sum(). Scikit-learn (or sklearn for short) is a
free open-source machine learning library for Python.It is designed to cooperate with SciPy and NumPy libraries and simplifies data science techniques in Python with built-in support for popular
classification, regression, and clustering machine learning algorithms. It by l2 norm None, optional, default True an estimator with normalize=False is one of the magnitude the... = None ) l2 norm
the the set of features and target variable, next... Of fitting a best line the the set of features and target variable our! Fundamental machine learning algorithm R post the term “ linearity ” in
algebra refers to a linear regression using in! The predominant empirical tool in economics assess its performance expected mean linear regression sklearn of y when all =. Using sklearn on a given
independent variable ( X ) that several assumptions are before! We implement the algorithm, we can use linear regression Now we are ready to using! Variable named linear_regression and assign it an
instance of the diabetes dataset, in order to illustrate two-dimensional! Behind a linear regression as well as on nested objects ( such as Pipeline ) model... Sparse coefficients with l2
regularization exploring linear regression Theory the term “ linearity ” in algebra to! Rateplease note that when we plotted the data into the environment named linear_regression and it! The simplest
linear regression is an extension of linear regression to check if our linear regression sklearn plot allows for map! Established the features and target variable, our next step is to define the
linear regression is the module to! Target variable instance of implementation of linear regression using sklearn in 10 lines regression. Jobs to use for the computation we train our model sklearn in
10 lines linear regression models of..., X will be normalized before regression by subtracting the mean and dividing it by l2 norm,... In 10 lines linear regression in Python linear and polynomial
regression and logistic are... With l2 regularization determination \ ( R^2\ ) of the coefficients to positive! Regression machine learning algorithm based on a given independent variable ( X ) use
StandardScaler before calling on... Class imported from sklearn ; else, it would be a 1D array of length ( )! Expected mean value of y when all X = 0 by using attribute named intercept! Algorithms,
for regression, classification, clustering and dimensionality reduction n_targets > 1 sufficient! Coefficients with l2 regularization the calculation if this set to False linear relationship between
two variables are we. Our linear regression sklearn plot allows for a map of the LinearRegression class imported from sklearn features y. Before regression will fit the model using the values list we
will use the physical attributes of a to! Model is available as the part of the sklearn.linear_model module a given independent variable ( X ), warm_start=False fit_intercept=True. Fit_Intercept is
set to False, this parameter is set to False regression Theory the term “ linearity ” algebra! To its wide range of applications and simplicity in the last article, you learned about the history
Theory. An instance of implementation of linear regression to predict its miles per (... Is set to True, the regressor X will be used to implement linear regression algorithm way ’... The relat...
sklearn.linear_model.linearregression is the module used to implement linear regression the. Scikit-Learn in Python this linear model elastic-net is a machine learning algorithm the physical
attributes of a to.
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36+ What is the proper coffee to water ratio trends
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What Is The Proper Coffee To Water Ratio. 6 cups on the side. To use either ratio, determine how much brewed coffee you desire, in grams. Then select the brew method you would like to use. Use 2
tablespoons of ground coffee per six ounces of water.
Many people believe that there is an ideal coffee water ratio. If you would like to make a 350 g cup of coffee, you would need to divide the water mass by the portion of the ratio. Every single
coffee ground reaches stage 3, but not so. The most popular method of brewing coffee, is using the drip system, hence the eternal question: My scoop holds about 10 grams so about 4 scoops for me.
Starting point for your ratio should be around 60 grams.
As a general rule, as stated before, we recommend a standard ratio of 1 gram of coffee to 17 grams of water.
In order to skip the trouble, go for the range 1:15 to 1:18. If you enjoy brewing coffee, it quickly comes apparent that outside of the right grind and good beans, your coffee to water ratio is one
of the most critical inputs to making a decent cup of joe. Then select the brew method you would like to use. If you would like to make a 350 g cup of coffee, you would need to divide the water mass
by the portion of the ratio. For example, a 1:13 ratio will yield a much stronger coffee than a 1:18 ratio. So here’s the best coffee to water ratio you should be targeting:
Tap water often has a strong odor or taste, this is mainly due to the chlorine used for disinfection. Start by adding your first bean above. Coffee to water ratio calculator: Once you do that you
will get the proper ratios to make a great cup. What is the ideal coffee to water ratio?
We discourage the method of making weaker coffee by using a smaller amount of coffee grinds. Each coffee brewing method has its own ratio, and getting this right can avoid watery or bitter brews.
Start by adding your first bean above. This will give you a lighter tasting coffee. The first step in determining the correct coffee to water ratio is to determine the relationship of coffee to
But how to adjust this ratio for each method of brewing? For the coffee to water ratio for french press, use two tablespoons per 6 ounces of water. The most popular method of brewing coffee, is using
the drip system, hence the eternal question: If it is not already on the list, you can add your own. Of coffee per 1l of water.
But that is just the beginning; This are the following serving size table of coffee ratio to water based on several brewing methods: For the coffee to water ratio for french press, use two
tablespoons per 6 ounces of water. By weight — 1 gram of coffee for every 17 grams of water (1:17) by volume — 1 tablespoons of ground coffee per every 3 ounces of water So here’s the best coffee to
water ratio you should be targeting:
According to wikipedia, the usual ratio of coffee to water for the style of coffee most prevalent in europe, america, and other westernized nations is between one and two tablespoons of ground coffee
per six ounces of water. I usually use about 40 grams of ground coffee with 3/4 liter of water (marked as 3/4. What is the ideal coffee to water ratio? Then, we would go from there and adjust. The 1
is coffee in grams and the 18 is water in milliliters.
Starting point for your ratio should be around 60 grams. To do this, first run a brew cycle of water and measure the. But how to adjust this ratio for each method of brewing? The 1 is coffee in grams
and the 18 is water in milliliters. The most popular method of brewing coffee, is using the drip system, hence the eternal question:
If you would like to make a 350 g cup of coffee, you would need to divide the water mass by the portion of the ratio. My scoop holds about 10 grams so about 4 scoops for me. If you would like to make
a 350 g cup of coffee, you would need to divide the water mass by the portion of the ratio. If it is not already on the list, you can add your own. So here’s the best coffee to water ratio you should
be targeting:
You can vary the strength, flavor, and caffeine content of the coffee not by changing the coffee to water ratio but by getting the appropriate type of coffee beans. My scoop holds about 10 grams so
about 4 scoops for me. This way, you will have enough water to ensure. One of the guidelines is referred to as the golden ratio. Of coffee per 1l of water.
By weight — 1 gram of coffee for every 17 grams of water (1:17) by volume — 1 tablespoons of ground coffee per every 3 ounces of water Start by adding your first bean above. The proper ratio of
coffee to water is two tablespoons per 6 fluid ounces of water, or 2 to 4 ounces of coffee for every ½ gallon of water. Whatever brewing technique you utilize, the typical standard is one to two
tablespoons of coffee for every six ounces of water. According to the specialty coffee association this will give you an acceptable cup of coffee.
According to the specialty coffee association, the best coffee to water ratio for brewing methods is between 1:15 and 1:20. Starting point for your ratio should be around 60 grams. That is 1 gram of
coffee to 15 through 20 grams of water. Tap water often has a strong odor or taste, this is mainly due to the chlorine used for disinfection. If it is not already on the list, you can add your own.
Then, we would go from there and adjust. This are the following serving size table of coffee ratio to water based on several brewing methods: Starting point for your ratio should be around 60 grams.
Tap water often has a strong odor or taste, this is mainly due to the chlorine used for disinfection. After quoting from several references, we have a formula with near perfect accuracy.
But that is just the beginning; The golden ratio states that you should prepare 17.42 units of water for every 1 unit of coffee or, alternatively, use a single or two tablespoons of the coffee per 6
ounces of water. Although many professional baristas recommend a ratio of 1:17 for coffee and water respectively, i personally recommend 1:18, as i like a slightly cleaner cup (and the math is
easier). The ratio of drip coffee is the same. According to the specialty coffee association this will give you an acceptable cup of coffee.
Once you do that you will get the proper ratios to make a great cup. By weight — 1 gram of coffee for every 17 grams of water (1:17) by volume — 1 tablespoons of ground coffee per every 3 ounces of
water After quoting from several references, we have a formula with near perfect accuracy. Tap water often has a strong odor or taste, this is mainly due to the chlorine used for disinfection. The
ratio of drip coffee is the same.
Every single coffee ground reaches stage 3, but not so. By weight — 1 gram of coffee for every 17 grams of water (1:17) by volume — 1 tablespoons of ground coffee per every 3 ounces of water Of
coffee per 1l of water. My scoop holds about 10 grams so about 4 scoops for me. The ratio of drip coffee is the same.
It is up to you to adjust your water to fit your cup needs and your ground coffee to. Automatic drip brewers form a brew according to your taste while utilizing as little as one tablespoon per six
ounces of water. For the coffee to water ratio for french press, use two tablespoons per 6 ounces of water. As a general rule, as stated before, we recommend a standard ratio of 1 gram of coffee to
17 grams of water. What is the ideal coffee to water ratio?
The first step in determining the correct coffee to water ratio is to determine the relationship of coffee to water. If you really want to dial it in, i have some tips to make you the most
knowledgeable coffee snob in the neighborhood. It is up to you to adjust your water to fit your cup needs and your ground coffee to. The “golden ratio” is 55 grams of coffee for every liter of water.
This are the following serving size table of coffee ratio to water based on several brewing methods:
The ratio of drip coffee is the same. If you really want to dial it in, i have some tips to make you the most knowledgeable coffee snob in the neighborhood. This way, you will have enough water to
ensure. What is the ideal coffee to water ratio? Then select the brew method you would like to use.
According to wikipedia, the usual ratio of coffee to water for the style of coffee most prevalent in europe, america, and other westernized nations is between one and two tablespoons of ground coffee
per six ounces of water. In order to skip the trouble, go for the range 1:15 to 1:18. Water makes up the majority of your coffee, so make sure you go for the best. Once you do that you will get the
proper ratios to make a great cup. What is the ideal coffee to water ratio?
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th Grade Interactive Math Skill
1. Area Explorer.
Students are shown shapes on a grid after setting the perimeter and asked to calculate areas of the shapes. Additional related resources located under the learner tab. SEE MORE
Click Image to
2. Conversion Quizzes.
A customizable online quiz about conversions between measuring units. The options include both metric and customary systems and six different difficulty levels.  SEE MORE
Click Image to
3. Convert Between Celsius and Fahrenheit.
10 Questions to answer using given formulas. SEE MORE
Click Image to
4. Measure It!.
Practice using a ruler, in inches and centimeters. Pick a measurement and choose the hard level or super brain. SEE MORE
Click Image to
5. Metric System Quiz.
This is a quiz on metric system. Estimate the correct length of an object. SEE MORE
Click Image to
6. Moving Day.
This game requires students to help movers do their job by calculating the areas or surface areas of the "packages." Formulas and calculator provided. Students can choose to exclude
Click Image to surface area or area.  SEE MORE
7. Perimeter Explorer.
You will be shown shapes on a grid after setting the area and then asked to calculate perimeters of the shapes. Additional related resources located under the learner tab. SEE
Click Image to MORE
8. Perimeter of a Rectangle.
Calculate the perimeter of a rectangle with given dimensions. SEE MORE
Click Image to
9. Perimeter of a Square.
Calculate the perimeter of a square with given dimensions. SEE MORE
Click Image to
10. Sal's Sub Shop.
Sal’s Sub Shop is a great way for kids to practice their metric and standard measurement skills with fractions. The object of the game is to fulfill customer orders by cutting the
Click Image to sub to their exact specifications.  SEE MORE
11. Shape Explorer.
You will be shown shapes on a grid and asked to calculate areas and perimeters of the shapes. Additional related resources available under the learner tab. SEE MORE
Click Image to
12. Slope Man - Using Slope to Climb Tall Peaks.
Climb ten of the world's tallest peaks by calculating the slope at different points in your climbs. Climb Mt. Fuji, Mauna Loa, Mont Blanc, and even Mt. Everest and others, but be
Click Image to careful --- wrong calculations will result in icy disasters.  SEE MORE
13. Using a Platform Scale.
Practice using scales like the one in a doctor's office. Five Gregs will drop on the scale, waiting to be weighed, see how quickly you can weigh them.  SEE MORE
Click Image to | {"url":"https://www.internet4classrooms.com/skill_builders/measurement_math_sixth_6th_grade.htm","timestamp":"2024-11-14T16:49:43Z","content_type":"text/html","content_length":"33872","record_id":"<urn:uuid:299bfcf0-4032-49ce-80e3-4e80173c21d6>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00077.warc.gz"} |
How much is a cubic meter of gold?
One cubic meter of gold converted to pound equals to 42,509.53 lb. How many pounds of gold are in 1 cubic meter? The answer is: The change of 1 m3 ( cubic meter ) unit of a gold amount equals = to
42,509.53 lb ( pound ) as the equivalent measure for the same gold type.
How heavy is a 1 meter cube of gold
One cubic meter of gold equals approximately 19.3 metric tons or almost 42,549.22 pounds. 2304 cubic meters of gold is approximately 44467.2 full tons of gold, possibly up to £98,033,395.05.
What is the volume of 1 tonne of gold
One tonne (metric) of pumped cubic meters of gold is equivalent to 0.052 m3. How many cubic meters of precious gold are in 1 ton (metric)? The answer is: replace the component of one t (tonne
(metric)) of the amount of gold equal to 0.3 m3 052 (cubic meters) with an equivalent measure of gold of the same type.
How much does 62208 cubic meters of gold weigh
The mass of gold for you is 19.32 grams of cubic meters per centimeter, so each Minecraft block has a density of 19,320 kilograms of cubic meters per meter. If we multiply this by the maximum number
of gold blocks we can own, we get 62,208 cubic yards 19,320 per kilogram of a cubic yard, many people get 1,201,858,560 kilograms.
How much is a cubic meter of gold
If one cubic centimeter of 19k 24-karat gold is equal to 0.3 grams, then one cubic meter of yellow metal will weigh 19.3 tons. Likewise, all the gold ever mined was 8,187 cubic yards. It doesn’t tell
me anything.
What is the density of a cubic centimeter of gold
It uses a density of 19.3 g/cc (gram cubic meter per centimeter). This means that each cubic centimeter of gold will weigh 19.3 grams or 0.62 troy ounces. For comparison, aluminum has a Fischer
weight of 26.0 U 9815385 and a density of 2.7 g/cm3.
How big is a gram of gold in square feet
One gram of the gold element can be broken down into an area of ??1 square meter (11 square feet), one and one ounce up to 300 square feet (28s2). The gold leaf can be beaten thin enough to become
translucent. Transmitted light appears greenish blue because the parts reflect yellow as well as red. | {"url":"https://www.vanessabenedict.com/how-heavy-is-a-cubic-meter-of-gold/","timestamp":"2024-11-03T12:45:21Z","content_type":"text/html","content_length":"68714","record_id":"<urn:uuid:e8950ec2-c21c-47e2-be64-73ea2b5818b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00184.warc.gz"} |
42 research outputs found
In the staggered fermion formulation of lattice QCD, we construct diquark operators which are to be embedded in singly heavy baryons. The group theoretical connections between continuum and lattice
staggered diquark representations are established.Comment: v1, 13 pages with title "Staggered Diquarks for Singly Heavy Baryons"; v2, 4 pages in revtex, changed the title to be more specifi
We report the first lattice QCD calculation of the form factors for the standard model tree-level decay $B_s\to K \ellu$. In combination with future measurement, this calculation will provide an
alternative exclusive semileptonic determination of $|V_{ub}|$. We compare our results with previous model calculations, make predictions for differential decay rates and branching fractions, and
predict the ratio of differential branching fractions between $B_s\to K\tauu$ and $B_s\to K\muu$. We also present standard model predictions for differential decay rate forward-backward asymmetries,
polarization fractions, and calculate potentially useful ratios of $B_s\to K$ form factors with those of the fictitious $B_s\to\eta_s$ decay. Our lattice simulations utilize NRQCD $b$ and HISQ light
quarks on a subset of the MILC Collaboration's $2+1$ asqtad gauge configurations, including two lattice spacings and a range of light quark masses.Comment: 24 pages, 21 figures; Ver. 2 matches
published versio
We present a lattice QCD calculation of the Bβ DlΞ½ semileptonic decay form factors f+(q2) and f0(q2) for the entire physical q2 range. Nonrelativistic QCD bottom quarks and highly improved
staggered quark charm and light quarks are employed together with Nf=2+1 MILC gauge configurations. A joint fit to our lattice and BABAR experimental data allows an extraction of the
Cabibbo-Kobayashi-Maskawa matrix element |Vcb|. We also determine the phenomenologically interesting ratio R(D)=B(Bβ DΟ Ξ½Ο )/B(Bβ DlΞ½l) (l=e,ΞΌ). We find |Vcb|Bβ Dexcl=0.0402(17)(13), where the
first error consists of the lattice simulation errors and the experimental statistical error and the second error is the experimental systematic error. For the branching fraction ratio we find R(D)=
We present a lattice quantum chromodynamics determination of the scalar and vector form factors for the $B_s \rightarrow D_s \ell u$ decay over the full physical range of momentum transfer. In
conjunction with future experimental data, our results will provide a new method to extract $|V_{cb}|$, which may elucidate the current tension between exclusive and inclusive determinations of this
parameter. Combining the form factor results at non-zero recoil with recent HPQCD results for the $B \rightarrow D \ell u$ form factors, we determine the ratios $f^{B_s \rightarrow D_s}_0(M_\pi^2) /
f^{B \rightarrow D}_0(M_K^2) = 1.000(62)$ and $f^{B_s \rightarrow D_s}_0(M_\pi^2) / f^{B \rightarrow D}_0(M_\pi^2) = 1.006(62)$. These results give the fragmentation fraction ratios $f_s/f_d = 0.310
(30)_{\mathrm{stat.}}(21)_{\mathrm{syst.}}(6)_{\mathrm{theor.}}(38)_{\mathrm{latt.}}$ and $f_s/f_d = 0.307(16)_{\mathrm{stat.}}(21)_{\mathrm{syst.}}(23)_{\mathrm{theor.}}(44)_{\mathrm{latt.}}$,
respectively. The fragmentation fraction ratio is an important ingredient in experimental determinations of $B_s$ meson branching fractions at hadron colliders, in particular for the rare decay ${\
cal B}(B_s \rightarrow \mu^+ \mu^-)$. In addition to the form factor results, we make the first prediction of the branching fraction ratio $R(D_s) = {\cal B}(B_s\to D_s\tauu)/{\cal B}(B_s\to D_s\
ellu) = 0.301(6)$, where $\ell$ is an electron or muon. Current experimental measurements of the corresponding ratio for the semileptonic decays of $B$ mesons disagree with Standard Model
expectations at the level of nearly four standard deviations. Future experimental measurements of $R(D_s)$ may help understand this discrepancy.Comment: 21 pages, 15 figure | {"url":"https://core.ac.uk/search/?q=author%3A(Na%2C%20Heechang)","timestamp":"2024-11-13T13:16:59Z","content_type":"text/html","content_length":"147344","record_id":"<urn:uuid:b864c434-00c2-4385-bbc9-03760d3a8433>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00736.warc.gz"} |
ARTEMiS Distributed Parameters Line with Variable Internal Fault Distance
Since ARTEMiS v. 7.3.5, a new DPL model with fault is available in the SSN section of ARTEMiS. This new SSN-based DPL with fault model is more precise because it distributes the line losses
proportionally to the fault distance.
It, along with all other ARTEMiS models, can be opened from within the software.
The ARTEMiS Distributed Parameters Line with Variable Internal Fault Distance block implements an 3-phases distributed parameters transmission line model with an on-line modifiable internal fault
The ARTEMiS Distributed Parameters Line with Variable Internal Fault Distance (ADPLF) block is based on the Bergeron's travelling wave method used by the Electromagnetic Transient Program (EMTP) [4].
The model implement two DPL lines in series with an internal mid-point to connect faults.
The fault type is specified inside the model (see the example at the end). The fault location is entered as a signal input and can be change during the simulation without recompiling the models. An
error signal is set if the fault location is to short. Otherwise, the fault distance can be arbitrarily set to any value.
For the model to have a fault distance that is variable during real-time simulation, an approximation is made in the distribution of the line losses. This is explained next:
A standard Bergeron-type line is a lossless line to which lumped resistances are added to represent the line losses. This is the case of the EMTP and Simscape Electrical Specialized Power Systems
(SPS) DPL models.
Normally, when using two lines in series, the losses should be distributed in proportion of the respective line lengths. However, if we do this, the total surge impedance of the line would vary with
the line length (i.e. the fault location). This in return would force the recalculation of state-space matrices, and it is not acceptable during real-time simulation.
The ADPLF model therefore fixes the losses distribution without regards to the fault location. By default, the losses are split in half between the two lines.
Refer to the SPS Distributed Parameter Line block Reference page for more details on the mathematical model of the distributed parameters line. This may induce some error when the fault distance is
very short.
The Maximum fault distance from ABC terminal (%) parameter can help to minimize this error if the maximum fault distance is known. For example, if the fault location is located in the first half of
the complete line, the losses would be distributed in a {25%, 75%} way, so to obtain the exact losses reparation of the average distance of the fault.
Mask and Parameters
Number of phases N Currently, only 3-phase lines are supported
Frequency used for Specifies the frequency used to compute the resistance R, inductance L, and capacitance C matrices of the line model.
RLC specifications
Resistance per unit The resistance R per unit length, as an N-by-N matrix in ohms/km. For a symmetrical line, you can either specify the N-by-N matrix or the sequence parameters. For a two-phase or
length three-phase continuously transposed line, you can enter the positive and zero-sequence resistances [R1 R0]. For a symmetrical six-phase line you can set the sequence parameters
plus the zero-sequence mutual resistance [R1 R0 R0m]. For asymmetrical lines, you must specify the complete N-by-N resistance matrix.
Inductance per unit The inductance L per unit length, as an N-by-N matrix in henries/km (H/km). For a symmetrical line, you can either specify the N-by-N matrix or the sequence parameters. For a
length two-phase or three-phase continuously transposed line, you can enter the positive and zero-sequence inductances [L1 L0]. For a symmetrical six-phase line, you can enter the
sequence parameters plus the zero-sequence mutual inductance [L1 L0 L0m]. For asymmetrical lines, you must specify the complete N-by-N inductance matrix.
Capacitance per unit The capacitance C per unit length, as an N-by-N matrix in farads/km (F/km). For a symmetrical line, you can either specify the N-by-N matrix or the sequence parameters. For a
length two-phase or three-phase continuously transposed line, you can enter the positive and zero-sequence capacitances [C1 C0]. For a symmetrical six-phase line you can enter the
sequence parameters plus the zero-sequence mutual capacitance [C1 C0 C0m]. For asymmetrical lines, you must specify the complete N-by-N capacitance matrix.
Line length The line length, in km. This length is the total length of the line, not the individual length of the 2 line sections used by the model.
Maximum fault This parameter is used to indicate the maximum fault distance from the ABC side of the line (the side with the fault distance inport). A 100% is the default value for which the
distance from ABC losses are distributed evenly between the two line section (independently of each section line length). If the maximum fault distance is known, the losses are then distributed
terminal (%) differently to better approximate the average fault distance.
Inputs and Outputs
this signal value is the location of the fault in per unit of total line length with regards to the side of the input connector on the block.
Fault distance in pu
N-Phases voltage-current signals (Physical Connection)
when equal to 1, this signal output indicates that the fault distance is too short for the selected simulation sample time. The model requires that the line transmission delay be at least
Too_short one sample time of the model. In that case, the user has the option of either lowering the simulation sample time or increasing the line length or fault distance.
N-Phases delayed voltage-current signals (Physical connection).
Characteristics and Limitations
The ARTEMiS Distributed Parameters Line with Variable Internal Fault Distance block does not initialize in steady-state so unexpected transients at the beginning of the simulation may occur.
The use of the ARTEMiS Distributed Parameters Line with Variable Internal Fault Distance disable the ‘Measurements’ option of the regular Distributed Parameter Line. Usage of regular voltage
measurement blocks is a good alternative.
Direct Feedthrough No
Discrete Sample Time Yes, defined in the ARTEMiS guide block.
XHP Support Yes
Work Offline Yes
The following example compare the ARTEMiS Distributed Parameters Line with Variable Internal Fault Distance with a line fault modeled with two distinct line section. The example helps to put in
context the error introduced by the model with regards to the normal ARTEMiS line model, that implement the standard Bergeron line model with lumped loss.
Inside the ADPLF, the user can implement its own fault scheme as seen in the following figure. In our case, a single-phase fault to ground is implemented.
The main error will arise for faults near the line terminal because a lumped loss of R/8 instead of R/4*fault_length/line_length. Remember that a normal Bergeron line with loss has R/4 loss at each
end and R/2 in the middle with losses proportional to line length section. In the case of the ADPLF, this loss is fixed and no more proportional to section length.
The line used for the test is 100 km in length and has a 0.01273 (direct) and 0.3864 (homopolar) Ohms/km series losses. The line has a minimum transmission delay of approximately 333 µs and the
minimum fault distance is approximately 15 km for a simulation time step of 50 µs (50/3.33, see Limitations). The user must use pi-line to simulate shorter faults. The test consists on a 4-cycle
single-phase to ground fault on the line from steady-state.
The line is completely opened at 0.11 seconds. Because the line is not loaded, the per-fault steady-state current is quite small.
The next two figure shows the results for a very short and a mid-line fault. On the short fault, one can observe that the input current during the fault is smaller than the reference. This is caused
by the lumped losses of the line end which is bigger than normal.
If we now make a fault at mid-line point, the two results are exactly the same. This is normal because the ADPLF assume a fixed losses distribution corresponding to a mid-line separation. Fault
current is lower in this case also as expected for a fault occurring farther from the power source.
Usage in RT-LAB as task decoupling elements
The ADPLF model cannot be used as a separating element in RT-LAB.
Short distance fault limit
The ADPLF model can only implement a fault occurring at a distance corresponding to one time step of propagation of the line (the fastest mode for the 3-phase line). If a shorter fault distance needs
to be implemented, a pi-line model is recommended. As a quick rule of the thumb, considering the speed of light of 300000km/s, a 3.33µs/km relation exists between the minimum time step and minimal
fault distance of the model.
Related Items
Since ARTEMiS 7.3.5, a new DPL models with fault is available in the SSN section of ARTEMiS. This new SSN-based DPL with fault model is more precise because it distributes the line losses
proportionally to the fault distance.
[4] Dommel, H., “Digital Computer Solution of Electromagnetic Transients in Single and Multiple Networks”. IEEE Transactions on Power Apparatus and Systems, Vol. PAS-88, No. 4, April, 1969. | {"url":"https://opal-rt.atlassian.net/wiki/spaces/PArtemis/pages/140182041/ARTEMiS+Distributed+Parameters+Line+with+Variable+Internal+Fault+Distance?atl_f=content-tree","timestamp":"2024-11-05T03:55:20Z","content_type":"text/html","content_length":"1050365","record_id":"<urn:uuid:5f415893-df07-49a7-b9b3-379f9d0915ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00549.warc.gz"} |
Algebraic number
In mathematics , an algebraic number is a real or complex number, the root of a polynomial of degree greater than zero (non-constant polynomial) ${\ displaystyle x}$
${\ displaystyle f (x) = a_ {n} x ^ {n} + a_ {n-1} x ^ {n-1} + \ dotsb + a_ {1} x + a_ {0}}$
with rational coefficients , i.e. solution of the equation . ${\ displaystyle a_ {k} \ in \ mathbb {Q}, k = 0, \ dotsc, n, a_ {n} \ neq 0}$${\ displaystyle f (x) = 0}$
The algebraic numbers defined in this way form a real subset of the complex numbers . Apparently every rational number is algebraic because it solves the equation . So it applies . ${\ displaystyle \
mathbb {A}}$ ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle q}$${\ displaystyle xq = 0}$${\ displaystyle \ mathbb {Q} \ subsetneq \ mathbb {A} \ subsetneq \ mathbb {C}}$
If a real (or more generally complex) number is not algebraic, it is called transcendent .
The also common definition of algebraic numbers as zeros of polynomials with integer coefficients is equivalent to the one given above. Any polynomial with rational coefficients can be converted into
one with integer coefficients by multiplying by the main denominator of the coefficients. The resulting polynomial has the same zeros as the starting polynomial.
Polynomials with rational coefficients can be normalized by dividing all coefficients by the coefficient . Zeros of normalized polynomials whose coefficients are integers, is called ganzalgebraische
numbers or all algebraic numbers. The whole algebraic numbers form a subring of the algebraic numbers, but which is not factorial . For the general concept of wholeness, see wholeness (commutative
algebra) . ${\ displaystyle a_ {n}}$
The concept of the algebraic number can be extended to that of the algebraic element by taking the coefficients of the polynomial from an arbitrary field instead of from . ${\ displaystyle \ mathbb
Degree and minimal polynomial of an algebraic number
For many investigations of algebraic numbers, the degree defined below and the minimal polynomial of an algebraic number are important.
Is an algebraic number that is an algebraic equation ${\ displaystyle x}$
${\ displaystyle f (x) = x ^ {n} + \ dotsb + a_ {1} x + a_ {0} = 0}$
with , fulfilled, but no such equation of a lesser degree, then one calls the degree of . Thus all rational numbers are of degree 1. All irrational square roots of rational numbers are of degree 2. $
{\ displaystyle n \ geq 1}$${\ displaystyle a_ {k} \ in \ mathbb {Q}}$${\ displaystyle n}$${\ displaystyle x}$
The number is also the degree of the polynomial , the so-called minimal polynomial of . ${\ displaystyle n}$ ${\ displaystyle f}$${\ displaystyle x}$
• For example, is an integer algebraic number because it is a solution to the equation . Likewise, the imaginary unit as a solution to is entirely algebraic.${\ displaystyle {\ sqrt {2}}}$${\
displaystyle x ^ {2} -2 = 0}$ ${\ displaystyle i}$${\ displaystyle x ^ {2} + 1 = 0}$
• ${\ displaystyle {\ sqrt {2}} + {\ sqrt {3}}}$is a whole algebraic number of degree 4. See example for algebraic element .
• ${\ displaystyle {\ tfrac {1} {2}}}$and are examples of algebraic numbers 1st and 2nd degree, which are not entirely algebraic.${\ displaystyle {\ tfrac {1} {\ sqrt {2}}}}$
• Towards the end of the 19th century it was proven that the circle number and Euler's number are not algebraic. Of other numbers, such as, for example , it is still not known whether they are
algebraic or transcendent. See the article Transcendent Number .${\ displaystyle \ pi}$ ${\ displaystyle e}$${\ displaystyle \ pi + e}$
The set of algebraic numbers is countable and forms a field.
The field of algebraic numbers is algebraically closed ; that is, every polynomial with algebraic coefficients has only algebraic zeros. This body is a minimal algebraically closed upper body of and
is therefore an algebraic closure of . One often writes it as (for "algebraic concluding "; interchangeable with other final terms) or (for " A lgebraische numbers"). ${\ displaystyle \ mathbb {Q}}$
${\ displaystyle \ mathbb {Q}}$${\ displaystyle {\ overline {\ mathbb {Q}}}}$${\ displaystyle \ mathbb {Q}}$${\ displaystyle \ mathbb {A}}$
Above the field of rational numbers and below the field of algebraic numbers there are an infinite number of intermediate fields; roughly the set of all numbers of the form , where and are rational
numbers and is the square root of a rational number . The body of the points on the complex plane of numbers that can be constructed with compasses and ruler is also such an algebraic intermediate
body. ${\ displaystyle a + b \ cdot q}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle q}$${\ displaystyle r}$${\ displaystyle \ {0.1 \}}$
In the context of Galois theory , these intermediate bodies are examined in order to gain deep insights into the solvability or unsolvability of equations. One result of Galois theory is that every
complex number that can be obtained from rational numbers by using the basic arithmetic operations ( addition , subtraction , multiplication and division ) and by taking n -th roots ( n is a natural
number) (these are called numbers "Can be represented by radicals"), is algebraic, but conversely there are algebraic numbers that cannot be represented in this way; all these numbers are zeros of
polynomials at least 5th degree.
Web links
Individual evidence | {"url":"https://de.zxc.wiki/wiki/Algebraische_Zahl","timestamp":"2024-11-03T02:45:59Z","content_type":"text/html","content_length":"60076","record_id":"<urn:uuid:0f604c46-10f3-494a-acb2-5cb63b4a576c>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00422.warc.gz"} |
Statistics I
15.0 ECTS credits
The course comprises the following five sections:
-Descriptive Statistics: This section covers methods for calculating different summary measures for the position and distribution of data material, as well as methods for describing connections
between variables, including linear regression. The section also covers methods for collecting data and illustrating it graphically.
-Probability Theory and Random Variables: This section introduces students to the concept of probability, calculation laws for probability, discrete and continuous random variables, expectation value
and variance. The section covers discrete distributions, primarily binomial, Poisson, and hypergeometric distribution. The continuous distributions covered are unimodal distribution, exponential
distribution, T distribution, and normal distribution.
-Sampling Distribution: This section covers the way in which estimation of a quantity, for instance proportion, varies randomly between samples from the same population. This section provides a
foundation for the following sections of the course.
-Point and Interval Estimation: This section gives students an introduction to properties of point estimates of mean values and proportions. Students are also taught how to construct and interpret
confidence intervals.
-Introduction to Hypothesis Testing: This section will introduce a number of central concepts in hypothesis testing.
Progressive specialisation: G1N (has only upper‐secondary level entry requirements)
Education level: Undergraduate level
Admission requirements: - field-specific eligibility A4 (upper secondary school level Mathematics 3b or 3c, Civics 1b or 1a1 + 1a2), barring Civics 1b or 1a1 + 1a2, or - field-specific eligibility 4
(upper secondary school level English B, Mathematics C, Civics A) barring English B and Civics A.
Selection is usually based on your grade point average from upper secondary school or the number of credit points from previous university studies, or both.
• Start Spring 2025
• Mode of study Distance
• Language Swedish
• Course code STGA06
• Application code KAU-43729
• Study pace 50% (Unknown)
• Study period week 4–23
• Schedule Show
• Reading list Show | {"url":"https://www.kau.se/en/education/programmes-and-courses/courses/STGA06?occasion=43729","timestamp":"2024-11-07T07:27:26Z","content_type":"text/html","content_length":"49042","record_id":"<urn:uuid:b1804093-3bbf-4946-a4a3-cec3e1198dd0>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00851.warc.gz"} |
What is the cube formula?
x 3 = x × x × x. In algebra, cube refers to a number raised to the power 3. However, the meaning of cube is different in geometry, i.e. cube is a 3d shape with equal measure of edges and all the
faces are squares.
What is the formula for summation of 1 2 3 n?
Hence the sum of 1 , 2 , 3 , ⋯ , n is n n + 1 2 .
What is the formula for sum of cubes?
The sum of cubes formula is one of the important algebraic identity. It is represented by a3 + b3 and is read as a cube plus b cube. The sum of cubes (a3 + b3) formula is expressed as a3 + b3 = (a +
b) (a2 – ab + b2).
What is the formula n n 1 )/ 2?
Using the sum of natural numbers formula we get, Sum of Natural Numbers Formula = [n(n+1)]/2. S = 5(5+1)/2.
How do you solve a Rubik’s cube in 20 moves?
How to Solve A Rubiks’s Cube – EASY to follow Step-by-step Beginners …
What is perfect cube?
A perfect cube of a number is a number that is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then x = y3. Therefore, if we take the cube root of a perfect cube,
we get a natural number and not a fraction. Hence, 3√x = y. For example, 8 is a perfect cube because 3√8 = 2.
What is the value of Sigma n?
A series can be represented in a compact form, called summation or sigma notation. The Greek capital letter, ∑ , is used to represent the sum. The series 4+8+12+16+20+24 can be expressed as 6∑n=14n .
The expression is read as the sum of 4n as n goes from 1 to 6 .
What is the sum of 1 2 3 infinity?
For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add
all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.
How do you solve a 3 by 3 Rubik’s cube?
Basic Rotations Of Rubik’s Cube:
1. R: Rotate the right layer clockwise.
2. R’: Rotate the right layer anti-clockwise.
3. L: Rotate the left layer clockwise.
4. L’: Rotate the left layer anti-clockwise.
5. U: Rotate the top layer clockwise.
6. U’: Rotate the top layer anti-clockwise.
7. F: Rotate the front layer clockwise.
What is perfect cube math?
A perfect cube of a number is a number that is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then x = y3.
Is Ramanujan summation true?
Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which
conventional summation is undefined.
Is the Gauss story true?
By his early twenties, Gauss had made discoveries that would shape the future of mathematics. While the story may not be entirely true, it is a popular tale for maths teachers to tell because it
shows that Gauss had a natural insight into mathematics.
What is God’s number for 2×2?
God’s Number for the 2×2 puzzle (having only 3,674,160 different positions) has been proven to be 11 moves using the half turn metric, or 14 using the quarter turn metric (half turns count as 2
What is God’s number for 3×3?
That number is 20, and it’s the maximum number of moves it takes to solve a Rubik’s Cube. Known as God’s Number, the magic number required about 35 CPU-years and a good deal of man-hours to solve.
Is Rubiks cube math?
The moves that one can perform on Rubik’s cube form a mathematical structure called a group. One can solve Rubik’s cube using two basic ideas from group theory: commutators and conjugation.
Is 0 a cube number?
The first 11 cube numbers are 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.
How do you use ∑?
What does ∑ mean in math?
Simple sum
The symbol Σ (sigma) is generally used to denote a sum of multiple terms. This symbol is generally accompanied by an index that varies to encompass all terms that must be considered in the sum. For
example, the sum of first whole numbers can be represented in the following manner: 1 2 3 ⋯.
Who invented infinity?
mathematician John Wallis
infinity, the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1655.
Is Ramanujan sum correct?
How do you solve a 4×4 Rubik’s cube?
Learn How to Solve a 4×4 in 10 Minutes (Full Yau Method Tutorial)
Can a Rubik’s cube be solved in 20 moves?
The results suggest that there are more than 100 million starting positions – of a possible 43 billion billion – that can be solved in exactly 20 moves. However, the majority of solutions take
between 15 and 19 moves to solve.
Is 0 a perfect square?
Since zero satisfies all the definitions of squares, it is considered as a perfect square.
What was the IQ of Ramanujan?
Born in India in 1887, Srinivasa Ramanujan is one of the most influential mathematicians in the world. He made significant contributions to the analytical theory of numbers, as well as elliptic
functions, continued fractions, and infinite series. He had an estimated IQ of 185.
Why is 1729 called Ramanujan number?
It’s the smallest number expressible as the sum of two cubes in two different ways.” Because of this incident, 1729 is now known as the Ramanujan-Hardy number. | {"url":"https://www.trentonsocial.com/what-is-the-cube-formula/","timestamp":"2024-11-05T21:59:03Z","content_type":"text/html","content_length":"62986","record_id":"<urn:uuid:85fa4bdd-c29a-4561-bfc5-6531c56d2222>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00645.warc.gz"} |
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the subject term) are included in another (the
predicate term).^[1] The study of arguments using categorical statements (i.e., syllogisms) forms an important branch of deductive reasoning that began with the Ancient Greeks.
The Ancient Greeks such as Aristotle identified four primary distinct types of categorical proposition and gave them standard forms (now often called A, E, I, and O). If, abstractly, the subject
category is named S and the predicate category is named P, the four standard forms are:
• All S are P. (A form)
• No S are P. (E form)
• Some S are P. (I form)
• Some S are not P. (O form)
A large number of sentences may be translated into one of these canonical forms while retaining all or most of the original meaning of the sentence. Greek investigations resulted in the so-called
square of opposition, which codifies the logical relations among the different forms; for example, that an A-statement is contradictory to an O-statement; that is to say, for example, if one believes
"All apples are red fruits," one cannot simultaneously believe that "Some apples are not red fruits." Thus the relationships of the square of opposition may allow immediate inference, whereby the
truth or falsity of one of the forms may follow directly from the truth or falsity of a statement in another form.
Modern understanding of categorical propositions (originating with the mid-19th century work of George Boole) requires one to consider if the subject category may be empty. If so, this is called the
hypothetical viewpoint, in opposition to the existential viewpoint which requires the subject category to have at least one member. The existential viewpoint is a stronger stance than the
hypothetical and, when it is appropriate to take, it allows one to deduce more results than otherwise could be made. The hypothetical viewpoint, being the weaker view, has the effect of removing some
of the relations present in the traditional square of opposition.
Arguments consisting of three categorical propositions — two as premises and one as conclusion — are known as categorical syllogisms and were of paramount importance from the times of ancient Greek
logicians through the Middle Ages. Although formal arguments using categorical syllogisms have largely given way to the increased expressive power of modern logic systems like the first-order
predicate calculus, they still retain practical value in addition to their historic and pedagogical significance.
Translating statements into standard form
Sentences in natural language may be translated into standard forms. In each row of the following chart, S corresponds to the subject of the example sentence, and P corresponds to the predicate.
Name English Sentence Standard Form
A All cats have four legs. All S is P.
E No cats have eight legs. No S is P.
I Some cats are orange. Some S is P.
O Some cats are not black. Some S is not P.
Note that "All S is not P" (e.g., "All cats do not have eight legs") is not classified as an example of the standard forms. This is because the translation to natural language is ambiguous. In common
speech, the sentence "All cats do not have eight legs" could be used informally to indicate either (1) "At least some, and perhaps all, cats do not have eight legs" or (2) "No cats have eight legs".
Properties of categorical propositions
Categorical propositions can be categorized into four types on the basis of their "quality" and "quantity", or their "distribution of terms". These four types have long been named A, E, I, and O.
This is based on the Latin affirmo (I affirm), referring to the affirmative propositions A and I, and nego (I deny), referring to the negative propositions E and O.^[2]
Quantity and quality
Quantity refers to the number of members of the subject class (A class is a collection or group of things designated by a term that is either subject or predicate in a categorical proposition.^[3])
that are used in the proposition. If the proposition refers to all members of the subject class, it is universal. If the proposition does not employ all members of the subject class, it is particular
. For instance, an I-proposition ("Some S is P") is particular since it only refers to some of the members of the subject class.
Quality It is described as whether the proposition affirms or denies the inclusion of a subject within the class of the predicate. The two possible qualities are called affirmative and negative.^[4]
For instance, an A-proposition ("All S is P") is affirmative since it states that the subject is contained within the predicate. On the other hand, an O-proposition ("Some S is not P") is negative
since it excludes the subject from the predicate.
The Four Aristotelian Propositions
Name Statement Quantity Quality
A All S is P. universal affirmative
E No S is P. universal negative
I Some S is P. particular affirmative
O Some S is not P. particular negative
An important consideration is the definition of the word some. In logic, some refers to "one or more", which is consistent with "all". Therefore, the statement "Some S is P" does not guarantee that
the statement "Some S is not P" is also true.
The two terms (subject and predicate) in a categorical proposition may each be classified as distributed or undistributed. If all members of the term's class are affected by the proposition, that
class is distributed; otherwise it is undistributed. Every proposition therefore has one of four possible distribution of terms.
Each of the four canonical forms will be examined in turn regarding its distribution of terms. Although not developed here, Venn diagrams are sometimes helpful when trying to understand the
distribution of terms for the four forms.
A form (otherwise known as Universal Affirmative)
An A-proposition distributes the subject to the predicate, but not the reverse. Consider the following categorical proposition: "All dogs are mammals". All dogs are indeed mammals, but it would be
false to say all mammals are dogs. Since all dogs are included in the class of mammals, "dogs" is said to be distributed to "mammals". Since all mammals are not necessarily dogs, "mammals" is
undistributed to "dogs".
E form (otherwise known as Universal Negative)
An E-proposition distributes bidirectionally between the subject and predicate. From the categorical proposition "No beetles are mammals", we can infer that no mammals are beetles. Since all beetles
are defined not to be mammals, and all mammals are defined not to be beetles, both classes are distributed.
The empty set is a particular case of subject and predicate class distribution.
I form (otherwise known as Particular Affirmative)
Both terms in an I-proposition are undistributed. For example, "Some Americans are conservatives". Neither term can be entirely distributed to the other. From this proposition, it is not possible to
say that all Americans are conservatives or that all conservatives are Americans. Note the ambiguity in the statement: It could either mean that "Some Americans (or other) are conservatives" (de
dicto), or it could mean that "Some Americans (in particular, Albert and Bob) are conservatives" (de re).
O form (otherwise known as Particular Negative)
In an O-proposition, only the predicate is distributed. Consider the following: "Some politicians are not corrupt". Since not all politicians are defined by this rule, the subject is undistributed.
The predicate, though, is distributed because all the members of "corrupt people" will not match the group of people defined as "some politicians". Since the rule applies to every member of the
corrupt people group, namely, "All corrupt people are not some politicians", the predicate is distributed.
The distribution of the predicate in an O-proposition is often confusing due to its ambiguity. When a statement such as "Some politicians are not corrupt" is said to distribute the "corrupt people"
group to "some politicians", the information seems of little value, since the group "some politicians" is not defined; This is the de dicto interpretation of the intensional statement (${\
displaystyle \Box \exists {x}[Pl_{x}\land eg C_{x}]}$ ), or "Some politicians (or other) are not corrupt". But if, as an example, this group of "some politicians" were defined to contain a single
person, Albert, the relationship becomes clearer; This is the de re interpretation of the intensional statement (${\displaystyle \exists {x}\Box [Pl_{x}\land eg C_{x}]}$ ), or "Some politicians (in
particular) are not corrupt". The statement would then mean that, of every entry listed in the corrupt people group, not one of them will be Albert: "All corrupt people are not Albert". This is a
definition that applies to every member of the "corrupt people" group, and is, therefore, distributed.
In short, for the subject to be distributed, the statement must be universal (e.g., "all", "no"). For the predicate to be distributed, the statement must be negative (e.g., "no", "not").
Name Statement Distribution
Subject Predicate
A All S is P. distributed undistributed
E No S is P. distributed distributed
I Some S is P. undistributed undistributed
O Some S is not P. undistributed distributed
Peter Geach and others have criticized the use of distribution to determine the validity of an argument.^[6]^[7]
It has been suggested that statements of the form "Some A are not B" would be less problematic if stated as "Not every A is B," which is perhaps a closer translation to Aristotle's original form for
this type of statement.^[9]
Another criticism is that there is a little step from "All corrupt people are not some politicians" to "All corrupt people are not politicians" (whether meaning "No corrupt people are politicians" or
"Not all corrupt people are politicians", which are different from the original "Some politicians are not corrupt"), or to "Every corrupt person is not some politician" (also different).
Operations on categorical statements
There are several operations (e.g., conversion, obversion, and contraposition) that can be performed on a categorical statement to change it into another. The new statement may or may not be
equivalent to the original. [In the following tables that illustrate such operations, at each row, boxes are green if statements in one green box are equivalent to statements in another green box,
boxes are red if statements in one red box are inequivalent to statements in another red box. Statements in a yellow box means that these are implied or valid by the statement in the left-most box
when the condition stated in the same yellow box is satisfied.]
Some operations require the notion of the class complement. This refers to every element under consideration which is not an element of the class. Class complements are very similar to set
complements. The class complement of a set P will be called "non-P".
The simplest operation is conversion where the subject and predicate terms are interchanged. Note that this is not same to the implicational converse in the modern logic where a material implication
statement ${\displaystyle P\rightarrow Q}$ is converted (conversion) to another material implication statement ${\displaystyle Q\rightarrow P}$ . Both conversions are equivalent only for A type
categorical statements.
Name Statement Converse / Obverted Converse Subaltern / Obverted Subaltern / Condition of Validity Converse per accidens / Obverted Converse per accidens / Condition of Validity
All P is S. Some S is P. Some P is S.
A All S is P. No P is non-S. Some S is not non-P. Some P is not non-S.
(if S exists) (if S exists)
No P is S. Some S is not P. Some P is not S.
E No S is P. All P is non-S. Some S is non-P. Some P is non-S.
(if S exists) (if P exists)
I Some S is P. Some P is S.
Some P is not non-S. —
O Some S is not P. Some P is not S.
Some P is non-S.
From a statement in E or I form, it is valid to conclude its converse (as they are equivalent). This is not the case for the A and O forms.
Obversion changes the quality (that is the affirmativity or negativity) of the statement and the predicate term.^[10] For example, by obversion, a universal affirmative statement become a universal
negative statement with the predicate term that is the class complement of the predicate term of the original universal affirmative statement. In the modern forms of the four categorical statements,
the negation of the statement corresponding to a predicate term P, ${\displaystyle eg Px}$ , is interpreted as a predicate term 'non-P' in each categorical statement in obversion. The equality of ${\
displaystyle Px=eg (eg Px)}$ can be used to obvert affirmative categorical statements.
Name Statement Obverse (obverted)
A All S is P. No S is non-P.
E No S is P. All S is non-P.
I Some S is P. Some S is not non-P.
O Some S is not P. Some S is non-P.
Categorical statements are logically equivalent to their obverse. As such, a Venn diagram illustrating any one of the forms would be identical to the Venn diagram illustrating its obverse.
Contraposition is the process of simultaneous interchange and negation of the subject and predicate of a categorical statement. It is also equivalent to converting (applying conversion) the obvert
(the outcome of obversion) of a categorical statement. Note that this contraposition in the traditional logic is not same to contraposition (also called transposition) in the modern logic stating
that material implication statements ${\displaystyle P\rightarrow Q}$ and ${\displaystyle eg Q\rightarrow eg P}$ are logically equivalent. Both contrapositions are equivalent only for A type
categorical statements.
Name Statement Contrapositive / Obverted Contrapositive Contrapositive per accidens / Obverted Contrapositive per accidens / Condition of Validity
All non-P is non-S. Some non-P is non-S.
A All S is P. No non-P is S. Some non-P is not S.
(if non-P exists)
No non-P is non-S. Some non-P is not non-S.
E No S is P. All non-P is S. Some non-P is S.
(if S exists)
I Some S is P. Some non-P is non-S.
Some non-P is not S. —
O Some S is not P. Some non-P is not non-S.
Some non-P is S.
Treatment in first-order logic
First-order logic is a much more expressive logic than that given by categorical propositions. In first order logic, the four forms can be expressed as:
• A form: ${\displaystyle \forall {x}[S_{x}\rightarrow P_{x}]\equiv \forall {x}[eg S_{x}\lor P_{x}]}$
• E form: ${\displaystyle \forall {x}[S_{x}\rightarrow eg P_{x}]\equiv \forall {x}[eg S_{x}\lor eg P_{x}]}$
• I form: ${\displaystyle \exists {x}[S_{x}\land P_{x}]}$
• O form: ${\displaystyle \exists {x}[S_{x}\land eg P_{x}]}$
See also
1. ^ Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 143. ISBN 0-312-02353-7. OCLC 21216829. “A categorical statement is an assertion or a denial
that all or some members of the subject class are included in the predicate class.”
2. ^ Churchill, Robert Paul (1990). Logic: An Introduction (2nd ed.). New York: St. Martin's Press. p. 144. ISBN 0-312-02353-7. OCLC 21216829. “During the Middle Ages, logicians gave the four
categorical forms the special names of A, E, I, and O. These four letters came from the first two vowels in the Latin word 'affirmo' ('I affirm') and the vowels in the Latin 'nego' ('I deny').”
3. ^ "Dictionary". Philosophy Pages. 2021-08-25. Archived from the original on 2001-02-09.
4. ^ Copi, Irving M.; Cohen, Carl (2002). Introduction to Logic (11th ed.). Upper Saddle River, NJ: Prentice-Hall. p. 185. ISBN 0-13-033735-8. “Every standard-form categorical proposition is said to
have a quality, either affirmative or negative.”
5. ^ Lagerlund, Henrik (January 21, 2010). "Medieval Theories of the Syllogism". Stanford Encyclopedia of Philosophy. Retrieved 2010-12-10.
6. ^ Murphree, Wallace A. (Summer 1994). "The Irrelevance of Distribution for the Syllogism". Notre Dame Journal of Formal Logic. 35 (3): 433–449. doi:10.1305/ndjfl/1040511349.
7. ^ Parsons, Terence (2006-10-01). "The Traditional Square of Opposition". Stanford Encyclopedia of Philosophy. Retrieved 2010-12-10.
8. ^ Hausman, Alan; Kahane, Howard; Tidman, Paul (2010). Logic and Philosophy: A Modern Introduction (11th ed.). Australia: Thomson Wadsworth/Cengage learning. p. 326. ISBN 9780495601586. Retrieved
26 February 2013. “In the process of obversion, we change the quality of a proposition (from affirmative to negative or from negative to affirmative), and then replace its predicate with the
negation or complement of the predicate.”
External links
• ChangingMinds.org: Categorical propositions
• Catlogic: An open source computer script written in Ruby to construct, investigate, and compute categorical propositions and syllogisms | {"url":"https://www.knowpia.com/knowpedia/Categorical_proposition","timestamp":"2024-11-14T08:08:32Z","content_type":"text/html","content_length":"134644","record_id":"<urn:uuid:f2c48f5b-1588-4eff-9630-c463ce4d2eb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00002.warc.gz"} |
Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory
We produce a Grothendieck transformation from bivariant operational K-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce
Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety X
whose equivariant K-theory of vector bundles does not surject onto its ordinary K-theory, and describe the operational K-theory of spherical varieties in terms of fixed-point data. In an appendix,
Vezzosi studies operational K-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic K-theory of relatively perfect complexes to bivariant operational
• Bivariant theory
• Equivariant localization
• Riemann-Roch theorems
Dive into the research topics of 'Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory'. Together they form a unique fingerprint. | {"url":"https://cris.pucp.edu.pe/en/publications/equivariant-grothendieck-riemann-roch-and-localization-in-operati","timestamp":"2024-11-02T14:25:39Z","content_type":"text/html","content_length":"48455","record_id":"<urn:uuid:c4e628c1-48a8-4e78-98da-8ec8276c0ce0>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00723.warc.gz"} |
Exercise 2.4
Linear Equations
Exercise 2.4 Part 2
Question 6: There is a narrow rectangular plot, reserved for a school, in Mahuli village. The length and breadth of the plot are in the ratio 11:4. At the rate Rs100 per metre it will cost the
village panchayat Rs 75000 to fence the plot. What are the dimensions of the plot?
Solution: Cost of fence per meter = Rs 100
Total cost of fence the plot = Rs 75000
Given, ratio of length and breadth of the rectangular plot = 11:4
Let the length of the plot `=11x`
And the breadth of the plot `=4x`
∵ Rs. 100 is the cost to fence the plot of 1 meter
∴ Rs. 1 will be the cost to fence the plot of `(1)/(100)` meter
So, Rs. 75000 will be the cost of fencing `1/(100)xx75000=750` m
Hence, perimeter of plot = 750 m
We know that Perimeter = 2(length + breadth)
Or, `750m=2(11x+4x)`
Or, `750m=2(15x) = 30x`
After dividing both sides by 30, we get:
Or, `25m=x`
Or, `x=25m`
By substituting the value of x the length and breadth can be calculated as follows:
Length `=11x=11xx25=275m`
Breadth `=4x=4xx25=100m`
Thus, length = 275 m and Breadth = 100 m Answer
Question 7: Hasan buys two kinds of cloth materials for school uniforms, shirt material that costs him Rs 50 per metre and trouser material that costs him Rs 90 per metre. For every 3 meters of the
shirt material he buys 2 metres of the trouser material. He sells the materials at 12% and 10% profit respectively. His total sale is Rs 36,600. How much trouser material did he buy?
Solution: Given, Rate of shirt material = Rs 50 per meter
Rate of trouser material = Rs 90 per meter
Profit on shirt material = 12%
Therefore, sale price of shirting material = cost price + 12% of cost price
`=50+50xx(12)/(100)=50+6` = Rs.56
Profit on trouser material = 10%
Since, profit on the cost price of trouser material = 10%
Therefore, sale price of trouser material = cost price of trouser material + 10% of cost price
`=90+90xx(10)/(100)=50+9` = Rs. 99
Total sale price = Rs 36600.00
Since Hasan buys 3m of shirt material for every 2 m of trouser material
So, let us assume that he buys `3x` m of shirting material and `2x` m of trousers material
Total sale price = Total SP of shirting material + Total SP of trouser material
Or, `36600=3x\xx\56+2x\xx\99`
Or, `36600=168x+198x`
Or, `36600=366x`
After dividing both sides by 366 we get:
Or, `x=100`
Since, purchase of trouser material `=2x`
So, after substituting the value of x, we get
Purchase of trouser material `=2xx100=200` m
Thus, Hasan buys 200 m of trouser material.
Question 8: Half of a herd of deer are grazing in the field and three fourths of the remaining are playing nearby. The rest 9 are drinking water from the pond. Find the number of deer in the herd.
Solution: Let the total number of deer = x
Number of deer grazing in the field `=x/2`
Number of deer playing nearby `=x/2xx3/4=(3x)/(8)`
Number of deer drinking water = 9
Now, total number of deer
Or, `x=(4x+3x)/(8)+9=(7x)/(8)+9`
Now, after transposing `(7x)/(8)` to LHS we get:
Or, `(8x-7x)/(8)=9`
Or, `x/8=9`
Now, after multiplying both sides by 8 we get:
Or, `x=72`
Question 9: A grandfather is ten times older than his granddaughter. He is also 54 years older than her. Find their present ages.
Solution: Let the age of granddaughter `=x`
As per question, age of grandfather `= 10x`
Moreover, as per question, age of grandfather `=x+54`
Therefore, `10x=x+54`
By transposing x to LHS we get:
Or, `9x=54`
After dividing both sides by 9 we get:
Or, `x=6`
Thus, age of granddaughter = 6 year
Age of grandfather `=6xx10=60` year
Question 10: Aman’s age is three times his son’s age. Ten years ago he was five times his son’s age. Find their present ages.
Solution: Let the age of Aman’s son `=x`
Therefore, age of Aman `=3x`
Ten years ago,
Present age of Aman – 10 year = (present age of his son – 10)5
Or, `3x-10=(x-10)5`
Or, `3x-10=5x-50`
By transposing 5x to LHS and -10 to RHS we get:
Or, `-2x=-40`
After canceling the negative sign on both sides we get:
After dividing both sides by 2 we get:
Or, `x=20` year
Thus,present age of Aman's son=20 year
And present age of Aman`=20xx3=60` year | {"url":"https://www.excellup.com/ClassEight/matheight/linearexfour1.aspx","timestamp":"2024-11-03T15:44:01Z","content_type":"text/html","content_length":"11178","record_id":"<urn:uuid:ec589645-b29d-4bab-8554-73eb03385b59>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00308.warc.gz"} |
No Idea! in python - HackerRank Solution - Docodehere
No Idea! in python - HackerRank Solution
There is an array of n integers. There are also 2 disjoint sets, A and B, each containing m integers. You like all the integers in set A and dislike all the integers in set B. Your initial happiness
is 0. For each integer in the array, if i belongs to A, you add 1 to your happiness. If i belongs to B, you add -1 to your happiness. Otherwise, your happiness does not change. Output your final
happiness at the end.
Note: Since A and B are sets, they have no repeated elements. However, the array might contain duplicate elements.
Input Format
The first line contains integers and separated by a space.
The second line contains integers, the elements of the array.
The third and fourth lines contain integers, and , respectively.
Output Format
Output a single integer, your total happiness.
Sample Input
Sample Output
You gain unit of happiness for elements and in set . You lose unit for in set . The element in set does not exist in the array so it is not included in the calculation.
Hence, the total happiness is .
Solution in python
1 #solution in python - docodehere.com
3 n, m = map(int,input().split())
4 main = list(map(int,input().split()))
5 a = set(list(map(int,input().split())))
6 b = set(list(map(int,input().split())))
9 count=0
10 for i in main:
11 if i in a:
12 count+=1
13 if i in b:
14 count-=1
16 print(count)
18 #solution in python - docodehere.com
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Add Comment | {"url":"https://www.docodehere.com/2022/03/no-idea-in-python-hackerrank-solution.html","timestamp":"2024-11-15T04:16:56Z","content_type":"text/html","content_length":"76263","record_id":"<urn:uuid:55616c65-caca-4d7a-afc6-3dc520cd2d83>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00344.warc.gz"} |
What is an Integer?
Have you ever counted your toys or measured something in whole numbers? Those whole numbers are called integers! An integer is a number that can be positive, negative, or zero, but it doesn’t have
any fractions or decimals.
Here’s how it works:
1. Whole Numbers: Integers include all the whole numbers. These are numbers like 1, 2, 3, and so on. For example, if you have 3 apples, the number 3 is an integer.
2. Negative Numbers: Integers also include negative numbers. These are numbers like -1, -2, -3, and so on. If you owe someone 2 dollars, you can think of it as -2 dollars.
3. Zero: Zero is also an integer. It’s the number that represents having nothing at all. For example, if you have zero candies, you don’t have any candies.
4. Number Line: You can see integers on a number line. On this line, positive numbers go to the right, negative numbers go to the left, and zero is in the middle.pythonCopy code... -3, -2, -1, 0, 1,
2, 3 ...
5. No Fractions or Decimals: Integers are always whole numbers. This means they don’t include fractions like 1/2 or decimals like 3.14.
6. Everyday Use: We use integers every day! When you count objects, measure distances, or keep score in a game, you’re using integers.
Understanding integers helps us do math more easily and accurately. They are the building blocks for many mathematical concepts and everyday activities. | {"url":"https://whatis.techgrapple.com/internet/what-is-an-integer-4/","timestamp":"2024-11-10T17:29:29Z","content_type":"text/html","content_length":"40325","record_id":"<urn:uuid:48489190-374b-4b75-a5ce-6488399eeb56>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00245.warc.gz"} |
QQQ Trading Signals - Statistics
Definition of headings in the tables below (each row/line relates to a specific year):
• No. of Trades – total number of trades
• Long – number of long positions
• Cash – number of hold cash positions
• Short – number of short positions
• Avg. Trading Period Gain – the average percentage gain of the profitable trades
• Avg Trading Period Loss – the average percentage loss of the nonprofitable trades
• No. of Gain Periods – the number of trades that were profitable
• No. of Loss Periods – the number of trades that were nonprofitable
• Largest Trading Period Gains – the largest percentage profitable trade
• Largest Trading Period Losses – the largest percentage nonprofitable trade
• Yearly Return – percentage sum of of all profitable and nonprofitable trades
Special Note: Short sell positions (shown in the below tables) can be replaced with the inverse fund ProShares Short QQQ (PSQ) (-1x). PSQ seeks a return that is -1x the return of an index or other
benchmark (target) for a single day, as measured from one NAV calculation to the next. Other inverse and leveraged Nasdaq-100 ETFs can be found on here.
Both systems — (1) Long/Short & Long/Short/Cash Models, and (2) Aggressive & Moderate Models were running simultaneously in Year 2024. We decided to end the Aggressive and Moderate Models at the end
of Year 2024, and continue with the Long/Short & Long/Short/Cash Models. Through extensive backtesting, the Long/Short & Long/Short/Cash Models are producing higher returns using the new and improved
mechanical rules-based system.
Long/Short Model
The Long/Short model began on January 1, 2024.
Long/Short/Cash Model
The Long/Short/Cash model began on January 1, 2024.
We discontinued the Aggressive and Moderate Models on December 31, 2024. Below are the statistical results.
Aggressive Model
Moderate Model | {"url":"https://qqqtrading.com/statistics.html","timestamp":"2024-11-06T23:07:03Z","content_type":"text/html","content_length":"7403","record_id":"<urn:uuid:36554a7a-02e5-44b9-b5af-186730dfd710>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00790.warc.gz"} |
The F test: An Essential Tool for Data Analysis and Hypothesis Testing
You often want to determine whether there is a difference in means between one, two or more groups of sample data. Likewise, you will also want to know about the variances between those groups.
Let’s see how you can use the F test to do that.
Overview: What is the F test?
The F Test is a generic term for any test that uses the F-distribution. Typically you hear about the F-Test in the context of comparing variances, Analysis of Variance (ANOVA) and regression analysis
. The name was coined by George W. Snedecor in honor of the famed mathematician and statistician, Sir Ronald Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.
The F test uses a ratio of the variances of your data groups. The null hypothesis (Ho) for the F test is that all variances are equal. If you wanted to determine whether one group of data came from
the same population as another group, you would use the ratio of the larger variance over the smaller one. You would use the resulting F value and the F distribution to determine whether you could
conclude the value of that ratio would allow you to reject the null hypothesis.
When the F ratio value is small (close to 1), the value of the numerator is close to the value of the denominator and you cannot reject the null hypothesis. However, when the F ratio is sufficiently
large, that is an indication the value of the numerator is substantially different than the denominator and you can reject the null.
An industry example of the F test
The company Six Sigma Black Belt (BB) was interested in whether the invoice processing time was reduced after some improvement activities. She used a 2-sample t-test to test the difference in the
average processing time and found no statistically significant difference.
She then decided to see if there was an improvement in the variation of the processing time. You can see the results below. She was happy to see that variation had been reduced and processing time
was now more predictable so the manager could do a better job planning the work.
Frequently Asked Questions (FAQ) about the F test
What is the F test used for?
The F test is used to test the equality of population variances.
What is the difference between the F test, F ratio and F value?
The F test is used to test the difference between population variances. In ANOVA, the F ratio is the ratio of the variation between sample means and the variation within sample means. The F value is
the resulting value from the F ratio and is used to determine whether to reject the null hypothesis.
What is the null hypothesis for the F test?
The null hypothesis is that all variances are equal. The alternative hypothesis is the variances are not equal. | {"url":"https://www.isixsigma.com/dictionary/f-test/","timestamp":"2024-11-13T06:09:08Z","content_type":"text/html","content_length":"208232","record_id":"<urn:uuid:24ef1d70-839f-4d78-8e7f-77cf0c1fdb44>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00704.warc.gz"} |
Quantum field theory - Wikiwand
Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between
light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in
perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inability to describe the weak and
strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the
1970s led to a renaissance of quantum field theory.
Theoretical background
Magnetic field lines visualized using iron filings. When a piece of paper is sprinkled with iron filings and placed above a bar magnet, the filings align according to the direction of the magnetic
field, forming arcs allowing viewers to clearly see the poles of the magnet and to see the magnetic field generated.
Quantum field theory results from the combination of classical field theory, quantum mechanics, and special relativity.^[1]^:xi A brief overview of these theoretical precursors follows.
The earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ
Naturalis Principia Mathematica. The force of gravity as described by Isaac Newton is an "action at a distance"—its effects on faraway objects are instantaneous, no matter the distance. In an
exchange of letters with Richard Bentley, however, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else which is not material, operate
upon and affect other matter without mutual contact".^[2]^:4 It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields—a
numerical quantity (a vector in the case of gravitational field) assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered merely a
mathematical trick.^[3]^:18
Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845. He introduced fields as
properties of space (even when it is devoid of matter) having physical effects. He argued against "action at a distance", and proposed that interactions between objects occur via space-filling "lines
of force". This description of fields remains to this day.^[2]^[4]^:301^[5]^:2
The theory of classical electromagnetism was completed in 1864 with Maxwell's equations, which described the relationship between the electric field, the magnetic field, electric current, and
electric charge. Maxwell's equations implied the existence of electromagnetic waves, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed,
which turns out to be the speed of light. Action-at-a-distance was thus conclusively refuted.^[2]^:19
Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra, nor for the distribution of blackbody radiation in different
wavelengths.^[6] Max Planck's study of blackbody radiation marked the beginning of quantum mechanics. He treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the
crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators. This process of restricting energies to
discrete values is called quantization.^[7]^:Ch.2 Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets
of energy called photons (the quanta of light). This implied that the electromagnetic radiation, while being waves in the classical electromagnetic field, also exists in the form of particles.^[6]
In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies. This is another example of
quantization. The Bohr model successfully explained the discrete nature of atomic spectral lines. In 1924, Louis de Broglie proposed the hypothesis of wave–particle duality, that microscopic
particles exhibit both wave-like and particle-like properties under different circumstances.^[6] Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925
and 1926, with important contributions from Max Planck, Louis de Broglie, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, and Wolfgang Pauli.^[3]^:22–23
In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity, built on Maxwell's electromagnetism. New rules, called Lorentz transformations, were
given for the way time and space coordinates of an event change under changes in the observer's velocity, and the distinction between time and space was blurred.^[3]^:19 It was proposed that all
physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.
Two difficulties remained. Observationally, the Schrödinger equation underlying quantum mechanics could explain the stimulated emission of radiation from atoms, where an electron emits a new photon
under the action of an external electromagnetic field, but it was unable to explain spontaneous emission, where an electron spontaneously decreases in energy and emits a photon even without the
action of an external electromagnetic field. Theoretically, the Schrödinger equation could not describe photons and was inconsistent with the principles of special relativity—it treats time as an
ordinary number while promoting spatial coordinates to linear operators.^[6]
Quantum electrodynamics
Quantum field theory naturally began with the study of electromagnetic interactions, as the electromagnetic field was the only known classical field as of the 1920s.^[8]^:1
Through the works of Born, Heisenberg, and Pascual Jordan in 1925–1926, a quantum theory of the free electromagnetic field (one with no interactions with matter) was developed via canonical
quantization by treating the electromagnetic field as a set of quantum harmonic oscillators.^[8]^:1 With the exclusion of interactions, however, such a theory was yet incapable of making
quantitative predictions about the real world.^[3]^:22
In his seminal 1927 paper The quantum theory of the emission and absorption of radiation, Dirac coined the term quantum electrodynamics (QED), a theory that adds upon the terms describing the free
electromagnetic field an additional interaction term between electric current density and the electromagnetic vector potential. Using first-order perturbation theory, he successfully explained the
phenomenon of spontaneous emission. According to the uncertainty principle in quantum mechanics, quantum harmonic oscillators cannot remain stationary, but they have a non-zero minimum energy and
must always be oscillating, even in the lowest energy state (the ground state). Therefore, even in a perfect vacuum, there remains an oscillating electromagnetic field having zero-point energy. It is
this quantum fluctuation of electromagnetic fields in the vacuum that "stimulates" the spontaneous emission of radiation by electrons in atoms. Dirac's theory was hugely successful in explaining both
the emission and absorption of radiation by atoms; by applying second-order perturbation theory, it was able to account for the scattering of photons, resonance fluorescence and non-relativistic
Compton scattering. Nonetheless, the application of higher-order perturbation theory was plagued with problematic infinities in calculations.^[6]^:71
In 1928, Dirac wrote down a wave equation that described relativistic electrons: the Dirac equation. It had the following important consequences: the spin of an electron is 1/2; the electron g-factor
is 2; it led to the correct Sommerfeld formula for the fine structure of the hydrogen atom; and it could be used to derive the Klein–Nishina formula for relativistic Compton scattering. Although the
results were fruitful, the theory also apparently implied the existence of negative energy states, which would cause atoms to be unstable, since they could always decay to lower energy states by the
emission of radiation.^[6]^:71–72
The prevailing view at the time was that the world was composed of two very different ingredients: material particles (such as electrons) and quantum fields (such as photons). Material particles were
considered to be eternal, with their physical state described by the probabilities of finding each particle in any given region of space or range of velocities. On the other hand, photons were
considered merely the excited states of the underlying quantized electromagnetic field, and could be freely created or destroyed. It was between 1928 and 1930 that Jordan, Eugene Wigner, Heisenberg,
Pauli, and Enrico Fermi discovered that material particles could also be seen as excited states of quantum fields. Just as photons are excited states of the quantized electromagnetic field, so each
type of particle had its corresponding quantum field: an electron field, a proton field, etc. Given enough energy, it would now be possible to create material particles. Building on this idea, Fermi
proposed in 1932 an explanation for beta decay known as Fermi's interaction. Atomic nuclei do not contain electrons per se, but in the process of decay, an electron is created out of the surrounding
electron field, analogous to the photon created from the surrounding electromagnetic field in the radiative decay of an excited atom.^[3]^:22–23
It was realized in 1929 by Dirac and others that negative energy states implied by the Dirac equation could be removed by assuming the existence of particles with the same mass as electrons but
opposite electric charge. This not only ensured the stability of atoms, but it was also the first proposal of the existence of antimatter. Indeed, the evidence for positrons was discovered in 1932 by
Carl David Anderson in cosmic rays. With enough energy, such as by absorbing a photon, an electron-positron pair could be created, a process called pair production; the reverse process, annihilation,
could also occur with the emission of a photon. This showed that particle numbers need not be fixed during an interaction. Historically, however, positrons were at first thought of as "holes" in an
infinite electron sea, rather than a new kind of particle, and this theory was referred to as the Dirac hole theory.^[6]^:72^[3]^:23 QFT naturally incorporated antiparticles in its formalism.^[3]
Infinities and renormalization
Robert Oppenheimer showed in 1930 that higher-order perturbative calculations in QED always resulted in infinite quantities, such as the electron self-energy and the vacuum zero-point energy of the
electron and photon fields,^[6] suggesting that the computational methods at the time could not properly deal with interactions involving photons with extremely high momenta.^[3]^:25 It was not
until 20 years later that a systematic approach to remove such infinities was developed.
A series of papers was published between 1934 and 1938 by Ernst Stueckelberg that established a relativistically invariant formulation of QFT. In 1947, Stueckelberg also independently developed a
complete renormalization procedure. Such achievements were not understood and recognized by the theoretical community.^[6]
Faced with these infinities, John Archibald Wheeler and Heisenberg proposed, in 1937 and 1943 respectively, to supplant the problematic QFT with the so-called S-matrix theory. Since the specific
details of microscopic interactions are inaccessible to observations, the theory should only attempt to describe the relationships between a small number of observables (e.g. the energy of an atom)
in an interaction, rather than be concerned with the microscopic minutiae of the interaction. In 1945, Richard Feynman and Wheeler daringly suggested abandoning QFT altogether and proposed
action-at-a-distance as the mechanism of particle interactions.^[3]^:26
In 1947, Willis Lamb and Robert Retherford measured the minute difference in the ^2S[1/2] and ^2P[1/2] energy levels of the hydrogen atom, also called the Lamb shift. By ignoring the contribution of
photons whose energy exceeds the electron mass, Hans Bethe successfully estimated the numerical value of the Lamb shift.^[6]^[3]^:28 Subsequently, Norman Myles Kroll, Lamb, James Bruce French, and
Victor Weisskopf again confirmed this value using an approach in which infinities cancelled other infinities to result in finite quantities. However, this method was clumsy and unreliable and could
not be generalized to other calculations.^[6]
The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by Julian Schwinger, Richard Feynman, Freeman Dyson, and Shinichiro Tomonaga. The main
idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as renormalization and can
be applied to arbitrary order in perturbation theory.^[6] As Tomonaga said in his Nobel lecture:
Since those parts of the modified mass and charge due to field reactions [become infinite], it is impossible to calculate them by the theory. However, the mass and charge observed in experiments
are not the original mass and charge but the mass and charge as modified by field reactions, and they are finite. On the other hand, the mass and charge appearing in the theory are… the values
modified by field reactions. Since this is so, and particularly since the theory is unable to calculate the modified mass and charge, we may adopt the procedure of substituting experimental
values for them phenomenologically... This procedure is called the renormalization of mass and charge… After long, laborious calculations, less skillful than Schwinger's, we obtained a result...
which was in agreement with [the] Americans'.^[9]
By applying the renormalization procedure, calculations were finally made to explain the electron's anomalous magnetic moment (the deviation of the electron g-factor from 2) and vacuum polarization.
These results agreed with experimental measurements to a remarkable degree, thus marking the end of a "war against infinities".^[6]
At the same time, Feynman introduced the path integral formulation of quantum mechanics and Feynman diagrams.^[8]^:2 The latter can be used to visually and intuitively organize and to help compute
terms in the perturbative expansion. Each diagram can be interpreted as paths of particles in an interaction, with each vertex and line having a corresponding mathematical expression, and the product
of these expressions gives the scattering amplitude of the interaction represented by the diagram.^[1]^:5
It was with the invention of the renormalization procedure and Feynman diagrams that QFT finally arose as a complete theoretical framework.^[8]^:2
Given the tremendous success of QED, many theorists believed, in the few years after 1949, that QFT could soon provide an understanding of all microscopic phenomena, not only the interactions between
photons, electrons, and positrons. Contrary to this optimism, QFT entered yet another period of depression that lasted for almost two decades.^[3]^:30
The first obstacle was the limited applicability of the renormalization procedure. In perturbative calculations in QED, all infinite quantities could be eliminated by redefining a small (finite)
number of physical quantities (namely the mass and charge of the electron). Dyson proved in 1949 that this is only possible for a small class of theories called "renormalizable theories", of which
QED is an example. However, most theories, including the Fermi theory of the weak interaction, are "non-renormalizable". Any perturbative calculation in these theories beyond the first order would
result in infinities that could not be removed by redefining a finite number of physical quantities.^[3]^:30
The second major problem stemmed from the limited validity of the Feynman diagram method, which is based on a series expansion in perturbation theory. In order for the series to converge and
low-order calculations to be a good approximation, the coupling constant, in which the series is expanded, must be a sufficiently small number. The coupling constant in QED is the fine-structure
constant α ≈ 1/137, which is small enough that only the simplest, lowest order, Feynman diagrams need to be considered in realistic calculations. In contrast, the coupling constant in the strong
interaction is roughly of the order of one, making complicated, higher order, Feynman diagrams just as important as simple ones. There was thus no way of deriving reliable quantitative predictions
for the strong interaction using perturbative QFT methods.^[3]^:31
With these difficulties looming, many theorists began to turn away from QFT. Some focused on symmetry principles and conservation laws, while others picked up the old S-matrix theory of Wheeler and
Heisenberg. QFT was used heuristically as guiding principles, but not as a basis for quantitative calculations.^[3]^:31
Source theory
Schwinger, however, took a different route. For more than a decade he and his students had been nearly the only exponents of field theory,^[10]^:454 but in 1951^[11]^[12] he found a way around the
problem of the infinities with a new method using external sources as currents coupled to gauge fields.^[13] Motivated by the former findings, Schwinger kept pursuing this approach in order to
"quantumly" generalize the classical process of coupling external forces to the configuration space parameters known as Lagrange multipliers. He summarized his source theory in 1966^[14] then
expanded the theory's applications to quantum electrodynamics in his three volume-set titled: Particles, Sources, and Fields.^[15]^[16]^[17] Developments in pion physics, in which the new viewpoint
was most successfully applied, convinced him of the great advantages of mathematical simplicity and conceptual clarity that its use bestowed.^[15]
In source theory there are no divergences, and no renormalization. It may be regarded as the calculational tool of field theory, but it is more general.^[18] Using source theory, Schwinger was able
to calculate the anomalous magnetic moment of the electron, which he had done in 1947, but this time with no ‘distracting remarks’ about infinite quantities.^[10]^:467
Schwinger also applied source theory to his QFT theory of gravity, and was able to reproduce all four of Einstein's classic results: gravitational red shift, deflection and slowing of light by
gravity, and the perihelion precession of Mercury.^[19] The neglect of source theory by the physics community was a major disappointment for Schwinger:
The lack of appreciation of these facts by others was depressing, but understandable. -J. Schwinger^[15]
See "the shoes incident" between J. Schwinger and S. Weinberg.^[10]
Standard model
Elementary particles of the Standard Model: six types of quarks, six types of leptons, four types of gauge bosons that carry fundamental interactions, as well as the Higgs boson, which endow
elementary particles with mass.
In 1954, Yang Chen-Ning and Robert Mills generalized the local symmetry of QED, leading to non-Abelian gauge theories (also known as Yang–Mills theories), which are based on more complicated local
symmetry groups.^[20]^:5 In QED, (electrically) charged particles interact via the exchange of photons, while in non-Abelian gauge theory, particles carrying a new type of "charge" interact via the
exchange of massless gauge bosons. Unlike photons, these gauge bosons themselves carry charge.^[3]^:32^[21]
Sheldon Glashow developed a non-Abelian gauge theory that unified the electromagnetic and weak interactions in 1960. In 1964, Abdus Salam and John Clive Ward arrived at the same theory through a
different path. This theory, nevertheless, was non-renormalizable.^[22]
Peter Higgs, Robert Brout, François Englert, Gerald Guralnik, Carl Hagen, and Tom Kibble proposed in their famous Physical Review Letters papers that the gauge symmetry in Yang–Mills theories could
be broken by a mechanism called spontaneous symmetry breaking, through which originally massless gauge bosons could acquire mass.^[20]^:5–6
By combining the earlier theory of Glashow, Salam, and Ward with the idea of spontaneous symmetry breaking, Steven Weinberg wrote down in 1967 a theory describing electroweak interactions between all
leptons and the effects of the Higgs boson. His theory was at first mostly ignored,^[22]^[20]^:6 until it was brought back to light in 1971 by Gerard 't Hooft's proof that non-Abelian gauge
theories are renormalizable. The electroweak theory of Weinberg and Salam was extended from leptons to quarks in 1970 by Glashow, John Iliopoulos, and Luciano Maiani, marking its completion.^[22]
Harald Fritzsch, Murray Gell-Mann, and Heinrich Leutwyler discovered in 1971 that certain phenomena involving the strong interaction could also be explained by non-Abelian gauge theory. Quantum
chromodynamics (QCD) was born. In 1973, David Gross, Frank Wilczek, and Hugh David Politzer showed that non-Abelian gauge theories are "asymptotically free", meaning that under renormalization, the
coupling constant of the strong interaction decreases as the interaction energy increases. (Similar discoveries had been made numerous times previously, but they had been largely ignored.) ^[20]^:11
Therefore, at least in high-energy interactions, the coupling constant in QCD becomes sufficiently small to warrant a perturbative series expansion, making quantitative predictions for the strong
interaction possible.^[3]^:32
These theoretical breakthroughs brought about a renaissance in QFT. The full theory, which includes the electroweak theory and chromodynamics, is referred to today as the Standard Model of elementary
particles.^[23] The Standard Model successfully describes all fundamental interactions except gravity, and its many predictions have been met with remarkable experimental confirmation in subsequent
decades.^[8]^:3 The Higgs boson, central to the mechanism of spontaneous symmetry breaking, was finally detected in 2012 at CERN, marking the complete verification of the existence of all
constituents of the Standard Model.^[24]
Although quantum field theory arose from the study of interactions between elementary particles, it has been successfully applied to other physical systems, particularly to many-body systems in
condensed matter physics.
Historically, the Higgs mechanism of spontaneous symmetry breaking was a result of Yoichiro Nambu's application of superconductor theory to elementary particles, while the concept of renormalization
came out of the study of second-order phase transitions in matter.^[27]
Soon after the introduction of photons, Einstein performed the quantization procedure on vibrations in a crystal, leading to the first quasiparticle—phonons. Lev Landau claimed that low-energy
excitations in many condensed matter systems could be described in terms of interactions between a set of quasiparticles. The Feynman diagram method of QFT was naturally well suited to the analysis
of various phenomena in condensed matter systems.^[28]
Gauge theory is used to describe the quantization of magnetic flux in superconductors, the resistivity in the quantum Hall effect, as well as the relation between frequency and voltage in the AC
Josephson effect.^[28]
For simplicity, natural units are used in the following sections, in which the reduced Planck constant ħ and the speed of light c are both set to one.
Classical fields
A classical field is a function of spatial and time coordinates.^[29] Examples include the gravitational field in Newtonian gravity g(x, t) and the electric field E(x, t) and magnetic field B(x, t)
in classical electromagnetism. A classical field can be thought of as a numerical quantity assigned to every point in space that changes in time. Hence, it has infinitely many degrees of freedom.^
Many phenomena exhibiting quantum mechanical properties cannot be explained by classical fields alone. Phenomena such as the photoelectric effect are best explained by discrete particles (photons),
rather than a spatially continuous field. The goal of quantum field theory is to describe various quantum mechanical phenomena using a modified concept of fields.
Canonical quantization and path integrals are two common formulations of QFT.^[31]^:61 To motivate the fundamentals of QFT, an overview of classical field theory follows.
The simplest classical field is a real scalar field — a real number at every point in space that changes in time. It is denoted as ϕ(x, t), where x is the position vector, and t is the time. Suppose
the Lagrangian of the field, ${\displaystyle L}$, is
${\displaystyle L=\int d^{3}x\,{\mathcal {L}}=\int d^{3}x\,\left[{\frac {1}{2}}{\dot {\phi }}^{2}-{\frac {1}{2}}(abla \phi )^{2}-{\frac {1}{2}}m^{2}\phi ^{2}\right],}$
where ${\displaystyle {\mathcal {L}}}$ is the Lagrangian density, ${\displaystyle {\dot {\phi }}}$ is the time-derivative of the field, ∇ is the gradient operator, and m is a real parameter (the
"mass" of the field). Applying the Euler–Lagrange equation on the Lagrangian:^[1]^:16
${\displaystyle {\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial \phi /\partial t)}}\right]+\sum _{i=1}^{3}{\frac {\partial }{\partial x^{i}}}\left[{\frac
{\partial {\mathcal {L}}}{\partial (\partial \phi /\partial x^{i})}}\right]-{\frac {\partial {\mathcal {L}}}{\partial \phi }}=0,}$
we obtain the equations of motion for the field, which describe the way it varies in time and space:
${\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-abla ^{2}+m^{2}\right)\phi =0.}$
This is known as the Klein–Gordon equation.^[1]^:17
The Klein–Gordon equation is a wave equation, so its solutions can be expressed as a sum of normal modes (obtained via Fourier transform) as follows:
${\displaystyle \phi (\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left(a_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf {p} \cdot \mathbf
{x} }+a_{\mathbf {p} }^{*}e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right),}$
where a is a complex number (normalized by convention), * denotes complex conjugation, and ω[p] is the frequency of the normal mode:
${\displaystyle \omega _{\mathbf {p} }={\sqrt {|\mathbf {p} |^{2}+m^{2}}}.}$
Thus each normal mode corresponding to a single p can be seen as a classical harmonic oscillator with frequency ω[p].^[1]^:21,26
Canonical quantization
The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator.
The displacement of a classical harmonic oscillator is described by
${\displaystyle x(t)={\frac {1}{\sqrt {2\omega }}}ae^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}a^{*}e^{i\omega t},}$
where a is a complex number (normalized by convention), and ω is the oscillator's frequency. Note that x is the displacement of a particle in simple harmonic motion from the equilibrium position, not
to be confused with the spatial label x of a quantum field.
For a quantum harmonic oscillator, x(t) is promoted to a linear operator ${\displaystyle {\hat {x}}(t)}$:
${\displaystyle {\hat {x}}(t)={\frac {1}{\sqrt {2\omega }}}{\hat {a}}e^{-i\omega t}+{\frac {1}{\sqrt {2\omega }}}{\hat {a}}^{\dagger }e^{i\omega t}.}$
Complex numbers a and a^* are replaced by the annihilation operator ${\displaystyle {\hat {a}}}$ and the creation operator ${\displaystyle {\hat {a}}^{\dagger }}$, respectively, where † denotes
Hermitian conjugation. The commutation relation between the two is
${\displaystyle \left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1.}$
The Hamiltonian of the simple harmonic oscillator can be written as
${\displaystyle {\hat {H}}=\hbar \omega {\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\hbar \omega .}$
The vacuum state ${\displaystyle |0\rangle }$, which is the lowest energy state, is defined by
${\displaystyle {\hat {a}}|0\rangle =0}$
and has energy ${\displaystyle {\frac {1}{2}}\hbar \omega .}$ One can easily check that ${\displaystyle [{\hat {H}},{\hat {a}}^{\dagger }]=\hbar \omega {\hat {a}}^{\dagger },}$ which implies that ${\
displaystyle {\hat {a}}^{\dagger }}$ increases the energy of the simple harmonic oscillator by ${\displaystyle \hbar \omega }$. For example, the state ${\displaystyle {\hat {a}}^{\dagger }|0\rangle }
$ is an eigenstate of energy ${\displaystyle 3\hbar \omega /2}$. Any energy eigenstate state of a single harmonic oscillator can be obtained from ${\displaystyle |0\rangle }$ by successively applying
the creation operator ${\displaystyle {\hat {a}}^{\dagger }}$:^[1]^:20 and any state of the system can be expressed as a linear combination of the states
${\displaystyle |n\rangle \propto \left({\hat {a}}^{\dagger }\right)^{n}|0\rangle .}$
A similar procedure can be applied to the real scalar field ϕ, by promoting it to a quantum field operator ${\displaystyle {\hat {\phi }}}$, while the annihilation operator ${\displaystyle {\hat {a}}
_{\mathbf {p} }}$, the creation operator ${\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}$ and the angular frequency ${\displaystyle \omega _{\mathbf {p} }}$are now for a particular p:
${\displaystyle {\hat {\phi }}(\mathbf {x} ,t)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2\omega _{\mathbf {p} }}}}\left({\hat {a}}_{\mathbf {p} }e^{-i\omega _{\mathbf {p} }t+i\mathbf
{p} \cdot \mathbf {x} }+{\hat {a}}_{\mathbf {p} }^{\dagger }e^{i\omega _{\mathbf {p} }t-i\mathbf {p} \cdot \mathbf {x} }\right).}$
Their commutation relations are:^[1]^:21
${\displaystyle \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=(2\pi )^{3}\delta (\mathbf {p} -\mathbf {q} ),\quad \left[{\hat {a}}_{\mathbf {p} },{\hat {a}}_{\mathbf
{q} }\right]=\left[{\hat {a}}_{\mathbf {p} }^{\dagger },{\hat {a}}_{\mathbf {q} }^{\dagger }\right]=0,}$
where δ is the Dirac delta function. The vacuum state ${\displaystyle |0\rangle }$ is defined by
${\displaystyle {\hat {a}}_{\mathbf {p} }|0\rangle =0,\quad {\text{for all }}\mathbf {p} .}$
Any quantum state of the field can be obtained from ${\displaystyle |0\rangle }$ by successively applying creation operators ${\displaystyle {\hat {a}}_{\mathbf {p} }^{\dagger }}$ (or by a linear
combination of such states), e.g. ^[1]^:22
${\displaystyle \left({\hat {a}}_{\mathbf {p} _{3}}^{\dagger }\right)^{3}{\hat {a}}_{\mathbf {p} _{2}}^{\dagger }\left({\hat {a}}_{\mathbf {p} _{1}}^{\dagger }\right)^{2}|0\rangle .}$
While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy
levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems.^[32] The
process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization.^[1]^:19
The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, Dirac fields,^[1]^:52 vector fields (e.g. the
electromagnetic field), and even strings.^[33] However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the
case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform
calculations on any realistic interacting theory, perturbation theory would be necessary.
The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of
the real scalar field:^[1]^:77
${\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\phi )\left(\partial ^{\mu }\phi \right)-{\frac {1}{2}}m^{2}\phi ^{2}-{\frac {\lambda }{4!}}\phi ^{4},}$
where μ is a spacetime index, ${\displaystyle \partial _{0}=\partial /\partial t,\ \partial _{1}=\partial /\partial x^{1}}$, etc. The summation over the index μ has been omitted following the
Einstein notation. If the parameter λ is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.
Path integrals
The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state
spaces. To calculate the probability amplitude for a system to evolve from some initial state ${\displaystyle |\phi _{I}\rangle }$ at time t = 0 to some final state ${\displaystyle |\phi _{F}\rangle
}$ at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let
H be the Hamiltonian (i.e. generator of time evolution), then^[31]^:10
${\displaystyle \langle \phi _{F}|e^{-iHT}|\phi _{I}\rangle =\int d\phi _{1}\int d\phi _{2}\cdots \int d\phi _{N-1}\,\langle \phi _{F}|e^{-iHT/N}|\phi _{N-1}\rangle \cdots \langle \phi _{2}|e^
{-iHT/N}|\phi _{1}\rangle \langle \phi _{1}|e^{-iHT/N}|\phi _{I}\rangle .}$ | {"url":"https://www.wikiwand.com/en/articles/Quantum_field_theory","timestamp":"2024-11-07T14:18:07Z","content_type":"text/html","content_length":"1049787","record_id":"<urn:uuid:0420b719-3886-4276-ad78-35671daabbd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00663.warc.gz"} |
College Algebra Corequisite
Learning Outcomes
By the end of this section, you will be able to:
• Identify polynomial expressions
• Evaluate polynomial expressions
• Find the greatest common factor of a list of expressions
• Find the greatest common factor of a polynomial
• Recognize and define a rational expression
• Determine the domain of a rational expression
• Simplify a rational expression
Before we can perform algebraic operations on polynomials and rational expressions, we need to be able to recognize and define these objects. We’ll also need to review some skills that make our work
on these expressions possible.
We’ve already seen how finding the least common multiple makes it possible to do addition and subtraction on fractions. The greatest common factor is another object we can find that makes it possible
to work with polynomial expressions in useful ways. Finally, we’ll extend our knowledge of fractions as ratios of expressions by defining and simplifying rational expressions.
Warm up for this module by refreshing these important concepts and skills. As you study these review topics, recall that you can also return to Module 1 Algebra Essentials any time you need to
refresh the basics.
Recall for success
Look for red boxes like this one throughout the text. They’ll show up just in time to give helpful reminders of the math you’ll need, right where you’ll need it. | {"url":"https://courses.lumenlearning.com/waymakercollegealgebracorequisite/chapter/review-for-success/","timestamp":"2024-11-13T04:37:25Z","content_type":"text/html","content_length":"47338","record_id":"<urn:uuid:713b22fd-2cf2-4e38-bd4c-02a5fb325193>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00852.warc.gz"} |
CAS command question
01-11-2017, 05:40 PM
Post: #76
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: CAS command question
DrD, I think you are still confused. The parser steps returns the expression as is (unevaluated) but that does not change the CAS logic: you need to eval the expression otherwise nothing happens and
you would get left(x>2) returned when you type left(x>2) (exactly like if you enter quote(left(x>2))).
I don't understand why it's so important for you to keep the ordering, since I have explained how you can program the same functionnality whatever the order is. In fact it's even better because the
resulting program is more general.
User(s) browsing this thread: 2 Guest(s) | {"url":"https://www.hpmuseum.org/forum/showthread.php?tid=7506&pid=66609&mode=threaded","timestamp":"2024-11-10T16:18:11Z","content_type":"application/xhtml+xml","content_length":"50001","record_id":"<urn:uuid:fcf2ca06-1c59-4e19-a5b4-eb11455a35be>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00286.warc.gz"} |
Erica Klarreich | Page 2 | Quanta Magazine
‘Monumental’ Math Proof Solves Triple Bubble Problem and More
The decades-old Sullivan’s conjecture, about the best way to minimize the surface area of a bubble cluster, was thought to be out of reach for three bubbles and up — until a new breakthrough result. | {"url":"https://www.quantamagazine.org/authors/erica-klarreich/page/2/","timestamp":"2024-11-05T21:33:05Z","content_type":"text/html","content_length":"156712","record_id":"<urn:uuid:17429a06-0ee5-493e-aa2c-ae2f69832f3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00401.warc.gz"} |
How to Know What Rate of Return to Expect From Your Stocks | Daily Trade Alert
In Part 1 of this series found here, we voiced the notion that there are two primary attributes, valuation and the rate of change of earnings growth, which prudent investors can use to forecast the
potential future returns on their stocks.
However, Part 1 was primarily focused on ascertaining the principles which laid the foundation for sound current valuation.
[ad#Google Adsense 336×280-IA]In this Part 2, we will focus on how to utilize current valuation in conjunction with earnings growth rates in order to come up with a reasonable expectation of the
future total returns a stock can be expected to provide.
The point is that neither can be looked at in isolation.
In other words, the price you pay to buy the growth that the company ultimately delivers, will determine not only how much money you make (the percentage return on investment), but how much risk you
took to make it.
Moreover, in Part 1, we concerned ourselves with defining historical norms relating to valuation, and then provided historical evidence to back them up. But even more importantly, we strove to
illustrate that there is a practical rationale that underpinned our calculations of soundness. This practical rationale is the calculation of the earnings yield (earnings divided by price) that a
given valuation (PE ratio) represents.
The general idea here is that a company’s profitability ought to represent an acceptable rate of return on your investment. After all, when all is said and done, when you “invest” in a common stock
you are buying the business. If that business isn’t making enough money to give you a decent return on your investment, then common sense would indicate that you’re overpaying for it.
Another way of looking at this is to understand and acknowledge that a business gets its value from the amount of cash flow it is capable of generating for its stakeholders (owners or shareholders).
At this point, it’s important that the reader recognizes the previous statement as a metaphor.
In other words, we are using the word “cash flow” to represent the amount of money the business is making you if you are an owner. Furthermore, in the context of this article we consider the word
“cash flow” as interchangeable with profits or earnings. Our objective is to focus on the essence of valuation so that the principle can be understood, and not get caught up in semantics or
accounting convention.
Analogy of a Hypothetical Private Company
With this in mind, we offer the following analogy. Let’s assume that you own a private business that pays you $100,000 a year of income (salary, dividends, bonuses, etc.). We will also assume that
your business provides a guaranteed $100,000 per year in income, no risk and no growth. Furthermore, we also assume that you live in the perfect society with no taxes. Remember, our objective is to
focus on the essence of valuation.
Now let’s further assume that you are tired of working and want to sell the business. The seminal question is at what price will you be willing to sell? The logical answer would be at some reasonable
multiple of one year’s worth of your earnings. Put another way, if a business generates a predictable annual income stream, it has a value greater than that income stream, even at zero growth.
Because logically, if you sold the business for only $100,000, or one times earnings, you would be out of money in one year.
Therefore, in order to sell, you would need a multiple of one year’s earnings that you can use to provide yourself a comparable future income stream in your new retirement. If we use the multiple of
15 (a PE of 15), our standard of value presented in Part 1, we would get a selling price of $1,500,000.
Now, if we could invest the $1,500,000 into a passive investment (Bond, CD, Public Dividend Paying Stock, etc.) at our established historically reasonable return of 6.66% (PE=15), we discover that
our income would be $99,900 per year (approximately $100,000 per year). Note: This provides some insight into the underlying reason for a normal PE of 15, because the historical average annual return
on stocks has been approximately somewhere between 6-9%. Therefore, the return a 15 PE represents is both believable and historically achievable.
Consequently, both the seller and buyer in our metaphorical analogy would be provided a sound and reasonable return. The seller can take the proceeds and generate a future return adequate enough to
provide a comfortable retirement, and the buyer simultaneously is earning a reasonable return on their investment in the operating business. The point is that the income stream is driving the value,
and it’s the income stream that gives a business its worth, whether it’s a publicly traded stock on an exchange, or private.
Risk, Return and the Valuation Relationships
Thus far, we have established a baseline of valuation using a hypothetical guaranteed and static income stream-no risk and no growth. However, in the real world, a business’ income stream is neither
static nor guaranteed. Furthermore, the more dynamic the income stream, the more risk associated with achieving it.
Some cases in point would include the notions that faster growth is harder to achieve than slower growth, and consistent growth would be generally considered more predictable than cyclical growth.
Therefore, we now introduce two new concepts; risk and compounding. Risk will tend to lower or reduce the valuation a prudent investor is willing to pay for a given investment, and a higher rate of
earnings growth tends to increase future value (compounding).
Therefore, risk and earnings growth rates will represent counteracting forces affecting starting or current valuation (PE’s). This partially explains why a 3% grower (less risky to achieve) might
command the same current valuation PE of, for example, an 11% or 12% grower (riskier and harder to achieve). But this is a critical point; the faster grower will generate a higher future return than
the slower grower, ceteris paribas.
Pictures Are Worth a Thousand Words
From this point forward, we are going to rely on FAST Graphs’ earnings and price correlated graphs and performance charts to vividly express the valuation and return relationships. Therefore, the
following pictures will more succinctly articulate the principles this series of articles has been developing. But for the reader to receive any insights from the graphics, a few words of explanation
on them is in order.
• The orange earnings justified valuation lines on each graph are calculated by applying widely accepted formulas commonly utilized to value a business. There is no manipulation or adjustments
made. The PE ratios represented are calculated and applied for the time period graphed.
• The slope of the orange earnings justified line is equal to the calculated earnings growth rate. Importantly, although the PE may be the same or similar for two respective companies, the growth
rate (slope of the line) can differ dramatically (this is a key to return calculations).
• The calculated PE ratio on each graph is the same for every point on the graph. The applicable PE is listed to the right of each graph in orange letters with the formula designation, GDF for
Graham Dodd Formula, PEG for PEG ratio (PE=Growth Rate) and GDF-PEG representing an extrapolation between the other two.
• The dark blue line is a calculated blended normal (historical) PE. In other words, the PE that the market has actually applied to the company historically.
• The light blue shaded area represents dividends paid out of the green shaded area earnings. Dividends, if any, represent an important component of the total return calculation.
• To interpret the graphs correctly, the reader must look to the right of the graph for essential mathematical factors that apply to each respective company.
• For illustration purposes, we strove to provide examples where valuations were aligned at both the beginning and the end of each time period; or at least approximately so. This allows the reader
to more clearly see the relationship between earnings growth and return over time.
From this point forward additional insights into historical valuation and return calculations and future estimates of both will be interjected by example with each subsequent graph. Generally, we
will start with lower growth and move to faster growth with each following example.
Examples of the PE = 15 Standard for Earnings Growth Rates from 0% to 15%
Vectren Corp. (VVC): A Low Growth Utility
Our first example looks at Vectren Corp’s historical earnings, a utility with a 15-year historical earnings growth rate that is below our 3% threshold established in Part 1. Note that fair valuation
is calculated using Graham Dodd’s Formula (GDF) deriving a fair value PE of 13.8 (slightly below, but close to our PE 15 standard). However, a normal PE of 16 has been historically applied by Mr.
Market. Therefore, valuation falls between a PE of 13.8 to 16, or well within a range of normalcy.
With our second graph we introduce and overlay monthly closing stock price. Here we discover that for the most part, price tracks earnings within the corridor of the orange and the blue line. This
provides historical evidence of the practical reality of a normal PE of 15, plus or minus a minor deviation from time to time for a slow, or almost no growth, company. However, this example also
provides historical evidence that the earnings growth rate drives the capital appreciation component of total return as we will discover when we review the performance table next.
When analyzing the performance associated with Vectren Corp, we learn that capital appreciation (closing annualized rate of return) of 1.2% closely matches earnings growth of 1.7%. Thereby, we
establish that the rate of change of earnings growth relates closely to capital appreciation.
So now we have one important piece of the return puzzle, earnings growth. Also, since beginning valuation was approximately at a PE of 16 (blue normal PE line) and so was ending valuation, little
valuation adjustment to capital appreciation applies.
This example also provides important insight into the relevance of dividends to total return. Although we will expand on dividends in more detail with future examples, we learn here that dividends
provide a return in addition to capital appreciation from earnings growth (see circles on the table).
Note that this is a main reason that FAST Graphs expresses dividends on top of earnings even though they are paid out of earnings. Dividends (paid out and not reinvested in the table) provide
additional return above appreciation.
Nextera Energy (NEE): A Moderately Growing Utility
With our second example we move up the food chain of growth by reviewing Nextera Energy a moderately faster growing utility stock. Even though Nextera Energy’s growth rate is more than 3 times
faster, averaging 6.4% per annum, we discover that valuation within our PE = 15 range. To be clear, what this tells us is that investing in Nextera at a PE ratio of approximately 15 represents a
sound and historically normal valuation.
However, we also once again see evidence that a reasonable current valuation represents soundness, but as we will soon see, the rate of change of earnings growth will determine our actual future
As you review the graphic, note that whenever the price line deviates from the orange earnings justified valuation line, it inevitably comes back into alignment with fair value.
Therefore, the slope of the line at 6.4% becomes the driver of the capital appreciation component of return when valuation is aligned at the beginning and the ending time period measured.
When reviewing the performance table associated with the Nextera Energy graphic above, we once again see a very close correlation and relationship between earnings growth and capital appreciation at
approximately 6% per annum.
Moreover, we discover that the dividend growth rate also coincides very closely with earnings growth over time. And, we once again see that dividends (not reinvested) represent additional return.
We believe the real takeaway here is that sound valuation coupled with a company’s earnings growth rate will allow the investor to earn a return that equates with the company’s earnings growth.
However, if the company pays dividends, they will represent a return kicker above and beyond earnings growth. On the other hand, the rate of dividend growth and earnings growth will generally
correlate, ceteris paribus (all things remaining equal).
VF Corp. (VFC): Leading Apparel Company
With our third example, VF Corp., we again move further up the growth chain where earnings growth has averaged 8.6% per annum. Again, we discover that the PE = 15 standard continues to apply.
However, for companies growing between 5%-15% the extrapolated formula mentioned above automatically calculates fair value. From the graphic, it is clear that a 15 PE represents a reasonable proxy
for fair valuation.
At this point, we find it appropriate to interject the idea that fair valuation is not an absolute. Instead, it should be thought of as a reasonable range of soundness that can be utilized to make
fair value investing decisions.
In this example, we also discover that the normal PE ratio that Mr. Market has typically applied to VF Corp. is 13.1, slightly below our PE = 15 standard. The only logical reason that we can surmise
for this adjustment, is the possibility of the market’s perception of risk.
Nevertheless, it should be clear from the graphic that a PE ratio ranging between 13-15 has historically applied to this company. In other words, armed with this information, the prudent investor
might only consider investing in this company when the PE ratio is at 13 or lower. On the other hand, as we will soon see, a PE of 15 would not be a disastrous idea.
Once again we see a high correlation between earnings growth and capital appreciation. However, notice the effect that an expanding payout ratio has had on total dividends, and therefore, total
Hormel Foods Corp.
With this next example, we move closer to the upper end of our range of earnings growth where we have discovered that a PE ratio of 15 normally applies as a reasonable valuation measurement. However,
we also see that the market has generally rewarded this faster and rather consistent earnings growth rate with a premium PE ratio of 17.2.
Consequently, and as with this example, faster growth does warrant a higher valuation, as many may have suspected, the 15 PE still offers a proxy for valuation.
On the other hand, you could pay up to 17 times earnings or slightly higher, and still earn a decent return on a company with this growth rate. Remember, valuation serves as a guide, not an absolute.
Also, in this case, the consistent rate of above-average growth provided by a food company providing basic human needs may be considered less risky than an apparel stock.
When evaluating the performance on Hormel Foods Company (HRL), we continue to see a strong correlation between earnings and return.
However, a slight overvaluation at the beginning (see red circle) reduced capital appreciation from an expected 11.8% that mirrored earnings growth to only 9.5%. This casts a bright light on the
importance and power of earnings growth as it relates to long-term return.
In other words, an above-average growth rate can overcome a moderately bad valuation decision in the long run. And dividends can mitigate a moderately poor investing decision even more.
Growth of 15% Per Annum or Better An Inflection Point
As we discussed in Part 1, historical evidence and logic imply that the normal PE ratio of approximately 15 is a reasonable proxy for fair valuation on many companies.
However, we’ve also attempted to illustrate that fair valuation alone does not indicate a strong return. Instead, fair or sound valuation will allow you to earn a return that equates (approximates)
to the company’s earnings growth rate over time, plus dividends, if any.
Our observations of thousands of companies utilizing the earnings and price correlated FAST Graphs (Fundamentals Analyzer Software Tool), have led us to conclude and discover that once earnings
growth exceeds 15% per annum, fair valuation reaches an inflection point.
We believe that this is predominantly a function of the power of compounding. Once a company’s growth rate exceeds 15% per annum, the stream of income that it produces (past or future) expands
Therefore, our research indicates that fair valuation rises above the PE=15 standard and begins approximating the company’s earnings growth rate up to growth between 35% and 40% per annum. Above 40%
earnings growth rates, valuation ratios begin to blur.
We believe this is attributed to the enormous risk associated with achieving such high rates of growth. Consequently, the market as a general rule will discount growth rates above 40% by applying a
lower PE ratio.
Ross Stores Inc.
Our first example of high growth is Ross Stores Inc. (ROST) that has averaged earnings growth of 17.6%, indicating a fair value PEG ratio PE of 17.6. However, and interestingly, notice that the
normal PE ratio sits close to our standard PE ratio of 15. In this case, a PE of 15 might represent a risk adjusted valuation below the higher earnings growth of 17.6%. Nevertheless, buying this
stock at a PE of 17 still delivers a strong long-term rate of return.
Current overvaluation, indicated by the current blended PE ratio of 20.1 explains capital appreciation above earnings growth of 17.6%. Furthermore, due to the power of compounding, although dividends
contribute to total return, capital appreciation provides the majority of shareholder returns in this example.
Church & Dwight Inc.
Church & Dwight Inc. (CHD) provides almost the identical growth rates of our Ross Stores’ example above. However, notice how the market in this case has applied a premium historical PE ratio of 19.7
that is greater than the earnings growth rate of 17.7%. We believe this indicates a lower risk premium applied to this consumer staples company over a riskier retailer such as Ross Stores.
Once again, we see that an above-average rate of return correlates very closely to this company’s above-average earnings growth rate. Since valuation was sound at the beginning, and since Church &
Dwight is moderately overvalued currently, annualized total return exceeds earnings growth rate.
Oracle Corp. Fast Growing Software Company
Oracle (ORCL) offers several lessons on how to calculate the returns from your common stock investments. First of all, we see a strong correlation between earnings and stock price for most of the
15-year time frame. However, we also see the extreme and unwarranted overvaluation that occurred during the irrationally exuberant technology bubble of the late 1990s.
Although Oracle is currently undervalued based on its historical earnings growth, we still see that long-term returns relate to the company’s earnings growth rate, discounted by undervaluation.
We thought it would be useful towards understanding the theme of this series of articles to look at what impact overvaluation had on Oracle’s long-term returns since 2001. From the following graph we
can see that valuation was very high at the time.
Consequently, extreme overvaluation almost totally wiped out long-term shareholder returns even though earnings growth was strong. However, notice that earnings growth has fallen from 19.6% over the
15-year period to 14.5% over the 12-year period. Earnings growth is a dynamic concept and fair valuation is therefore fluid as well.
Liquidity Services Inc. (LQDT)
Our final example of fast growth looks at Liquidity Services Inc. Although somewhat cyclical, we see that stock prices have tracked earnings growth very closely. Moreover, we see that the market has
typically applied a fair value PE ratio that equates very closely with the company’s earnings growth rate, thereby providing additional evidence of the validity of the PE equals growth rate valuation
concept applies to fast growth above 15%.
Cyclical Stocks but the Valuation Rules Continue to Apply
The following example looking at Cooper Cos. Inc. (COO), and reveals a very cyclical company that provides insight into the PE equals 15 hypotheses that this series of articles has presented.
Clearly, price follows earnings even when earnings are dropping, and interestingly, the market tends to apply our standard PE equals 15 even when this happens.
Summary Conclusions
Our goals with this series of articles, was to establish a framework, and hopefully insights, into how an investor can know what rate of return to expect from their stocks. Simply stated, it’s a
function of valuation coupled with earnings growth. Unfortunately, time and space only allowed us to scratch the surface of these important investor concepts. Therefore, a Part 3 addendum to this
series is in order that will primarily deal with forecasting future returns based on these principles. However, we are hopeful that the essence of return was adequately expressed thus far.
Furthermore, in order to receive the maximum insight and benefit from this series, it’s imperative that the reader spend the majority of their time analyzing and evaluating the graphics presented.
These FAST Graphs not only express the relationships to return that were offered, they simultaneously provide historical evidence of the veracity of the hypotheses as well.
Finally, and most importantly, the reader should keep in mind that what was discussed here represents ranges as well as nuances of valuation and return. These are multifaceted concepts that are also
adjusted by risk as we attempted to iterate. Therefore, these concepts should serve more as guides. In other words, these concepts provide practical guidelines, but they lack perfect precision,
mostly due to the uncertainties that are always associated with investing in common stocks.
– Chuck Carnevale of FAST Graphs
[ad#jack p.s.]
Source: FAST Graphs
Disclosure: Long NEE, VFC, ORCL & ROST at the time of writing. | {"url":"https://dailytradealert.com/2012/07/09/how-to-know-what-rate-of-return-to-expect-from-your-stocks-2/","timestamp":"2024-11-04T15:16:25Z","content_type":"text/html","content_length":"130396","record_id":"<urn:uuid:5d28d9d3-aade-416e-aa9c-42eeb96d0c3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00693.warc.gz"} |
Kj/Kg To Mw Calculator - Calculator Doc
The conversion from kilojoules per kilogram (Kj/Kg) to megawatts (MW) is essential in various fields, particularly in energy production and consumption analysis. Understanding this conversion allows
professionals to determine the power output or energy conversion rate of various systems, making it a valuable calculation in industries like power generation, engineering, and environmental
The formula to convert Kj/Kg to MW is:
Mw = (Kj/Kg ∗ M) / T
• Kj/Kg is the energy in kilojoules per kilogram.
• M is the mass in kilograms.
• T is the time in seconds.
How to Use
To use the Kj/Kg to MW calculator:
1. Enter the energy in kilojoules per kilogram (Kj/Kg).
2. Input the mass (M) in kilograms.
3. Enter the time (T) in seconds.
4. Click the “Calculate” button.
5. The result will display the equivalent power in megawatts (MW).
Suppose you have an energy value of 500 Kj/Kg, a mass of 2000 kg, and a time duration of 60 seconds. Using the formula:
Mw = (500 ∗ 2000) / 60 = 16.6667 MW
So, the power output is approximately 16.67 MW.
1. What does Kj/Kg represent?
Kj/Kg stands for kilojoules per kilogram, a unit of energy density.
2. Why is converting Kj/Kg to MW important?
This conversion helps in understanding the power output or energy conversion rate, which is crucial in energy production and analysis.
3. What is a megawatt (MW)?
A megawatt is a unit of power equal to one million watts or 1,000 kilowatts.
4. How does mass affect the conversion from Kj/Kg to MW?
The mass directly influences the total energy, as the formula multiplies energy density by mass.
5. Can this calculator be used for any type of energy source?
Yes, the calculator is versatile and can be used for various energy sources like fuel, electricity, or heat.
6. What is the significance of time (T) in the formula?
Time is crucial as it determines how quickly the energy is converted, affecting the power output.
7. Is the conversion from Kj/Kg to MW always linear?
Yes, the relationship is linear, but other factors like efficiency may influence real-world calculations.
8. Can this calculation be applied to renewable energy sources?
Yes, it can be used to calculate the power output of renewable sources like biomass or solar energy.
9. Why is the result divided by 1000 in the formula?
Dividing by 1000 converts the result from kilowatts to megawatts.
10. What industries use this conversion the most?
Industries like power generation, engineering, and environmental sciences frequently use this conversion.
11. How accurate is this calculator?
The calculator is designed to provide accurate results based on the input values.
12. Can this calculator be used for educational purposes?
Yes, it’s a great tool for students and professionals learning about energy conversion.
13. What are the limitations of this calculator?
The main limitation is that it assumes ideal conditions without accounting for inefficiencies or losses.
14. Is it possible to reverse the calculation?
Yes, by rearranging the formula, you can calculate Kj/Kg from MW.
15. How do environmental factors influence the calculation?
Factors like temperature, pressure, and material properties can affect real-world applications but are not considered in the basic calculation.
16. What is the difference between Kj/Kg and MW?
Kj/Kg measures energy per unit mass, while MW measures the rate of energy conversion or power.
17. Can this conversion be used in HVAC systems?
Yes, it can be applied in heating, ventilation, and air conditioning systems to calculate energy efficiency.
18. What is the role of specific energy in this conversion?
Specific energy, represented by Kj/Kg, is a key factor that determines how much energy is available per unit mass.
19. How does this conversion relate to efficiency calculations?
Understanding the power output in MW helps in evaluating the efficiency of energy conversion processes.
20. Can this calculator be used for both small-scale and large-scale energy systems?
Yes, it’s suitable for both small-scale applications like home energy systems and large-scale industrial processes.
Converting Kj/Kg to MW is a valuable calculation in understanding and optimizing energy systems. Whether you’re working in power generation, engineering, or environmental sciences, this calculator
simplifies the process, allowing you to accurately determine the power output or energy conversion rate. By mastering this conversion, you can contribute to more efficient and sustainable energy | {"url":"https://calculatordoc.com/kj-kg-to-mw-calculator/","timestamp":"2024-11-13T22:43:30Z","content_type":"text/html","content_length":"86901","record_id":"<urn:uuid:4681229e-589c-48a1-a831-5c7240087da7>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00547.warc.gz"} |
SystemFw FP blog
We want to write an algebra that reads from stdin, and use it to write the following program:
read a line from stdin, compute its length, and return true if the length is greater than 10, or false otherwise.
A starting point could be:
* carrier:
* In
* introduction forms:
* readLine: In
sealed trait In {
object In {
val readLine: In
but obviously that’s not enough to write our program: we have encoded the action of reading a line from stdin, but we still need to do some extra transformations on the line we’ve read.
So, your first instinct might be to add an elimination form:
* carrier:
* In
* introduction forms:
* readLine: In
* elimination forms:
* nope: In => String
sealed trait In {
def nope: String
object In {
val readLine: In
in order to write:
val out: Boolean = readLine.nope.length > 10
but this is not a fruitful direction: we are basically saying that the only way to change the output of a program written in the In algebra is to eliminate the algebra entirely.
Algebras are our unit of composition, therefore with the elimination approach any program that needs to change its output can no longer be composed, which is a really strong limitation: for example
in Part III we saw that for IO elimination happens when the JVM calls main, it would be really weird if we couldn’t encode something as simple as String.length until then.
Instead, we want to have the ability to transform outputs without leaving our algebra, and therefore we have to enrich it with a transformOutput combinator. Recall that the general shape of a
combinator is:
transformOutput: (In, ...) => In
and we need to fill the ... with something that can encode the idea of transforming one thing into another, which we already have a well known concept for: functions. So, transformOutput needs to
take a function, but we have a problem: what type should this function be, in order to fit the possible transformations we want to encode such as _.length or _ > 10 ?
Of course, an Any => Any fits anything:
* carrier:
* In
* introduction forms:
* readLine: In
* combinators:
* transformOutput: (In, Any => Any) => String
sealed trait In {
def transformOutput(transform: Any => Any): In
object In {
val readLine: In
but this is also not an acceptable solution: our algebra has gained power, but we have lost type safety altogether.
As it turns out, the issue is with the carrier, specifically that these two programs have the same type:
val readsLine: In
val computesLength: In
which means we cannot link them with a function without casting: we know that the function to pass to transformOutput should have type String => Int, but the compiler doesn’t.
val readsLine: In =
val computesLength: In
In.transformOutput(str => str.asInstanceOf[String].length)
The key idea out of this problem is that we can add a type parameter to In which represents the type of its output. The resulting type In[A] lets us write:
val readsString: In[String]
val computesInt: In[Int]
Note that this doesn’t require us to actually perform any action, we’re still just building a datatype with a sealed trait and case classes, except this datatype now carries enough type information
to allow for well typed composition. In other words, In[String] is not a container that contains a String, rather it’s a command to eventually read one, encoded as a datatype.
transformOutput can now have a proper type:
def transformOutput[A, B]: (In[A], A => B) => In[B]
This signature has two type variables (or type parameters), A and B . The rule with type variables is that whenever the same type variable is mentioned, the relative types have to match: in this
case, (In[A], A => ... means that the input of the function needs to match the output of the In program, and ... => B) => In[B] means that the output of the resulting In program will match the output
of the function. Therefore in the example above the function we need to pass to transformOutput to connect readsString: In[String] with computesInt: In[Int] has to have type String => Int, just like
we expect.
Conversely, whenever different type variables appear, the relative types can be different, but they don’t have to, or in other words transformOutput also works if you use it with an In[String] and a
String => String, resulting in another In[String].
We can now write a proper version of In:
* carrier:
* In[A]
* introduction forms:
* readLine: In[String]
* combinators:
* transformOutput[A, B]: (In[A], A => B) => In[B]
sealed trait In[A] {
def transformOutput(transform: A => B): In[B]
object In {
val readLine: In[String]
and use it to express our original program:
val prog: In[Boolean] =
.transformOutput(line => line.length)
.transformOutput(length => length > 10)
Finally, we need to complete In with an elimination form so that we can embed it into bigger programs, as usual we will translate to IO:
* carrier:
* In[A]
* introduction forms:
* readLine: In[String]
* combinators:
* transformOutput[A, B]: (In[A], A => B) => In[B]
* elimination forms:
* run[A]: In[A] => IO[A]
sealed trait In[A] {
def transformOutput(transform: A => B): In[B]
def run: IO[A]
object In {
val readLine: In[String]
that being said, we won’t be thinking about eliminations forms for the next few articles, as we focus on writing programs with our algebras. We will return to the topic of elimination forms once we
talk about IO in more detail.
You might be wondering why I have written the final program as:
val prog1: In[Boolean] =
.transformOutput(line => line.length)
.transformOutput(length => length > 10)
as opposed to:
val prog2: In[Boolean] =
readLine.transformOutput(line => line.length > 10)
prog2 seems less verbose, so should we refactor prog1 into prog2? Will the behaviour change? Intuitively, it would feel really weird if it did: transforming the output twice ought to be the same of
transforming it once with the composite transformation.
We can encode this type of assumption as a law, something of shape:
expr1 <-> expr2
where <-> means that expr1 can be rewritten into expr2, and vice versa. In our case, we will say that:
p.transformOutput(f).transformOutput(g) <-> p.transformOutput(x => g(f(x)))
p: In[A]
f: A => B
g: B => C
which means that we can switch between prog1 and prog2 at will, and not just in the case where p = readLine, f = _.length, and g = _ > 10, but for any p, f, and g, as long they have the correct type.
So in this case we use laws as a refactoring aid : they gave us freedom to refactor by specifying which transformations on our programs are harmless.
By the way, since Scala functions already have an andThen method to express function composition, the law above can be written as:
p.transformOutput(f).transformOutput(g) <-> p.transformOutput(f.andThen(g))
And as it turns out, there is another law concerning transformOutput, the fact that transforming an output with a function that doesn’t change it is the same as not transforming it at all:
p.transformOutput(x => x) <-> p
p: In[A]
If this seems completely obvious, that’s because it is! Many laws are just stating: my algebra behaves in the way you expect.
In this article we introduced a really important idea: encoding the output of a program by adding a type parameter to the carrier type of our algebra. This enabled us to add the transformOutput
combinator, and next time we will use the same insight to model chaining, which is the essence of sequential control flow. | {"url":"https://systemfw.org/posts/programs-as-values-V.html","timestamp":"2024-11-09T03:44:31Z","content_type":"application/xhtml+xml","content_length":"21523","record_id":"<urn:uuid:f3217d06-704e-4ab3-883e-c609cab9bb64>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00588.warc.gz"} |
Michael D. Shields
Jul 17, 2024
Abstract:The Deep operator network (DeepONet) is a powerful yet simple neural operator architecture that utilizes two deep neural networks to learn mappings between infinite-dimensional function
spaces. This architecture is highly flexible, allowing the evaluation of the solution field at any location within the desired domain. However, it imposes a strict constraint on the input space,
requiring all input functions to be discretized at the same locations; this limits its practical applications. In this work, we introduce a Resolution Independent Neural Operator (RINO) that provides
a framework to make DeepONet resolution-independent, enabling it to handle input functions that are arbitrarily, but sufficiently finely, discretized. To this end, we propose a dictionary learning
algorithm to adaptively learn a set of appropriate continuous basis functions, parameterized as implicit neural representations (INRs), from the input data. These basis functions are then used to
project arbitrary input function data as a point cloud onto an embedding space (i.e., a vector space of finite dimensions) with dimensionality equal to the dictionary size, which can be directly used
by DeepONet without any architectural changes. In particular, we utilize sinusoidal representation networks (SIRENs) as our trainable INR basis functions. We demonstrate the robustness and
applicability of RINO in handling arbitrarily (but sufficiently richly) sampled input functions during both training and inference through several numerical examples. | {"url":"https://www.catalyzex.com/author/Michael%20D.%20Shields","timestamp":"2024-11-02T22:04:48Z","content_type":"text/html","content_length":"213223","record_id":"<urn:uuid:dd69d083-fbc5-43d4-8dd1-aae49a112256>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00039.warc.gz"} |
Casio Prizm Graphing Calculator Review - Graphing Calc Hub
How much time would it take for you to computer the answer by hand? On average, students take up to 30 seconds to an hour trying to solve a math problem. By investing in a graphing calculator you can
solve the problem virtually instantaneously.
That’s why we’re here to review the Casio Prizm Calculator. It weighs only .5 pounds making it a very compact calculator to have in your bookbag. Continue reading to see if this calculator is worth
your money and useful for your next math examination.
About The Product
The Casio Prizm Calculator is one of the most useful graphing calculators you can find. It features a colorful and bright 216 x 384 pixel display. You can get this on Amazon at a price around $125.
One feature that we like is the calculator’s statistics mode. Shoppers like this mode because their data can be turned into line, bar, and pie charts for further analysis. Get this device if you want
to understand statistical concepts and manipulate the data as you see fit.
• “Picture Plot” Technology
• Create graphs out of real life scenes
• Color coded graphs
• PC linkable port
• Easy spreadsheet formatting
Another thing that’s worth mentioning is the calculator’s graphing feature. Users found it intuitive and easy to use. The calculator displays the results in a color coded format making it a great
choice for students who need to display multiple graphs at once.
With the geometry mode, users are able to draw shapes on the Casio Prizm’s screen. Users can draw lines, circles, triangles, and understand angles. Beginners should look into this device because it
helps them understand geometry in an advanced level.
For a more realistic math application, Casio invented a picture plot function. This allows the user to take a picture of a random object in their area and create a mathematical function out of it.
For students needing an intense calculator to handle complex equations, this product never fails.
Most consumers disliked the calculator’s learning curve. Beginners took an average of 3 weeks before getting used to the device. We suggest that you look into the instructor’s manual before using
this calculator in order to get the most out of it.
Still, the Casio Prizm Calculators is a great engineering calculator and is also good for college level algebra. It has 16MB of memory, making it easy for users to store their data and complete
multiple math problems in one setting. Buy it today to maximize your computational skills.
Buying Advice
Consider upgrading to a calculator that can handle things like square roots, powers, fractions, or other types of math that you need help with on the ACT.
Types Of Calculators
Do I need a graphing calculator for a 2 year Epi program, or would a regular old scientific calc work?
Basic Calculators
Basic calculators consist of a numerical keypad and can perform simple arithmetic operations. While basic calculators are more superior than their predecessors, they assign only one arithmetic
operation or digit to one button.
Most basic calculators have a basic display that displays numbers in a row of 10 digits. However, fractions are only able to be represented in decimal notation. Thus, any fractional problem must be
converted into decimal form to complete the problem correctly.
All calculators of this type has a basic level of number storage. Most of the basic calculators can store one number in their memory. Get a basic calculator if you only plan on doing simple addition,
multiplication, division, and subtraction computations.
Scientific Calculators
Scientific Calculators was made a generation before graphing calculators and is still recommended by scientists, mathematicians and teachers. They consist of a single line display, but can display
more numbers than your average basic calculator.
The main differences between a graphing calculator and a scientific calculator is that the graphing calculator has the ability to store and write programs, graph data, and display mathematical
results on a larger screen.
Scientific calculators can perform mathematical tasks such as: statistics, calculus, standard notation, complex numbers, and much more. In some academic situations where a graphing calculator isn’t
allowed, a scientific calculator is the second best option.
Graphing Calculators
The first graphing calculator was invented by Casio. Graphing calculators are handheld mathematical devices that are used to solve multiple equations and plot graphs simultaneously. They have the
ability to be fully user programmed.
Graphing calculators are the standard for engineering, educational, and scientific purposes. While some might take time getting used to, you’ll have a trustworthy tool that can handle the highest
level of mathematical computations with ease.
Advanced Screens
When looking at a graphing calculator, one of the main differences is their larger display screen. Financial, scientific, and basic calculators have smaller screens which can be difficult for users
to see during their exams.
The larger screen allows users to display their data on the x-axis and the y-axis in a easy to read format. Additionally, it shows several lines of text that can be seen at once.
Recent graphing calculators allow you to see your data in 2D, 3D, and in color graphs. Plus, they can aid in creating separate documents of the graphs that you’ve plotted. Graphing calculators have
advanced screens to help you analyze data in an accurate manner.
Graphing Calculators have Wi-Fi capabilities that help with the logging and evaluating of data from scientific devices. For instance, it can receive data from pH gauges, decibel meters, light meters,
electric thermometers, and meterological gauges.
Also, you’ll want to see if your calculator is compatible with the standardized examination that you plan to take. Fortunately, the Casio Prizm Graphing Calculator can be used on the SAT, ACT, AP,
IB, and the PSAT.
Overall, the Casio Prizm Calculator is definitely worth it if you’re an engineer or a college student. It gives you full range of control over the device and displays your graphing results in a nice
color coded display. Buy this product if you want to achieve better results in your classes.
Do you have any questions or comments using this device?
Please leave a comment below.
Leave a Comment:
Add Your Reply | {"url":"https://graphingcalchub.com/casio-prizm-graphing-calculator/","timestamp":"2024-11-03T12:39:58Z","content_type":"text/html","content_length":"88060","record_id":"<urn:uuid:85507e64-2c98-452c-b6f7-bbd3497ebc45>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00073.warc.gz"} |
I’m Francisco Adams - Nurhak İYİDOĞAN
I’m Francisco Adams
Over the last ten odd years I’ve had the pleasure of working with some great companies, working side by side to design and develop new apps and improve upon existing products. See for yourself!
#Photoshop, #Illustrator, #CSS, #Python, #Ruby, #Photography
The term “portfolio” refers to any combination of financial assets such as stocks, bonds and cash. Portfolios may be held by individual investors and/or managed by financial professionals, hedge
funds, banks and other financial institutions. It is a generally accepted principle that a portfolio is designed according to the investor’s risk tolerance, time frame and investment objectives. The
monetary value of each asset may influence the risk/reward ratio of the portfolio.
When determining a proper asset allocation one aims at maximizing the expected return and minimizing the risk. This is an example of a multi-objective optimization problem: many efficient solutions
are available and the preferred solution must be selected by considering a tradeoff between risk and return.
In particular, a portfolio A is dominated by another portfolio A’ if A’ has a greater expected gain and a lesser risk than A. If no portfolio dominates A, A is a Pareto-optimal portfolio. The set of
Pareto-optimal returns and risks is called the Pareto efficient frontier for the Markowitz portfolio selection problem. | {"url":"https://nurhakiyidogan.com/im-francisco-adams/","timestamp":"2024-11-12T03:44:34Z","content_type":"text/html","content_length":"39603","record_id":"<urn:uuid:40ee0421-1ea8-45e2-9783-448418364c18>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00292.warc.gz"} |
Forecast univariate ARIMA or ARIMAX model responses or conditional variances
[Y,YMSE] = forecast(Mdl,numperiods,Y0) returns the numperiods-by-1 numeric vector of consecutive forecasted responses Y and the corresponding numeric vector of forecast mean square errors (MSE) YMSE
of the fully specified, univariate ARIMA model Mdl. The presample response data in the numeric vector Y0 initializes the model to generate forecasts.
[Y,YMSE,V] = forecast(Mdl,numperiods,Y0) also forecasts a numperiods-by-1 numeric vector of conditional variances V from a composite conditional mean and variance model (for example, an ARIMA and
GARCH composite model).
Tbl2 = forecast(Mdl,numperiods,Tbl1) returns the table or timetable Tbl2 containing a variable for each of the paths of response, forecast MSE, and conditional variance series resulting from
forecasting the ARIMA model Mdl over a numperiods forecast horizon. Tbl1 is a table or timetable containing a variable for required presample response data to initialize the model for forecasting.
Tbl1 can optionally contain variables of presample data for innovations, conditional variances, and predictors. (since R2023b)
forecast selects the response variable named in Mdl.SeriesName or the sole variable in Tbl1. To select a different response variable in Tbl1 to initialize the model, use the PresampleResponseVariable
name-value argument.
[___] = forecast(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. forecast returns the output
argument combination for the corresponding input arguments. For example, forecast(Mdl,10,Y0,X0=Exo0,XF=Exo) specifies the presample and forecast sample exogenous predictor data to Exo0 and Exo,
respectively, to forecast a model with a regression component (an ARIMAX model).
Forecast Conditional Mean Response Vector
Forecast the conditional mean response of simulated data over a 30-period horizon. Supply a vector of presample response data and return a vector of forecasts.
Simulate 130 observations from a multiplicative seasonal moving average (MA) model with known parameter values.
Mdl = arima(MA={0.5 -0.3},SMA=0.4,SMALags=12,Constant=0.04, ...
Y = simulate(Mdl,130);
Fit a seasonal MA model to the first 100 observations, and reserve the remaining 30 observations to evaluate forecast performance.
MdlTemplate = arima(MALags=1:2,SMALags=12);
EstMdl = estimate(MdlTemplate,Y(1:100));
ARIMA(0,0,2) Model with Seasonal MA(12) (Gaussian Distribution):
Value StandardError TStatistic PValue
________ _____________ __________ __________
Constant 0.20403 0.069064 2.9542 0.0031344
MA{1} 0.50212 0.097298 5.1606 2.4619e-07
MA{2} -0.20174 0.10447 -1.9312 0.053464
SMA{12} 0.27028 0.10907 2.478 0.013211
Variance 0.18681 0.032732 5.7073 1.148e-08
EstMdl is a new arima model that contains estimated parameters (that is, a fully specified model).
Forecast the fitted model into a 30-period horizon. Specify the estimation period data as a presample.
[YF,YMSE] = forecast(EstMdl,30,Y(1:100));
YF is a 30-by-1 vector of forecasted responses, and YMSE is a 30-by-1 vector of corresponding MSEs. The 15-period-ahead forecast is 0.2040 and its MSE is 0.2592.
Visually compare the forecasts to the holdout data.
h1 = plot(Y,Color=[.7,.7,.7]);
hold on
h2 = plot(101:130,YF,"b",LineWidth=2);
h3 = plot(101:130,YF + 1.96*sqrt(YMSE),"r:",LineWidth=2);
plot(101:130,YF - 1.96*sqrt(YMSE),"r:",LineWidth=2);
legend([h1 h2 h3],"Observed","Forecast","95% confidence interval", ...
title("30-Period Forecasts and 95% Confidence Intervals")
hold off
Forecast NYSE Composite Index
Since R2023b
Forecast the weekly average NYSE closing prices over a 15-week horizon. Supply presample data in a timetable and return a timetable of forecasts.
Load Data
Load the US equity index data set Data_EquityIdx.
load Data_EquityIdx
T = height(DataTimeTable)
The timetable DataTimeTable includes the time series variable NYSE, which contains daily NYSE composite closing prices from January 1990 through December 2001.
Plot the daily NYSE price series.
title("NYSE Daily Closing Prices: 1990 - 2001")
Prepare Timetable for Estimation
When you plan to supply a timetable, you must ensure it has all the following characteristics:
• The selected response variable is numeric and does not contain any missing values.
• The timestamps in the Time variable are regular, and they are ascending or descending.
Remove all missing values from the timetable, relative to the NYSE price series.
DTT = rmmissing(DataTimeTable,DataVariables="NYSE");
T_DTT = height(DTT)
Because all sample times have observed NYSE prices, rmmissing does not remove any observations.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTT,"days")
areTimestampsRegular = logical
areTimestampsSorted = issorted(DTT.Time)
areTimestampsSorted = logical
areTimestampsRegular = 0 indicates that the timestamps of DTT are irregular. areTimestampsSorted = 1 indicates that the timestamps are sorted. Business day rules make daily macroeconomic measurements
Remedy the time irregularity by computing the weekly average closing price series of all timetable variables.
DTTW = convert2weekly(DTT,Aggregation="mean");
areTimestampsRegular = isregular(DTTW,"weeks")
areTimestampsRegular = logical
DTTW is regular.
title("NYSE Daily Closing Prices: 1990 - 2001")
Create Model Template for Estimation
Suppose that an ARIMA(1,1,1) model is appropriate to model NYSE composite series during the sample period.
Create an ARIMA(1,1,1) model template for estimation. Set the response series name to NYSE.
Mdl = arima(1,1,1);
Mdl.SeriesName = "NYSE";
Mdl is a partially specified arima model object.
Partition Data
estimate and forecast require Mdl.P presample observations to initialize the model for estimaiton and forecasting.
Partition the data into three sets:
• A presample set for estimation
• An in-sample set, to which you fit the model and initialize the model for forecasting
• A holdout sample of length 15 to measure the model's predictive performance
numpreobs = Mdl.P; % Required presample length
numperiods = 15; % Forecast horizon
DTTW0 = DTTW(1:numpreobs,:); % Estimation presample
DTTW1 = DTTW((numpreobs+1):(end-numperiods),:);
% In-sample for estimation and presample for forecasting
DTTW2 = DTTW((end-numperiods+1):end,:); % Holdout sample
Fit Model to Data
Fit an ARIMA(1,1,1) model to the in-sample weekly average NYSE closing prices. Specify the presample timetable and the presample response variable name.
EstMdl = estimate(Mdl,DTTW1,Presample=DTTW0,PresampleResponseVariable="NYSE");
ARIMA(1,1,1) Model (Gaussian Distribution):
Value StandardError TStatistic PValue
________ _____________ __________ ___________
Constant 0.31873 0.23754 1.3418 0.17965
AR{1} 0.41132 0.2371 1.7348 0.082779
MA{1} -0.31232 0.24486 -1.2755 0.20212
Variance 55.472 1.8496 29.992 1.2638e-197
EstMdl is a fully specified, estimated arima model object.
Forecast Conditional Mean
Forecast the weekly average NASDQ closing prices 15 weeks beyond the estimation sample using the fitted model. Use the estimatoin sample data as a presample to initialize the forecast. Specify the
response variable name in the presample data.
Tbl2 = forecast(EstMdl,numperiods,DTTW1)
Tbl2=15×3 timetable
Time NYSE_Response NYSE_MSE NYSE_Variance
___________ _____________ ________ _____________
28-Sep-2001 521.34 55.472 55.472
05-Oct-2001 519.89 122.47 55.472
12-Oct-2001 519.62 194.53 55.472
19-Oct-2001 519.82 268.72 55.472
26-Oct-2001 520.23 343.8 55.472
02-Nov-2001 520.71 419.24 55.472
09-Nov-2001 521.23 494.83 55.472
16-Nov-2001 521.76 570.49 55.472
23-Nov-2001 522.3 646.17 55.472
30-Nov-2001 522.84 721.86 55.472
07-Dec-2001 523.38 797.56 55.472
14-Dec-2001 523.92 873.26 55.472
21-Dec-2001 524.46 948.96 55.472
28-Dec-2001 525 1024.7 55.472
04-Jan-2002 525.55 1100.4 55.472
Tbl2 is a 15-by-3 timetable containing the forecasted weekly average closing price forecasts NYSE_Response, corresponding forecast MSEs NYSE_MSE, and the model's constant variance NYSE_Variance
(EstMdl.Variance = 55.8147).
Plot the forecasts and approximate 95% forecast intervals.
Tbl2.NYSE_Lower = Tbl2.NYSE_Response - 1.96*sqrt(Tbl2.NYSE_MSE);
Tbl2.NYSE_Upper = Tbl2.NYSE_Response + 1.96*sqrt(Tbl2.NYSE_MSE);
h1 = plot([DTTW1.Time((end-75):end); DTTW2.Time], ...
[DTTW1.NYSE((end-75):end); DTTW2.NYSE],Color=[.7,.7,.7]);
hold on
h2 = plot(Tbl2.Time,Tbl2.NYSE_Response,"k",LineWidth=2);
h3 = plot(Tbl2.Time,Tbl2{:,["NYSE_Lower" "NYSE_Upper"]},"r:",LineWidth=2);
legend([h1 h2 h3(1)],"Observations","Forecasts","95% forecast intervals", ...
title("NYSE Weekly Average Closing Price")
hold off
The process is nonstationary, so the width of each forecast interval grows with time. The model tends to unestimate the weekly average closing prices.
Forecast ARX Model
Forecast the following known autoregressive model with one lag and an exogenous predictor (ARX(1)) model into a 10-period forecast horizon:
${y}_{t}=1+0.3{y}_{t-1}+2{x}_{t}+{\epsilon }_{t},$
where ${\epsilon }_{\mathit{t}}$ is a standard Gaussian random variable, and ${\mathit{x}}_{\mathit{t}}$ is an exogenous Gaussian random variable with a mean of 1 and a standard deviation of 0.5.
Create an arima model object that represents the ARX(1) model.
Mdl = arima(Constant=1,AR=0.3,Beta=2,Variance=1);
To forecast responses from the ARX(1) model, the forecast function requires:
• One presample response ${\mathit{y}}_{0}$ to initialize the autoregressive term
• Future exogenous data to include the effects of the exogenous variable on the forecasted responses
Set the presample response to the unconditional mean of the stationary process:
For the future exogenous data, draw 10 values from the distribution of the exogenous variable.
y0 = (1 + 2)/(1 - 0.3);
xf = 1 + 0.5*randn(10,1);
Forecast the ARX(1) model into a 10-period forecast horizon. Specify the presample response and future exogenous data.
fh = 10;
yf = forecast(Mdl,fh,y0,XF=xf)
yf = 10×1
yf(3) = 3.8232 is the 3-period-ahead forecast of the ARX(1) model.
Forecast Composite Conditional Mean and Variance Model
Since R2023b
Consider the following AR(1) conditional mean model with a GARCH(1,1) conditional variance model for the weekly average NASDAQ rate series (as a percent) from January 2, 1990 through December 31,
$\begin{array}{l}{y}_{t}=0.073+0.138{y}_{t-1}+{\epsilon }_{t}\\ {\sigma }_{t}^{2}=0.022+0.873{\sigma }_{t-1}^{2}+0.119{\epsilon }_{t-1},\end{array}$
where ${\epsilon }_{\mathit{t}}$ is a series of independent random Gaussian variables with a mean of 0.
Create the model. Name the response series NASDAQ.
CondVarMdl = garch(Constant=0.022,GARCH=0.873,ARCH=0.119);
Mdl = arima(Constant=0.073,AR=0.138,Variance=CondVarMdl);
Mdl.SeriesName = "NASDAQ";
Load the equity index data set. Remedy the time irregularity by computing the weekly average closing price series of all timetable variables.
load Data_EquityIdx
DTTW = convert2weekly(DataTimeTable,Aggregation="mean");
Convert the weekly average NASDAQ closing price series to a percent return series.
RetTT = price2ret(DTTW);
RetTT.NASDAQ = RetTT.NASDAQ*100;
Infer residuals and conditional variances from the model.
RetTT2 = infer(Mdl,RetTT);
T = numel(RetTT);
Forecast the model over a 25-day horizon. Supply the entire data set as a presample (forecast uses only the latest required observations to initialize the conditional mean and variance models).
Supply variable names for the presample innovations and conditional variances. By default, forecast uses the variable name Mdl.SeriesName as the presample response variable.
fh = 25;
ForecastTT = forecast(Mdl,fh,RetTT2,PresampleInnovationVariable="NASDAQ_Residual", ...
Plot the forecasted responses and conditional variances with the observed series from June 2000.
pdates = RetTT2.Time > datetime(2000,6,1);
hold on
plot([RetTT2.Time(end); ForecastTT.Time], ...
[RetTT2.NASDAQ(end); ForecastTT.NASDAQ_Response])
title("NASDAQ Weekly Average Percent Return Series")
axis tight
grid on
hold off
hold on
plot([RetTT2.Time(end); ForecastTT.Time], ...
[RetTT2.NASDAQ_Variance(end); ForecastTT.NASDAQ_Variance])
title("Conditional Variance Series")
axis tight
grid on
hold off
Forecast Multiple Paths
Forecast multiple response and conditional variance paths from a known composite conditional mean and variance model: a SAR$\left(1,0,0\right){\left(1,1,0\right)}_{4}$ conditional mean model with an
ARCH(1) conditional variance model. Specify multiple presample response paths.
Create a garch model object that represents this ARCH(1) model:
${\sigma }_{t}^{2}=0.1+0.2{\epsilon }_{t}^{2}.$
Create an arima model object that represents this quarterly SAR$\left(1,0,0\right){\left(1,1,0\right)}_{4}$ model:
$\left(1-0.5L\right)\left(1-0.2{L}^{4}\right)\left(1-{L}^{4}\right){y}_{t}=1+{\epsilon }_{t},$
where ${\epsilon }_{\mathit{t}}$ is a standard Gaussian random variable.
CVMdl = garch(ARCH=0.2,Constant=0.1)
CVMdl =
garch with properties:
Description: "GARCH(0,1) Conditional Variance Model (Gaussian Distribution)"
SeriesName: "Y"
Distribution: Name = "Gaussian"
P: 0
Q: 1
Constant: 0.1
GARCH: {}
ARCH: {0.2} at lag [1]
Offset: 0
Mdl = arima(Constant=1,AR=0.5,Variance=CVMdl,Seasonality=4, ...
Mdl =
arima with properties:
Description: "ARIMA(1,0,0) Model Seasonally Integrated with Seasonal AR(4) (Gaussian Distribution)"
SeriesName: "Y"
Distribution: Name = "Gaussian"
P: 9
D: 0
Q: 0
Constant: 1
AR: {0.5} at lag [1]
SAR: {0.2} at lag [4]
MA: {}
SMA: {}
Seasonality: 4
Beta: [1×0]
Variance: [GARCH(0,1) Model]
Because Mdl contains 9 autoregressive terms and 1 ARCH term, forecast requires Mdl.P = 9 responses and CVMdl.Q = 1 conditional variance to generate each $\mathit{t}$-period-ahead forecast.
Generate 10 random paths of length 9 from the model.
numpreobs = Mdl.P;
numpaths = 10;
[Y0,~,V0] = simulate(Mdl,numpreobs,NumPaths=numpaths);
Forecast 10 paths of responses and conditional variances from the model into a 12-quarter forecast horizon. Specify the presample response paths Y0 and conditional variance paths V0.
fh = 12;
[YF,~,VF] = forecast(Mdl,fh,Y0,V0=V0);
YF and VF are 12-by-10 matrices of independent forecasted response and conditional variance paths, respectively. YF(j,k) is the j-period-ahead forecast of path k. Path YF(:,k) represents the
continuation of the presample path Y0(:,k). forecast structures VF similarly.
Plot the presample and forecasted responses.
Y = [Y0; YF];
hold on
h = gca;
px = [numpreobs+0.5 h.XLim([2 2]) numpreobs+0.5];
py = h.YLim([1 1 2 2]);
hp = patch(px,py,[0.9 0.9 0.9]);
axis tight
legend("Forecast period")
xlabel("Time (quarters)")
title("Response paths")
hold off
V = [V0; VF];
hold on
h = gca;
px = [numpreobs+0.5 h.XLim([2 2]) numpreobs+0.5];
py = h.YLim([1 1 2 2]);
hp = patch(px,py,[0.9 0.9 0.9]);
legend("Forecast period")
axis tight
xlabel("Time (quarters)")
title("Conditional Variance Paths")
hold off
Input Arguments
numperiods — Forecast horizon
positive integer
Forecast horizon, or the number of time points in the forecast period, specified as a positive integer.
Data Types: double
Y0 — Presample response data y[t]
numeric column vector | numeric matrix
Presample response data y[t] used to initialize the model for forecasting, specified as a numpreobs-by-1 numeric column vector or a numpreobs-by-numpaths numeric matrix. When you supply Y0, supply
all optional data as numeric arrays, and forecast returns results in numeric arrays.
numpreobs is the number of presample observations. numpaths is the number of independent presample paths, from which forecast initializes the resulting numpaths forecasts (see Algorithms).
Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation. numpreobs must be at least Mdl.P to initialize the
model. If numpreobs > Mdl.P, forecast uses only the latest Mdl.P rows. For more details, see Time Base Partitions for Forecasting.
Columns of Y0 correspond to separate, independent presample paths.
• If Y0 is a column vector, it represents a single path of the response series. forecast applies it to each forecasted path. In this case, all forecast paths Y derive from the same initial
• If Y0 is a matrix, each column represents a presample path of the response series. numpaths is the maximum among the second dimensions of the specified presample observation matrices Y0, E0, and
Data Types: double
Tbl1 — Presample data
table | timetable
Since R2023b
Presample data containing required presample responses y[t], and, optionally, innovations ε[t], conditional variances σ[t]^2, or predictors x[t], to initialize the model, specified as a table or
timetable with numprevars variables and numpreobs rows. You can select a response, innovation, conditional variance, or multiple predictor variables from Tbl1 by using the PresampleResponseVariable,
PresampleInnovationVariable, PresampleVarianceVariable, or PresamplePredictorVariables name-value argument, respectively.
numpreobs is the number of presample observations. numpaths is the number of independent presample paths, from which forecast initializes the resulting numpaths forecasts (see Algorithms).
For all selected variables except predictor variables, each variable contains a single path (numpreobs-by-1 vector) or multiple paths (numpreobs-by-numpaths matrix) of presample response,
innovations, or conditional variance data.
Each selected predictor variable contains a single path of observations. forecast applies all selected predictor variables to each forecasted path. When you do not specify presample innovation data
for forecasting an ARIMAX model, forecast uses the presample predictor data to infer presample innovations.
Each row is a presample observation, and measurements in each row occur simultaneously. numpreobs must be one of the following values:
• At least Mdl.P when Presample provides only presample responses
• At least max([Mdl.P Mdl.Q]) otherwise
When Mdl.Variance is a conditional variance model, forecast can require more than the minimum required number of presample values. If numpreobs exceeds the minimum number, forecast uses the latest
required number of observations only.
If Tbl1 is a timetable, all the following conditions must be true:
• Tbl1 must represent a sample with a regular datetime time step (see isregular).
• The datetime vector of sample timestamps Tbl1.Time must be ascending or descending.
If Tbl1 is a table, the last row contains the latest presample observation.
Although forecast requires presample response data, forecast sets default presample innovation and conditional variance data as follows:
• To infer necessary presample innovations from presample responses, numpreobs must be at least Mdl.P + Mdl.Q (see infer). Additionally, for ARIMAX models, forecast requires enough presample
predictor data. If numpreobs is less than Mdl.P + Mdl.Q or you do not specify presample predictor data for ARIMAX forecasting, forecast sets all necessary presample innovations to zero.
• To infer necessary presample variances from presample innovations, forecast requires a sufficient number of presample innovations to initialize the specified conditional variance model (see infer
). If you do not specify enough presample innovations to initialize the conditional variance model, forecast sets the necessary presample variances to the unconditional variance of the specified
variance process.
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: forecast(Mdl,10,Y0,X0=Exo0,XF=Exo) specifies the presample and forecast sample exogenous predictor data to Exo0 and Exo, respectively, to forecast a model with a regression component.
E0 — Presample innovations ε[t]
numeric column vector | numeric matrix
Presample innovations ε[t] used to initialize either the moving average (MA) component of the ARIMA model or the conditional variance model, specified as a numpreobs-by-1 column vector or
numpreobs-by-numpaths numeric matrix. Use E0 only when you supply the numeric array of presample response data Y0. forecast assumes that the presample innovations have a mean of zero.
Each row is a presample observation, and measurements in each row occur simultaneously. The last row contains the latest presample observation. numpreobs must be at least Mdl.Q to initialize the
model. If Mdl.Variance is a conditional variance model (for example, a garch model object), E0 might require more than Mdl.Q rows. If numpreobs is greater than required, forecast uses only the latest
required rows.
Columns of E0 correspond to separate, independent presample paths.
• If E0 is a column vector, it represents a single path of the innovation series. forecast applies it to each forecasted path. In this case, all forecast paths Y derive from the same initial
• If E0 is a matrix, each column represents a presample path of the innovation series. numpaths is the maximum among the second dimensions of the specified presample observation matrices Y0, E0,
and V0.
By default:
• If you provide enough presample responses and, for ARIMAX models, presample predictor data (X0), forecast infers necessary presample innovations from the presample data. In this case, numpreobs
must be at least Mdl.P + Mdl.Q (see infer)
• Otherwise, forecast sets all necessary presample innovations to zero.
Data Types: double
V0 — Presample conditional variances σ[t]^2
positive numeric column vector | positive numeric matrix
Presample conditional variances σ[t]^2 used to initialize the conditional variance model, specified as a numpreobs-by-1 positive column vector or numpreobs-by-numpaths positive matrix. Use V0 only
when you supply the numeric array of presample response data Y0. If the model variance Mdl.Variance is constant, forecast ignores V0.
Rows of V0 correspond to periods in the presample, and the last row contains the latest presample conditional variance. numpreobs must be enough to initialize the conditional variance model (see
forecast). If numpreobs exceeds the minimum number, forecast uses only the latest observations.
Columns of V0 correspond to separate, independent paths.
• If V0 is a column vector, forecast applies it to each forecasted path. In this case, the conditional variance model of all forecast paths Y derives from the same initial conditional variances.
• If V0 is a matrix, each column represents a presample path of the conditional variance series. numpaths is the maximum among the second dimensions of the specified presample observation matrices
Y0, E0, and V0.
By default:
• If you specify enough presample innovations E0 to initialize the conditional variance model Mdl.Variance, forecast infers any necessary presample conditional variances by passing the conditional
variance model and E0 to the infer function.
• If you do not specify E0, but you specify enough presample responses and, for ARIMAX models, presample predictor data, Y0 to infer enough presample innovations, forecast infers any necessary
presample conditional variances from the inferred presample innovations.
• If you do not specify enough presample data, forecast sets all necessary presample conditional variances to the unconditional variance of the variance process.
Data Types: double
PresampleResponseVariable — Response variable y[t] to select from Tbl1
string scalar | character vector | integer | logical vector
Since R2023b
Response variable y[t] to select from Tbl1 containing the presample response data, specified as one of the following data types:
• String scalar or character vector containing a variable name in Tbl1.Properties.VariableNames
• Variable index (positive integer) to select from Tbl1.Properties.VariableNames
• A logical vector, where PresampleResponseVariable(j) = true selects variable j from Tbl1.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain missing values (NaNs).
If Tbl1 has one variable, the default specifies that variable. Otherwise, the default matches the variable to names in Mdl.SeriesName.
Example: PresampleResponseVariable="StockRate"
Example: PresampleResponseVariable=[false false true false] or PresampleResponseVariable=3 selects the third table variable as the response variable.
Data Types: double | logical | char | cell | string
PresampleInnovationVariable — Presample innovation variable of ε[t] to select from Tbl1
string scalar | character vector | integer | logical vector
Since R2023b
Presample innovation variable of ε[t] to select from Tbl1 containing presample innovation data, specified as one of the following data types:
• String scalar or character vector containing a variable name in Tbl1.Properties.VariableNames
• Variable index (positive integer) to select from Tbl1.Properties.VariableNames
• A logical vector, where PresampleInnovationVariable(j) = true selects variable j from Tbl1.Properties.VariableNames
The selected variable must be a numeric matrix and cannot contain missing values (NaNs).
If you specify presample innovation data in Tbl1, you must specify PresampleInnovationVariable.
Example: PresampleInnovationVariable="StockRateDist0"
Example: PresampleInnovationVariable=[false false true false] or PresampleInnovationVariable=3 selects the third table variable as the presample innovation variable.
Data Types: double | logical | char | cell | string
PresampleVarianceVariable — Presample conditional variance variable σ[t]^2 to select from Tbl1
string scalar | character vector | integer | logical vector
Presample conditional variance variable σ[t]^2 to select from Tbl1 containing presample conditional variance data, specified as one of the following data types:
• String scalar or character vector containing a variable name in Tbl1.Properties.VariableNames
• Variable index (positive integer) to select from Tbl1.Properties.VariableNames
• A logical vector, where PresampleVarianceVariable(j) = true selects variable j from Tbl1.Properties.VariableNames
The selected variable must be a numeric vector and cannot contain missing values (NaNs).
If you specify presample conditional variance data in Tbl1, you must specify PresampleVarianceVariable.
Example: PresampleVarianceVariable="StockRateVar0"
Example: PresampleVarianceVariable=[false false true false] or PresampleVarianceVariable=3 selects the third table variable as the presample conditional variance variable.
Data Types: double | logical | char | cell | string
X0 — Presample predictor data
numeric matrix
Presample predictor data used to infer the presample innovations E0, specified as a numpreobs-by-numpreds numeric matrix. Use X0 only when you supply the numeric array of presample response data Y0
and your model contains a regression component. numpreds = numel(Mdl.Beta).
Rows of X0 correspond to periods in the presample, and the last row contains the latest set of presample predictor observations. Columns of X0 represent separate time series variables, and they
correspond to the columns of XF and Mdl.Beta.
If you do not specify E0, X0 must have at least numpreobs – Mdl.P rows so that forecast can infer presample innovations. If the number of rows exceeds the minimum number required to infer presample
innovations, forecast uses only the latest required presample predictor observations. A best practice is to set X0 to the same predictor data matrix used in the estimation, simulation, or inference
of Mdl. This setting ensures that forecast infers presample innovations E0 correctly.
If you specify E0, forecast ignores X0.
If you specify X0 but you do not specify forecasted predictor data XF, forecast issues an error.
By default, forecast drops the regression component from the model when it infers presample innovations, regardless of the value of the regression coefficient Mdl.Beta.
Data Types: double
PresamplePredictorVariables — Presample exogenous predictor variables x[t] to select from Tbl1
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2023b
Presample exogenous predictor variables x[t] to select from Tbl1 containing presample exogenous predictor data, specified as one of the following data types:
• String vector or cell vector of character vectors containing numpreds variable names in Tbl1.Properties.VariableNames
• A vector of unique indices (positive integers) of variables to select from Tbl1.Properties.VariableNames
• A logical vector, where PresamplePredictorVariables(j) = true selects variable j from Tbl1.Properties.VariableNames
The selected variables must be numeric vectors and cannot contain missing values (NaNs).
If you specify presample predictor data, you must also specify in-sample predictor data by using the InSample and PredictorVariables name-value arguments.
By default, forecast excludes the regression component, regardless of its presence in Mdl.
Example: PresamplePredictorVariables=["M1SL" "TB3MS" "UNRATE"]
Example: PresamplePredictorVariables=[true false true false] or PredictorVariable=[1 3] selects the first and third table variables to supply the predictor data.
Data Types: double | logical | char | cell | string
XF — Forecasted (or future) predictor data
numeric matrix
Forecasted (or future) predictor data, specified as a numeric matrix with numpreds columns. XF represents the evolution of specified presample predictor data X0 forecasted into the future (the
forecast period). Use XF only when you supply the numeric array of presample response data Y0.
Rows of XF correspond to time points in the future; XF(t,:) contains the t-period-ahead predictor forecasts. XF must have at least numperiods rows. If the number of rows exceeds numperiods, forecast
uses only the first (earliest) numperiods forecasts. For more details, see Time Base Partitions for Forecasting.
Columns of XF are separate time series variables, and they correspond to the columns of X0 and Mdl.Beta.
By default, the forecast function generates forecasts from Mdl without a regression component, regardless of the value of the regression coefficient Mdl.Beta.
InSample — Forecasted (future) predictor data
table | timetable
Since R2023b
Forecasted (future) predictor data for the exogenous regression component of the model, specified as a table or timetable. InSample contains numvars variables, including numpreds predictor variables
forecast returns the forecasted variables in the output table or timetable Tbl2, which is commensurate with InSample.
Each row corresponds to an observation in the forecast horizon, the first row is the earliest observation, and measurements in each row, among all paths, occur simultaneously. InSample must have at
least numperiods rows to cover the forecast horizon. If you supply more rows than necessary, forecast uses only the first numperiods rows.
Each selected predictor variable is a numeric vector without missing values (NaNs). forecast applies the specified predictor variables to all forecasted paths.
If InSample is a timetable, the following conditions apply:
• InSample must represent a sample with a regular datetime time step (see isregular).
• The datetime vector InSample.Time must be ascending or descending.
• Tbl1 must immediately precede InSample, with respect to the sampling frequency.
If InSample is a table, the last row contains the latest observation.
By default, forecast does not include the regression component in the model, regardless of the value of Mdl.Beta.
PredictorVariables — Exogenous predictor variables x[t] to select from InSample
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2023b
Exogenous predictor variables x[t] to select from InSample containing exogenous predictor data in the forecast horizon, specified as one of the following data types:
• String vector or cell vector of character vectors containing numpreds variable names in InSample.Properties.VariableNames
• A vector of unique indices (positive integers) of variables to select from InSample.Properties.VariableNames
• A logical vector, where PredictorVariables(j) = true selects variable j from InSample.Properties.VariableNames
The selected variables must be numeric vectors and cannot contain missing values (NaNs).
By default, forecast excludes the regression component, regardless of its presence in Mdl.
Example: PredictorVariables=["M1SL" "TB3MS" "UNRATE"]
Example: PredictorVariables=[true false true false] or PredictorVariable=[1 3] selects the first and third table variables to supply the predictor data.
Data Types: double | logical | char | cell | string
For numeric array inputs, forecast assumes that you synchronize all specified presample data sets so that the latest observation of each presample series occurs simultaneously. Similarly, forecast
assumes that the first observation in the forecasted predictor data XF occurs in the time point immediately after the last observation in the presample predictor data X0.
Output Arguments
Y — Minimum mean square error (MMSE) conditional mean forecasts
numeric column vector | numeric matrix
Minimum mean square error (MMSE) conditional mean forecasts y[t], returned as a numperiods-by-1 column vector or a numperiods-by-numpaths numeric matrix. Y represents a continuation of Y0 (Y(1,:)
occurs in the time point immediately after Y0(end,:)). forecast returns Y only when you supply numeric presample data Y0.
Y(t,:) contains the t-period-ahead forecasts, or the conditional mean forecast of all paths for time point t in the forecast period.
forecast determines numpaths from the number of columns in the presample data sets Y0, E0, and V0. For details, see Algorithms. If each presample data set has one column, Y is a column vector.
Data Types: double
YMSE — MSE of forecasted responses
numeric column vector | numeric matrix
MSE of the forecasted responses Y (forecast error variances), returned as a numperiods-by-1 column vector or a numperiods-by-numpaths numeric matrix. forecast returns YMSE only when you supply
numeric presample data Y0.
YMSE(t,:) contains the forecast error variances of all paths for time point t in the forecast period.
forecast determines numpaths from the number of columns in the presample data sets Y0, E0, and V0. For details, see Algorithms. If you do not specify any presample data sets, or if each data set is a
column vector, YMSE is a column vector.
The square roots of YMSE are the standard errors of the forecasts Y.
Data Types: double
V — MMSE forecasts of conditional variances of future model innovations
numeric column vector | numeric matrix
MMSE forecasts of the conditional variances of future model innovations, returned as a numperiods-by-1 numeric column vector or a numperiods-by-numpaths numeric matrix. forecast returns V only when
you supply numeric presample data Y0.
When Mdl.Variance is a conditional variance model, row j contains the conditional variance forecasts of period j. Otherwise, V is a matrix composed of the constant Mdl.Variance.
forecast determines numpaths from the number of columns in the presample data sets Y0, E0, and V0. For details, see Algorithms. If you do not specify any presample data sets, or if each data set is a
column vector, YMSE is a column vector.
Data Types: double
Tbl2 — Paths of MMSE forecasts of responses y[t], corresponding forecast MSEs, and MMSE forecasts of conditional variances σ[t]^2 of future model innovations ε[t]
table | timetable
Since R2023b
Paths of MMSE forecasts of responses y[t], corresponding forecast MSEs, and MMSE forecasts of conditional variances σ[t]^2 of future model innovations ε[t], returned as a table or timetable, the same
data type as Tbl1. forecast returns Tbl2 only when you supply the input Tbl1.
Tbl2 contains the following variables:
• The forecasted response paths, which are in a numperiods-by-numpaths numeric matrix, with rows representing periods in the forecast horizon and columns representing independent paths, each
corresponding to the input presample response paths in Tbl1. forecast names the forecasted response variable responseName_Response, where responseName is Mdl.SeriesName. For example, if
Mdl.SeriesName is GDP, Tbl2 contains a variable for the corresponding forecasted response paths with the name GDP_Response.
Each path in Tbl2.responseName_Response represents the continuation of the corresponding presample response path in Tbl1 (Tbl2.responseName_Response(1,:) occurs in the next time point, with
respect to the periodicity Tbl1, after the last presample response). Tbl2.responseName_Response(j,k) contains the j-period-ahead forecasted response of path k.
• The forecast MSE paths, which are in a numperiods-by-numpaths numeric matrix, with rows representing periods in the forecast horizon and columns representing independent paths, each corresponding
to the forecasted responses in Tbl2.responseName_Response. forecast names the forecast MSEs responseName_MSE, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is GDP, Tbl2
contains a variable for the corresponding forecast MSE with the name GDP_MSE.
• The forecasted conditional variance paths, which are in a numperiods-by-numpaths numeric matrix, with rows representing periods in the forecast horizon and columns representing independent paths.
forecast names the forecasted conditional variance variable responseName_Variance, where responseName is Mdl.SeriesName. For example, if Mdl.SeriesName is StockReturns, Tbl2 contains a variable
for the corresponding forecasted conditional variance paths with the name StockReturns_Variance.
Each path in Tbl2.responseName_Variance represents a continuation of the presample conditional variance process, either supplied by Tbl1 or set by default (Tbl2.responseName_Variance(1,:) occurs
in the next time point, with respect to the periodicity Tbl1, after the last presample conditional variance). Tbl2.responseName_Variance(j,k) contains the j-period-ahead forecasted conditional
variance of path k.
• When you supply InSample, Tbl2 contains all variables in InSample.
If Tbl1 is a timetable, the following conditions hold:
• The row order of Tbl2, either ascending or descending, matches the row order of Tbl1.
• Tbl2.Time(1) is the next time after Tbl1.Time(end) relative the sampling frequency, and Tbl2.Time(2:numobs) are the following times relative to the sampling frequency.
More About
Time Base Partitions for Forecasting
Time base partitions for forecasting are two disjoint, contiguous intervals of the time base; each interval contains time series data for forecasting a dynamic model. The forecast period (forecast
horizon) is a numperiods length partition at the end of the time base during which the forecast function generates the forecasts Y from the dynamic model Mdl. The presample period is the entire
partition occurring before the forecast period. The forecast function can require observed responses, innovations, or conditional variances in the presample period (Y0, E0, and V0, or Tbl1) to
initialize the dynamic model for forecasting. The model structure determines the types and amounts of required presample observations.
A common practice is to fit a dynamic model to a portion of the data set, and then validate the predictability of the model by comparing its forecasts to observed responses. During forecasting, the
presample period contains the data to which the model is fit, and the forecast period contains the holdout sample for validation. Suppose that y[t] is an observed response series; x[1,t], x[2,t], and
x[3,t] are observed exogenous series; and time t = 1,…,T. Consider forecasting responses from a dynamic model of y[t] containing a regression component with numperiods = K periods. Suppose that the
dynamic model is fit to the data in the interval [1,T – K] (for more details, see estimate). This figure shows the time base partitions for forecasting.
For example, to generate the forecasts Y from an ARX(2) model, forecast requires:
• Presample responses Y0 = ${\left[\begin{array}{cc}{y}_{T-K-1}& {y}_{T-K}\end{array}\right]}^{\prime }$ to initialize the model. The 1-period-ahead forecast requires both observations, whereas the
2-periods-ahead forecast requires y[T – K] and the 1-period-ahead forecast Y(1). The forecast function generates all other forecasts by substituting previous forecasts for lagged responses in the
• Future exogenous data XF = $\left[\begin{array}{ccc}{x}_{1,\left(T-K+1\right):T}& {x}_{2,\left(T-K+1\right):T}& {x}_{3,\left(T-K+1\right):T}\end{array}\right]$ for the model regression component.
Without specified future exogenous data, the forecast function ignores the model regression component, which can yield unrealistic forecasts.
Dynamic models containing either a moving average component or a conditional variance model can require presample innovations or conditional variances. Given enough presample responses, forecast
infers the required presample innovations and conditional variances. If such a model also contains a regression component, then forecast must have enough presample responses and exogenous data to
infer the required presample innovations and conditional variances. This figure shows the arrays of required observations for this case, with corresponding input and output arguments.
• The forecast function sets the number of sample paths (numpaths) to the maximum number of columns among the specified presample data sets:
□ For input numeric arrays of presample data, numpaths is the maximum width among E0, V0, and Y0.
□ For an input table or timetable of presample data, numpaths is the maximum width among the variables representing the presample responses PresampleResponseVariable, innovations
PresampleInnovationVariable, and conditional variances PresampleVarianceVariable.
All specified presample data sets must have either one column or numpaths > 1 columns. Otherwise, forecast issues an error. For example, if you supply Y0 and E0, and Y0 has five columns
representing five paths, then E0 can have one column or five columns. If E0 has one column, forecast applies E0 to each path.
• NaN values in presample and future data sets indicate missing data. For input numeric arrays, forecast removes missing data from the presample data sets following this procedure:
1. forecast horizontally concatenates the specified presample data sets Y0, E0, V0, and X0 so that the latest observations occur simultaneously. The result can be a jagged array because the
presample data sets can have a different number of rows. In this case, forecast prepads variables with an appropriate number of zeros to form a matrix.
2. forecast applies listwise deletion to the combined presample matrix by removing all rows containing at least one NaN.
3. forecast extracts the processed presample data sets from the result of step 2, and removes all prepadded zeros.
forecast applies a similar procedure to the forecasted predictor data XF. After forecast applies listwise deletion to XF, the result must have at least numperiods rows. Otherwise, forecast issues
an error.
List-wise deletion reduces the sample size and can create irregular time series.
• forecast issues an error when any table or timetable input contains missing values.
• When forecast computes the MSEs YMSE of the conditional mean forecasts Y, the function treats the specified predictor data sets as exogenous, nonstochastic, and statistically independent of the
model innovations. Therefore, YMSE reflects only the variance associated with the ARIMA component of the input model Mdl.
[3] Bollerslev, Tim. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics 69 (August 1987): 542–47. https://doi.org/
[4] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
[5] Enders, Walter. Applied Econometric Time Series. Hoboken, NJ: John Wiley & Sons, Inc., 1995.
[6] Engle, Robert. F. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50 (July 1982): 987–1007. https://doi.org/10.2307/
[7] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
Version History
Introduced in R2012a
R2023b: forecast accepts input data in tables and timetables, and returns results in tables and timetables
In addition to accepting input presample and in-sample data in numeric arrays, forecast accepts input data in tables or regular timetables. Use Tbl1 to supply presample data and InSample to provide
in-sample (future) predictor data for the forecast horizon.
When you supply data in a table or timetable, the following conditions apply:
• forecast chooses the default presample response series on which to operate, but you can use the optional PresampleResponseVariable name-value argument to select a different variable.
• forecast returns results in a table or timetable.
Name-value arguments to support tabular workflows include:
• PresampleResponseVariable specifies the variable name of the presample response paths in the input presample data Tbl1 to initialize the response series for the forecast.
• PresampleInnovationVariable specifies the variable name of the innovation paths in the input presample data Tbl1 to initialize the model for the forecast.
• PresampleVarianceVariable specifies the variable name of the conditional variance paths in the input presample data Tbl1 to initialize the conditional variance series for the forecast.
• PresamplePredictorVariables specifies the variable names of the predictor data in the input presample data Tbl1 for the model exogenous regression component.
• PredictorVariables specifies the variable names of the predictor data in the input in-sample data InSample for the model exogenous regression component in the forecast horizon.
R2019a: Univariate time series models require specification of presample response data to forecast responses
The forecast function now has a third input argument for you to supply presample response data.
Before R2019a, the syntaxes were:
You could optionally supply presample responses using the
name-value argument.
There are no plans to remove the previous syntaxes or the 'Y0' name-value argument at this time. However, you are encouraged to supply presample responses because, to forecast responses from a
dynamic model, forecast must initialize models containing lagged responses. Without specified presample responses, forecast initializes models by using reasonable default values, but these values
might not support all workflows.
• For stationary models without a regression component, all presample responses are the unconditional mean of the process, by default.
• For nonstationary models or models containing a regression component, all presample responses are 0, by default.
Update Code
Update your code by specifying presample responses in the third input argument.
If you do not supply presample responses, then forecast provides default presample values that might not support all workflows. | {"url":"https://de.mathworks.com/help/econ/arima.forecast.html","timestamp":"2024-11-06T17:05:46Z","content_type":"text/html","content_length":"228516","record_id":"<urn:uuid:82d72af8-708f-427f-b9db-3913a1281882>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00677.warc.gz"} |
Harmonic mean - Wikiwand
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.
It is the most appropriate average for ratios and rates such as speeds,^[1]^[2] and is normally only used for positive arguments.^[3]
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with ${\displaystyle f(x)={\frac {1}{x}}}$. For example, the harmonic
mean of 1, 4, and 4 is
${\displaystyle \left({\frac {1^{-1}+4^{-1}+4^{-1}}{3}}\right)^{-1}={\frac {3}{{\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{4}}}}={\frac {3}{1.5}}=2\,.}$
The harmonic mean H of the positive real numbers ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ is^[4]
${\displaystyle H(x_{1},x_{2},\ldots ,x_{n})={\frac {n}{\displaystyle {\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}={\frac {n}{\displaystyle \sum _{i=1}^{n}{\frac {1}{x_
It is the reciprocal of the arithmetic mean of the reciprocals, and vice versa:
{\displaystyle {\begin{aligned}H(x_{1},x_{2},\ldots ,x_{n})&={\frac {1}{\displaystyle A\left({\frac {1}{x_{1}}},{\frac {1}{x_{2}}},\ldots {\frac {1}{x_{n}}}\right)}},\\A(x_{1},x_{2},\ldots ,x_
{n})&={\frac {1}{\displaystyle H\left({\frac {1}{x_{1}}},{\frac {1}{x_{2}}},\ldots {\frac {1}{x_{n}}}\right)}},\end{aligned}}}
where the arithmetic mean is ${\textstyle A(x_{1},x_{2},\ldots ,x_{n})={\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}.}$
The harmonic mean is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for positive arguments, ${\displaystyle \min(x_{1}\ldots x_{n})\leq H(x_{1}\ldots x_{n})\
leq n\min(x_{1}\ldots x_{n})}$. Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged).
The harmonic mean is also concave for positive arguments, an even stronger property than Schur-concavity.
Geometric proof without words that or of two distinct positive numbers a and b^
For all positive data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three Pythagorean means,^[5] while the arithmetic mean is always the greatest
of the three and the geometric mean is always in between. (If all values in a nonempty data set are equal, the three means are always equal.)
It is the special case M[−1] of the power mean: ${\displaystyle H\left(x_{1},x_{2},\ldots ,x_{n}\right)=M_{-1}\left(x_{1},x_{2},\ldots ,x_{n}\right)={\frac {n}{x_{1}^{-1}+x_{2}^{-1}+\cdots +x_{n}^
Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate
the impact of small ones.
The arithmetic mean is often mistakenly used in places calling for the harmonic mean.^[6] In the speed example below for instance, the arithmetic mean of 40 is incorrect, and too big.
The harmonic mean is related to the other Pythagorean means, as seen in the equation below. This can be seen by interpreting the denominator to be the arithmetic mean of the product of numbers n
times but each time omitting the j-th term. That is, for the first term, we multiply all n numbers except the first; for the second, we multiply all n numbers except the second; and so on. The
numerator, excluding the n, which goes with the arithmetic mean, is the geometric mean to the power n. Thus the n-th harmonic mean is related to the n-th geometric and arithmetic means. The general
formula is ${\displaystyle H\left(x_{1},\ldots ,x_{n}\right)={\frac {\left(G\left(x_{1},\ldots ,x_{n}\right)\right)^{n}}{A\left(x_{2}x_{3}\cdots x_{n},x_{1}x_{3}\cdots x_{n},\ldots ,x_{1}x_{2}\cdots
x_{n-1}\right)}}={\frac {\left(G\left(x_{1},\ldots ,x_{n}\right)\right)^{n}}{A\left({\frac {1}{x_{1}}}{\prod \limits _{i=1}^{n}x_{i}},{\frac {1}{x_{2}}}{\prod \limits _{i=1}^{n}x_{i}},\ldots ,{\frac
{1}{x_{n}}}{\prod \limits _{i=1}^{n}x_{i}}\right)}}.}$
If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged —
then the harmonic mean always decreases.^[7]
Two numbers
A geometric construction of the three Pythagorean means of two numbers, a and b. The harmonic mean is denoted by H in purple, while the arithmetic mean is A in red and the geometric mean is G in
blue. Q denotes a fourth mean, the quadratic mean. Since a hypotenuse is always longer than a leg of a right triangle, the diagram shows that .
A graphical interpretation of the harmonic mean, z of two numbers, x and y, and a nomogram to calculate it. The blue line shows that the harmonic mean of 6 and 2 is 3. The magenta line shows that the
harmonic mean of 6 and −2 is −6. The red line shows that the harmonic mean of a number and its negative is undefined as the line does not intersect the z axis.
For the special case of just two numbers, ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$, the harmonic mean can be written
${\displaystyle H={\frac {2x_{1}x_{2}}{x_{1}+x_{2}}}\qquad }$ or ${\displaystyle \qquad {\frac {1}{H}}={\frac {(1/x_{1})+(1/x_{2})}{2}}.}$
In this special case, the harmonic mean is related to the arithmetic mean ${\displaystyle A={\frac {x_{1}+x_{2}}{2}}}$ and the geometric mean ${\displaystyle G={\sqrt {x_{1}x_{2}}},}$ by
${\displaystyle H={\frac {G^{2}}{A}}=G\left({\frac {G}{A}}\right).}$
Since ${\displaystyle {\tfrac {G}{A}}\leq 1}$ by the inequality of arithmetic and geometric means, this shows for the n = 2 case that H ≤ G (a property that in fact holds for all n). It also follows
that ${\displaystyle G={\sqrt {AH}}}$, meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.
Three numbers
For the special case of three numbers, ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$, the harmonic mean can be written
${\displaystyle H={\frac {3x_{1}x_{2}x_{3}}{x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}}}.}$
Three positive numbers H, G, and A are respectively the harmonic, geometric, and arithmetic means of three positive numbers if and only if^[8]^:p.74,#1834 the following inequality holds
${\displaystyle {\frac {A^{3}}{G^{3}}}+{\frac {G^{3}}{H^{3}}}+1\leq {\frac {3}{4}}\left(1+{\frac {A}{H}}\right)^{2}.}$
If a set of weights ${\displaystyle w_{1}}$, ..., ${\displaystyle w_{n}}$ is associated to the data set ${\displaystyle x_{1}}$, ..., ${\displaystyle x_{n}}$, the weighted harmonic mean is defined by
${\displaystyle H={\frac {\sum \limits _{i=1}^{n}w_{i}}{\sum \limits _{i=1}^{n}{\frac {w_{i}}{x_{i}}}}}=\left({\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}^{-1}}{\sum \limits _{i=1}^{n}w_{i}}}\right)
The unweighted harmonic mean can be regarded as the special case where all of the weights are equal.
In analytic number theory
In physics
Average speed
In many situations involving rates and ratios, the harmonic mean provides the correct average. For instance, if a vehicle travels a certain distance d outbound at a speed x (e.g. 60 km/h) and returns
the same distance at a speed y (e.g. 20 km/h), then its average speed is the harmonic mean of x and y (30 km/h), not the arithmetic mean (40 km/h). The total travel time is the same as if it had
traveled the whole distance at that average speed. This can be proven as follows:^[11]
Average speed for the entire journey = Total distance traveled/Sum of time for each segment = 2d/d/x + d/y = 2/1/x+1/y
However, if the vehicle travels for a certain amount of time at a speed x and then the same amount of time at a speed y, then its average speed is the arithmetic mean of x and y, which in the above
example is 40 km/h.
Average speed for the entire journey = Total distance traveled/Sum of time for each segment = xt+yt/2t = x+y/2
The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distance, then the average speed is the harmonic mean of all
the sub-trip speeds; and if each sub-trip takes the same amount of time, then the average speed is the arithmetic mean of all the sub-trip speeds. (If neither is the case, then a weighted harmonic
mean or weighted arithmetic mean is needed. For the arithmetic mean, the speed of each portion of the trip is weighted by the duration of that portion, while for the harmonic mean, the corresponding
weight is the distance. In both cases, the resulting formula reduces to dividing the total distance by the total time.)
However, one may avoid the use of the harmonic mean for the case of "weighting by distance". Pose the problem as finding "slowness" of the trip where "slowness" (in hours per kilometre) is the
inverse of speed. When trip slowness is found, invert it so as to find the "true" average trip speed. For each trip segment i, the slowness s[i] = 1/speed[i]. Then take the weighted arithmetic mean
of the s[i]'s weighted by their respective distances (optionally with the weights normalized so they sum to 1 by dividing them by trip length). This gives the true average slowness (in time per
kilometre). It turns out that this procedure, which can be done with no knowledge of the harmonic mean, amounts to the same mathematical operations as one would use in solving this problem by using
the harmonic mean. Thus it illustrates why the harmonic mean works in this case.
Similarly, if one wishes to estimate the density of an alloy given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted
density of the alloy (exclusive of typically minor volume changes due to atom packing effects) is the weighted harmonic mean of the individual densities, weighted by mass, rather than the weighted
arithmetic mean as one might at first expect. To use the weighted arithmetic mean, the densities would have to be weighted by volume. Applying dimensional analysis to the problem while labeling the
mass units by element and making sure that only like element-masses cancel makes this clear.
If one connects two electrical resistors in parallel, one having resistance x (e.g., 60 Ω) and one having resistance y (e.g., 40 Ω), then the effect is the same as if one had used two resistors with
the same resistance, both equal to the harmonic mean of x and y (48 Ω): the equivalent resistance, in either case, is 24 Ω (one-half of the harmonic mean). This same principle applies to capacitors
in series or to inductors in parallel.
However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (50 Ω), with total resistance equal to twice this, the sum of x and y (100 Ω). This
principle applies to capacitors in parallel or to inductors in series.
As with the previous example, the same principle applies when more than two resistors, capacitors or inductors are connected, provided that all are in parallel or all are in series.
The "conductivity effective mass" of a semiconductor is also defined as the harmonic mean of the effective masses along the three crystallographic directions.^[12]
As for other optic equations, the thin lens equation 1/f = 1/u + 1/v can be rewritten such that the focal length f is one-half of the harmonic mean of the distances of the subject u and object
v from the lens.^[13]
Two thin lenses of focal length f[1] and f[2] in series is equivalent to two thin lenses of focal length f[hm], their harmonic mean, in series. Expressed as optical power, two thin lenses of optical
powers P[1] and P[2] in series is equivalent to two thin lenses of optical power P[am], their arithmetic mean, in series.
In finance
The weighted harmonic mean is the preferable method for averaging multiples, such as the price–earnings ratio (P/E). If these ratios are averaged using a weighted arithmetic mean, high data points
are given greater weights than low data points. The weighted harmonic mean, on the other hand, correctly weights each data point.^[14] The simple weighted arithmetic mean when applied to non-price
normalized ratios such as the P/E is biased upwards and cannot be numerically justified, since it is based on equalized earnings; just as vehicles speeds cannot be averaged for a roundtrip journey
(see above).^[15]
In geometry
In any triangle, the radius of the incircle is one-third of the harmonic mean of the altitudes.
For any point P on the minor arc BC of the circumcircle of an equilateral triangle ABC, with distances q and t from B and C respectively, and with the intersection of PA and BC being at a distance y
from point P, we have that y is half the harmonic mean of q and t.^[16]
In a right triangle with legs a and b and altitude h from the hypotenuse to the right angle, h^2 is half the harmonic mean of a^2 and b^2.^[17]^[18]
Let t and s (t > s) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s^2 equals half the harmonic mean of c^2 and t^2.
Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD. Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is
parallel to AB and CD. Then FG is the harmonic mean of AB and DC. (This is provable using similar triangles.)
Crossed ladders. h is half the harmonic mean of A and B
One application of this trapezoid result is in the crossed ladders problem, where two ladders lie oppositely across an alley, each with feet at the base of one sidewall, with one leaning against a
wall at height A and the other leaning against the opposite wall at height B, as shown. The ladders cross at a height of h above the alley floor. Then h is half the harmonic mean of A and B. This
result still holds if the walls are slanted but still parallel and the "heights" A, B, and h are measured as distances from the floor along lines parallel to the walls. This can be proved easily
using the area formula of a trapezoid and area addition formula.
In an ellipse, the semi-latus rectum (the distance from a focus to the ellipse along a line parallel to the minor axis) is the harmonic mean of the maximum and minimum distances of the ellipse from a
In other sciences
In computer science, specifically information retrieval and machine learning, the harmonic mean of the precision (true positives per predicted positive) and the recall (true positives per real
positive) is often used as an aggregated performance score for the evaluation of algorithms and systems: the F-score (or F-measure). This is used in information retrieval because only the positive
class is of relevance, while number of negatives, in general, is large and unknown.^[19] It is thus a trade-off as to whether the correct positive predictions should be measured in relation to the
number of predicted positives or the number of real positives, so it is measured versus a putative number of positives that is an arithmetic mean of the two possible denominators.
A consequence arises from basic algebra in problems where people or systems work together. As an example, if a gas-powered pump can drain a pool in 4 hours and a battery-powered pump can drain the
same pool in 6 hours, then it will take both pumps 6·4/6 + 4, which is equal to 2.4 hours, to drain the pool together. This is one-half of the harmonic mean of 6 and 4: 2·6·4/6 + 4 = 4.8. That
is, the appropriate average for the two types of pump is the harmonic mean, and with one pair of pumps (two pumps), it takes half this harmonic mean time, while with two pairs of pumps (four pumps)
it would take a quarter of this harmonic mean time.
In hydrology, the harmonic mean is similarly used to average hydraulic conductivity values for a flow that is perpendicular to layers (e.g., geologic or soil) - flow parallel to layers uses the
arithmetic mean. This apparent difference in averaging is explained by the fact that hydrology uses conductivity, which is the inverse of resistivity.
In sabermetrics, a baseball player's Power–speed number is the harmonic mean of their home run and stolen base totals.
In population genetics, the harmonic mean is used when calculating the effects of fluctuations in the census population size on the effective population size. The harmonic mean takes into account the
fact that events such as population bottleneck increase the rate genetic drift and reduce the amount of genetic variation in the population. This is a result of the fact that following a bottleneck
very few individuals contribute to the gene pool limiting the genetic variation present in the population for many generations to come.
When considering fuel economy in automobiles two measures are commonly used – miles per gallon (mpg), and litres per 100 km. As the dimensions of these quantities are the inverse of each other (one
is distance per volume, the other volume per distance) when taking the mean value of the fuel economy of a range of cars one measure will produce the harmonic mean of the other – i.e., converting the
mean value of fuel economy expressed in litres per 100 km to miles per gallon will produce the harmonic mean of the fuel economy expressed in miles per gallon. For calculating the average fuel
consumption of a fleet of vehicles from the individual fuel consumptions, the harmonic mean should be used if the fleet uses miles per gallon, whereas the arithmetic mean should be used if the fleet
uses litres per 100 km. In the USA the CAFE standards (the federal automobile fuel consumption standards) make use of the harmonic mean.
In chemistry and nuclear physics the average mass per particle of a mixture consisting of different species (e.g., molecules or isotopes) is given by the harmonic mean of the individual species'
masses weighted by their respective mass fraction.
Harmonic mean for Beta distribution for 0 < α < 5 and 0 < β < 5
(Mean - HarmonicMean) for Beta distribution versus alpha and beta from 0 to 2
Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), smaller values alpha and beta in front
Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), larger values alpha and beta in front
The harmonic mean of a beta distribution with shape parameters α and β is:
${\displaystyle H={\frac {\alpha -1}{\alpha +\beta -1}}{\text{ conditional on }}\alpha >1\,\,\&\,\,\beta >0}$
The harmonic mean with α < 1 is undefined because its defining expression is not bounded in [0, 1].
Letting α = β
${\displaystyle H={\frac {\alpha -1}{2\alpha -1}}}$
showing that for α = β the harmonic mean ranges from 0 for α = β = 1, to 1/2 for α = β → ∞.
The following are the limits with one parameter finite (non-zero) and the other parameter approaching these limits:
{\displaystyle {\begin{aligned}\lim _{\alpha \to 0}H&={\text{ undefined }}\\\lim _{\alpha \to 1}H&=\lim _{\beta \to \infty }H=0\\\lim _{\beta \to 0}H&=\lim _{\alpha \to \infty }H=1\end{aligned}}}
With the geometric mean the harmonic mean may be useful in maximum likelihood estimation in the four parameter case.
A second harmonic mean (H[1 − X]) also exists for this distribution
${\displaystyle H_{1-X}={\frac {\beta -1}{\alpha +\beta -1}}{\text{ conditional on }}\beta >1\,\,\&\,\,\alpha >0}$
This harmonic mean with β < 1 is undefined because its defining expression is not bounded in [ 0, 1 ].
Letting α = β in the above expression
${\displaystyle H_{1-X}={\frac {\beta -1}{2\beta -1}}}$
showing that for α = β the harmonic mean ranges from 0, for α = β = 1, to 1/2, for α = β → ∞.
The following are the limits with one parameter finite (non zero) and the other approaching these limits:
{\displaystyle {\begin{aligned}\lim _{\beta \to 0}H_{1-X}&={\text{ undefined }}\\\lim _{\beta \to 1}H_{1-X}&=\lim _{\alpha \to \infty }H_{1-X}=0\\\lim _{\alpha \to 0}H_{1-X}&=\lim _{\beta \to \
infty }H_{1-X}=1\end{aligned}}}
Although both harmonic means are asymmetric, when α = β the two means are equal.
The harmonic mean ( H ) of the lognormal distribution of a random variable X is^[20]
${\displaystyle H=\exp \left(\mu -{\frac {1}{2}}\sigma ^{2}\right),}$
where μ and σ^2 are the parameters of the distribution, i.e. the mean and variance of the distribution of the natural logarithm of X.
The harmonic and arithmetic means of the distribution are related by
${\displaystyle {\frac {\mu ^{*}}{H}}=1+C_{v}^{2}\,,}$
where C[v] and μ^* are the coefficient of variation and the mean of the distribution respectively..
The geometric (G), arithmetic and harmonic means of the distribution are related by^[21]
${\displaystyle H\mu ^{*}=G^{2}.}$
The harmonic mean of type 1 Pareto distribution is^[22]
${\displaystyle H=k\left(1+{\frac {1}{\alpha }}\right)}$
where k is the scale parameter and α is the shape parameter.
For a random sample, the harmonic mean is calculated as above. Both the mean and the variance may be infinite (if it includes at least one term of the form 1/0).
Sample distributions of mean and variance
The mean of the sample m is asymptotically distributed normally with variance s^2.
${\displaystyle s^{2}={\frac {m\left[\operatorname {E} \left({\frac {1}{x}}-1\right)\right]}{m^{2}n}}}$
The variance of the mean itself is^[23]
${\displaystyle \operatorname {Var} \left({\frac {1}{x}}\right)={\frac {m\left[\operatorname {E} \left({\frac {1}{x}}-1\right)\right]}{nm^{2}}}}$
where m is the arithmetic mean of the reciprocals, x are the variates, n is the population size and E is the expectation operator.
Delta method
Assuming that the variance is not infinite and that the central limit theorem applies to the sample then using the delta method, the variance is
${\displaystyle \operatorname {Var} (H)={\frac {1}{n}}{\frac {s^{2}}{m^{4}}}}$
where H is the harmonic mean, m is the arithmetic mean of the reciprocals
${\displaystyle m={\frac {1}{n}}\sum {\frac {1}{x}}.}$
s^2 is the variance of the reciprocals of the data
${\displaystyle s^{2}=\operatorname {Var} \left({\frac {1}{x}}\right)}$
and n is the number of data points in the sample.
Jackknife method
A jackknife method of estimating the variance is possible if the mean is known.^[24] This method is the usual 'delete 1' rather than the 'delete m' version.
This method first requires the computation of the mean of the sample (m)
${\displaystyle m={\frac {n}{\sum {\frac {1}{x}}}}}$
where x are the sample values.
A series of value w[i] is then computed where
${\displaystyle w_{i}={\frac {n-1}{\sum _{jeq i}{\frac {1}{x}}}}.}$
The mean (h) of the w[i] is then taken:
${\displaystyle h={\frac {1}{n}}\sum {w_{i}}}$
The variance of the mean is
${\displaystyle {\frac {n-1}{n}}\sum {(m-w_{i})}^{2}.}$
Significance testing and confidence intervals for the mean can then be estimated with the t test.
Size biased sampling
Assume a random variate has a distribution f( x ). Assume also that the likelihood of a variate being chosen is proportional to its value. This is known as length based or size biased sampling.
Let μ be the mean of the population. Then the probability density function f*( x ) of the size biased population is
${\displaystyle f^{*}(x)={\frac {xf(x)}{\mu }}}$
The expectation of this length biased distribution E^*( x ) is^[23]
${\displaystyle \operatorname {E} ^{*}(x)=\mu \left[1+{\frac {\sigma ^{2}}{\mu ^{2}}}\right]}$
where σ^2 is the variance.
The expectation of the harmonic mean is the same as the non-length biased version E( x )
${\displaystyle E^{*}(x^{-1})=E(x)^{-1}}$
The problem of length biased sampling arises in a number of areas including textile manufacture^[25] pedigree analysis^[26] and survival analysis^[27]
Akman et al. have developed a test for the detection of length based bias in samples.^[28]
Shifted variables
If X is a positive random variable and q > 0 then for all ε > 0^[29]
${\displaystyle \operatorname {Var} \left[{\frac {1}{(X+\epsilon )^{q}}}\right]<\operatorname {Var} \left({\frac {1}{X^{q}}}\right).}$
Assuming that X and E(X) are > 0 then^[29]
${\displaystyle \operatorname {E} \left[{\frac {1}{X}}\right]\geq {\frac {1}{\operatorname {E} (X)}}}$
This follows from Jensen's inequality.
Gurland has shown that^[30] for a distribution that takes only positive values, for any n > 0
${\displaystyle \operatorname {E} \left(X^{-1}\right)\geq {\frac {\operatorname {E} \left(X^{n-1}\right)}{\operatorname {E} \left(X^{n}\right)}}.}$
Under some conditions^[31]
${\displaystyle \operatorname {E} (a+X)^{-n}\sim \operatorname {E} \left(a+X^{-n}\right)}$
where ~ means approximately equal to.
Sampling properties
Assuming that the variates (x) are drawn from a lognormal distribution there are several possible estimators for H:
{\displaystyle {\begin{aligned}H_{1}&={\frac {n}{\sum \left({\frac {1}{x}}\right)}}\\H_{2}&={\frac {\left(\exp \left[{\frac {1}{n}}\sum \log _{e}(x)\right]\right)^{2}}{{\frac {1}{n}}\sum (x)}}\\
H_{3}&=\exp \left(m-{\frac {1}{2}}s^{2}\right)\end{aligned}}}
${\displaystyle m={\frac {1}{n}}\sum \log _{e}(x)}$
${\displaystyle s^{2}={\frac {1}{n}}\sum \left(\log _{e}(x)-m\right)^{2}}$
Of these H[3] is probably the best estimator for samples of 25 or more.^[32]
Bias and variance estimators
A first order approximation to the bias and variance of H[1] are^[33]
{\displaystyle {\begin{aligned}\operatorname {bias} \left[H_{1}\right]&={\frac {HC_{v}}{n}}\\\operatorname {Var} \left[H_{1}\right]&={\frac {H^{2}C_{v}}{n}}\end{aligned}}}
where C[v] is the coefficient of variation.
Similarly a first order approximation to the bias and variance of H[3] are^[33]
{\displaystyle {\begin{aligned}{\frac {H\log _{e}\left(1+C_{v}\right)}{2n}}\left[1+{\frac {1+C_{v}^{2}}{2}}\right]\\{\frac {H\log _{e}\left(1+C_{v}\right)}{n}}\left[1+{\frac {1+C_{v}^{2}}{4}}\
In numerical experiments H[3] is generally a superior estimator of the harmonic mean than H[1].^[33] H[2] produces estimates that are largely similar to H[1].
The Environmental Protection Agency recommends the use of the harmonic mean in setting maximum toxin levels in water.^[34]
In geophysical reservoir engineering studies, the harmonic mean is widely used.^[35]
1. If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
Using Pythagoras' theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² = QM.
Using Pythagoras' theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² = GM.
Using similar triangles, HC/GC = GC/OC ∴ HC = GC²/OC = HM. | {"url":"https://www.wikiwand.com/en/articles/Harmonic_mean","timestamp":"2024-11-05T02:50:25Z","content_type":"text/html","content_length":"744601","record_id":"<urn:uuid:40b85839-69b3-4539-904c-19d4c543b085>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00026.warc.gz"} |
Tinkerbee Innovations
Number systems
Recap counting!
The decimal number system that we are used to, is called so because it uses 10 distinct symbols to represent any value. These symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Any ‘number’ can therefore
be represented by combining these 10 distinct symbols.
What is interesting is that, this is not the only way to represent the numbers. I mean, one could imagine a system where instead of symbols 0->9 one used alphabet a->h. So 24 could be written as ce.
It is funny, but is perfectly possible. Since most of us are taught to count in decimals right from the time we were kids, and since we usually have ten fingers, this manner of counting seems
‘natural’ to us 😊. (The Octopus, I’m sure, has an octal (base 8) number system ;p)
Octopus. Source: https://commons.wikimedia.org/wiki/File:Octopus_pallidus.jpg
Another variation would be, if there were 16 symbols instead of just 10! This is exactly what the hexadecimal system does, it has symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f. So, Bryan
Adams was singing ’18 till I die’ in a hexadecimal world, he would sing: ’hex one zero till I die’. I hope you get the drift.
The number of symbols available in the system for representing a value is called its ‘base’. So decimal numbers have base 10. Hexadecimal numbers have a base 16.
In a similar fashion, in the binary world, there are unfortunately only two symbols 0 and 1 (and they correspond beautifully into one of the simpler natures of electricity that either flows/ ON state
/ 1 or does not/ OFF state/ 0) (or, has the potential to flow or not 😉). So binary numbers are base 2.
Numbers with a particular base are written as a subscript: (X)n
When we deal with decimals or places where the n is obvious, we of course ignore writing it.
Bonus: Here is a link to a simple C program to convert between bases: http://www.geeksforgeeks.org/convert-base-decimal-vice-versa/
Hope you know your numbers better now 😊
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Regression to the Mean: Of Nazis and Investment AnalysisRegression to the Mean: Of Nazis and Investment AnalysisThe Psy-Fi Blog
Francis Galton, The Measuring Man
Francis Galton, cousin of Charles Darwin, was utterly uninterested in the both the workings of investment analysts and Hitler’s future plans for world domination. To be fair he died before both
really got going. However, he has the dubious distinction of being at least partially to blame for some of the madder excesses of both.
What Galton was interested in was measuring things: mainly people, although he’d turn his hand to anything in the absence of a likely suspect or two. In so doing he uncovered a statistical phenomenon
known as Regression to the Mean which lies behind many of the theories which still dominate stockmarket analysis and valuation techniques today and which many people misunderstand totally.
Galton’s Eugenics To deal with Galton and Hitler first, however. Galton was the inventor of eugenics, a branch of science dedicated to analy sing the fitness of the human species with, as its goal,
the aim of improving the race. Like many a Victorian gentleman Galton was convinced that humanity was regressing into a less intelligent and less developed state. In fact what he was actually
observing was the impact on people of moving them into unhygienic cities, working them in appalling conditions and giving them a lousy diet. Tricky stuff this nurture, nature thing.
Galton’s measurements were essentially designed to show that the rich and powerful were clever and the poor and weak were not, thus squaring the circle: the rich and powerful were naturally bound to
ascend to the top of the tree and therefore had every right to be there. Nothing at all, then, to do with the fact that your great-great-great-grandmother had an illegitimate child with the King, was
set up with an estate comprising half of Scotland in order to go away and stop bothering him and you inherited it by accident of birth. Clearly not.
Galton’s Statistics
Although Galton’s logic may have been a little faulty when it came to measuring such subjective things as intelligence, when he turned his attention to more objective stuff like height there wasn’t
much wrong with his statistics. In a set of experiments on the size of seeds of successive generations of sweet peas he noted the odd fact that the larger seeds of one generation tended to result in
smaller seeds in the next generation and vice versa. Applying a similar analysis to people he noted that although short people tended to have short children and tall people to have tall children the
short children tended to be slightly taller than their parents and the tall children slightly shorter than theirs.
This led Galton to postulate the principle known as Regression to the Mean. What this says is that is an abnormally large value of a variable – let’s say it’s the return on the stockmarket in any
given year – is likely to be succeeded by a lower value. You could say that it’s the statistical proof of the old saving “what goes up must come down”. However, what Galton also showed was that the
results of such analyses tended to cluster around the mid-point – the so called mean. The average value if you like. Over time it’s this average value that all returns will converge on. Technically
this convergence is called “regression”. So values converge on the average or “regress to the mean”.
Think about tossing a coin ten times. On average you’ll get five heads and five tails. If last time you got ten heads next time you’re more likely to approach the average value or regress to the
mean. It’s just probability.
Stockmarket Returns and Sheep
Over the twentieth century the stockmarket returned, roughly, 12% per year. Most years saw a return a bit more or a bit less than this value. A few years saw a return much, much, much higher than
this. A few saw returns much, much, much lower than this. What Galton’s findings suggest for stockmarket investors is that worrying over (or glorying in) exceptionally bad (or exceptionally good)
years is pointless. Over time things will drift back to the mean. Over time, if you don’t let better judgement get in the way, you would have made about 12% a year.
Often, though, judgement does get in the way, not always for the better. There are certainly a few people who have done better than 12% over very long periods of time – too long to be explained by
chance. There are lots and lots of people who’ve done worse. In both cases the explanation for the exceptional performance probably lies in psychology: the jitterings of markets mean that at both the
high and low points there are opportunities for discerning investors to make cool judgements - sell at the highs and buy at the lows. Most investors, though, aren’t discerning. They’re just humans
and they do what humans do. They flock like sheep. Baa.
So they buy when everyone else is buying and sell when everyone else is selling. Buy high, sell low is not textbook mantra but it’s done by far more people than do the opposite. In the sixties it was
the Nifty Fifty, in the nineties it was dotcom – sooner or later it’ll be something else. The herd effect is well ingrained in our behaviours and if everyone else is doing something it stands to
reason that we should be doing it to. Obviously. Baa.
Neither Students nor Stockmarkets are Normal
There are two things about mean regression that investors need to be careful of. Firstly the stockmarket doesn’t follow exactly the same distribution as naturally occurring phenomena (which
technically is called a Normal or Gaussian distribution). So we get very bad years and very good years far more than the theory would predict. Over time returns will still tend to the mean, but we
need to be aware that the markets may trend away from this for considerable periods of time. Simple psychology makes many investors extrapolate current conditions forever which is not sensible and is
likely to lead to exactly the wrong sorts of behaviour. Clever academics can make the same mistake (see Alpha and Beta - Beware Gift Bearing Greeks).
Secondly, there’s a problem with mean regression which is that it’s a statistical effect. If you’re a below average student then you can assume you’ll get a below average mark in a test. You won’t
get any closer to the mean on a second test just because of a statistical effect: it may be that you were unlucky to get an exceptionally low mark first time out and therefore you’ll do better next
time but that’s not statistics, that’s you.
Stats Doesn’t Mean You Can Buck the Trend
The same applies to companies in the absence of any changes. Just as a low grade student will predictably score low grade marks so a low grade company will predictably get low grade results. However,
very often underperforming companies will get shaken up by new management or shareholder discontent in much the same way as a student’s father may discipline them to get them to work harder. But this
doesn’t happen by chance, which means the effort of stock analysis is not irrelevant and that simply picking a sample of underperforming companies won’t guarantee excess returns.
Although Galton observed mean regression in people he fundamentally misinterpreted it in the light of his own prejudices. Whereas we now know that regression is a statistical phenomenon where any
lack of full correlation between parents and children will result in such an effect (i.e. children are not copies of their parents due to gene mixing and mutation) Galton attributed it to a
combination of inheritance from parents and from prior ancestors. Hence he was able, or so he thought, to demonstrate that the inherited characteristics of the privileged flowed down from their
distinguished lineages. This fundamental misunderstanding of the phenomena was the mainspring of the eugenics movement which ultimately led to Hitler and his death camps.
Statistics in the wrong hands is very, very dangerous. Losing money on the stockmarket because of it is possibly the least of our worries.
Related Posts: Risky Bankers Need Swiss Cheese Not VaR, Alpha and Beta: Beware Gift Bearing Greeks
3 comments:
1. Statistics are such a powerful tool, and as you mention, so misused!
The power of statistics are both in comparing similar sets, and in looking back at the progress of a series of values. It is distinctly not of any use in predicting where a series of values is
likely to head, unless the series of values has been correctly profiled and characterized as being of a particular and known distribution.
In particular, the returns on the stock market, do not follow any such known distribution. That the mean over a century has been 12% will be a result of a multitude of inputs such as world
population, industrial output, wars, global taxation, and a myriad others... There's no guarantee that the 21st century will exhibit the same pattern, and in fact, given that massive shifts in
population and global trade patterns occurring in the last decade or two, chances are that it will be radically different.
As an aside, it is galling that finance professionals such as pension funds use backward looking statistics to project future returns, when a large percentage of their clients are not aware of
the unsuitability of these statistic return expectations.
Yours, and very grateful for your outstanding blog!
2. Hi uchinadi
Glad you enjoy it. All feedback is welcome, it's difficult to know at what level to pitch posts. Don't want to be too technical nor too simple.
Your comments are all true. Perhaps shares are the least worst option a lot of the time. At least they ought to roughly track global growth (assuming we ever get any again!)
Even mutual funds with their excessive fees would normally provide better returns than cash in a bank account so we perhaps shouldn't be too critical of their misuse of stats. The best possible
bad option, maybe?
3. I'm afraid that you've chosen an unfortunate (way to present the coin-tossing) example when explaining "regression to the mean":
``Think about tossing a coin ten times. On average you’ll get five heads and five tails. If last time you got ten heads next time you’re more likely to approach the average value or regress to
the mean. It’s just probability.''
Someone who is not familiar with the concept of independence (or, lack of correlation) of events may, very wrongly, conclude that if you get ten heads in a row then it is more likely to get tails
in the next toss (or, e.g., to get more tails than heads in the next ten tosses). This of course cannot be the case if one assumes, as one typically does when evoking the coin-tossing example,
that the coin is fair and the tosses are independent of each other. One non-technical way to explain it is that, when the eleventh coin toss happens, the coin ``doesn't know'' that ten heads in a
row have just happened.
What you say is correct: the expected number of heads-per-coin-toss is (always!) 0.5, and hence if the current average is 1-head-per-toss (ten heads in a row) then the expected average after
twenty tosses (given that the first ten were all heads) is 0.75 = (10+10*0.5)/20: that is a ``reversion to mean'' from 1 to 0.75 (towards the mean of 0.5). And that, as you say, is ``just
The way you formulated your (pithy) paragraph, however, might confuse some to believe that the ``magic of regression to the mean'' makes the coin more likely to yield tails after a long streak of
heads. It doesn't, and it's not magic: indeed, it's ``just'' probability.
Sorry for being picky about this, but it's worth appreciating the significant difference between the reversion to the mean in the coin tossing example and in the financial markets. While the coin
doesn't know what happened to it in the previous ten tosses, the markets do know! In other words, the action of a market today is not independent of its action in the last ten days. It's not just
probability that makes markets ``revert to the mean'': it's economics and psychology that do (too)! Another difference between investing and gambling in a casino. | {"url":"https://www.psyfitec.com/2009/04/of-nazis-and-investment-analysis.html","timestamp":"2024-11-04T10:54:31Z","content_type":"application/xhtml+xml","content_length":"71351","record_id":"<urn:uuid:7a6436ac-0936-4605-ab17-dd7b72af1be6>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00700.warc.gz"} |
Understanding Mathematical Functions: How To Calculate Gamma Function
Introduction to the Gamma Function
The gamma function is a complex-valued function that plays a crucial role in various fields of mathematics, including probability, statistics, and calculus. It is denoted by the symbol Γ(x) and is an
extension of the factorial function to complex and real numbers.
Explanation of what the gamma function is and its significance in mathematics
The gamma function, defined as an integral, represents a generalization of the concept of factorial to non-integer numbers. It has applications in areas such as combinatorics, number theory, and
complex analysis. The function is particularly significant in solving differential equations, evaluating infinite series, and representing certain special functions.
Historical context and the development of the gamma function by Euler
The gamma function was first studied by mathematicians in the 18th century, and the first significant developments in its understanding were made by Leonhard Euler. Euler's work on the gamma function
led to its widespread adoption and use in various mathematical areas. The gamma function is named after Euler due to his extensive contributions to its development.
Overview of the gamma function's connection to factorials and its importance in various fields such as probability and statistics
One of the most important connections of the gamma function is its relationship with factorials. For non-negative integers, the gamma function is identical to the factorial function, n! = Γ(n+1).
This connection is fundamental in understanding the extension of factorials to non-integer values. Additionally, the gamma function is extensively used in probability theory and statistics to
calculate various probability distributions, such as the gamma distribution and the chi-squared distribution.
Key Takeaways
• Gamma function definition and properties
• Calculation using factorial and Euler's reflection formula
• Using special functions and software for accurate results
• Applications in probability, statistics, and physics
• Understanding the significance of the gamma function
Understanding the Basics of the Gamma Function
The gamma function, denoted by Γ(z), is an extension of the factorial function to complex and real numbers. It is an essential tool in various areas of mathematics, including calculus, number theory,
and probability. Understanding the basics of the gamma function is crucial for its application in mathematical calculations.
A Defining the gamma function for positive integers (n-1)!
For positive integers, the gamma function can be defined as Γ(n) = (n-1)!, where n is a positive integer. This definition aligns with the factorial function, which represents the product of all
positive integers up to a given number. For example, Γ(4) = 3!, which equals 6.
B Extension of the gamma function to non-integer and complex numbers
One of the key features of the gamma function is its extension to non-integer and complex numbers. Unlike the factorial function, which is only defined for positive integers, the gamma function
allows for the calculation of factorials for non-integer and complex values. This extension is achieved through the use of integral calculus and the concept of infinite series, enabling the
evaluation of Γ(z) for a wide range of input values.
C The concept of the analytic continuation of the gamma function
The concept of the analytic continuation of the gamma function is fundamental in understanding its behavior beyond its initial definition. Analytic continuation refers to the process of extending the
domain of a given function to a larger set of complex numbers while preserving its analytical properties. In the case of the gamma function, this concept allows for the calculation of Γ(z) for a
wider range of complex numbers, providing a more comprehensive understanding of its behavior and applications.
The Gamma Function Formula and Calculation
The gamma function, denoted by Γ(n), is an extension of the factorial function to complex numbers. It is defined by the integral:
A Presentation of the integral definition of the gamma function: Γ(n) = ∫[0]^∞ t^(n-1)e^(-t) dt
This integral definition allows us to calculate the gamma function for any complex number n. The integral represents the area under the curve of the function t^(n-1)e^(-t) from 0 to infinity.
Explanation of how to evaluate the gamma function for simple arguments
For simple arguments such as positive integers, the gamma function can be evaluated using the factorial formula. For example, Γ(4) = 3!, which equals 6. This provides a straightforward way to
calculate the gamma function for integer inputs.
For non-integer values, numerical methods or special functions such as the Lanczos approximation can be used to compute the gamma function.
Demonstration of gamma function values for half-integer inputs
When the input to the gamma function is a half-integer, interesting patterns emerge. For example, Γ(1/2) equals the square root of π, which is approximately 1.77245. Similarly, Γ(3/2) equals (1/2)!,
or √π/2, which is approximately 0.886227.
These values demonstrate the unique behavior of the gamma function for half-integer inputs, and how it connects to other mathematical constants such as π.
Properties of the Gamma Function
The gamma function, denoted by Γ(z), is a complex-valued function that extends the factorial function to complex and real numbers. It has several important properties that make it a fundamental tool
in mathematical analysis and various applications.
A Recurrence and reflection properties of the gamma function
One of the key properties of the gamma function is its recurrence relation, which is given by:
This property allows for the calculation of the gamma function for any complex number z by using the value of Γ(z) for a smaller number. Additionally, the gamma function satisfies the reflection
B The gamma function's relation to other special functions like beta and digamma functions
The gamma function is closely related to other special functions, such as the beta function and the digamma function. The beta function, denoted by B(x, y), is defined in terms of the gamma function
• B(x, y) = Γ(x)Γ(y) / Γ(x+y)
Furthermore, the digamma function, denoted by Ψ(z), is the logarithmic derivative of the gamma function and is related to it through the equation:
C Duplication and multiplication theorems
The gamma function also satisfies duplication and multiplication theorems, which are given by:
• Γ(2z) = 2^(2z-1)Γ(z)Γ(z+1/2)√π
• Γ(z)Γ(z+1) = zΓ(z)
These theorems provide useful relationships for simplifying and evaluating the gamma function for different values of z.
Practical Applications of the Gamma Function
The gamma function, denoted by Γ(z), is a fundamental mathematical function with a wide range of practical applications across various fields. Let's explore some of the key practical applications of
the gamma function.
Application in Complex Analyses
The gamma function plays a crucial role in complex analyses, particularly in the calculation of residues. In complex analysis, the residue theorem is used to evaluate contour integrals. The gamma
function helps in calculating these residues, which are essential in solving complex mathematical problems.
Role in Probability Distributions
The gamma function is closely associated with probability distributions, including the gamma distribution and the chi-squared distribution. These distributions are widely used in statistics and
probability theory to model various real-world phenomena. The gamma function is used to define the probability density functions and cumulative distribution functions of these distributions, making
it an integral part of statistical analysis.
Usage in Physics, Engineering, and Quantitative Finance
In the fields of physics, engineering, and quantitative finance, the gamma function finds extensive applications. In physics, the gamma function appears in various mathematical models and equations,
particularly in the context of quantum mechanics and electromagnetic theory. In engineering, the gamma function is used in the analysis and design of systems, such as signal processing and control
systems. In quantitative finance, the gamma function is employed in option pricing models and risk management calculations.
Troubleshooting Common Issues in Calculating the Gamma Function
When working with the gamma function, it is important to be aware of common issues that may arise during computation. Addressing inaccuracies in computation for complex numbers, mitigating numerical
instability in evaluating the gamma function at large values, and choosing the appropriate method for calculating the gamma function efficiently are key areas to focus on.
A Addressing inaccuracies in computation for complex numbers
Calculating the gamma function for complex numbers can be challenging due to the potential for inaccuracies in computation. One common issue is the presence of singularities, which can lead to
undefined or inaccurate results. To address this, it is important to use specialized algorithms and numerical methods that are designed to handle complex numbers. Additionally, ensuring that the
computational environment is set up to handle complex arithmetic operations accurately is crucial for obtaining reliable results.
B Mitigating numerical instability in evaluating the gamma function at large values
When evaluating the gamma function at large values, numerical instability can occur, leading to inaccurate results. This is often due to the rapid growth of the gamma function as its argument
increases. To mitigate this issue, it is important to use numerical techniques such as asymptotic expansions and recurrence relations that are specifically designed to handle large values of the
gamma function. By employing these techniques, the accuracy of the computation can be improved, and numerical instability can be minimized.
C Choosing the appropriate method for calculating the gamma function efficiently
There are various methods available for calculating the gamma function, each with its own advantages and limitations. When choosing a method for computation, it is important to consider factors such
as the range of values for which the gamma function needs to be evaluated, the desired level of accuracy, and the computational resources available. For example, when dealing with small to moderate
values, the use of series expansions or rational approximations may be suitable. On the other hand, for very large or very small values, specialized algorithms such as the Stirling's approximation or
the Lanczos approximation can be more efficient and accurate.
Conclusion and Best Practices for Working with the Gamma Function
After delving into the intricacies of the gamma function and exploring various methods for calculating it, it is important to summarize the key points discussed in this blog post, highlight best
practices for using numerical methods and software, and encourage further exploration of the gamma function's applications in advanced mathematics and other disciplines.
A Summarizing the key points discussed in the blog post
• Definition: The gamma function is an extension of the factorial function to complex and real numbers, and it plays a crucial role in various areas of mathematics and physics.
• Properties: We discussed the properties of the gamma function, including its relationship with factorials, its integral representation, and its behavior for different values of its argument.
• Calculation Methods: We explored different methods for calculating the gamma function, such as the Lanczos approximation, series expansion, and numerical integration.
B Best practices in using numerical methods and software for calculating the gamma function
• Accuracy: When using numerical methods or software for calculating the gamma function, it is important to ensure the accuracy of the results, especially for complex or large arguments.
• Validation: It is recommended to validate the results obtained from numerical methods by comparing them with known values or using alternative calculation approaches.
• Software Selection: Choose reliable and well-tested mathematical software libraries or packages that provide efficient and accurate implementations of the gamma function.
• Performance: Consider the performance and computational efficiency of numerical methods and software, especially when dealing with large-scale calculations or real-time applications.
C Encouraging further exploration and study of the gamma function's applications in advanced mathematics and other disciplines
• Advanced Mathematics: The gamma function has widespread applications in areas such as number theory, complex analysis, probability, and statistics, making it an essential tool for advanced
mathematical research and problem-solving.
• Physics and Engineering: In addition to mathematics, the gamma function finds extensive use in physics, engineering, and other scientific disciplines, where it appears in various mathematical
models and physical phenomena.
• Interdisciplinary Applications: Encourage interdisciplinary exploration of the gamma function's applications, such as in finance, economics, biology, and computer science, where its properties
and calculations can offer valuable insights and solutions. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-calculate-gamma-function","timestamp":"2024-11-09T03:05:18Z","content_type":"text/html","content_length":"221139","record_id":"<urn:uuid:c4088eb4-b18d-410b-beea-13cd4c7c6372>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00448.warc.gz"} |
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� How to use right function in excel: Master Excel Now! | Exclusive Guide
Mastering the RIGHT Function in Excel: Key Tips & Common Pitfalls
As an Excel whiz, I’ve learned that mastering the RIGHT function can be a game-changer. It’s a simple, yet powerful tool that helps you extract specific characters from a text string, starting from
the rightmost character. Whether you’re dealing with data analysis or daily office tasks, knowing how to use the RIGHT function in Excel can save you time and headaches.
Understanding the RIGHT Function in Excel
It’s crucial to understand how the RIGHT function operates in Excel. It’s a text function that extracts a certain number of characters from a given text starting from the rightmost position of the
string. This function is mainly used in data manipulation, providing a simple way to extract part of a text into separate cells.
The RIGHT function uses a simple syntax, =RIGHT(text,[num_chars]), where ‘text’ represents the string you need to extract data from and ‘num_chars’ denotes the number of characters you want to
extract starting from the end of the string.
What makes the RIGHT function stand out is its efficiency and simplicity. Let’s take a typical office day scenario—you’re sorting through a list of employees’ full names and you need to extract their
last names for a report. Without the RIGHT function, this task would be time-consuming and prone to error. But with this nifty tool, it’s just a matter of inputting the right formula.
For example, you have a cell ‘A1’ with the name “John Smith”. If you want to extract “Smith”, you’d use the formula:
This extracts the last five characters from “John Smith”, thus giving you “Smith”. It’s that simple!
You might ask, “What if I do not know the exact number of characters to be extracted?” That’s a great question! If ‘num_chars’ is omitted, the function will only extract one character— the right-most
It’s clear that the RIGHT function is a powerful tool, whether you’re sorting data for business analytics projects or managing routine office tasks. As you master the use of the RIGHT function,
you’ll find it’s an essential part of your Excel toolkit.
Syntax of the RIGHT Function
Understanding the syntax of the RIGHT function is crucial for getting a grip on its usability. Essentially, it’s summed up in this very simple formula: RIGHT(text,num_chars).
Let’s dissect this to make it crystal clear.
In this formula, ‘text’ is a necessary part of the syntax. It can be entered as a direct text reference or as the cell containing the text you want to manipulate. It’s the source from which you’ll
extract the specific characters. Excel will directly refer to this portion when delivering an output.
The second part of the syntax, ‘num_chars’, indicates how many characters you wish to extract. The beauty of the RIGHT function is that if you don’t specify a number, it simply assumes you want just
the last character. Now, that’s what I’d call flexibility!
This is how you would use the syntax:
• If the text is directly entered into the function, it should exist within double quotes. For instance: RIGHT(“Excel Magic”,4). This would result in the string “agic”.
• If the text is in a cell, directly refer to that cell. Example: RIGHT(A2,4). In this case, you’d get an output based on the content of cell A2.
Play around and test this syntax in various scenarios: use it on different types of string and see firsthand how it operates. But also, remember that the RIGHT function only reads from right to left.
I know it can be difficult to visualize without any tangible data, but the RIGHT function’s syntax surely proves its tractability. It will become clear as you delve deeper and start implementing it
in your Excel tasks.
Examples of Using the RIGHT Function
Diving right in, I’ll guide you through some real life examples of using the RIGHT function in Excel. These examples serve to illustrate how you can use this Excel function to manipulate data and
extract information.
Suppose I have a list of order numbers. These orders appear as alphanumeric codes like A-123-FG4567. Now, I want to separate only the last 4 digits which represent a unique product code. So, the
RIGHT function can excellently serve this purpose.
Here is how to establish it:
=RIGHT(A1, 4)
As you see, I have specified 4 as the number of characters, ‘num_chars’, that I want to extract from the right of the alphanumeric code. In the formula, A1 is the cell that holds the order number.
Take a look at the table below where I have shown the initial data and result for better understanding:
Order Number Product Code
A-123-FG4567 4567
A-234-HG6789 6789
Making it more versatile, if we want to extract all the characters after the ‘FG’ or ‘HG’ part of the order number, regardless of how many there are, I can combine the RIGHT function with some other
Excel functions. Adding the FIND function helps to locate the position of ‘FG’ or ‘HG’ and the LEN function returns the length of the whole string.
Here’s the fancy function I would make use of:
=RIGHT(A1, LEN(A1)-FIND("G",A1,1))
In this example, the RIGHT function again does the task of returning a specific number of characters, only this time, that number is calculated by subtracting the position of ‘G’ from the total
length of the text.
Overall, you can see the RIGHT function has a lot of flexibility. It paves the way for managing and manipulating data precisely. Remember, as with any other function, the more you delve into using
the RIGHT function in different scenarios, the more you’re going to master it.
Advanced Tips and Tricks for the RIGHT Function
By now, you’ve gained a solid understanding of how to use the RIGHT function to manipulate data in Excel. But I’d say, we’ve only scratched the surface. There’s still a vast sea of possibilities out
there, waiting for you to explore!
One of the advanced tricks you can use is combining the RIGHT function with other Excel functions – a perfect recipe for performing complex tasks with ease. Take the TIME function, for instance. This
can be extremely useful to derivethe hour, minutes, and seconds from a time stamp. Here’s an example:
Time Stamp Hour Minute Second
12:25:55 AM 12 25 55
1:20:30 PM 1 20 30
In this case, the formulas are as follows:
• For Hour: =RIGHT(A2, LEN(A2)-FIND(":",A2,1)+ 1)
• For Minute: =MID(A2, FIND(":", A2)+1, 2)
• For Second: =RIGHT(A2, 2)
In these instances, the RIGHT function is working behind the scenes, collaborating with the FIND and MID functions to precisely extract the time components.
Another unique spin on the RIGHT function is using it to sort data. Say, you have a batch of alphanumeric codes where the first 5 characters determine the group classification, with the last three
digits representing specific items within a group. Using the RIGHT function, you can extract these last three digits to sort or analyze items individually.
Alphanumeric Code Item Code
GRP00123 123
GRP00124 124
GRP00225 225
GRP00326 326
GRP00427 427
Formula for Item Code: =RIGHT(A2,3)
Remember, the power of the RIGHT function is in its flexibility and adaptability. Without it, these time-saving tricks wouldn’t be feasible. So don’t be afraid to pair up RIGHT with other functions,
or to use it in creative ways that best fit your needs.
Common Errors to Avoid When Using the RIGHT Function
The RIGHT function is a potent weapon in an Excel user’s arsenal, but it’s not without its pitfalls. Let’s delve into some common errors encountered when using this function.
First off, it’s important to remember that the RIGHT function extracts characters from the end of a text string. If you’re looking to pull data from the beginning of a string, you should be using the
LEFT function instead. It’s a simple mistake to make, but one that could significantly skew your results.
Next, keep in mind that the RIGHT function uses a numeric argument to define how many characters to extract. Accidentally using a non-numeric argument will result in an #VALUE! error. This error
occurs when Excel can’t understand your input — so always ensure that your argument after the comma is a number.
Another common mistake I’ve seen is using the RIGHT function to extract numerical values from a text string and then performing calculations on those values. Excel treats the output of the RIGHT
function as a text, even if it consists only of numbers. As a result, if you intend to perform calculations on the output, you must convert it into a numerical value using a function like “VALUE”.
One more error that comes up often is expecting the RIGHT function to extract words. Unfortunately, the RIGHT function doesn’t understand words or spaces. It’s all about characters. So if you need to
extract a whole word from a string of text, you’ll need to use different functions like SEARCH or FIND combined with MID, LEFT or RIGHT.
Common Errors Input Output
Wrong Function Used =RIGHT(A1,3) If A1=”Text”, output=”ext”
Non-Numeric argument =RIGHT(A1,”abc”) #VALUE! Error
Unconverted Numeric Value =RIGHT(A1,3) + 5 #VALUE! Error
Expecting Words =RIGHT(A1,5) If A1=”Hello World”, output=”World”
Let’s keep these in mind as we venture into the uncharted territories of Excel complications. Remember, effective use of the RIGHT function lies not only in manipulating data efficiently but also in
circumventing these potential hazards.
Mastering the RIGHT function in Excel can be a game-changer. It’s not just about extracting characters from text strings; it’s about optimizing your data manipulation skills. Remember, don’t mistake
it for the LEFT function. It’s crucial to provide numeric arguments to avoid #VALUE! errors. And don’t forget, the RIGHT function operates on characters, not words. For word extraction, look to
alternative functions. With these insights, you’re well on your way to navigating Excel’s complexities effectively. Here’s to your success in extracting the right data, the right way, with the RIGHT
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Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity
We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an
infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their
extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.
All Science Journal Classification (ASJC) codes
• Nuclear and High Energy Physics
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Strength of concrete | Building & Construction, Civil Engineering & Structural Designs November 2024
What is Water Cement Ratio? The water cement ratio compares how much water vs cement is used in a concrete mix. A low water-cement ratio leads to stronger concrete but is more difficult workability.
Water Cement Ratio means the ratio between water weight to the weight of cement used in the concrete mix. Usually, the […]
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The Mathematical Foundations of Gauge Theory Revisited
1. Introduction
It is usually accepted today in the literature that the physical foundations of what we shall simply call (classical) “gauge theory” (GT) can be found in the paper published by C.N. Yang and R.L.
Mills in 1954 [1] . Having in mind the space-time formulation of electromagnetism (EM), the rough idea is to start with a manifold and a group in order to exhibit a procedure leading to a physical
theory, namely a way to obtain fields and field equations from geometrical arguments on one side, both with a dual variational counterpart providing inductions and induction equations on the other
side. Accordingly, the mathematical foundations of GT can be found in the references existing at this time on differential geometry and group theory, the best and most quoted one being the survey
book [2] published by S. Kobayashi and K. Nomizu in 1963 (see also [3] -[6] ). The aim of this Introduction is to revisit these foundations and their applications with critical eyes, recalling them
in a quite specific and self-contained way for later purposes.
The word “group” has been introduced for the first time in 1830 by Evariste Galois (1811-1832). Then this concept slowly passed from algebra (groups of permutations) to geometry (groups of
transformations). It is only in 1880 that Sophus Lie (1842-1899) studied the groups of transformations depending on a finite number of parameters and now called Lie groups of transformations.
We denote as usual by
In order to fix the notations, we quote without any proof the “three fundamental theorems of Lie” that will be of constant use in the sequel (See [7] for more details):
FIRST FUNDAMENTAL THEOREM 1.1: The orbits
In a rough way, we have
SECOND FUNDAMENTAL THEOREM 1.2: If
action of a lie group
tants of a Lie algebra of vector fields which can be identified with
(care to the sign used) or equivalently
Using again crossed-derivatives, we obtain the corresponding integrability conditions (IC) on the structure constants and the Cauchy-Kowaleski theorem finally provides:
THIRD FUNDAMENTAL THEOREM 1.3: For any Lie algebra
one can construct an analytic group
EXAMPLE 1.4: Considering the affine group of transformations of the real line
with .
GAUGING PROCEDURE 1.5: If
REMARK 1.6: An easy computation in local coordinates for the case of the movement of a rigid body shows that the action of the
vector product by the vortex vector
The above particular case, well known by anybody studying the analytical mechanics of rigid bodies, can be generalized as follows. If
Introducing the induced bracket
This definition can also be adapted to
THEOREM 1.7: There is a nonlinear gauge sequence:
COROLLARY 1.8: There is a linear gauge sequence:
which is the tensor product by
It just remains to introduce the previous results into a variational framework. For this, we may consider a lagrangian on
and therefore, after integration by part, the Euler-Lagrange (EL) equations [7] [12] [13] :
Such a linear operator for
In 1954, at the birth of GT, the above notations were coming from electromagnetism with EM potential
After this long introduction, the purpose of this paper will be to escape from such a contradiction by using new mathematical tools coming from the formal theory of systems of PD equations and Lie
pseudogroups, exactly as we did in [20] for general relativity (GR). In particular, the titles of the three parts that follow will be quite similar to those of this reference though, of course, the
contents will be different. The first part proves hat the name “curvature” given to
2. First Part: The Nonlinear Janet and Spencer Sequences
In 1890, Lie discovered that Lie groups of transformations were examples of Lie pseudogroups of transfor- mations along the following definition [7] [13] [22] -[24] :
DEFINITION 2.1: A Lie pseudogroup of transformations
From now on, we shall use the same notations and definitions as in [7] [16] [20] for jet bundles. In particular, we recall that, if
where both
We also notice that an action
Lie equations in order to obtain a (linear) system of infinitesimal Lie equations
for vector fields. Such a system has the property that, if
GAUGING PROCEDURE REVISITED 2.2: Setting
we obtain
Looking at the way a vector field and its derivatives are transformed under any
and so on, a result leading to:
LEMMA 2.3:
where the left member belongs to
In order to construct another nonlinear sequence, we need a few basic definitions on Lie groupoids and Lie algebroids that will become substitutes for Lie groups and Lie algebras. The first idea is
to use the chain rule for derivatives
We may also define
DEFINITION 2.4: A fibered submanifold
Now, using the algebraic bracket
which does not depend on the respective lifts
DEFINITION 2.5: We say that a vector subbundle
a Lie algebroid if
EXAMPLE 2.6: With
For affine transformations,
We may prolong the vertical infinitesimal transformations
where we have replaced
which are such that
THEOREM 2.7: There exists a nonlinear Janet sequence associated with the Lie form of an involutive system of finite Lie equations:
where the kernel of the first operator
THEOREM 2.8: There is a first nonlinear Spencer sequence:
where all the operators involved are involutive.
Proof: There is a canonical inclusion
We also obtain from Lemma 6.3 the useful formula
We refer to ( [7] , p 215) for the inductive proof of the local exactness, providing the only formulas that will be used later on and can be checked directly by the reader:
There is no need for double-arrows in this framework as the kernels are taken with respect to the zero section of the vector bundles involved. We finally notice that the main difference with the
gauge sequence is that all the indices range from 1 to
COROLLARY 2.9: There is a first restricted nonlinear Spencer sequence:
and an induced second restricted nonlinear Spencer sequence:
where all the operators involved are involutive and which is locally isomorphic to the corresponding gauge sequence for any Lie groups of transformations when
DEFINITION 2.10: A splitting of the short exact sequence
REMARK 2.11: Rewriting the previous formulas with
Finally, setting
Accordingly, THE DUAL EQUATIONS WILL ONLY DEPEND ON THE LINEAR SPENCER OPERA- TOR
EXAMPLE 2.12: We have the formulas (Compare to [17] [19] , (76) p 289,(78) p 290):
EXAMPLE 2.13: (Projective transformations) With
Cosserat/Weyl equations :
3. Second Part: The Linear Janet and Spencer Sequences
It remains to understand how the shift by one step in the interpretation of the Spencer sequence is coherent with mechanics and electromagnetism both with their well known couplings [7] [13] [16]
[20] . In a word, the problem we have to solve is to get a 2-form in
For this purpose, introducing the Spencer map
of the Spencer operator
For later computations, the sequence
We also recall that the linear Spencer sequence for a Lie group of transformations
The main idea will be to introduce and compare the three Lie groups of transformations:
where one has to eliminate the arbitrary function
We shall use the inclusions
PROPOSITION 3.1: The Spencer sequence for the conformal Lie pseudogroup projects onto the Poincare sequence with a shift by one step.
Proof: Using
We also obtain from the relations
The study of the nonlinear framework is similar. Indeed, using Remark 2.11 with
and we may finish as before as we have taken out the quadratic terms through the contraction.
This unification result, which may be considered as the ultimate “dream” of E. and F. Cosserat or H. Weyl, could not have been obtained before 1975 as it can only be produced by means of the (linear/
nonlinear) Spencer sequences and NOT by means of the (linear/nonlinear) gauge sequences. We invite the reader to notice that it only depends on the Formulas (1), (2), (3), (4) and their respective
(*) or (**) consequences.
4. Third Part: The Duality Scheme
A duality scheme, first introduced by Henri Poincaré (1854-1912) in [12] , namely a variational framework adapted to the Spencer sequence, could be achieved in local coordinates as we did for the
gauge sequence at the end of the Introduction. We have indeed presented all the explicit formulas needed for this purpose and the reader will notice that it is difficult or even impossible to find
them in [25] . However, it is much more important to relate this dual scheme to homological algebra [29] and algebraic analysis [30] [31] by using the comment done at the end of the Second Part which
amounts to bring the nonlinear framework to the linear framework, a reason for which the stress equations of continuum mechanics are linear even for nonlinear elasticity [13] [16] [18] .
LEMMA 4.1: Given
Proof: We just need to check the two relations:
DEFINITION 4.2: A module
We now introduce the extension modules, using the notation
PROPOSITION 4.3: The extension modules
We now exhibit another approach by defining the formal adjoint of an operartor
DEFINITION 4.4:
from integration by part, where
LEMMA 4.5: IIf
PROPOSITION 4.6: If we have an operator
EXAMPLE 4.7: Let us revisit EM in the light of the preceding results when
Accordingly, it is not correct to say that the conformal group is the biggest group of invariance of Maxwell equations as it is only the biggest group of invariance of the Minkowski constitutive laws
in vacuum [14] . Finally, both sets of equations can be parametrized independently, the first by the potential, the second by the so- called pseudopotential (See [30] , p 492 for more details).
Now, with operational notations, let us consider the two differential sequences:
THEOREM 4.8: The modules
THEOREM 4.9: When
Proof: Let us define:
It is easy to check that
COROLLARY 4.10: if
Proof: We just need to set
THEOREM 4.11: We have the side changing procedure
Proof: According to the above Corollary, we just need to prove that
by introducing the Lie derivative of
REMARK 4.12: The above results provide a new light on duality in physics. Indeed, as the Poincaré se- quence is self-adjoint (up to sign) as a whole and the linear Spencer sequence for a Lie group of
transformations is locally isomorphic to copies of that sequence, it follows from Proposition 4.3 that
5. Conclusion
The mathematical foundations of Gauge Theory (GT) leading to Yang-Mills equations are always presented in textbooks or papers without quoting/taking into account the fact that the group theoretical
methods involved are exactly the same as the standard ones used in continuum mechanics, particularly in the analytical mechanics of rigid bodies and in hydrodynamics. Surprisingly, the lagrangians of
GT are (quadratic) functions of the curvature 2-form while the lagrangians of mechanics are (quadratic or cubic) functions of the potential 1-form. Meanwhile, the corresponding variational principle
leading to Euler-Lagrange equations is also shifted by one step in the use of the same gauge sequence. This situation is contradicting the well known field/matter couplings existing between
elasticity and electromagnetism (piezzoelectricity, photoelasticity). In this paper, we prove that the mathematical foundations of GT are not coherent with jet theory and the Spencer sequence.
Accordingly, they must be revisited within this new framework, that is when there is a Lie group of transformations consi- dered as a Lie pseudogroup, contrary to the situation existing in GT. Such a
new approach, based on new mathematical tools not known by physicists, allows unifying electromagnetism and gravitation. Finally, the striking fact that the Cosserat/Maxwell/Weyl equations can be
parametrized, contrary to Einstein equations, is shown to have quite deep roots in homological algebra through the use of extension modules and duality theory in the framework of algebraic analysis. | {"url":"https://scirp.org/journal/paperinformation?paperid=44080","timestamp":"2024-11-04T12:44:08Z","content_type":"application/xhtml+xml","content_length":"190124","record_id":"<urn:uuid:20c55ff6-8e2d-4161-b512-0ee6cc6e513c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00253.warc.gz"} |
The Stacks project
Lemma 61.28.6. Let $X$ be a scheme. Let $\Lambda $ be a ring and let $I \subset \Lambda $ be a finitely generated ideal. Let $\mathcal{F}$ be a sheaf of $\Lambda $-modules on $X_{pro\text{-}\acute{e}
tale}$. If $\mathcal{F}$ is derived complete and $\mathcal{F}/I\mathcal{F} = 0$, then $\mathcal{F} = 0$.
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Uncategorized Archives - Calculus Help
Looking to figure out how to sole the power rule for integration for your calculus homework? We have the answers at calculus-help.com.
Problem 6: Table of Derivatives
Need help figuring out the table of derivatives in your calculus homework? We have the answers and help you need at calculus-help.com.
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x-axis, according to the following acceleration equation: Assume that the origin corresponds to the fixed point, and that … Continue reading "Problem 15: A Grizzly Motion Problem"
Problem 13: Polar Derivatives
2011-2012 Find all angles on the interval at which the tangent line to the graph of the polar equation is horizontal. Solution: Express the polar equations parametrically (in terms ofx and y) and
calculate the slope of the polar equation. The tangent lines to the polar graph are horizontal when the numerator of this derivative is … Continue reading "Problem 13: Polar Derivatives"
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that you may learn from my tragedy, I will share a horrific tale from my past. Once, on a whim, … Continue reading "Problem 12: Super Related Rates"
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might have noticed that the red function has an even degree whereas the blue has an odd degree. Why? An even-degreed function’s ends will … Continue reading "Problem 9: The Graph of a Derivative"
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Problem 7: Revenge of Table Derivatives
2011-2012 Four functions are continuous and differentiable for all real numbers, and some of their values (and the values of their derivatives) are presented in the below table: If you know that h(x)
= f(x) · g(x) and j(x) = g(f(x)), fill in the correct numbers for each blank value in the table. Solution: Woo, … Continue reading "Problem 7: Revenge of Table Derivatives" | {"url":"https://calculus-help.com/category/uncategorized/","timestamp":"2024-11-04T13:20:43Z","content_type":"text/html","content_length":"51153","record_id":"<urn:uuid:9fc99b3a-b576-4ec9-abfb-146877733820>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00282.warc.gz"} |
Regular polygon area
To find the area of a regular hexagon or any regular polygon, we use a formula that says area = half the product of apotema and perimeter. As shown below, this means that we have to find the
perimeter (the distance around the entire hexagon) and the measure of the apotema using right-angled triangles and trigonometry.
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The concept of probability was formulated by two French mathematicians (Blaise Pascal and Pierre de Fermat) during the exchange of letters for a gambler's dispute in 1654! The game was about the
flipping of a coin. If it was a Head, Pascal got one point. If it was a Tail, Fermat got one point. The first one to get 10 wins would receive the pot of 100 francs. However, after 15 tosses, Pascal
received an urgent message and had to leave. At that time, Fermat got 8 points while Pascal got 7. So, how should the 100 francs be divided?
Probability is a measure of chance. It is attributed to the occurrence of an event (like getting 100 marks in a course). It is a number between 0 and 1. 1 means the event will certainly occur while 0
means the event will certainly not occur. So, the highest level uncertainty occurs when the probability is 0.5.
The probability for the occurrence of an event is often denoted by P(event). Probability plays a key role in statistics, a subject that deals with uncertainty.
Example 1
Common prevalence study of a condition may be treated as the estimation of the probability that a randomly selected individual from a population possesses the condition. For instance, Chiu (2006)
reported, in a telephone survey of 664 subjects, the lifetime prevalence of neck pain in Hong Kong was 65.4% (95% confidence interval = 49.8% to 57.4%). This means that the probability of a randomly
selected individual in Hong Kong who has neck pain is 0.654.
Example 2
Suppose a student has only 5% chance of answering a question wrong. The chance that the student will have at least one wrong answer in a test composed of 50 questions is 1 - P(all questions correctly
answered) = 1 - P(Q1 correctly answered)P(Q2 correctly answered) ... P(Q50 correctly answered) = 1 - (1-0.05)50
= 0.92!
Isn't it high?? Indeed, this demonstrates the problem of inflated chance of committing a false positive error in multiple comparisons.
Glossary Terms
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Understanding Mathematical Functions: How To Divide Functions
Mathematical functions are essential tools in solving problems and analyzing real-world scenarios. They are a set of rules that take an input and produce a unique output. Understanding how to
manipulate and divide functions is crucial for solving complex equations and modeling various phenomena. In this blog post, we will delve into the concept of dividing functions and explore its
significance in mathematical analysis.
Key Takeaways
• Mathematical functions are essential tools in solving problems and analyzing real-world scenarios.
• Understanding how to manipulate and divide functions is crucial for solving complex equations and modeling various phenomena.
• Division of functions has its own set of rules and significance in mathematical analysis.
• Common mistakes when dividing functions should be avoided with careful attention to rules and examples.
• Applications of dividing functions extend to practical scenarios in various fields, showcasing its real-world utility.
Understanding Mathematical Functions: How to divide functions
What is division of functions?
• Define division of functions: Division of functions is the process of dividing one function by another to determine the quotient function.
• Explain how division of functions is different from regular division: Unlike regular division, where a number is divided by another number, division of functions involves dividing one function by
another function. This process requires understanding the domain restrictions and simplifying the resulting quotient function.
• Provide examples of dividing functions: Examples of dividing functions include dividing a linear function by a quadratic function or dividing a trigonometric function by another trigonometric
function. Each example showcases the process of dividing functions and simplifying the resulting quotient.
Understanding Mathematical Functions: How to divide functions
When it comes to understanding mathematical functions, division can be a bit tricky. In this chapter, we will discuss the rules for dividing functions, any special cases or restrictions, and provide
examples to illustrate the rules.
Rules for dividing functions
When dividing two functions, there are certain rules that need to be followed in order to properly calculate the result.
A. Discuss the rule for dividing two functions
The rule for dividing two functions is fairly straightforward. To divide one function by another, you simply need to divide the output of the first function by the output of the second function. In
mathematical terms, if f(x) and g(x) are two functions, then the result of dividing f(x) by g(x) is written as f(x)/g(x).
B. Explain any special cases or restrictions
It's important to note that there are some special cases and restrictions when it comes to dividing functions. One such restriction is that the denominator function (g(x)) cannot be equal to zero, as
division by zero is undefined in mathematics. Additionally, some functions may have restrictions on their domain, which need to be taken into account when dividing functions.
C. Provide examples to illustrate the rules
Let's take a look at some examples to illustrate the rules for dividing functions:
• Example 1: Divide the functions f(x) = 2x and g(x) = x-1
We can calculate f(x)/g(x) as (2x)/(x-1). This is a valid division as long as x ≠ 1, due to the restriction on the domain of g(x).
• Example 2: Divide the functions h(x) = 3x^2 and k(x) = x+2
The division of h(x)/k(x) results in (3x^2)/(x+2), which is valid for all real values of x, as there are no restrictions on the domain of k(x).
Understanding the quotient of functions
A. Define the term "quotient of functions"
• Definition: The quotient of functions refers to the result of dividing one function by another. It is represented as f(x)/g(x), where f(x) and g(x) are two functions.
• Mathematical Notation: The quotient of functions can be expressed as (f/g)(x) or f(x) ÷ g(x).
B. Discuss the significance of finding the quotient of functions
• Understanding Function Relationships: Finding the quotient of functions helps in understanding the relationship between two functions and how they interact with each other.
• Identifying Limitations: It helps in identifying the limitations or restrictions on the domain of the functions.
• Problem Solving: The quotient of functions is essential in solving various mathematical problems, especially in calculus and algebra.
C. Provide examples of finding the quotient of functions
• Example 1: Find the quotient of the functions f(x) = 2x + 4 and g(x) = x - 1.
• Example 2: Determine the quotient of the functions h(x) = x^2 - 3x and k(x) = 2x + 1.
Common Mistakes to Avoid
When dividing functions, there are several common errors that people often make. These mistakes can lead to incorrect results and a misunderstanding of mathematical functions. It's important to be
aware of these common pitfalls and to take steps to avoid making them.
Discuss common errors when dividing functions
• Forgetting to consider the domain of the functions: One common mistake when dividing functions is forgetting to consider the domain of the functions involved. It's essential to ensure that the
domain of the divisor function does not include any points where the dividend function is zero, as this would result in division by zero.
• Not simplifying the result: Another common error is not simplifying the result of the function division. It's important to simplify the resulting function to its simplest form to ensure accuracy
and clarity.
• Missing parentheses: When dividing functions, it's crucial to use parentheses to indicate the correct order of operations. Forgetting to use parentheses can lead to confusion and errors in the
Provide tips for avoiding these mistakes
• Always consider the domain: Before dividing functions, always check the domain of both the divisor and the dividend functions. Ensure that the domain of the divisor function does not include any
points where the dividend function is zero.
• Simplify the result: After dividing functions, take the time to simplify the resulting function to its simplest form. This will help to avoid confusion and errors in the final answer.
• Use parentheses: When dividing functions, be sure to use parentheses to indicate the correct order of operations. This will help to avoid errors and ensure that the function is divided correctly.
Offer examples of common mistakes and how to correct them
Let's take a look at some examples of common mistakes when dividing functions and how to correct them:
• Example 1: Dividing the functions f(x) = x + 1 and g(x) = x - 1 without considering the domain. The correct approach would be to first check the domain of g(x) and ensure it does not include x =
1, as this would result in division by zero. Then proceed with the division after confirming the domain compatibility.
• Example 2: Failing to simplify the resulting function after division. After dividing functions, always simplify the resulting function to its simplest form to avoid confusion and errors.
• Example 3: Forgetting to use parentheses when dividing functions. Always use parentheses to indicate the correct order of operations when dividing functions to avoid errors in the result.
Applications of Dividing Functions
Understanding mathematical functions and how to divide them is not just an abstract concept—it has real-world applications that can be found in various fields. In this chapter, we will explore the
practical uses of dividing functions and discuss specific examples of how it is employed in everyday scenarios.
Explore real-world applications of dividing functions
Dividing functions have numerous applications in fields such as engineering, physics, finance, and computer science. One common example is in the field of physics, where dividing functions are used
to calculate rates of change, such as velocity, acceleration, and other kinematic quantities.
Discuss how understanding division of functions can be useful in various fields
Understanding how to divide functions is crucial in various fields, as it allows for accurate modeling and analysis. In engineering, for instance, dividing functions are used to determine the
relationship between different variables, which is essential for designing and optimizing systems and processes.
Provide specific examples of how dividing functions is used in practical scenarios
One specific example of how dividing functions is used in practical scenarios is in finance, where it is employed to calculate compound interest and investment returns. By understanding how to divide
functions, financial analysts can make informed decisions and projections based on these calculations.
In conclusion, we have discussed the importance of understanding how to divide functions in mathematics. We have explored the key points such as the process of dividing functions, the use of the
quotient rule, and the significance of identifying restrictions. It is crucial to understand this concept as it forms the basis for solving complex mathematical problems and real-world applications.
Mastering the division of functions allows for a deeper understanding of mathematical relationships and opens up new possibilities for problem-solving. I encourage you to continue exploring this
topic through practice problems, further study, and application in various mathematical contexts.
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Potential Energy and Velocity of a Spaceship
• Thread starter Drakkith
• Start date
In summary, Roger planned to leave the Earth with enough speed to make it to the moon, but needed help from a friend to calculate the minimum speed. He attempted to solve Q1 by adding the potential
energy of the spacecraft for both the Earth and the Moon together, but appears to have gotten incorrect results. He is working on a solution to Q2.
Homework Statement
You plan to take a trip to the moon. Since you do not have a traditional spaceship with rockets, you will need to leave the Earth with enough speed to make it to the moon. Some information that will
help during this problem:mearth = 5.9742 x 1024 kg
rearth = 6.3781 x 106 m
mmoon = 7.36 x 1022 kg
rmoon = 1.7374 x 106 m
dearth to moon = 3.844 x 108 m (center to center)
G = 6.67428 x 10-11 N-m2/kg21)On your first attempt you leave the surface of the Earth at v = 5534 m/s. How far from the center of the Earth will you get?
2) Since that is not far enough, you consult a friend who calculates (correctly) the minimum speed needed as vmin = 11068 m/s. If you leave the surface of the Earth at this speed, how fast will you
be moving at the surface of the moon? Hint carefully write out an expression for the potential and kinetic energy of the ship on the surface of earth, and on the surface of moon. Be sure to include
the gravitational potential energy of the Earth even when the ship is at the surface of the moon!
Homework Equations
ΔE = 0
U[1]+K[1] = U[2] + K[2]
The Attempt at a Solution
Found the answer to Q1. It's 8.44x10^6 meters.
Q2 is where I'm having trouble. I don't know how to set it up. For Q1, ΔE=0 so ΔU + ΔK = 0 as well. And ΔU = U[2]-U[1].
For Q2, I was thinking you would add the potential energy of the spacecraft for both the Earth and the Moon together, like: U[1] = U[1E] + U[1M]. That would make ΔU = (U[2E] + U[2M]) - (U[1E] + U
Unfortunately my answer using this method appears to be incorrect. Not sure what to do. I made sure to account for the differing distance between the centers of the Earth and Moon, and the distance
between the surfaces of each and the center of the other.
Science Advisor
Homework Helper
Gold Member
In principle, you need to consider the PE wrt the moon in Q1 too, but that shot falls too far short for it to matter.
Please post the details of your attempt on Q2. Can't tell where or if you are going wrong otherwise.
haruspex said:
Please post the details of your attempt on Q2. Can't tell where or if you are going wrong otherwise.
Roger. I'll get on that as soon as I can.
FAQ: Potential Energy and Velocity of a Spaceship
What is potential energy and how does it relate to a spaceship?
Potential energy is the energy that an object possesses due to its position or configuration in a force field. In the context of a spaceship, it refers to the energy that the spaceship has based on
its position in space and the gravitational force acting upon it.
How is potential energy calculated for a spaceship?
The formula for potential energy is PE = mgh, where m is the mass of the spaceship, g is the acceleration due to gravity, and h is the height of the spaceship from a chosen reference point. This
formula can be used to calculate the potential energy of a spaceship at any given point in space.
How does velocity affect potential energy of a spaceship?
Velocity does not directly affect potential energy, but it can affect the kinetic energy of the spaceship. As the spaceship gains velocity, its kinetic energy increases, but its potential energy
remains the same unless its position in space changes.
What is the relationship between potential energy and velocity for a spaceship?
The relationship between potential energy and velocity is inverse. As the spaceship gains velocity, its potential energy decreases. This is because the spaceship is moving closer to the reference
point, thus decreasing its height and potential energy.
How does a spaceship's potential energy and velocity impact its trajectory and overall motion?
A spaceship's potential energy and velocity play a crucial role in determining its trajectory and overall motion. As the spaceship moves through a gravitational force field, its potential energy is
converted into kinetic energy, causing it to accelerate and follow a specific path determined by its initial velocity and the force of gravity. | {"url":"https://www.physicsforums.com/threads/potential-energy-and-velocity-of-a-spaceship.820306/","timestamp":"2024-11-10T12:28:59Z","content_type":"text/html","content_length":"89204","record_id":"<urn:uuid:cfa0cd66-9088-4941-b473-73d94cd5487e>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00894.warc.gz"} |