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Diffusion probabilistic models (DPMs) have recently emerged as the state of the art for generative modelling. At their core, they rely on a very simple idea: given a training sample (e.g. an image),
one first iteratively and gradually adds gaussian noise (on each pixel) over a series of time steps, and secondly trains a neural network to reverse this process. Removing the noise from the
“corrupted” samples, the NN learns to recover the initial image, thus becoming a high quality generative model. So, given a completely noisy image, it can recursively de-noise it towards a brand-new
sample, not present in the training set.
Despite their success in sample quality and stability compared to VAEs and GANs, DPMs suffer from very slow inference due to the need to iterate over thousands of samples. One of the key reasons why
so many de-noising steps are necessary is related to the estimation of the pixel variance in the reverse (de-noising) process. Previous works have used handcrafted values or trained a separate
estimator, e.g. a neural network, thus slowing down training.
[Bao22A] presents the surprising result that the optimal reverse variance of a DPM has an analytic form w.r.t. their score function. This means that, with a fast Monte Carlo based approach, one can
increase the efficiency of the backward de-noising process considerably while keeping comparable or even superior performance.
The table above shows the comparison in sample quality using the Frechet Inception Distance (FID) score. Introduced in [Heu18G], the FID score calculates the Wasserstein distance between the
distribution of activations for generated and training images in one of the deeper layers of the Inception v3 network. Lower scores (lower distances) have been shown to correlate well with higher
quality of generated images.
The newly proposed method, based on optimal-variance calculation, is used in the models Analytic-DDPM and Analytic DDIM. Columns are the number of steps used in inference, and the four sections
represent different datasets and/or different noise schedules (how much noise to input at each training time step). The new model consistently improves sample quality when using the same number of
inference steps, and, considering also the log-likelihood of the reverse process, it is comparable to the scores of samples created with 20 to 80 times more steps. This result represents a major
advancement in sample-efficient image generation using diffusion probabilistic models, which projects them even more as the go-to approach for generative modelling. Code is available at github. | {"url":"https://transferlab.ai/pills/2022/analytic-dpm/","timestamp":"2024-11-13T16:10:08Z","content_type":"text/html","content_length":"24131","record_id":"<urn:uuid:c43a70ee-4672-4cbf-9246-73292e706768>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00733.warc.gz"} |
civil-engineering-surveying Related Question Answers
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Faith Club
• 2005.06.20, 13:27
• Ir viens teksts, kas man lika mazliet tuvāk apjēgt kāpēc lietas, kas liekas pavisam vienkāršas, īstenībā ir ļoti sarežģītas. (Precīzāk, kāpēc reālās pasaules problēmas ir tik grūti risināmas.
Nevar teikt ka es to tekstu izprastu, bet... ;)
A friend of mine over in Microsoft Research pointed out to me the other day that high-dimensional spaces are really counterintuitive. He'd just attended a lecture by the research guys who wrote
this excellent paper and we were geeking out at a party about it. I found this paper quite eye-opening and I thought I might talk a bit about some of the stuff that's in here at a slightly less
highfalutin level -- the paper assumes a pretty deep understanding of high-dimensional spaces from the get-go. **************************
It's hard to have a geometrical picture in your head of a more than three-dimensional space. I usually have to use one of two analogies. The first analogy I like goes like this: think of a line
-- one dimensional. Imagine that you have a slider control that determines your position on that line, from, say, -1 to 1, left-to-right. That's pretty visually clear. Add another slider that
determines your up-and-down position, and you've got a square. Each point on the square has a unique set of slider positions. Add another slider that determines your out-and-in position, and
you've got a cube. Again, these are easy to picture. Every point in the cube has a unique combination of slider positions that gets you to that point. Now think of a cube with a slider control
below it that lets you slide from intense red on one end through dark red and to black on the other. Now you've got four axes you can move around -- height, width, depth and redness. The top
right front corner of the bright red cube is a certain "colour distance" from the corner of the top right front black cube. That this is not a spatial dimension isn't particularly relevant; we're
just picturing a dimension as redness for convenience. Every time you want to add another dimension, add another slider -- just make sure that whatever is sliding is completely independent of
every other dimension. Once you've added green and blue sliders then you've got a six-dimensional hypercube. The "distance" between any two 6-d points is a function of how much you have to move
how many sliders to get from one to the other.
That analogy gets across one of the key ideas of multi-dimensional spaces: that each dimension is simply another independent degree of freedom through which you can move. But this is a quite
mathematical and not very geometric way of thinking about dimensionality, and I want to think about the geometry of these objects. Let's abandon this analogy. The second analogy is a little bit
more geometrical. Think of a line, say two units long. Now associate every point on that line with another line, also two units long "crossing" it at the new line's center. Clearly that's a
filled-in square -- after all, every point along one side of a square has a straight line coming out from it perpendicularly. In our slider analogy, one slider determines the point along the
"main line", and the second determines how far to go along its associated line.
Think of another line, but this time, associate every point on it with a square. That's a solid cube.
Now think of yet another line. Associate every point on it with a cube, and you've got a 4-cube. At this point it gets hard to visualize, but just as a cube is an infinite number of equally-sized
squares stuck together along a line, so is a 4-cube an infinite number of 3-cubes stuck together along a line. Similarly, a 5-cube is a line of 4-cubes, and so on. Where things get weird is when
you start to think about hyperspheres instead of hypercubes. Hyperspheres have some surprising properties that do not match our intuition, given that we only have experience with two and three
dimensional spheres. (2-spheres are of course normally called "circles".) The definition of a hypersphere is pretty simple -- like a 2-sphere or 3-sphere, a hypersphere is the collection of
points that are all the same distance from a given center point. (But distance works strangely in higher dimensions, as we'll see in future episodes!) Its hard to picture a hypercube; it's even
hard to picture a hypersphere. The equivalent analogy for n-spheres requires us to think about size. Again, imagine a line two units long. Associate with each point on the line another line
crossing at the middle. But this time, the associated lines are of different lengths. The lines associated with the end points are tiny, and the lines associated with the middle are longer. This
describes a circular disk -- for each point along the diameter of a circle you can draw a perpendicular line through the point extending to the boundaries of the disk on each side.
Now do the same thing again. Take a line, and associate each point on the line with a circle. If the circles are all the same size, you have a cylinder. But if they vary from small at the ends to
big in the middle, you've got a sphere. Successive cross-sections of a sphere are all circles, but they start small and get big and then get small again. Now do the same thing again. Take a line
and associate each point on the line with a sphere, small at the ends and big in the middle, and you've got a 4-sphere. Successive "cross sections" of a 4-sphere are 3-spheres of varying size.
Keep going to 5-, 6-, etc, spheres. A circle of diameter 2 fits into a square of edge length 2, and a sphere of diameter 2 fits into a cube of edge length 2. Clearly an n-sphere of diameter two
fits exactly into an n-cube of edge length two -- the n-sphere "kisses" the center of each face of the n-cube. You can't make the n-cube smaller without the n-sphere poking out of it somewhere.
But things start getting weird when you consider the volume of an n-sphere. Tomorrow we'll compare the volume of an n-sphere to the volume of an n-cube, and discover some surprising and
counterintuitive things about where that volume is.
The volume of an n-cube of edge length s is easy to work out. A 2-cube has s^2 units of area. A 3-cube has s^3 units of volume. A 4-cube has s^4 units of 4-volume, and so on -- an n-cube has s^n
units of n-volume. If the n-cube has edge of s>1, say s=2, then clearly the n-volume dramatically increases as the dimensionality increases -- each dimension adds a lot more "room" to the n-cube.
A 2-sphere (ie, circle) is pretty close in area to the smallest 2-cube (ie, square) that encloses it -- sure, you lose some area at the four corners, but not a whole lot. Though the circle is far
from the square at the four corner, it is very close to the square at the four sides. A circle has about 75% the area of its enclosing square. But a 3-sphere inside a the smallest 3-cube that
encloses it is far from eight corners and close to only six sides. A 3-sphere is about half the volume of the 3-cube. As you go up in dimensions, you get more and more corners that are far from
the n-sphere -- there are 2^n corners and only 2n sides so the comparative volume of the sphere goes down.
In fact, you don't even need to compare n-spheres to n-cubes -- after you reach 5 dimensions, the n-volume of an n-sphere starts going down, not up, as dimensionality increases. With some pretty
easy calculus you can show that the n-volume of an n-sphere of radius r is:
V[1] = 2 r
V[2] = π r^2
V[n] = V[n-2] 2 π r^2 / n
For any fixed radius this rapidly approaches zero as n gets big. Pick a big n, say 100. The volume of a 100-sphere is going to be (π r^2 /50) x (π r^2 /49) x (π r^2/48) ... (π r^2/1). Suppose r
is 1 -- then all of those terms except for the last few are going to be quite a bit smaller than 1. Every time you add more dimensions, the n-volume of the unit n-sphere gets smaller and smaller
even as the n-volume of the smallest n-cube that encloses the unit n-sphere gets exponentially larger and larger!
Here's another weird fact about the volume of hypersolids. Consider two squares, one inside the other. How big does the small square have to be in order to have, say, 1% the area of the larger
square? That's pretty easy. If the inner square has 10% the edge length of the outer square, then it has 1% of the area of the outer square.
What about nested 3-cubes? An inner 3-cube with edges 10% the length of the edge of the outer 3-cube would have 0.1% the volume, too small. Rather, it needs to havean edge about 21% of the edge
of the outer 3-cube, because .21 x .21 x .21 = about 0.01.
What about a nested n-cube? In order to have 1% the n-volume of the outer n-cube, the inner n-cube needs to have an edge of (0.01)^1/n of the outer n-cube side. For n=100, that's 0.955. Think
about that for a moment. You've got two 100-cubes, one has edges 2 units long, the other has edges 1.91 units long. The larger n-cube contains ONE HUNDRED TIMES more volume.
Try to visualize the smaller n-cube being entirely inside the larger n-cube, the two n-cubes having the same central point. Now wrap your mind around the fact that the smaller n-cube is 1% the
volume of the larger. The conclusion is unavoidable: in high dimensions the vast majority of the volume of a solid is concentrated in a thin shell near its surface! Remember, there's 2^100
corners in a 100-cube, and that makes for a lot of space to put stuff.
It's counterintuitive because the very idea of "near" is counterintuitive in higher dimensions. Every time you add another dimension, there's more room for points to be farther apart.
The distance between opposite corners of a square of edge 2 is 2√2. The distance between opposite corners of a 3-cube is 2√3, quite a bit bigger. The distance between opposite corners of a
100-cube is 2√100 = 20 units! There are a whole lot of dimensions to move through, and that adds distance.
We could make the same argument for an n-sphere and show that the vast majority of its (comparatively tiny) volume is also in a thin shell near the surface; I'm sure you can see how the argument
would go, so I won't bother repeating myself.
Because distance is so much more "expensive" in higher dimensions, this helps explain why n-spheres have so much less volume than n-cubes. Consider a 100-cube of edge 2 centered on the origin
enclosing a 100-sphere of diameter 2, also centered on the origin. The point (1,0,0,0,0,0...,0) is on both the 100-cube and the sphere, and is 1 unit from the origin. The point (1,1,1,...,1) is
on the 100-cube and is ten units away from the origin. But a 100-sphere by definition is the set of points equidistance from the origin, and distance is expensive in high dimensions. The nearest
point on the 100-sphere to that corner is (0.1, 0.1, 0.1, ..., 0.1), 9 units away from the corner of the 100-cube. Now its clear just how tiny the 100-sphere is compared to the 100-cube.
OK, so far we've been considering n-cubes that entirely enclose n-spheres, ie, an n-cube of edge length 2 that encloses a unit n-sphere, kissing the sphere at 2n points. But we know that this
n-cube has ginormously more volume than the n-sphere it encloses and that most of that volume is near the edges and corners. What if we abandon the constraint that the n-cube contains 100% of the
n-sphere's volume. After all, there are only 200 points where the 100-sphere kisses the 100-cube, and that's not very many at all.
Suppose we want an 100-cube that contains 99.9% of the volume of the unit 100-sphere. We can cover virtually all of the volume of the 100-sphere with an 100-cube of edge 0.7 instead of 2. Sure,
we're missing (1,0,0,0,...,0), but we're still hitting (0.1,0.1,0.1,...) with huge amounts of room to spare. Most of the volume inside the 100-sphere isn't near the 200 points with coordinates
near the axes.
How much do we reduce the volume of the 100-cube by shrinking it from 2 on the edge to 0.7? We go from 2^100 n-units of volume to 0.7^100, a factor of around 4x10^45 times smaller volume! And yet
we still enclose virtually all the volume of the 100-sphere. The corner of the smaller 100-cube at (0.35, 0.35, 0.35, ...) is now only 2.5 units away from (0.1, 0.1, ...) instead of 9 units away.
This is a much better approximation of the unit 100-sphere. It's still hugely enormous compared to the unit 100-sphere in terms of sheer volume, but look at how much volume we save by
approximating the 100-sphere as a small 100-cube!
Feeling dizzy yet? Next time we'll see that these facts
□ n-spheres are tiny compared to n-cubes
□ hypersolids have most of their volume close to their surfaces
□ you can enclose almost all the volume of an n-sphere with a small n-cube
Last time we noticed a few interesting facts:
□ most of the volume of a high-dimensional object is very near its surface.
□ small movements away from a point in many dimensions add up to a long total distance.
Suppose you have a bunch of points in space scattered randomly around some average. You would expect that if the points are more or less evenly distributed then the number of points in a given
volume of the space would be proportional to the volume. (In fact, that’s probably how we’d define "more or less evenly distributed.")
Let’s think of a low-dimensional example – points on a line. For example, suppose you've got a thousand 30-year-old women and you measure their heights. If the average is 150 cm, then you'd
expect that the number of women that fall into the 2-cm range from 145-147 is roughly the same as from 150-152. There will probably be more in the range closer to the mean, but you'd expect that
the order of magnitude would be roughly the same. It's not like there's going to be 100 in one range and 10 in the other. I'm assuming here that we're considering regions close to the mean so
that the distribution is pretty even throughout.
Now suppose you take a 2-cm range and a 1-cm range, both close to the mean. You would similarly expect that the 1-cm range would have roughly half as many points in it as the 2-cm range, right?
The number of points in a region is roughly proportional to the size of the region.
What if the distribution is 2-dimensional? Suppose now you plot both the heights and hair lengths of these 1000 women. Again, there will be some average around which the points cluster, say (150
cm, 30 cm). Now imagine that you draw two circles of the same area close to the mean. Again, you'd expect that two circles of the same area would have roughly the same number of points inside
them. The circle with its center closer to the mean probably has a few more, but you'd expect roughly the same number of points in each circular region.
Now what if you make one of those circles half the radius of the other? Then it will have only one quarter of the area, and therefore only on average one quarter as many points.
Now suppose that you have a bunch of points scattered around a 100-dimensional space, all clustered around a particular average point. Pick a hyperspherical region centered on the average, with
radius large enough to contain most of the points.
We expect to find more points in regions of the space with more volume. We know that 99% of the volume of this n-sphere is concentrated in a thin shell near its surface. The conclusion is
inescapable: points clustered around a mean in high dimensions are far more likely to be found in the high-volume narrow shell far from the mean, not in the low-volume region near the mean
This again seems really counterintuitive at first but its not if you think about it for a while. In our example above, most of the women are of near average height. And most of the women are of
near average hair length. But clearly the number of women that are near-average height AND near-average hair length is considerably smaller than either taken separately.
As we add more and more dimensions to the mix, the number of points where all measurements are near the mean point gets very small indeed. Of the six billion people in the world, how many of them
are within 1% of average height and average hair length and average age and average income and average cholesterol level and average IQ? Probably a very, very small number. Most everyone is
slightly abnormal somehow! And in a high-dimensional space, many small deviations from the mean adds up to a large distance from the mean. If you plotted 100 characteristics of 6 billion people
in a 100-dimensional space, the number of people very near the middle of every axis is quite small; there's hardly any volume there.
What the heck does this have to do with computer science? Suppose you have all that personal data in a database, you have a dead guy in the morgue with no ID, and you want to find a match in your
database, if there is one? What if you have 100 variables about a song and you want to identify whether a given scrap of music is in your database? There are lots of real-world search problems
that have sparse, large data sets spread out over a high-dimensional space.
It's reasonably common to have a database containing a few thousand or million points in some high-dimensional space. If you have some "query point" in that space you might like to know whether
there is a match in your database. If you're looking for an exact match then the problem is pretty easy -- you just come up with a suitable hash algorithm that hashes points in your space and
build a big old hash table. That's extremely fast lookup. But what if you're not looking for an exact match, you're looking for a close match? Perhaps there is some error, either in the
measurements that went into the database or the measurement of the query point, that needs to be accounted for. Now the problem is "for a given query point, what is the closest point in the
database to it?" That's not a problem that's amenable to hashing. (Well, you could use a Locality Sensitive Hash algorithm, but we'll rule that out later.) What if the closest point to the query
point is so far away from the query point that it's more likely that the query point simply has no match at all in the database? In that case we don't really want the closest point, because
that's not really relevant. Essentially you can think of every point in the database as being surrounded by a "probability sphere". As you move farther away from a given point in the database,
the probability that you have a match to that point gets smaller and smaller. Eventually its small enough that the probability that the query point is "junk" -- not a match to any point in the
database at all -- gets larger than the probability that the closest point is a match. To sum up the story so far: we've got a query point, which may or may not correspond to a point in our
database of points. We need a way to say "is this point junk? If not, what are some reasonably close points in the database that might be matches?" Here's an idea for an algorithm: compute the
distance between the query point and every point in the database. Discard all points where the distance is outside of a certain tolerance. Geometrically, we're constructing a sphere of a certain
radius around each point in the database. We check to see whether the query point is inside each sphere.
We know that the volume of an n-sphere is tiny. That's good. If we do get a match then we know that the match is probably in a very small volume and therefore likely to be correct. However, we
also know that such a system is not tolerant of small errors in many dimensions -- because distances grow so fast, a small deviation in many dimensions leads to a big distance. That means that we
probably will have to construct a sphere with a fairly large radius in order to allow for measurement error in many dimensions.
But that's not really so bad. What's bad about this scheme is that if there are a million points in the database then we have to calculate one million Cartesian distances for every query. In a
100-dimensional space that means computing 100 differences, 100 multiplications, 100 additions and one comparison a million times just to do one query.
We could build some optimizations in -- we could check at each addition whether the total radius computed so far exceeds the tolerance and automatically discard the point. Maybe we could cut it
down to on average 50 differences, multiplications and additions per point -- but then we've just added in 50 comparison operations. No matter how you slice it, we're doing a lot of math.
A hash table works by automatically discarding all but a tiny number of points, so that you just have to check a small number rather than the whole database. A Locality Sensitive Hash algorithm
(which I might write about in another series) would work well here except that LSH algorithms have lousy performance if a large percentage of the queries are junk. Let's assume that there are
going to be a lot of junk queries. Is there some way that we can more rapidly find valid points?
We learned earlier that 99.9% of the volume of an n-sphere is contained by an n-cube with the same center as the sphere and an edge length fairly close to that of the radius.
Determining if a point is inside a cube is a lot less math than determining if a point is inside a sphere. With a cube you don't need to compute any distance. You just compare the upper and lower
boundaries of each dimension to the position of the point in that dimension and see if they overlap. For a 100-dimensional space, that's 100 to 200 comparisons per point, and as soon as even one
of the dimensions is bad, you can skip this cube and move on.
Maybe we can get it down to around a few dozen comparisons per point for a straightforward linear search. That's pretty good compared to the distance metric.
We know that a cube of a given side is much, much more voluminous than a sphere of similar radius. We don't really care about the 0.1% false negative rate caused by clipping off the bits of the
sphere that are close to the axes. But what about the false positive rate of the huge volume of points that are inside the cube but not inside the sphere? These are easily dealt with: once we
have winnowed the likely matches down to a small subset through the cheap cube method, we can do our more expensive spherical check on the small number of remaining candidates to eliminate false
By this point your bogosity sense should be tingling.
This armchair performance analysis is completely bogus. Yes, ~70 comparisons is cheaper than ~100 subtractions, multiplications and additions. Who cares? That's not the most expensive thing.
Performance analysis always has to be about what the most expensive thing is.
Think about this database for a moment -- a million records, each record is a 100-dimensional point in some space. Let's suppose for the sake of argument that it's a space of 8-byte
double-precision floats. That's 800 megabytes of memory. Now, there are certainly database servers out there that can keep an 800 meg file in physical memory, but we're clearly pushing an
envelope here. What if the database is large enough that there isn't enough physical memory to keep the whole thing in all at once? At that point, the cost of iterating over all one million
records becomes the cost of swapping big chunks of the database to disk and back. Now consider what happens if there are multiple threads doing searches at the same time. Unless you can
synchronize them somehow so that they all use the same bits of the database at the same time, you're just going to exacerbate the swapping problem as different threads access different parts of
the record set. (And if you can synchronize them like that, why even have multiple threads in the first place?) The fundamental problem with both the sphere and the cube methods isn't that the
math per record is expensive, it's that they must consider every record in the database every time you do a query. Getting those records into memory is the expensive part, not the math. What we
really need is some index that is small enough to be kept in memory that quickly eliminates lots of the records from consideration in the first place. Once we're down to just a few candidates,
they can be pulled into main memory and checked using as computationally-intensive algorithm as we like.
There's no obvious way to build an index of hyperspheres, but there might be things we can do with hypercubes. Stay tuned.
• 20.6.05 16:25 #
• nu, īstenībā, visa pasaules "vienkāršība" ir TIKAI tīra vienkāršošana līdz cilvēciskās sapratnes līmenim.
ja kaut kas (jebkas) šķiet vienkāršs = tas nozīmē, ka mēs esam, to apskatot, atmetuši lielu kaudzi ar īpašībām, kuras it kā neietekmē mūsu konkrēto mērķi.
• 20.6.05 19:23 #
• ja ieskatās "part 3", tur ir atbilde, kāpēc ir spēkā 10/90 % likums (tipa 90-99% of everything is shit)
• 20.6.05 22:04 #
• Tu mani piespiedīsi lasīt. Starpcitu vakar naktī jūs kopā ar Raiti K. mani piespiedāt klausīties Gridlock. Paši to neapzinoties. | {"url":"http://klab.lv/users/watt/273537.html?view=1271681","timestamp":"2024-11-11T00:51:13Z","content_type":"application/xhtml+xml","content_length":"34899","record_id":"<urn:uuid:2aeca207-4c8e-4de6-b402-401a8ffe69e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00009.warc.gz"} |
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• Stampa questa pagina, prendi una matita e prova a risolvere i seguenti rompicapo...
1. Mondrianize-it! This line of puzzles, inspired by the artworks of the Dutch artist Piet Mondrian, consists in dividing the large square board with straight lines along the small chequered
squares in order to partition the board into squares and rectangles so that each square holds exactly one red star and each rectangle, one red dot (see example below).
2. Reclone-it!
Using the grid dots you have to repartition the large gray shape within the board opposite into 2 smaller congruent shapes (that is shapes that coincide exactly when superimposed). Two congruent
shapes can be identical or mirrored, as illustrated in the examples below.
3. Pathfinder
Draw pathway lines on the pink board to join similar symbols together (triangle to triangle, pentagon to pentagon, etc.)
Rules of the game: 1. No line may cross another line. 2. No line can cross the black bands (see description at the bottom of the puzzle). 3. Only ONE line can cross the band with stripes. 4. The
wavy dotted band MUST be crossed by at least THREE different connecting pathway lines. 5. Respecting the previous points, every pathway line should be the SHORTEST possible.
4. Pythagoras
This game, involving square patterns, is a tribute to the Greek mathematician Pythagoras and was designed to stimulate your visuospatial skills! Blacken a white dot within the board so that,
starting from this dot, the largest number of perfect squares can be drawn by joining together their vertices. You can obviously join together only black dots as shown in the example below.
5. Calderize-it
"Calderize-it" is a balance puzzle line inspired by the kinetic artworks of the American artist Alexander Calder who was well known for his stunning "mobile sculptures".
torque of each mass is its weight times its distance from the fulcrum (= pivot point). The example opposite balances since:
2 x 7 = 1 x 2 + 3 x 4.
Note also that the total weight of any balanced rod equals the sum of all the hanging masses distributed underneath it, in the example: 7 + 2 + 4 = 13.
┌──┬──┬──┬──┬──┬┬──┬──┬──┬──┬──┐ 6. Mag-line
│ │24│25│18│16││16│26│18│19│ │ Mag-line is a neat Sudoku variant invented by the French puzzle designer Didier Faradji. Each column and each row of the puzzle must contain the digits from 1 to 9
├──┼──┼──┼──┼──┤├──┼──┼──┼──┼──┤ (except one) once. Moreover, each number in the green box is the sum of the first 4 digits of the respective row or column. (You can fill the empty boxes online)
│15│ │2 │ │ ││ │ │1 │ │21│
│22│ │ │ │ ││ │ │ │ │22│
│23│9 │ │ │1 ││2 │ │ │7 │19│
│23│2 │6 │ │ ││ │ │3 │4 │17│
│12│5 │4 │ │ ││ │ │9 │8 │27│
│25│8 │ │ │5 ││6 │ │ │1 │13│
│14│ │ │ │ ││ │ │ │ │23│
│27│ │7 │ │ ││ │ │5 │ │16│
│ │20│15│21│22││26│11│25│17│ │
7. Strimko
Strimko is a clever logic puzzle involving numbers invented by the Grabarchuk Family.
Fully fill in the given grid with missing numbers (1 through 6) observing three simple rules:
1. Each row must contain different numbers.
2. Each column should contain different numbers.
3. Each stream must contain different numbers.
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of your publishers and syndication agencies with puzzle contents integrating science and visual perception for recreational, Multicultural interactive puzzle activities & workshops for museums
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Puzzle of the Month.
Abracadabrain puzzles.
A range of
puzzles for
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Re^2: Algorithm: point with N distance of a line between two other points
> Like Limbic~Region already said if the distance is small, an approximation is good enough, i.e. use a straight line instead of an arc.
IMHO one can choose a spherical projection around the point in question which keeps distances¹ fix, no matter how big the region is.
Shouldn't be too difficult to find corresponding techniques in spherical geometry.
1) maybe its even better to choose gnomonic projection, where "great circles are mapped to straight lines" as long as bigger distances have bigger projections.
UPDATE: ah indeed "Thus the shortest route between two locations in reality corresponds to that on the map.". Of course the real distance on the sphere still has to be calculated after back
projecting the "nearest" point.
• Comment on Re^2: Algorithm: point with N distance of a line between two other points | {"url":"https://www.perlmonks.org/?node_id=869464","timestamp":"2024-11-04T08:20:04Z","content_type":"text/html","content_length":"20426","record_id":"<urn:uuid:a30b6748-fa76-4e55-afc6-c083779c63ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00068.warc.gz"} |
Definition of Rational and Irrational Real Numbers for Mathematics
Rational and Irrational Numbers Defined and Explained
Rational Numbers Definition
A real number that can be written or expressed as a ratio of two integers is rational.
All integers, positive or negative are rational numbers because they divide by 1.
Zero is a rational number that can be written as a ratio of two integer numbers, as 0/1.
Rational numbers can be written or expressed as either an ordinary (terminating) decimal or as an infinite repeating decimal.
Rational Numbers Examples:
• 5 can be written as a ratio of two numbers, 5/1.
It can be written as 5.0, an ordinary decimal.
• 1/11 is a ratio of two integers. Its decimal equivalent is
.090909… The decimal portion 09 repeats.
The decimal value never terminates.
• 2/3 is a ratio of two integer numbers. Its decimal
equivalent is .6666666… The decimal is infinite
repeating, but never terminates. It is a rational number.
• 0.5 is the decimal equivalent of two integer numbers, 1/2.
It is a decimal that terminates, but does not repeat.
It is a rational decimal.
• √4 can be written as a ratio of two numbers. The √4
is 2. 2 can be written as 2/1. There is no decimal place
value for 2 or 2/1. It is a rational square root number.
Some other rational square root numbers are:
√9, √16, √36, √49, √64.
• log10 x is 2 when x = 100. 2 is an integer, it is rational.
Irrational Numbers Definition
Real numbers that cannot be written as a ratio of two integers are irrational. They are decimal numbers that both do not terminate and do not repeat. Irrational numbers are infinite non-repeating
decimal numbers.
Irrational Numbers Examples:
• √2 is not a ratio of two integers. Its decimal equivalent
is 1.4142135… The decimal never repeats and never terminates. It cannot be written as a ratio of two integers.
It is an irrational square root number.
√3, √5, √7, √10, 2 + √2, 4 − √7.
• π is a math symbol with decimal value 3.1415926…
It cannot be written as a ratio of two integers. It is an
infinite non-repeating decimal number.
• e is a math symbol with decimal value 2.71828…
It cannot be written as a ratio of two integers. It is an
infinite non-repeating decimal number.
• log10 x when x = 2 is 0.30102999566… Is not a ratio
of two integers. It is irrational.
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The three internal bisectors AI, BI, CI of ABC are rotated about each corresponding vertex of ABC of a same angle t, all outwardly or all inwardly. The six rotated bisectors define a triangle NaNbNc
which is perspective to ABC at P. See figure below.
When t varies, the locus of P is the nodal cubic cK(#X1, X101) = K588.
K588 is a nodal isogonal non-pivotal cubic with node I = X(1). See Table 69. | {"url":"http://bernard-gibert.fr/Exemples/k588.html","timestamp":"2024-11-02T17:04:39Z","content_type":"text/html","content_length":"7503","record_id":"<urn:uuid:be76ebd9-ce27-46e8-8db1-21251ed3e020>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00447.warc.gz"} |
Integrable deformations of the O(3) sigma model. The sausage model
We consider two one-parameter families of two-dimensional relativistic factorized scattering theories, which are deformations of the ones characteristic for the two-dimensional O(3) sigma models with
θ = 0 and θ = π. The Bethe ansatz technique is applied to these two families to justify their interpretation as the factorized scattering theories of certain O(2)-symmetric deformations of the O(3)
sigma model (the sausage model). The result suggests that the sausage model is integrable at the topological angle values θ = 0 or θ = π.
Nuclear Physics B
Pub Date:
October 1993 | {"url":"https://ui.adsabs.harvard.edu/abs/1993NuPhB.406..521F","timestamp":"2024-11-01T19:52:59Z","content_type":"text/html","content_length":"34803","record_id":"<urn:uuid:3dd74e80-e5f8-4232-806c-373d4f0b064a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00755.warc.gz"} |
Test with permutations the significance of a classification score
Go to the end to download the full example code or to run this example in your browser via JupyterLite or Binder
Test with permutations the significance of a classification score¶
This example demonstrates the use of permutation_test_score to evaluate the significance of a cross-validated score using permutations.
# Authors: Alexandre Gramfort <alexandre.gramfort@inria.fr>
# Lucy Liu
# License: BSD 3 clause
We will use the Iris plants dataset, which consists of measurements taken from 3 types of irises.
We will also generate some random feature data (i.e., 20 features), uncorrelated with the class labels in the iris dataset.
import numpy as np
n_uncorrelated_features = 20
rng = np.random.RandomState(seed=0)
# Use same number of samples as in iris and 20 features
X_rand = rng.normal(size=(X.shape[0], n_uncorrelated_features))
Permutation test score¶
Next, we calculate the permutation_test_score using the original iris dataset, which strongly predict the labels and the randomly generated features and iris labels, which should have no dependency
between features and labels. We use the SVC classifier and Accuracy score to evaluate the model at each round.
permutation_test_score generates a null distribution by calculating the accuracy of the classifier on 1000 different permutations of the dataset, where features remain the same but labels undergo
different permutations. This is the distribution for the null hypothesis which states there is no dependency between the features and labels. An empirical p-value is then calculated as the percentage
of permutations for which the score obtained is greater that the score obtained using the original data.
from sklearn.model_selection import StratifiedKFold, permutation_test_score
from sklearn.svm import SVC
clf = SVC(kernel="linear", random_state=7)
cv = StratifiedKFold(2, shuffle=True, random_state=0)
score_iris, perm_scores_iris, pvalue_iris = permutation_test_score(
clf, X, y, scoring="accuracy", cv=cv, n_permutations=1000
score_rand, perm_scores_rand, pvalue_rand = permutation_test_score(
clf, X_rand, y, scoring="accuracy", cv=cv, n_permutations=1000
Original data¶
Below we plot a histogram of the permutation scores (the null distribution). The red line indicates the score obtained by the classifier on the original data. The score is much better than those
obtained by using permuted data and the p-value is thus very low. This indicates that there is a low likelihood that this good score would be obtained by chance alone. It provides evidence that the
iris dataset contains real dependency between features and labels and the classifier was able to utilize this to obtain good results.
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
ax.hist(perm_scores_iris, bins=20, density=True)
ax.axvline(score_iris, ls="--", color="r")
score_label = f"Score on original\ndata: {score_iris:.2f}\n(p-value: {pvalue_iris:.3f})"
ax.text(0.7, 10, score_label, fontsize=12)
ax.set_xlabel("Accuracy score")
_ = ax.set_ylabel("Probability density")
Random data¶
Below we plot the null distribution for the randomized data. The permutation scores are similar to those obtained using the original iris dataset because the permutation always destroys any feature
label dependency present. The score obtained on the original randomized data in this case though, is very poor. This results in a large p-value, confirming that there was no feature label dependency
in the original data.
fig, ax = plt.subplots()
ax.hist(perm_scores_rand, bins=20, density=True)
ax.axvline(score_rand, ls="--", color="r")
score_label = f"Score on original\ndata: {score_rand:.2f}\n(p-value: {pvalue_rand:.3f})"
ax.text(0.14, 7.5, score_label, fontsize=12)
ax.set_xlabel("Accuracy score")
ax.set_ylabel("Probability density")
Another possible reason for obtaining a high p-value is that the classifier was not able to use the structure in the data. In this case, the p-value would only be low for classifiers that are able to
utilize the dependency present. In our case above, where the data is random, all classifiers would have a high p-value as there is no structure present in the data.
Finally, note that this test has been shown to produce low p-values even if there is only weak structure in the data [1].
Total running time of the script: (0 minutes 11.248 seconds)
Related examples | {"url":"https://scikit-learn.org/1.4/auto_examples/model_selection/plot_permutation_tests_for_classification.html","timestamp":"2024-11-11T03:22:14Z","content_type":"text/html","content_length":"38850","record_id":"<urn:uuid:a8bb1e2d-340d-4a45-baf4-7847cec01477>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00648.warc.gz"} |
Equity Ratio | Formula | Calculator (Updated 2023)
This is an in-depth guide on how to calculate Equity Ratio with detailed analysis, interpretation, and example. You will learn how to utilize this ratio's formula to examine a company's current debt
situation by looking at its equity.
Definition - What is Equity Ratio?
All of a company’s assets are the result of shareholder equity, loans from creditors, or a combination of both.
The equity ratio, or shareholder’s equity ratio, is a simple calculation that can show you how much of a company’s assets are funded by owner shares.
When you evaluate a business as a potential investment, it’s important to find out as much as possible about its debt situation and its financial sustainability over the long-term.
This powerful ratio can provide you with information in both of these areas.
Because this ratio measures investor commitment to a company in the form of equity invested in assets, it also inversely demonstrates the amount of those assets that are supported and financed by
The lower the ratio value is; the more debt a company has used to fund its assets.
In terms of sustainability, the more capable a company is of servicing its debt load over the long run, the more financially stable it is.
The higher the ratio value, the more solvent a company is considered to be, since shareholder-owned assets are in excess of the firm’s liabilities.
To calculate the shareholder’s equity ratio for a given company, you would use the following formula:
Shareholders' Capital Ratio = Total Shareholders' Equity / Total Assets
In this ratio, the word “total” means exactly that, and ALL assets and equity reported on a company’s balance sheet must be included.
Read also: Time Interest Earned - Formula, Example & Analysis
Okay now let's dive into a quick example so you can understand clearly how to find this ratio.
As a potential investor, you’d like to further investigate Company K’s debt situation and financial sustainability by comparing its total assets with its shareholder equity.
Using Company K’s balance sheet as a reference, you come up with the following information:
• Total Assets = $1,000,000
• Total Liabilities = $250,000
• Total Shareholder's Equity = $750,000
Now you can calculate Company K’s stockholders' equity ratio value by plugging these figures into the formula, as follows:
This result shows you that 75% of Company K’s assets are financed by shareholder equity, while only 25% are attributed to funding from debt.
This means that if Company K were to sell all of its assets to pay off its liabilities, investors would retain ownership of ¾ of the company’s resources.
Interpretation & Analysis
The closer to 100% a firm’s shareholders' equity ratio is, the closer it is to financing all of its assets with shareholder equity.
As always, your interpretation of how high or low an acceptable shareholders' capital ratio value is for a specific company will hinge on other available information.
Comparing results with industry benchmarks is extremely important, since these dictate what level of equity to assets is considered standard for a particular type of business.
So what is a good equity ratio for a company?
While a higher ratio value is generally considered to be a good thing, that doesn’t necessarily mean that firms with a lower ratio are to be avoided.
When the equity ratio for a profitable company is relatively low, you’ll benefit from a higher return on investment because a smaller amount of overall equity is generating a greater level of
This lower ratio value can be relatively easy to sustain when a business is in an industry with inherently low levels of competition, and relatively stable sales and profits.
Just the same, investors usually prefer to see a higher ratio since it demonstrates a more conservative approach to debt management.
A higher ratio value shows that a large number of shareholders consider the company to be a worthwhile investment, and it lets potential creditors know that the company is a good credit risk.
It’s also worth noting that there are fewer financing costs associated with less debt, so a business with a higher ratio value will be much less expensive to operate.
Cautions & Further Explanation
There is no caution for this ratio. However, using this ratio alone may potentially lead to a less useful valuation result.
As a value investor, you should never rely on a single ratio or investing metric to make your investment decisions.
So it is worth considering to use this ratio with other debt ratios, such as the quick ratio, current ratio or debt to equity ratio when performing your financial ratio analysis. | {"url":"https://wealthyeducation.com/equity-ratio/","timestamp":"2024-11-14T10:35:41Z","content_type":"text/html","content_length":"395237","record_id":"<urn:uuid:cd20af1b-73bd-48b5-b701-c7dccf4749de>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00056.warc.gz"} |
Sum of squares lower bounds for refuting any CSP
Let P: {0, 1}^k → {0,1} be a nontrivial k-ary predicate. Consider a random instance of the constraint satisfaction problem CSP(P) on n variables with An constraints, each being P applied to k
randomly chosen literals. Provided the constraint density satisfes Δ ≥≥ 1, such an instance is unsatisfable with high probability. The refutation problem is to efficiently find a proof of
unsatisfability. We show that whenever the predicate P supports a t-wise uniform probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree d = Θ(n/Δ^2/(t-1)
longΔ) (which runs in time n^O(d)) cannot refute a random instance of CSP(P). In particular, the polynomial-time SOS algorithm requires Ω(n^(t+1)/2) constraints to refute random instances of CSP(P)
when P supports a t-wise uniform distribution on its satisfying assignments. Together with recent work of Lee, Raghavendra, Steurer (2015), our result also implies that any polynomial-size
semidefinite programming relaxation for refutation requires at least Ω(n^(t+1)/2) constraints. More generally, we consider the δ-refutation problem, in which the goal is to certify that at most a (1
- δ)-fraction of constraints can be simultaneously satisfed. We show that if P is δ-close to supporting a t-wise uniform distribution on satisfying assignments, then the degree-Ω(n/Δ^2/(t-1)logΔ) SOS
algorithm cannot (δ + o(1))- refute a random instance of CSP(P). This is the first result to show a distinction between the degree SOS needs to solve the refutation problem and the degree it needs to
solve the harder δ-refutation problem. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random
CSPs. For every constraint predicate P, they give a three-way hardness tradeof between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent
algorithmic results of Allen, O'Donnell, Witmer (2015) and Raghavendra, Rao, Schramm (2016), this full three-way tradeof is tight, up to lower-order factors.
Original language English (US)
Title of host publication STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
Editors Pierre McKenzie, Valerie King, Hamed Hatami
Publisher Association for Computing Machinery
Pages 132-145
Number of pages 14
ISBN (Electronic) 9781450345286
State Published - Jun 19 2017
Event 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 - Montreal, Canada
Duration: Jun 19 2017 → Jun 23 2017
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
Volume Part F128415
ISSN (Print) 0737-8017
Other 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017
Country/Territory Canada
City Montreal
Period 6/19/17 → 6/23/17
All Science Journal Classification (ASJC) codes
• Constraint satisfaction
• Lower bounds
• Sum-of-Squares semidefinite programming hierarchy
Dive into the research topics of 'Sum of squares lower bounds for refuting any CSP'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/sum-of-squares-lower-bounds-for-refuting-any-csp","timestamp":"2024-11-02T05:22:18Z","content_type":"text/html","content_length":"58331","record_id":"<urn:uuid:3318aba3-88cb-4d9a-b9ab-7cac05eef1c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00206.warc.gz"} |
Anisotropic Schrödinger Equation Quantum Corrections for 3D Finite Element Monte Carlo Simulations of Triangular SOI Fin-FET
Anisotropic 2-D Schrödinger equation-based quantum corrections dependent on valley orientation are incorporated into a 3-D finite-element Monte Carlo simulation toolbox. The new toolbox is then
applied to simulate nanoscale Si Silicon-on-Insulator FinFETs with a gate length of 8.1 nm to study the contributions of conduction valleys to the drive current in various FinFET architectures and
channel orientations. The 8.1 nm gate length FinFETs are studied for two cross sections: rectangular-like and triangular-like, and for two channel orientations: \langle 100\rangle and \langle 110\
rangle . We have found that quantum anisotropy effects play the strongest role in the triangular-like \langle 100\rangle channel device increasing the drain current by \sim 13 % and slightly
decreasing the current by 2% in the rectangular-like \langle 100\rangle channel device. The quantum anisotropy has a negligible effect in any device with the \langle 110\rangle channel orientation. | {"url":"https://citius.gal/research/publications/anisotropic-schrodinger-equation-quantum-corrections-for-3d-finite-element-monte-carlo-simulations-of-triangular-soi-fin-fet/","timestamp":"2024-11-05T12:18:40Z","content_type":"text/html","content_length":"136090","record_id":"<urn:uuid:47456185-c808-4d3a-a2e2-f8e99e7331d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00002.warc.gz"} |
★ ★ ★ ★ A List - Musical Symbol Names (UK)
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol
In UK musical notation, what is the name of this symbol | {"url":"https://quizlists.com/list.php?x=3580003","timestamp":"2024-11-06T06:20:11Z","content_type":"text/html","content_length":"37493","record_id":"<urn:uuid:2132dbb4-8a97-44a4-a321-13216461f28f>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00677.warc.gz"} |
Understanding the Debt Ratio: Definition and Formula (2024)
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The debt ratio is a financial metric that compares a business’ total debt to total assets. It’s a crucial ratio that analysts and finance professionals use to assess a company’s financial health. In
this article, we’ll review the debt ratio and why it is an essential concept for students interested in corporate finance.
What Is the Debt Ratio?
The debt ratio shows how leveraged a company is. It provides insights into the proportion of a company’s financing derived from debt compared to assets. There are variants of this ratio that consider
liabilities to equity. However, all leverage ratios measure how much a company relies on borrowed funds versus its own funds on some level.
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Build the confidence and practical skills that employers are looking for with Forage’s free job simulations.
How to Calculate the Debt Ratio
Mathematically, the debt ratio is calculated by dividing a company’s total debt by total assets and multiplying the result by 100 to express it as a percentage. The formula is as follows:
(Total debt / Total assets) x 100
For instance, if a company has $500,000 in total debt and $1,000,000 in total assets, the debt ratio would be 50%:
(500,000 / 1,000,000) x 100 = 50%
You can skip the multiplication at the end and express the ratio as a decimal. For instance, in this example, the debt ratio would be 0.5.
Interpreting the Debt Ratio
The debt ratio is valuable for evaluating a company’s financial structure and risk profile. If the ratio is over 1, a company has more debt than assets. If the ratio is below 1, the company has more
assets than debt. Broadly speaking, ratios of 60% (0.6) or more are considered high, while ratios of 40% (0.4) or less are considered low.
However, what constitutes a “good debt ratio” can vary depending on industry norms, business objectives, and economic conditions. For instance, startups or companies in rapid expansion phases, too,
may have higher ratios as they utilize debt to fund growth initiatives. While a higher ratio can be acceptable, carefully analyzing the company’s ability to generate sufficient cash flows to service
the debt is essential.
A good debt ratio should align with the company’s financial goals, risk tolerance, and industry standards. It should support the company’s ability to meet its financial obligations, maintain
financial stability, and enable sustainable growth. Comparing a company’s ratio to industry peers, historical performance, and industry averages can provide valuable insights to determine what is
considered favorable within a specific sector.
Learn how to do a comparable company analysis with this free JPMorgan Chase Investment Banking job simulation from Forage.
Who Uses the Debt Ratio?
This fundamental financial metric is used by various stakeholders in corporate finance, including:
• Financial analysts: Financial analysts play a crucial role in assessing a company’s financial performance and making investment recommendations. They rely on this metric to evaluate a company’s
risk profile and financial stability. By analyzing the debt ratio, analysts gain insights into the level of financial leverage and the potential impact of debt on the company’s profitability and
• Investors: Investors, including individual investors, institutional investors, and fund managers, closely examine the debt ratio when making investment decisions. A company with a favorable ratio
may be financially sound and capable of generating consistent returns. Conversely, a high ratio might raise concerns about a company’s ability to manage its debt and fulfill its financial
obligations. Investors consider the debt ratio as part of their overall risk assessment and investment strategy.
• Lenders and creditors: Lenders and creditors, such as banks and financial institutions, rely on this metric to evaluate a company’s creditworthiness and determine its borrowing capacity. A lower
ratio indicates a company is at a lower risk of defaulting on its loans and may be more likely to secure favorable financing terms. Lenders use this metric as one of the critical factors in
assessing the company’s ability to service its debt and make timely interest and principal payments.
• Management and executives: This metric is vital for management and executives in making informed financial decisions. It assists in determining the optimal capital structure for the company,
balancing the use of debt and equity financing. By monitoring changes in this ratio, management can assess the impact of financing decisions on the company’s risk profile, profitability, and
long-term sustainability.
• Regulatory bodies like the U.S. Securities and Exchange Commission (SEC) may require companies to disclose thia metric as part of their financial reporting obligations. Credit rating agencies
also use it as one of the factors in assessing a company’s credit rating. A higher ratio might lead to a lower credit rating, affecting the company’s ability to secure financing at favorable
>>MORE: Is Finance a Good Career Path?
Showcasing You Understand the Debt Ratio on Your Resume
You can convey that you understand this calculation by including any of the following items on your resume:
• In your skills section: Include “financial ratio analysis,” “debt ratio evaluation,” or “capital structure assessment” as skills to demonstrate your familiarity with financial metrics and your
ability to analyze and interpret financial data.
• Mention coursework: Highlight courses that cover financial analysis, financial management, or corporate finance, as these subjects typically delve into this and similar calculations.
• Highlight work experience: Describe specific projects or responsibilities that assessed a company’s capital structure or financial health.
Other necessary calculations to master if you’re interested in this career path include the following:
• Other leverage ratios, including the debt-to-equity (DE) ratio, the debt-to-capital ratio, and the asset-to-equity ratio.
• The current ratio, which compares a company’s current assets to its current liabilities.
• Quick ratio, which measures a company’s short-term liquidity against its short-term obligations.
• P/E (price-to-earnings) ratio, which compares a company’s share price to its annual net profits.
Learn these and other in-demand skills today with Forage’s free job simulations.
Image credit: alebloshka / Depositphotos.com | {"url":"https://bdacareerchoices.com/article/understanding-the-debt-ratio-definition-and-formula","timestamp":"2024-11-03T12:23:06Z","content_type":"text/html","content_length":"112901","record_id":"<urn:uuid:931405a9-251c-47ea-a9ed-738888d70d64>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00633.warc.gz"} |
Millimeters to Kiloyards Converter
Enter Millimeters
β Switch toKiloyards to Millimeters Converter
How to use this Millimeters to Kiloyards Converter π €
Follow these steps to convert given length from the units of Millimeters to the units of Kiloyards.
1. Enter the input Millimeters value in the text field.
2. The calculator converts the given Millimeters into Kiloyards in realtime β using the conversion formula, and displays under the Kiloyards label. You do not need to click any button. If the
input changes, Kiloyards value is re-calculated, just like that.
3. You may copy the resulting Kiloyards value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Millimeters to Kiloyards?
The formula to convert given length from Millimeters to Kiloyards is:
Length[(Kiloyards)] = Length[(Millimeters)] / 914399.9986101121
Substitute the given value of length in millimeters, i.e., Length[(Millimeters)] in the above formula and simplify the right-hand side value. The resulting value is the length in kiloyards, i.e.,
Calculation will be done after you enter a valid input.
Consider that a premium camera lens has a focal length of 85 millimeters.
Convert this focal length from millimeters to Kiloyards.
The length in millimeters is:
Length[(Millimeters)] = 85
The formula to convert length from millimeters to kiloyards is:
Length[(Kiloyards)] = Length[(Millimeters)] / 914399.9986101121
Substitute given weight Length[(Millimeters)] = 85 in the above formula.
Length[(Kiloyards)] = 85 / 914399.9986101121
Length[(Kiloyards)] = 0.0000929571305
Final Answer:
Therefore, 85 mm is equal to 0.0000929571305 kyd.
The length is 0.0000929571305 kyd, in kiloyards.
Consider that a luxury watch's thickness is 12 millimeters.
Convert this thickness from millimeters to Kiloyards.
The length in millimeters is:
Length[(Millimeters)] = 12
The formula to convert length from millimeters to kiloyards is:
Length[(Kiloyards)] = Length[(Millimeters)] / 914399.9986101121
Substitute given weight Length[(Millimeters)] = 12 in the above formula.
Length[(Kiloyards)] = 12 / 914399.9986101121
Length[(Kiloyards)] = 0.0000131233596
Final Answer:
Therefore, 12 mm is equal to 0.0000131233596 kyd.
The length is 0.0000131233596 kyd, in kiloyards.
Millimeters to Kiloyards Conversion Table
The following table gives some of the most used conversions from Millimeters to Kiloyards.
Millimeters (mm) Kiloyards (kyd)
0 mm 0 kyd
1 mm 0.00000109361 kyd
2 mm 0.00000218723 kyd
3 mm 0.00000328084 kyd
4 mm 0.00000437445 kyd
5 mm 0.00000546807 kyd
6 mm 0.00000656168 kyd
7 mm 0.00000765529 kyd
8 mm 0.00000874891 kyd
9 mm 0.00000984252 kyd
10 mm 0.00001093613 kyd
20 mm 0.00002187227 kyd
50 mm 0.00005468066 kyd
100 mm 0.00010936133 kyd
1000 mm 0.0010936133 kyd
10000 mm 0.010936133 kyd
100000 mm 0.1094 kyd
A millimeter (mm) is a unit of length in the International System of Units (SI). One millimeter is equivalent to 0.001 meters or approximately 0.03937 inches.
The millimeter is defined as one-thousandth of a meter, making it a precise measurement for small distances.
Millimeters are used worldwide to measure length and distance in various fields, including engineering, manufacturing, and everyday life. Many industries, especially those requiring high precision,
have adopted the millimeter as a standard unit of measurement for small lengths.
A kiloyard (ky) is a unit of length equal to 1,000 yards or approximately 914.4 meters.
The kiloyard is defined as one thousand yards, providing a convenient measurement for longer distances that are not as extensive as miles but larger than typical yard measurements.
Kiloyards are used in various fields to measure length and distance where a scale between yards and miles is appropriate. They offer a practical unit for certain applications, such as in land
measurement and engineering.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Millimeters to Kiloyards in Length?
The formula to convert Millimeters to Kiloyards in Length is:
Millimeters / 914399.9986101121
2. Is this tool free or paid?
This Length conversion tool, which converts Millimeters to Kiloyards, is completely free to use.
3. How do I convert Length from Millimeters to Kiloyards?
To convert Length from Millimeters to Kiloyards, you can use the following formula:
Millimeters / 914399.9986101121
For example, if you have a value in Millimeters, you substitute that value in place of Millimeters in the above formula, and solve the mathematical expression to get the equivalent value in | {"url":"https://convertonline.org/unit/?convert=millimeters-kiloyards","timestamp":"2024-11-08T22:32:40Z","content_type":"text/html","content_length":"91265","record_id":"<urn:uuid:08da741a-4459-4cb9-9451-4b8c2fe7b4f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00221.warc.gz"} |
Generalized Sørensen-Dice similarity coefficient for image segmentation
Since R2021a
The generalized Dice similarity coefficient measures the overlap between two segmented images. Generalized Dice similarity is based on Sørensen-Dice similarity and controls the contribution that each
class makes to the similarity by weighting classes by the inverse size of the expected region. When working with imbalanced data sets, class weighting helps to prevent the more prevalent classes from
dominating the similarity score.
similarity = generalizedDice(X,target) calculates the generalized Sørensen-Dice similarity coefficient between test image X and target image target.
similarity = generalizedDice(X,target,'DataFormat',dataFormat) also specifies the dimension labels, dataFormat, of unformatted image data. You must use this syntax when the input are unformatted
dlarray (Deep Learning Toolbox) objects.
Calculate Generalized Dice Similarity
Load a pretrained network.
data = load("triangleSegmentationNetwork");
net = data.net;
Load the triangle image data set using imageDatastore.
dataDir = fullfile(toolboxdir("vision"),"visiondata","triangleImages");
testImageDir = fullfile(dataDir,"testImages");
imds = imageDatastore(testImageDir);
Load ground truth labels for the triangle data set using pixelLabelDatastore.
labelDir = fullfile(dataDir,"testLabels");
classNames = ["triangle" "background"];
pixelLabelID = [255 0];
pxdsTruth = pixelLabelDatastore(labelDir,classNames,pixelLabelID);
Read a sample image and the corresponding ground truth labels.
I = readimage(imds,1);
gTruthLabels = readimage(pxdsTruth,1);
Run semantic segmentation on the image.
[predictions,scores] = semanticseg(I,net,Classes=classNames);
Encode the categorical predictions and targets using the onehotencode function.
featureDim = ndims(predictions) + 1;
encodedPredictions = onehotencode(predictions,featureDim);
encodedGroundTruthLabels = onehotencode(gTruthLabels,featureDim);
Ignore any undefined classes in the encoded data.
encodedPredictions(isnan(encodedPredictions)) = 0;
encodedGroundTruthLabels(isnan(encodedGroundTruthLabels)) = 0;
Compute generalized Dice similarity coefficient between the segmented image and the ground truth.
gDice = generalizedDice(encodedPredictions,encodedGroundTruthLabels)
Calculate Generalized Dice Loss of dlarray Input
Create input data as a formatted dlarray object containing 32 observations with unnormalized scores for ten output categories.
spatial = 10;
numCategories = 10;
batchSize = 32;
X = dlarray(rand(spatial,numCategories,batchSize),'SCB');
Convert unnormalized scores to probabilities of membership of each of the ten categories.
Create target values for membership in the second and sixth category.
targets = zeros(spatial,numCategories,batchSize);
targets(:,2,:) = 1;
targets(:,6,:) = 1;
targets = dlarray(targets,'SCB');
Compute the generalized Dice similarity coefficient between probability vectors X and targets for multi-label classification.
Z = generalizedDice(X,targets);
whos Z
Name Size Bytes Class Attributes
Z 1x1x32 262 dlarray
Calculate the generalized Dice loss.
loss =
1(S) x 1(C) x 1(B) dlarray
Input Arguments
X — Test image
numeric array | dlarray object
Test image to be analyzed, specified as one of these values.
• A numeric array of any dimension. The last dimension must correspond to classes.
• An unformatted dlarray (Deep Learning Toolbox) object. You must specify the data format using the dataFormat argument.
• A formatted dlarray object. The dlarray input must contain a channel dimension, 'C' and can contain a batch dimension, 'B'.
dlarray input requires Deep Learning Toolbox™.
target — Target image
numeric array | dlarray object
Target image, specified as a numeric array or a dlarray (Deep Learning Toolbox) object. The size and format of target must match the size and format of the test image, X. dlarray input requires Deep
Learning Toolbox.
dataFormat — Dimension labels
string scalar | character vector
Dimension labels for unformatted dlarray image input, specified as a string scalar or character vector. Each character in dataFormat must be one of these labels:
• S — Spatial
• C — Channel
• B — Batch observations
The format must include one channel label. The format cannot include more than one channel label or batch label. Do not specify the 'dataFormat' argument when the input images are formatted dlarray
Example: 'SSC' indicates that the array has two spatial dimensions and one channel dimension
Example: 'SSCB' indicates that the array has two spatial dimensions, one channel dimension, and one batch dimension
Output Arguments
similarity — Generalized Dice similarity coefficient
numeric scalar | dlarray object
Generalized Dice similarity coefficient, returned as a numeric scalar or a dlarray (Deep Learning Toolbox) object with values in the range [0, 1]. A similarity of 1 means that the segmentations in
the two images are a perfect match.
• If the input arrays are numeric images, then similarity is a numeric scalar.
• If the input arrays are dlarray objects, then similarity is a dlarray object of the same dimensionality as the input images. The spatial and channel dimensions of similarity are singleton
dimensions. There is one generalized Dice measurement for each element along the batch dimension.
More About
Generalized Dice Similarity
Generalized Dice similarity is based on Sørensen-Dice similarity for measuring overlap between two segmented images.
The generalized Dice similarity function S used by generalizedDice for the similarity between one image Y and the corresponding ground truth T is given by:
$S=\frac{2{\sum }_{k=1}^{K}{w}_{k}{\sum }_{m=1}^{M}{Y}_{km}{T}_{km}}{{\sum }_{k=1}^{K}{w}_{k}{\sum }_{m=1}^{M}{Y}_{km}^{2}+{T}_{km}^{2}}$
K is the number of classes, M is the number of elements along the first two dimensions of Y, and w[k] is a class specific weighting factor that controls the contribution each class makes to the
score. This weighting helps counter the influence of larger regions on the generalized Dice score. w[k] is typically the inverse area of the expected region:
${w}_{k}=\frac{1}{{\left({\sum }_{m=1}^{M}{T}_{km}\right)}^{2}}$
There are several variations of generalized Dice scores [1], [2]. The generalizedDice function uses squared terms to ensure that the derivative is 0 when the two images match [3].
[1] Crum, William R., Oscar Camara, and Derek LG Hill. "Generalized overlap measures for evaluation and validation in medical image analysis." IEEE Transactions on Medical Imaging. 25.11, 2006, pp.
[2] Sudre, Carole H., et al. "Generalised Dice overlap as a deep learning loss function for highly unbalanced segmentations." Deep Learning in Medical Image Analysis and Multimodal Learning for
Clinical Decision Support. Springer, Cham, 2017, pp. 240–248.
[3] Milletari, Fausto, Nassir Navab, and Seyed-Ahmad Ahmadi. "V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation". Fourth International Conference on 3D Vision (3DV)
. Stanford, CA, 2016: pp. 565–571.
Extended Capabilities
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2021a | {"url":"https://au.mathworks.com/help/vision/ref/generalizeddice.html","timestamp":"2024-11-10T06:02:08Z","content_type":"text/html","content_length":"101719","record_id":"<urn:uuid:e9a8136d-6be2-465e-9114-578eb2e72868>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00411.warc.gz"} |
Math Tools and Calculators
Enter two numbers, and see not only the percentages, but how they are calculated.
Convert between Binary, Decimal and Hexadecimal. Can handle fractions and negatives.
Want to know if a number is Prime? Or if not, which prime numbers are it's factors?
Find all the factors of a number
Find if two numbers are coprime (if they share anyprrime factors)
Enter formulas such as (3+2/9)^2 or sin(1/5), has history and even a slider for an "a" value.
Accepts formulas such as (3+2/9)^2 or sin(1/5) and you can also click on a keypad.
Can convert from/to Roman Numerals. What year was "MDMCLXXI" ?
Find the greatest common factor of 2 or 3 numbers. Useful for simplifying fractions.
Find the least common multiple of 2 or 3 numbers. Useful for adding or subtracting fractions.
Calculate answers to full precision. Want to multiply to hundreds of digits of accuracy? Done!
Enter the formula, see the result.
Find the right amounts of a mixture.
Converts Decimals to Fractional form, showing steps.
Calculate Combinations and Permutations
A neat yet powerful Unit Converter. Can convert length, mass, volume, temperature and more. Hundreds of imperial and metric units are supported
Calculate the day of the week for any date between 1582 and 4902
Do matrix calculations such as Determinant, Inverse and Multiplication
See what happens when you sum up a series of functions.
Convert between magnitude/angle and x,y. Add vectors. Dot product.
Have an equation like "ax^2 + bx + c = 0"? We can solve it!
You can plot two functions together, and even save the results as a web link.
Graph equations like "x^2+y^2=2"
Plot an f(x,y) style function like x^2-y^2
Calculate the areas of common shapes such as triangles, circles and ellipses
Spin, explode and see the nets of over 100 polyhedra.
Calculate the volume, area, diameter and radius of a sphere
If you have two points and need to know the straight line formula (y=mx+b) that connects them.
Calculate side length, area, diagonal or perimeter of a
width, height, area, diagonal or perimeter
of a
Shows you the results at every stage of the calculation
Print or save graph paper
Union, intersection and difference of two
A game that is also useful for adding up money.
Other Cool Stuff
A simulation of how suns, planets, moons, etc behave under the influence of gravity
Explore the interesting "interference patterns" created by overlapping grids, lines, etc
Try your estimation skills | {"url":"http://wegotthenumbers.org/math-tools.html","timestamp":"2024-11-08T12:20:01Z","content_type":"text/html","content_length":"14645","record_id":"<urn:uuid:9bd807ec-ade3-487d-ab48-a0886253903a>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00172.warc.gz"} |
03543 ASN GMO Memu Pgr Special Train Time Table
If we closely look at the 03543 train time table, travelling by 03543 ASN GMO Memu Pgr Special gives us a chance to explore the following cities in a quick view as they come along the route.
1. Asansol Jn
It is the 1st station in the train route of 03543 ASN GMO Memu Pgr Special. The station code of Asansol Jn is ASN. The departure time of train 03543 from Asansol Jn is 18:10. The next stopping
station is Barachak Jn at a distance of 5km.
2. Barachak Jn
It is the 2nd station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 5 Km from the source station Asansol Jn. The station code of Barachak Jn is BCQ. The arrival time of 03543
at Barachak Jn is 18:15. The departure time of train 03543 from Barachak Jn is 18:16. The total halt time of train 03543 at Barachak Jn is 1 minutes. The previous stopping station, Asansol Jn is 5km
away. The next stopping station is Sitarampur at a distance of 4km.
3. Sitarampur
It is the 3rd station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 9 Km from the source station Asansol Jn. The station code of Sitarampur is STN. The arrival time of 03543
at Sitarampur is 18:21. The departure time of train 03543 from Sitarampur is 18:22. The total halt time of train 03543 at Sitarampur is 1 minutes. The previous stopping station, Barachak Jn is 4km
away. The next stopping station is Kulti at a distance of 4km.
4. Kulti
It is the 4th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 13 Km from the source station Asansol Jn. The station code of Kulti is ULT. The arrival time of 03543 at
Kulti is 18:27. The departure time of train 03543 from Kulti is 18:28. The total halt time of train 03543 at Kulti is 1 minutes. The previous stopping station, Sitarampur is 4km away. The next
stopping station is Barakar at a distance of 3km.
5. Barakar
It is the 5th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 16 Km from the source station Asansol Jn. The station code of Barakar is BRR. The arrival time of 03543 at
Barakar is 18:32. The departure time of train 03543 from Barakar is 18:33. The total halt time of train 03543 at Barakar is 1 minutes. The previous stopping station, Kulti is 3km away. The next
stopping station is Kumardubi at a distance of 3km.
6. Kumardubi
It is the 6th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 19 Km from the source station Asansol Jn. The station code of Kumardubi is KMME. The arrival time of 03543
at Kumardubi is 18:37. The departure time of train 03543 from Kumardubi is 18:38. The total halt time of train 03543 at Kumardubi is 1 minutes. The previous stopping station, Barakar is 3km away. The
next stopping station is Mugma at a distance of 4km.
7. Mugma
It is the 7th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 23 Km from the source station Asansol Jn. The station code of Mugma is MMU. The arrival time of 03543 at
Mugma is 18:42. The departure time of train 03543 from Mugma is 18:43. The total halt time of train 03543 at Mugma is 1 minutes. The previous stopping station, Kumardubi is 4km away. The next
stopping station is Thapar Nagar at a distance of 5km.
8. Thapar Nagar
It is the 8th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 28 Km from the source station Asansol Jn. The station code of Thapar Nagar is TNW. The arrival time of
03543 at Thapar Nagar is 18:48. The departure time of train 03543 from Thapar Nagar is 18:49. The total halt time of train 03543 at Thapar Nagar is 1 minutes. The previous stopping station, Mugma is
5km away. The next stopping station is Kalubathan at a distance of 5km.
9. Kalubathan
It is the 9th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 33 Km from the source station Asansol Jn. The station code of Kalubathan is KAO. The arrival time of 03543
at Kalubathan is 18:53. The departure time of train 03543 from Kalubathan is 18:54. The total halt time of train 03543 at Kalubathan is 1 minutes. The previous stopping station, Thapar Nagar is 5km
away. The next stopping station is Chhota Ambana at a distance of 9km.
10. Chhota Ambana
It is the 10th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 42 Km from the source station Asansol Jn. The station code of Chhota Ambana is CAM. The arrival time of
03543 at Chhota Ambana is 19:01. The departure time of train 03543 from Chhota Ambana is 19:02. The total halt time of train 03543 at Chhota Ambana is 1 minutes. The previous stopping station,
Kalubathan is 9km away. The next stopping station is Pradhankhunta at a distance of 6km.
11. Pradhankhunta
It is the 11th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 48 Km from the source station Asansol Jn. The station code of Pradhankhunta is PKA. The arrival time of
03543 at Pradhankhunta is 19:12. The departure time of train 03543 from Pradhankhunta is 19:13. The total halt time of train 03543 at Pradhankhunta is 1 minutes. The previous stopping station, Chhota
Ambana is 6km away. The next stopping station is Dokra Halt at a distance of 4km.
12. Dokra Halt
It is the 12th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 52 Km from the source station Asansol Jn. The station code of Dokra Halt is DOKM. The arrival time of
03543 at Dokra Halt is 19:17. The departure time of train 03543 from Dokra Halt is 19:18. The total halt time of train 03543 at Dokra Halt is 1 minutes. The previous stopping station, Pradhankhunta
is 4km away. The next stopping station is Dhanbad Jn at a distance of 5km.
13. Dhanbad Jn
It is the 13th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 57 Km from the source station Asansol Jn. The station code of Dhanbad Jn is DHN. The arrival time of 03543
at Dhanbad Jn is 19:25. The departure time of train 03543 from Dhanbad Jn is 19:30. The total halt time of train 03543 at Dhanbad Jn is 5 minutes. The previous stopping station, Dokra Halt is 5km
away. The next stopping station is Bhuli at a distance of 4km.
14. Bhuli
It is the 14th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 61 Km from the source station Asansol Jn. The station code of Bhuli is BHN. The arrival time of 03543 at
Bhuli is 19:35. The departure time of train 03543 from Bhuli is 19:36. The total halt time of train 03543 at Bhuli is 1 minutes. The previous stopping station, Dhanbad Jn is 4km away. The next
stopping station is Tetulmari at a distance of 6km.
15. Tetulmari
It is the 15th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 67 Km from the source station Asansol Jn. The station code of Tetulmari is TET. The arrival time of 03543
at Tetulmari is 19:42. The departure time of train 03543 from Tetulmari is 19:43. The total halt time of train 03543 at Tetulmari is 1 minutes. The previous stopping station, Bhuli is 6km away. The
next stopping station is Nichitpur at a distance of 5km.
16. Nichitpur
It is the 16th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 72 Km from the source station Asansol Jn. The station code of Nichitpur is NPJE. The arrival time of 03543
at Nichitpur is 19:49. The departure time of train 03543 from Nichitpur is 19:50. The total halt time of train 03543 at Nichitpur is 1 minutes. The previous stopping station, Tetulmari is 5km away.
The next stopping station is Matari at a distance of 5km.
17. Matari
It is the 17th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 77 Km from the source station Asansol Jn. The station code of Matari is MRQ. The arrival time of 03543 at
Matari is 19:56. The departure time of train 03543 from Matari is 19:57. The total halt time of train 03543 at Matari is 1 minutes. The previous stopping station, Nichitpur is 5km away. The next
stopping station is Nsc Bose J Gomo at a distance of 10km.
18. Nsc Bose J Gomo
It is the 18th station in the train route of 03543 ASN GMO Memu Pgr Special at a distance of 87 Km from the source station Asansol Jn. The station code of Nsc Bose J Gomo is GMO. The arrival time of
03543 at Nsc Bose J Gomo is 20:15. The previous stopping station, Matari is 10km away.
Trainspnrstatus is one of the best website for checking trains running status. You can find the 03543 ASN GMO Memu Pgr Special running status here.
Dhanbad Junction are major halts where ASN GMO Memu Pgr Special halts for more than five minutes. Getting hotel accommodations and cab facilities in these cities is easy.
Trainspnrstatus is one stop best portal for checking pnr status. You can find the 03543 ASN GMO Memu Pgr Special IRCTC and Indian Railways PNR status here. All you have to do is to enter your 10
digit PNR number in the form. PNR number is printed on the IRCTC ticket.
Train number of ASN GMO Memu Pgr Special is 03543. You can check entire ASN GMO Memu Pgr Special train schedule here. with important details like arrival and departure time.
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Multiplication Chart 120 Printable - Dentopedia Blog
Multiplication Chart 120 Printable
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Subtraction Games
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Donut Shop - Subtraction
Making math fun and engaging for students is a top priority, and our Donut Shop: Subtraction printable game does just that! This free resource is designed for Year 3 and Year 4 students who are
learning about subtraction. With this game, students can practice their subtraction skills in an interactive and enjoyable way, all while running their very own donut shop.
About Donut Shop: Subtraction
Donut Shop: Subtraction is a hands-on
activity where students solve subtraction problems using colorful donut cards. The game combines the excitement of a pretend shop with valuable math practice, making it a perfect addition to any
How to Play
1. Prepare the Game: Print out pages 2 – 12 of the downloadable document. Cut and laminate the donut cards and shop cards for durability.
2. Set Up: Place the donut cards from pages 5 – 12 upside down in two separate piles – one for pink donuts and one for aqua donuts.
3. Choose a Shop Card: Students start by selecting a shop card from pages 2 – 4.
4. Pick Donut Cards: Students then pick one pink donut card and one aqua donut card and place them on their shop card.
5. Solve the Subtraction Problems: Students solve the subtraction problems displayed on their shop card using the numbers from the donut cards. They can use a whiteboard to show their working out.
6. Check Answers: Once solved, students can check their answers with their teacher or a peer to ensure accuracy.
Benefits of Playing Donut Shop: Subtraction
• Engaging and Fun: The game format makes subtraction practice enjoyable and interactive.
• Hands-On Learning: Students actively participate in the learning process, enhancing their understanding of subtraction.
• Improves Problem-Solving Skills: Solving different subtraction problems helps students develop critical thinking and problem-solving skills.
• Flexible Use: The game can be used in various settings, including classrooms, homeschool environments, and math centers.
• Free Resource: As always, this game is completely free to download and use, making it accessible for all educators and parents.
How to Get Started
1. Download and Print: Click the button below to download your free copy of the Donut Shop: Subtraction game. Print out the necessary pages and prepare the game pieces.
2. Introduce the Game: Explain the rules and objectives to your students, ensuring they understand how to play and solve the subtraction problems.
3. Play and Learn: Let your students dive into the game, enjoying the process of running their donut shop while practicing subtraction.
4. Review and Reflect: After playing, review the subtraction problems with your students, discussing any challenges they faced and celebrating their successes.
Download the Game
All resources on our website are free to use and can be downloaded by clicking the button at the bottom of the page. Explore our collection of worksheets and activities to support your students'
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Analysis of stiffness characteristics of a new fluid bag for axial shock protection
In view of large loads being needed to protect the axial from the shock situation under small displacement and deformation, a new fluid bag for axial protection was designed in Abaqus. Hydrostatic
fluid elements were used to simulate fluid. Interaction between the fluid and bag was simulated with the hydrostatic theory. Based on the finite element theory, the axial stiffness of fluid bag was
calculated. The results show that the stiffness had good linearity. The difference between the simulation and experiment results is small, proving the correctness of simulation. The effects of
initial bag pressure on the stiffness were discussed. The results indicate that different initial pressures have few impacts on the stiffness as well as tendency of bag pressure variations. Then the
effects of bag material properties and fluid bulk modulus on the stiffness were discussed. The results show that both of them are the key factors determining the stiffness. The effects of fluid bag
on the stress of a mechanism under axial shock load were discussed. The results show that the fluid bag has a good performance for axial protection.
1. Introduction
With the continuous development of industrial technology, the automation degree of mechanical equipments has become more and more advanced. The springs, a kind of mechanical part working through
elastic, have been vastly applied in many industry categories, such as automobile, railway, aerospace and so on. Currently the main springs being used are metal springs, rubber springs and air
springs, which has been demonstrated in many studies [1-9]. Metal springs have strong stability, high stiffness and reliability. Rubber springs and air springs have small size and light weight as
well as nonlinear stiffness characteristics. However, rubber springs and air springs cannot generate large loads under small displacement and deformation. In spite of high stiffness, metal springs
have weak acoustic attenuating performance. Meanwhile, they possibly cannot recover from long-term large loads.
In some special situations, large loads are needed for protection under small displacement and deformation. For example, large static displacement is not allowed when the adhesive joints are under
shock loads in the tensile direction, otherwise they will fail to work [10-13]. Currently an effective solution to improving their strength is using the weld-bonded joints [14-16]. However, if this
method is applied to the simulation, the consequent uneven stress distribution and minor reduction of loads in the joints may only lead to inaccurate results. Therefore, a new fluid bag, providing
axial protection under small displacement and deformation, was designed in the paper. Currently, the springs, a kind of retractable airtight container full of elastic medium, are mainly the air
springs taking the compressible gas as medium, which have been deeply researched and widely used [17-19]. But to the fluid bags taking the approximate incompressible fluid as medium, the studies are
very rare.
Based on the mentioned considerations, interaction between the fluid and bag has been simulated with the hydrostatic theory. Based on the finite element theory, the axial stiffness and pressure of
fluid bag have been calculated. Compared with the experimental data, the simulation results prove to be correct. According to the researches, the effects of different initial pressures on the axial
stiffness characteristics have been discussed, which demonstrate that the initial pressure almost make no difference. Then the effects of bag material properties and fluid bulk modulus have been
discussed, which show that both of them are the key factors determining the axial stiffness. The effects of fluid bag on the stress of a mechanism under axial shock load have been discussed. The
results show that the fluid bag have a good performance for axial protection.
2. The introduction of fluid bag
2.1. The fluid bag model
The fluid bag, which is a flexible sealed container, is full of fluid. It is a new kind of nonmetal spring working through the approximate incompressibility of fluid. The diagram of fluid bag is
shown in Fig. 1, the outside part is bag cloth, and the inside part is water. The pressure of fluid bag can be adjusted to the predetermined value.
Fig. 1The diagram of fluid bag model
2.2. The application of fluid bag
The diagram of a mechanism which requires axial protection is shown in Fig. 2. The inner and outer shell are glued together through the rubber joints. The steel and rubber are vulcanized to the
joints, which is similar to the rubber springs for rolling stock [20]. Large axial upward shock loads are imposed on the bottom of inner shell when the mechanism works. However, as for the inner
shell, large axial static displacement is not allowed, otherwise the adhesive joints will fail to work. In order to avoid it, a fluid bag which is able to decrease the loads transmitted to the
joints, was placed between the inner and outer shell, as is shown in Fig. 3.
Fig. 2The diagram of mechanism
Fig. 3The diagram of mechanism with a fluid bag
For the purpose of obtaining its axial protection performance, the axial stiffness characteristics of fluid bag will be analyzed alone. The simplified model for analyzing the axial stiffness is shown
in Fig. 4. The outer shell was fixed, and the inner shell was adjusted to a predetermined location to be fixed in the axle. The fluid bag pressure was adjusted to a predetermined value. Then the
inner shell was set up to be free and large axial loads were applied to its bottom region. The inner and outer shell models are shown in Figs. 5 and 6.
Fig. 4The diagram of simplified model
Fig. 5The diagram of inner shell model
Fig. 6The diagram of outer shell model
While the axial loads are transmitted from the inner shell to rubber joints, the fluid bag are compressed first. Due to the approximate incompressibility of fluid, large loads are generated by the
bag under small axial displacement and deformation, avoiding the large static displacement in rubber joints.
3. The analysis of the axial stiffness of fluid bag
3.1. The simulation of fluid bag
In Abaqus, there are a family of elements being able to represent fluid-filled cavities under hydrostatic conditions. The elements provide the coupling between the deformation of fluid-filled
structure and the pressure exerted by fluid on the boundary of cavity [21]. When the fluid bag is compressed, the volume and pressure of fluid will change with its deformation. The volume function of
fluid is given as:
$\stackrel{-}{V}=\stackrel{-}{V}\left(p,\mathrm{\theta },m\right),$
where $p$ is the initial fluid pressure, $\theta$ is the fluid temperature, $m$ is the fluid mass. Due to the bag being full of fluid, the fluid volume is always equal to the bag volume. So:
where $V$ is the volume of bag cavity.
When the bag deforms, the volume and pressure of fluid will change. When the fluid pressure equals to p, the virtual work contribution due to the fluid pressure is given as:
$\delta {T}^{*}=\delta T-p\delta V-\delta p\left(V-\stackrel{-}{V}\right),$
where $\delta {T}^{*}$ is the increment of virtual work and $\delta T$ is the virtual work for the bag without the cavity. The negative signs mean that an increase in the bag volume releases energy
from the fluid. This indicates a mixed formulation in which the bag displacement and fluid pressure are the two primary variables. The rate of the augmented virtual work expression is shown as:
$d\delta {T}^{*}=d\delta T-pd\delta V-\left(dV-d\stackrel{-}{V}\right)\delta p.$
Eq. (4) can be transformed into:
$d\delta {T}^{*}=d\delta T-pd\delta V-\left(dV-d\stackrel{-}{V}\right)\delta p+\frac{d\stackrel{-}{V}}{dp}dp\delta p,$
where $–pd\delta V$ indicates the pressure load stiffness, and $d\stackrel{-}{V}/dp$ represents the volume-pressure compliance of fluid.
Since the pressure is the same for all the elements of bag and fluid, the virtual work contribution can also be written as:
$\delta {T}^{*}=\delta T-p{\sum }_{a}\delta {V}^{a}-\delta p\left[{\sum }_{a}{V}^{a}-{\sum }_{a}{\stackrel{-}{V}}^{a}\right].$
Eq. (6) can be transformed into:
$\delta {T}^{*}={\sum }_{a}\left[\delta {T}^{a}-p\delta {V}^{a}-\delta p\left({V}^{a}-{\stackrel{-}{V}}^{a}\right)\right],$
where ${T}^{a}$ means the virtual strain energy of an element, ${V}^{a}$ means the volume a fluid element accounts for in the bag when it is compressed, and ${\stackrel{-}{V}}^{a}$ means the volume
of a fluid element.
Since the temperature is the same for all the elements of fluid, the volume of every fluid element can be expressed as:
${\stackrel{-}{V}}^{a}={\stackrel{-}{V}}^{a}\left(p,\theta ,{m}^{a}\right),$
where ${m}^{a}$ means the mass of an element.
When the external load or fluid pressure changes, so will the shape of bag and fluid. Therefore, the volume a fluid element occupies in the bag may be different from the actual volume of a fluid
element. Thus:
${V}^{a}-{\stackrel{-}{V}}^{a}e 0.$
However, the total fluid volume is always equal to the bag volume.
To simulate the real mechanical characteristics of fluid bag is to simulate the coupling between the bag and fluid. There are hydrostatic elements being approximate incompressible in Abaqus. These
elements can simulate the coupling between the bag deformation and fluid pressure. The fluid mass $m$ is related to its density$\mathrm{\rho }$, so the variable $m$ can be indicated by $\rho$.
Considering the density, temperature and pressure is respectively uniform, a reference node has been set for all the fluid elements, and the coupling between the reference node and every element has
been built. Then the density, temperature and pressure of fluid have been set in the reference node. Thus the real simulation of fluid bag has been completed.
3.2. The finite element model
Due to the axial and circumferential size of mechanism being much bigger than its thickness, shell elements were used for the simulation. The S4R finite-strain shell elements were used for the bag,
the inner and outer shell. In an attempt to simulate the coupling between the bag and fluid, the fluid elements shared the same nodes with the bag elements. To be specific, the bag elements were
duplicated. Then the duplicated elements were changed into F3D4 hydrostatic elements, the element labels being changed and node labels remaining unchanged. The finite element model is shown in Fig.
Fig. 7The finite element model of mechanism
3.3. The simulation of contact
While being compressed, the bag will contact with inner and outer shell. The contact states changes along with the variation of bag shape. However, the rule of this variation is beyond prediction.
Therefore, it is necessary to establish real contact pairs. Surface with coarse meshes, a large area and high stiffness are always the master surfaces, and soft surface with a small area and fine
meshes are always the slave surfaces [22]. There are two contact pairs in this paper. The first contact pair, the outer surface of inner shell being the master surface and inner diameter surface of
bag being the slave surface, is shown in Fig. 8. The second contact pair, the inner surface of outer shell being the master surface and outer diameter surface of bag being the slave surface, is shown
in Fig. 9. To ensure an easy convergence for the calculations, tolerance of 0.1 mm has been specified for the two contact pairs.
Fig. 8The master surface and slave surface for first contact pair
Fig. 9The master surface and slave surface for second contact pair
3.4. The boundary conditions and material properties
According to chapter 2.2, the outer shell is fixed. For the purpose of axial loads being imposed easily, the continuous distributing coupling was established between the coordinate origin and the
bottom surface of inner shell, making the loads imposed on the coordinate origin distribute on the bottom surface of inner shell. The boundary conditions and load coupling are shown in Fig. 10.
Fig. 10The diagram of boundary conditions and load coupling
The material of inner and outer shell is high-strength steel. The bag material is rubber fiber, and the bag is full of water. The equivalent elastic modulus, Poisson’s ratio and density are given in
Table 1.
Table 1The material properties of model
Material Density (kN/mm^3) Elastic modulus (MPa) Possion’s ratio Bulk modulus (MPa)
High-strength steel 7.8×10^-^9 2.1×10^5 0.3
Rubber fiber 1.4×10^-^9 7×10^3 0.3
Water 2200
3.5. The experimental verification of axial stiffness characteristics
Due to the fluid bag providing axial protection for rubber joints, the whole stiffness of fluid bag and joints were measured in experiments. Then the stiffness of fluid bag could be derived from the
stiffness of joints measured in advance. The inner shell would be restricted by the joints while moving down in the axle. Therefore, its position does not need to be adjusted while the bag are being
filled with fluid.
3.5.1. The introduction of experiment
The schematic is shown in Fig. 11, where the inner and outer shell are connected by the rubber joints as displayed. The fluid bag lies between the inner and outer shell. Four brackets, being made of
steel, are bolted with the outer shell and pinned to the ground uniformly. A piece of load transfer plate is attached and bolted to the inner shell, and is connected to the hydraulic cylinder.
Fig. 11The experimental schematic diagram
A pressure sensor is established at the orifice of the fluid bag for the purpose of measuring the pressure inside the fluid bag. Four displacement sensors are placed on the edge of the bottom of the
inner shell, which will offer an average displacement value for the experiment. The pressure sensor, displacement sensor, dynamic testing instrument and hydraulic cylinder is respectively shown in
Fig. 12 to Fig. 15.
The form of load in the experiment is shown in Fig. 16.
Fig. 12The pressure sensor
Fig. 13The displacement sensor
Fig. 14The dynamic testing instrument
Fig. 15The hydraulic cylinder
Fig. 16The curve of load varying with time
The experiment procedures are listed below.
Step 1, the stiffness of the rubber joints without the fluid bag was measured. The load was applied as required, and the displacement of the edge of the bottom of the inner shell along the axial
direction was measured.
Step 2, the fluid bag was placed between the inner and outer shell, and was pressurized to 1.5 MPa. The air in the bag being exhausted as far as possible, the load was applied according to the
requirement, and the displacement of the edge of the bottom of the inner shell along the axis direction as well as the pressure in the fluid bag was tested.
Step 3, the initial pressure of the bag was set to 2.0 MPa, then other procedures in Step 2 were repeated.
Step 4, according to the data from last three steps, the displacement-load curve and pressure-load curve were formed. Afterwards, the least square method was used to form the linear fitting chart and
the slopes of the lines were acquired.
Step 5, from Step 4 the axial stiffness of the rubber joints and the axial stiffness of the fluid bag together with the rubber joint were acquired.
3.5.2. The comparison between simulation and experiment results
When the initial bag pressure is 1.5 MPa, the simulation and experiment results are shown in Fig. 17 and Fig. 18.
When the initial bag pressure is 2.0 MPa, the simulation and experiment results are shown in Fig. 19 and Fig. 20.
Fig. 17The diagram of whole axial stiffness of simulations and experiments
Fig. 18The diagram of bag pressure of simulations and experiments
Fig. 19The diagram of whole axial stiffness of simulations and experiments
Fig. 20The diagram of bag pressure of simulations and experiments
According to Fig. 17 to Fig. 20, the curves of axial stiffness and pressure variations have good linearity. The difference of results between the simulations and experiments is small, proving the
correctness of simulations.
Table 2The whole axial stiffness under different initial bag pressures
The axial stiffness of inner shell (kN/mm)
The bag pressure / MPa
The simulation result The experimental result
1.5 224.3 234.7
2.0 222.8 231.6
As for the rubber joints, the elastic modulus of steel is 210000 MPa, and the equivalent modulus of rubber is 1.25 MPa. The measured axial stiffness of rubber joints is 74.2 kN/mm. Therefore, the
measured axial stiffness of fluid bag is 160.5 kN/mm under 1.5 MPa initial pressure, and the value would be 157.4 kN/mm under 2.0 MPa initial pressure.
According to Fig. 18 and Fig. 20, the fluid bag is compressed by the inner shell with the increasing axial load. Due to the bag being full of approximately incompressible fluid, the bag pressure
increases rapidly during the progress. Most of the loads on the inner shell could be transmitted to outer shell through the fluid bag, providing axial protection for the rubber joints. And the bag
pressure variations of simulations and experiments are very similar, which demonstrates that the simulations hold water.
4. The factors affecting the axial stiffness characteristics
The fluid bag was positioned between the inner and outer shell, and the axial displacement of inner shell was adjusted to –0.35 mm. Then the inner shell was fixed. The temperature of reference node
was set to 20°C, and the fluid density was 1×10^-9 kN/mm^3. Then the bag got filled with fluid: the bag pressure being set to ${P}_{1}$, the pressure ${P}_{2}$ was calculated when the fluid bag was
in balance. If ${P}_{2}$ had not reached the initial working pressure predetermined, ${P}_{1}$ would be adjusted until ${P}_{2}$ met it. Then all the constraints of inner shell got removed.
Meanwhile, the axial load was imposed on the bottom of inner shell, analyzing the axial stiffness characteristics of fluid bag.
According to the model in Chapter 2, the axial stiffness of fluid bag was calculated. When the initial pressure is 2.5 MPa and load is 250 kN, the results are shown in Fig. 21 and Fig. 22.
Fig. 21The stress nephogram of mechanism when the load is 250 kN
Fig. 22The axial displacement nephogram of mechanism when the load is 250 kN
According to Fig. 21 and Fig. 22, the stress and axial displacement nephogram are evenly distributed in the circumferential. According to Fig. 22, the fluid bag moves downward when the inner shell
moves upward along the axle.
4.1. The axial stiffness of fluid bag under different initial pressures
The results are displayed in Fig. 23 to Fig. 27 when the initial pressure is respectively 1.5 MPa, 2.0 MPa, 2.5 MPa, 3.0 MPa and 3.5 MPa.
Fig. 23The diagram of axial displacement and load of inner shell
Fig. 24The diagram of bag pressure and load
According to Fig. 23, the axial stiffness of fluid bag remains basically unchanged with load increasing when an initial pressure is exerted. The axial stiffness remains basically unchanged when the
initial pressure turned from 1.5 MPa to 3.5 MPa. According to the calculations, while the initial pressure varies from 1.5 to 3.5 MPa, the corresponding stiffness are 160.93, 158.95, 156.91, 154.98
and 153.12 kN/mm, which are extremely close to the experimental results.
According to Fig. 24, the initial loads are different, owing to the fluid bag having different balanced loads when different initial pressures are exerted. The bag pressure increases along with load,
and the pressure variation tendency has tiny nonlinearity. For example, when the load is 177.28 kN and initial pressure is respectively 1.5 to 3.5 MPa, the corresponding bag pressure is 3.23, 3.30,
3.38, 3.44 and 3.50 MPa, and the pressure difference of adjacent curves are respectively 0.079, 0.072, 0.067 and 0.058 MPa. When the load increases to 497.28 kN, the corresponding bag pressures are
7.91, 8.09, 8.27, 8.42 and 8.56 MPa, and the pressure difference of adjacent curves are respectively 0.179, 0.172, 0.156 and 0.141 MPa. The pressure difference of adjacent curves increases with load
when different initial pressures are exerted.
Fig. 25The diagram of inner contact area and load
Fig. 26The diagram of outer contact area and load
In Fig. 25 and Fig. 26, the inner contact area means the contact area between inner shell and fluid bag, and the outer contact area means the area between outer shell and fluid bag. According to the
figures, the inner and outer contact area both increase along with load, after reaching their respective maximum values, they are likely to almost remain unchanged under different initial pressures,
except for 1.5 and 2.0 MPa. Owing to being relatively soft, the bag is gradually compressed by the increasing load, making the contact area gradually expand. Due to being not able to be compressed
unlimitedly, the contact area will remain basically unchanged after reaching their peak figures. While the initial pressure is respectively 1.5 and 2.0 MPa, the contact area shrinks with load after
reaching their peak figures. According to Fig. 22, due to the fluid bag moving downward with load, the low part of bag would no longer contact with the inner and outer shell after the contact area
reaching peak figures, which eventually leads to the reduction of the area of contact.
The corresponding contact area decreases with the initial pressure. The higher initial pressure, as well as the large amount of fluid contained by the bag, is likely to reduce the initial contact
area. However, when the contact area reaches peak figures, the corresponding load would increase with initial pressure. It can be inferred that the higher the initial pressure is, the stronger the
bag’s ability to resist deformation is.
Fig. 27The diagram of bag volume and load
According to Fig. 27, the bag volume decreases with load, and the variation tendency remains basically unchanged under different initial pressures.
4.2. The axial stiffness of fluid bag under different material properties
The initial pressure of fluid bag is 2.5 MPa, and the fluid bulk modulus is 2200 MPa. The results are shown in Fig. 28 and Fig. 29 when the elastic modulus of bag is respectively 4000, 5000, 6000,
7000, 8000 and 9000 MPa.
Fig. 28The diagram of axial displacement and load of inner shell
Fig. 29The diagram of bag pressure and load
According to Fig. 28, the axial stiffness of fluid bag is evidently different under different elastic modulus. The axial stiffness increases with elastic modulus, but the trend of rising grows weaker
afterwards. For example, when the elastic modulus of bag varies from 4000 to 9000 MPa, the corresponding stiffness are 101.84, 121.74, 140.03, 156.91, 172.55 and 187.09 kN/mm, and the stiffness
differences between the adjacent curves are respectively 19.9, 18.29, 16.88, 15.64 and 14.54 kN/mm.
According to Fig. 29, all the curves almost coincide with one another, indicating that the difference of elastic modulus basically has no effects on the bag pressure variation.
4.3. The axial stiffness of fluid bag under different fluid bulk modulus
The initial pressure of fluid bag is 2.5 MPa, and the elastic modulus of bag is 7000 MPa. The results are shown in Fig. 30 and Fig. 31 when the fluid bulk modulus is respectively 200, 400, 600, 1000,
1400, 1600, 1800, 2000, 2200 and 2400 MPa.
Fig. 30The diagram of axial displacement and load of inner shell
Fig. 31The diagram of the pressure and load
According to Fig. 30 and Fig. 31, the axial stiffness and bag pressure increase with load, but the increasing trends are becoming weaker.
The axial stiffness and pressure variation of fluid bag have bad linearity when the fluid bulk modulus is small. It is indicated that the fluid is easy to be compressed under small bulk modulus,
making the bag have relatively large deformation. However, with the fluid bulk modulus increasing gradually, the fluid becomes difficult to be compressed, making the deformation progress harder to
occur on the bag. Thus the curves have relatively good linearity.
4.4. The finite element analysis of fluid bag for axial protection
According to Chapter 2.2, the inner and outer shell are glued together through the rubber joints. And the inner shell will have a big impact upward in the axle when the mechanism works. If the stress
in the rubber joints is too large, it may lead to the mechanism failing to work. Therefore, it is necessary to decrease the stress in the joints.
Fig. 32The finite element model of mechanism
Fig. 33The finite element model of mechanism with a fluid bag
In order to obtain the bag’s ability to decrease the stress, a simplified shock model has been established. Considering the exactitude and efficiency in calculation, the rubber joints are simplified
as a rubber pad. Due to the focus of research being the bag’s ability to decrease the stress, the modeling of adherend layer is ignored. The exact model is shown in Fig. 32. It is assumed that the
outer shell is stationary while the load is transmitted upward. Thus the outer shell is fixed. The finite element model of mechanism with a fluid bag is shown in Fig. 33. The joints, inner and outer
shell are joined together with tie constraints. The material properties are shown in Table 1, and the boundary conditions are shown in Fig. 10.
When the load is 250 kN, the stress distribution between the joints and inner shell is shown in Fig. 34 to Fig. 37.
According to Fig. 34 to Fig. 37, the stress distribution trends are basically the same whether mechanism has a bag or not. When it comes down to the mechanism without a bag, the maximum stress
between the bag and inner shell is 5.525 MPa, and the value between the bag and outer shell is 6.066 MPa. While the mechanism has a bag, the value between the bag and inner shell becomes 2.449 MPa,
and the value between the bag and outer shell becomes 2.689 MPa. It is indicated that the bag can decrease the stress in the joints effectively, having a good performance for axial protection.
Fig. 34The diagram of stress between the joints and inner shell without the fluid bag
Fig. 35The diagram of stress between the joints and outer shell without the fluid bag
Fig. 36The diagram of stress between the joints and inner shell with a fluid bag
Fig. 37The diagram of stress between the joints and outer shell with a fluid bag
5. Conclusions
1) The fluid bag is designed for axial protection under small displacement and deformation. The hydrostatic elements are used to simulate the incompressible fluid, and the hydrostatic theory is
applied to simulate the coupling between the bag and fluid. The axial stiffness and pressure variation of bag both has good linearity. The disparity between the simulations and experimental results
seems to be quite small, proving the accuracy of simulations.
2) The effects of initial pressure on the fluid bag have been discussed. When the initial pressure varies from 1.5 MPa to 3.5 MPa, the axial stiffness and pressure variation of bag would basically
maintain their original values.
3) The effects of material properties on the fluid bag are discussed. Despite having no effects on the pressure variation, the elastic modulus of bag appears to be one of the crucial factors
affecting its axial stiffness characteristics. The axial stiffness of bag increases with its elastic modulus.
4) The effects of fluid bulk modulus on the fluid bag are discussed. The bulk modulus is another crucial factor affecting the axial stiffness. The nonlinearity of axial stiffness and pressure
variation curves gradually becomes weak with the bulk modulus. The axial stiffness of bag increases with the fluid bulk modulus, and the slope of pressure varying with load increases with the fluid
bulk modulus.
5) The effects of fluid bag on a shock mechanism are discussed. The fluid bag can effectively decreases the stress in the rubber joints, having a good performance for axial protection.
• Chew A., Brewster B., Olsen I., et al. Developments in nEXT turbomolecular pumps based on compact metal spring damping. Vacuum, Vol. 85, Issue 12, 2011, p. 1156-1160.
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University, 2002.
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2014, p. 67-80.
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• Al-Samhan A., Darwish S. M. H. Finite element modeling of weld-bonded joints. Journal of Materials Processing Technology, Vol. 142, Issue 3, 2003, p. 587-598.
• Goncalves V. M., Martins P. A. F. Static and fatigue performance of weld-bonded stainless steel joints. Materials and Manufacturing Processes, Vol. 21, Issue 8, 2006, p. 774-778.
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About this article
Mechanical vibrations and applications
fluid bag
axial stiffness
factors affecting the stiffness
axial protection
Copyright © 2015 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/15321","timestamp":"2024-11-08T09:21:37Z","content_type":"text/html","content_length":"147261","record_id":"<urn:uuid:829cf7a2-484d-453b-8196-40b804d2e72e>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00712.warc.gz"} |
Performance Evaluation 6 (Question 5 & 6) - Physics Form 4 Chapter 6 - SPM Physics
Performance Evaluation 6 (Question 5 & 6) – Physics Form 4 Chapter 6
Question 5:
When light from a star travels into the Earth’s atmosphere, its direction of travel will change. This situation is shown in Figure 5. The change of direction is represented by the angle Δθ = i − r.
(a) Speed of light in air is 299 910 km s^−1 and speed of light in vacuum is 3.00 × 10^8 m s^−1.
(i) Calculate the refractive index of air.
(ii) Explain the value of refractive index obtained.
(b) Value of Δθ on a hot night is different from that on a cold night. State a logical reason for the difference.
(c) Rajiv returns from school in a school van on a hot and bright day. Rajiv can see a puddle of water on the surface of the road ahead. When the van reaches the location of the puddle of water,
Rajiv discovers that the puddle of water does not actually exist. Explain this phenomenon.
$$ \begin{aligned} n & =\frac{c}{v} \\ n & =\frac{3.0 \times 10^8 \mathrm{~m} \mathrm{~s}^{-1}}{2.9991 \times 10^8 \mathrm{~m} \mathrm{~s}^{-1}} \\ & =1.0003 \end{aligned} $$
(a)(ii) The value of the refractive index of air is almost equal to 1 , that is the speeds of light in air and in vacuum are almost the same.
(b) The value Δθ on a hot night is different from that on a cold night because the optical density of air depends on temperature.
Layers of air above the road have different optical density.
The layer of air just above the road surface is hotter than the upper layers.
The layer of hot air has smaller optical density than cold air.
Light which travels from the upper layer to the lower layer is refracted away from the normal repeatedly.
When the angle of incidence is larger than the critical angle of air, total internal reflection occurs.
Reflected light rays are then refracted towards the normal and reach Rajiv’s eyes.
Rajiv will see the image of clouds as a puddle of water on the road surface.
Question 6:
Figure 6 shows an object and its virtual image formed by a convex lens.
(a) One of the characteristics of image I in Figure 6 is that it is virtual. State the other characteristics of image I.
(b) Complete the ray diagram in Figure 6 and determine the position of the lens and focal point of the lens. Mark the position of the focal point of the lens with, F.
(c) If the object is slowly moved away from the lens, state two changes that might happen to the image without drawing a ray diagram.
(a) Upright and magnified
(c) The image will turn real, inverted, diminished and formed on the opposite side of the lens from the object. | {"url":"https://spmphysics.blog.onlinetuition.com.my/light/performance-evaluation-6-question-5-6-physics-form-4-chapter-6/","timestamp":"2024-11-10T01:38:28Z","content_type":"text/html","content_length":"69849","record_id":"<urn:uuid:1970486d-8430-489e-8b77-3494f72b1f38>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00509.warc.gz"} |
snowflake OA – snowflake OA String Formation – snowflake OA calculateMax-snowflake Coding Challenge Walkthrough-雪花OA - csOAhelp|代码代写|面试OA助攻|面试代面|作业实验代写|考试高分代考
String Formation
Given an array of strings, each of the same length, and a target string, construct the target string using characters from the strings in the given array such that the indices of the characters in
the order in which they are used form a strictly increasing sequence. Here the index of a character is the position at which it appears in the string. Note that it is acceptable to use multiple
characters from the same string.
Determine the number of ways to construct the target string. One construction is different from another if either the sequences of indices they use are different or the sequences are the same but
there exists a character at some index such that it is chosen from a different string in these constructions. Since the answer can be very large, return the value modulo (10^9 + 7).
Consider an example with n = 3 strings, each of length 3. Let the array of strings
The next part consists of multiple-choice questions for about 15 minutes. Don't stress about this section; just answer honestly according to your true thoughts, there's no need to try to guess the
correct answers.
If you're afraid that you can't solve the OA on your own, please scan the code to contact me or telegram | {"url":"https://csoahelp.com/2024/04/23/snowflake-oa-snowflake-oa-string-formation-snowflake-oa-calculatemax-snowflake-coding-challenge-walkthrough-%E9%9B%AA%E8%8A%B1oa/","timestamp":"2024-11-13T11:24:59Z","content_type":"text/html","content_length":"88146","record_id":"<urn:uuid:a36308f9-3fe8-42fa-bb11-f7f505a1d014>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00702.warc.gz"} |
Managing chaos: ensembles and forecasting the climate system | EarthSystemData
Forecasting weather, the climate or oceans is difficult for a host of reasons. Devising a set of equations that can describe how air and water circulate across the planet under different conditions
(height, temperature, changing density etc) to name but one. But beyond that, even with a good set of equations, predictions are plagued by another problem: that very small differences in the
starting numbers for the forecast equations – ‘initial conditions’ – lead to vastly-different end forecasts. How do we address this problem for end-users, especially for long-range forecasts?
The exact problem may be known to you already as the ‘Butterfly effect’ – the idea that the flapping of a butterfly’s wings can result in a particular sequence of chain reaction events leading to
vastly different outcomes elsewhere. This illustration, in fact, was coined by the mathematician and meteorologist, Edward Lorenz, himself who was central to discovering the ‘starting condition
problem’ together with researching its effects on how forecasts evolve through time and, ultimately, arrive at different predictions.
In the 1960s Lorenz created a simplified set of equations for an aspect of weather dynamics called ‘convection’ – describing how to simulate heat-driven rising and sinking of air masses. When
applying his equations into weather forecasting computer code, Lorenz was at a loss to explain how two simulations using the same computer arrived at radically different weather forecasts.
All he had done, for the second forecast, was to stop the computer half way through its run, print out the weather conditions at that point, and then re-input the numbers later to re-start, and
finish, the second forecast.
Unbeknown to Lorenz at the time, the printer used a slightly lower decimal precision for the numbers than used in the computer’s physical memory. In the first forecast (with no stopping involved) the
whole forecast proceeded at high numerical precision. The second forecast, however, the numbers had very slightly changed precision half way through.
Lorenz was still perplexed: both sets of numbers were reasonably high precision, what difference could five, rather than six, decimal precision make to the same numbers? In fact, and as his
subsequent research showed, a lot.
Equations describing a system have beautiful shapes - depicting the tendency of values of the system's properties through time - 'attractors'. Small differences in the starting co-ordinates within
the shape lead to different paths through the shape: predicted weather is similar initially, but increasingly divergent after a while - proportional to the space between the two paths. This
attractor: Nicolas Desprez, Chaos Scope http://www.chaoscope.org/
Lorenz had discovered the initial condition sensitivity of weather forecast equations, where even very small differences in the numbers, profoundly different forecasts emerged, and the differences
grew bigger the farther into the future the forecasts ran. This characteristic behaviour for solutions of this type of equations was eventually written up and gave rise to new research fields within
fluid dynamics: Lorenz systems, chaos theory, non-periodic flow, together with ideas about how this effect could be managed, especially given that we do not have perfect observations (starting
conditions) for the whole planet, when launching a forecast.
What, then, are ‘ensembles’ and how/why are they used? A good analogy is to consider the meaning of the word in a musical sense – a collection of components (instruments) all contributing together to
make a set. In our case, however, the ensemble is a collection of forecasts simulations, all identical but each with slightly varying starting conditions. An ensemble approach is needed for
forecasting because our real-time weather observations, whilst good, are nowhere near perfect – in terms of geographical coverage and accuracy. To account for this, an ensemble of forecasts, can
maximise as many runs as is computationally possible – to reflect our uncertainty of the observations. The end user is provided with a range of possible outcomes, that reflects how the forecasts vary
in each ensemble member.
Four forecasts – each identical, other than the starting observations are slightly different, accounting for the fact that weather observations are not 100% accurate. They agree well up to a point
and then diverge significantly. Each is as likely as the other. Is it better to give users the average of the four forecasts - or all four individual forecasts?
In part, this approach is why you hear phrases such as ‘10% chance’ of rainfall, or a ‘20% probability’ of a warmer than average summer: the numbers reflect how many of the ensemble members give rise
to that particular weather condition. Long range forecasts are especially susceptible to initial condition sensitivity because, as Lorenz showed, the differences due to small differences in the
starting conditions get very large the farther in time the simulations are run.
By averaging all ensemble members together and providing the user with ‘mean forecast conditions’ we can give users a general indication of forthcoming weather – up to a point. However, since all
ensemble members are equally possible – a particular ensemble member that predicts a very very warm month might be as equally as likely as an ensemble member predicting average conditions, it is
valuable to the user, if possible, to provide the full set of ensemble members.
This is especially important for organisations needing to plan for ‘worst’ or ‘best’ case scenarios. It may not be possible to say which forecast is likely, but stress-testing an organisation’s
strategy against the worst and best case conditions can identify shortcomings, and ways to improve resilience. In facing adverse weather, this preparedness can prove extremely valuable – especially
if the worst-case forecasts transpire.
Version 1.0 of ESD’s Seasonal Forecaster service currently provides water-industry specific ensemble-average data from five national numerical weather prediction systems. Our intention has been to
expand this to provide full sets of ensemble data – each transformed into the area and time frame of interest to our users – and we are pleased to announce that development of this feature has now
started. V2.0 will continue to give users bespoke, ensemble-average output for DWD, NCEP, UKMO and Meteo-France systems, but will offer ensemble products for the ECMWF system.
More soon as this feature is launched!
– Craig Wallace, EarthSystemData
EarthSystemData designs and deploys climate data solutions to meet organisational needs. We specialise in climate risk assessment, risk disclosure and adaptation projects assisting multi-national
firms through to local clients. To discuss using our research or consultancy services contact us here: info[at]earthsystemdata.com | {"url":"https://www.earthsystemdata.com/managing-chaos-ensembles-and-forecasting-the-climate-system/","timestamp":"2024-11-05T16:13:26Z","content_type":"text/html","content_length":"82682","record_id":"<urn:uuid:93855704-7368-40b3-91ba-a343c3a4cf97>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00473.warc.gz"} |
Go to the source code of this file.
Detailed Description
RTE Power Management via userspace ACPI cpufreq
Definition in file rte_power_acpi_cpufreq.h.
Function Documentation
int rte_power_acpi_cpufreq_init ( unsigned lcore_id )
Initialize power management for a specific lcore. It will check and set the governor to userspace for the lcore, get the available frequencies, and prepare to set new lcore frequency.
□ 0 on success.
□ Negative on error.
int rte_power_acpi_cpufreq_exit ( unsigned lcore_id )
Exit power management on a specific lcore. It will set the governor to which is before initialized.
□ 0 on success.
□ Negative on error.
uint32_t rte_power_acpi_cpufreq_freqs ( unsigned lcore_id,
uint32_t * freqs,
uint32_t num
Get the available frequencies of a specific lcore. The return value will be the minimal one of the total number of available frequencies and the number of buffer. The index of available frequencies
used in other interfaces should be in the range of 0 to this return value. It should be protected outside of this function for threadsafe.
lcore_id lcore id.
freqs The buffer array to save the frequencies.
num The number of frequencies to get.
The number of available frequencies.
uint32_t rte_power_acpi_cpufreq_get_freq ( unsigned lcore_id )
Return the current index of available frequencies of a specific lcore. It will return 'RTE_POWER_INVALID_FREQ_INDEX = (~0)' if error. It should be protected outside of this function for threadsafe.
The current index of available frequencies.
int rte_power_acpi_cpufreq_set_freq ( unsigned lcore_id,
uint32_t index
Set the new frequency for a specific lcore by indicating the index of available frequencies. It should be protected outside of this function for threadsafe.
lcore_id lcore id.
index The index of available frequencies.
□ 1 on success with frequency changed.
□ 0 on success without frequency changed.
□ Negative on error.
int rte_power_acpi_cpufreq_freq_up ( unsigned lcore_id )
Scale up the frequency of a specific lcore according to the available frequencies. It should be protected outside of this function for threadsafe.
□ 1 on success with frequency changed.
□ 0 on success without frequency changed.
□ Negative on error.
int rte_power_acpi_cpufreq_freq_down ( unsigned lcore_id )
Scale down the frequency of a specific lcore according to the available frequencies. It should be protected outside of this function for threadsafe.
□ 1 on success with frequency changed.
□ 0 on success without frequency changed.
□ Negative on error.
int rte_power_acpi_cpufreq_freq_max ( unsigned lcore_id )
Scale up the frequency of a specific lcore to the highest according to the available frequencies. It should be protected outside of this function for threadsafe.
□ 1 on success with frequency changed.
□ 0 on success without frequency changed.
□ Negative on error.
int rte_power_acpi_cpufreq_freq_min ( unsigned lcore_id )
Scale down the frequency of a specific lcore to the lowest according to the available frequencies. It should be protected outside of this function for threadsafe.
□ 1 on success with frequency changed.
□ 0 on success without frequency chnaged.
□ Negative on error. | {"url":"http://doc.dpdk.org/api-2.2/rte__power__acpi__cpufreq_8h.html","timestamp":"2024-11-01T22:09:38Z","content_type":"application/xhtml+xml","content_length":"16212","record_id":"<urn:uuid:31a364d9-867b-46ad-a6dd-055b4ef6c68a>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00238.warc.gz"} |
Data Structures
Learn Data Structures and Types, ADT, List, Graph, Tree, Traversal
A data structure is a way of storing and organizing data for easy and faster retrieval and maintenance.
• Faster retrieval includes easy reading and also faster searches.
• Faster maintenance includes faster insertion, faster updates and deletion of data in a large data
A data structure can either be linear or non-linear. Basic operations on data structures include insertion, deletion, searching, find, and sorting
Linear Data Structures
Storing data in a sequential manner. Lists, Stacks and Queues are linear data structures
Non-Linear Data Structures
Storing data in a non-sequential fashion. It can be in hiercharchial order or any other non-linear style
Example: Trees, Graphs
Linear Data Structures
Data can be linearly arranged in different ways
• Stack: Stack follows a Last-In -First-Out approach (LIFO). LIFO means that when you retrieve or remove data, you always remove the data that you recently added. You can add and remove data at one
end only. Stack is like a box in which you keep putting books one over the other. You can only pick up books one by one from the top. You cannot reach the bottom of the stack directly without
removing the top items one by one
• Queue: you can add data at the end and remove data at the other end. In this case, you service the data you added first. In a queue,you always serve the customer who came first and stood in the
• Priority Queue: A queue with each value having a priority.
• Double Ended Queue: You can add and remove data at the both ends in a double ended queue. Double ended queue is sometimes called as Dequeue or Deque which is misleading as it also refers to
remove element from a queue and hence the terminology "Dequeue" should never be used.
• Singly Linked List: A list in which element has a link to the next element
• Double Linked list: A list in which element has a link to the next element and a link to the previous element except that first node in the list does not have a previous link and last node does
not have a next link
• Circular Linked List: A single or double linked list in which the last node has a link to the first node
A tree can be structured in different ways. A Tree is a data structure with one root data. Root can have one or more children. Child node can have zero or more children
• Trie: A search tree in which each node has a prefix of its parent.
• Binary Tree: A tree in which each parent has maximum of two children
• Binary Search Tree: Sorted Binary Tree
Graph Shortest Path Algorithms
A graph is a data structure with nodes and edges with edges connecting the nodes. Graphs can be undirected or directed | {"url":"https://www.krivalar.com/data-structures","timestamp":"2024-11-03T00:54:23Z","content_type":"text/html","content_length":"33501","record_id":"<urn:uuid:dc978ebe-392a-44c3-9ebf-7b99ed8900d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00022.warc.gz"} |
Daydream: Hit ambitious revenue goals
Next: Let's find time to get you started.
Next: Let's find time to get you started!
Get ready for a better, results-oriented future.
Next: Let's find time to get you started.
Make the fruits of your labor riper and more delicious
Make the fruits of your labor riper and more delicious
Make the fruits of your labor riper and more delicious
© 2023 Daydream Factory Corporation
© 2023 Daydream Factory Corporation | {"url":"https://daydream.co/success-book-now","timestamp":"2024-11-05T03:57:06Z","content_type":"text/html","content_length":"174478","record_id":"<urn:uuid:9d60b8c2-0abc-4b7a-b5a2-cacb67e96f7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00852.warc.gz"} |
Chan's Algorithm Demo
Step: 0 of 5
First, let's add some points to our plane. You can click to the left to add your own points or press P to add random points.
In the next few steps we will be grouping our point set into smaller convex hulls and then finding the convex hull of those smaller hulls.
Hit the space bar when you are ready to continue.
Now that we have our points, we pick some small constant m. The choice of m will be explained later in the demo.
We partition our point set into groups of m points.
In this example m is set to 5 and each color represents a group of m points.
Given these groups of m points, we find the convex hull of each group with an O(nlogn) algorithm. In this demo we use Graham Scan.
Because each group has size m we can convex hull each group in O(mlogm). There are O(n/m) groups.
In total this step takes O(nlogm) time.
We then use Gift wrapping, an O(nh) algorithm on the small convex hulls.
To use gift wrapping on convex hulls rather than points we can perform a binary search to determine the tangent between an extreme point and a convex hull. A binary search on a small convex hull
takes O(logm).
We can compute tangents for all O(n/m) groups in O(n/m * logm) time. We use the tangent with the largest angle. By doing this we get one edge of the overall convex hull. We must do this for all h
hull points.
We can assume for now that m < h so this step is O(nlogh) like the last step.
We have to be careful that we do not exceed h iterations of gift wrapping given our O(n/m) input size. Therefore, we halt gift wrapping execution after m iterations.
We want to increase m until it equals h. If we increase m too slowly our gift wrapping time overall will surpass O(nlogh). On the other hand, if we increase m too quickly some gift wrapping iteration
will take much more than O(nlogh) on its own.
To solve this problem we can utilize a double exponential. We let m = 2^2^t where t is the current iteration number.
We throw away the work we do in each attempt at finding an m equal to h. Because iterations form a geometric series the total work is still O(nlogh).
We now have the convex hull of our orignal point set.
Press restart to try it again! | {"url":"https://static.chrisgregory.me/sub/chans/","timestamp":"2024-11-04T18:36:30Z","content_type":"text/html","content_length":"10738","record_id":"<urn:uuid:93410abd-10c6-40e4-b521-e8147267163f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00212.warc.gz"} |
Artificial Intelligence: Hidden Markov Model Classifiers and RADAR Objects Classification by Machine Learning 2/2
In the first part of this series, we introduced the general concepts needed for understanding the Hidden Markov Models Classifiers. Namely: Bayesian Logic, The concept of Bayesian Classifiers and
Bayesian networks. All these notions are now grouped to form a new type of classifier which can accurately model and classify time-series data such as the RADAR data.
What is a Markov model ?
A Markov model is a stochastic model designed to model systems which varies over time and change their states and parameters randomly (e.g., dynamical systems) . This can be for example:
• The price of a crypto-currency;
• Board games played with one or more dice;
• Some values from a stock market;
• The trajectory of a vehicle;
• Data from the weather of some given location (snow/sunshine/rain);
• RADAR data (our interest here!)
There are four types of Markov models:
• Markov chains,
• Markov decision processes,
• Partially observable Markov decision processes and
• Hidden Markov models.
Markov models relates to systems - Markov processes - where the future state is only dependent of the ‘most recent' values.
A stochastic process x(t[i]), i=1,2... is said to be a Markov process is for every n > 0 and every numerical value y:
P(x(t[n]) `<=` y | x(t[n-1]), ..., x(t[1])) = P(x(t[n]) `<=` y | x(t[n-1]))
In other terms, the value of the state of the system at the instant T=tn only conditionally depends on the value of the state of the system at the previous instant T=t[n-1].
This property of the Markov model is often referred to by the following axiom:
‘The future depends on past via the present’. [9]
A Markov process with a finite number of possible states (‘finite’ Markov process) can be described by a matrix, the ‘transition matrix’, which entries are conditional probabilities, e.g (P(X[i]|X
As an example, we consider the Markov process created by the movement of an insect in the air - say a fly. The process has four states: “North, South, East, West” with the following (constant)
conditional probabilities:
P(X[i]|X[j]) N S E W
N 0.7 0.05 0.125 0.125
S 0.05 0.7 0.125 0.125
E 0.125 0.125 0.7 0.05
W 0.125 0.125 0.05 0.7
A Markov process is usually represented by a graph where the relations between the states are coded by connections.
In our model, at any position, the insect have higher probability (70%) of keeping up with its trajectory, a small probability (5%) of going back where it came from and equal probabilities or turning
left or right (12.5%).
If P is the transition matrix, then the transition matrix from a state at T=t[n] to a state at T=t[n+k] is given by the new transition matrix Q=P^k.
The above model is a Markov chain model because it is fully observable. Not all models have such properties, instead often the Markov model is hidden from observation and is known to the observer
only via side-events.
In this article, we are only interested in the latter, the Hidden Markov models (HMMs). They occur for autonomous systems which states are partially observable. In the case of the example with the
insect, the hidden model would be the nature of the flying insect ( fly, mosquito, dragonfly…) and the observable model would be the directions taken by the insect.
As an example, we display the trajectory of such an insect with 300 RADAR ‘spots’.
Next we display an example of such insect trajectory with 2,500 spots : A HMM classifier would have to decide the nature of the moving flying insect (hidden model) from such trajectories (observable
It would be trained by a set of known classified trajectories obtained from existing dataset.
Overview of Hidden Markov models
In a hidden Markov model (also named Labelled Markov Chain) , the Markov chain - itself - is hidden (X[i]), only we see observable events (E[i]) depending on the states of the Markov chain. Note that
in Hidden Markov models, variables are discrete and not continuous. The continuous version of HMM is named the linear-Gaussian state-space model, also known as the Kalman filter, not to be considered
In terms of classification/recognition this means to be able to classify information given a series of ‘characteristic’ observations. For example, we could classify a moving insect(Xi) knowing only
its trajectory(Ei).
Per se, hidden Markov models are not Machine Learning algorithms at all. They are a probability model and bear no information on how to learn, how to be trained and how to classify, so they need in
addition algorithms to do so. Hidden Markov Models are usually seen as a special type of Bayesian networks, the Dynamical Bayesian networks.
In such a model, the input vectors are N values of the observable model (among n possible states in the finite case) .The Bayesian network is simplified regards to a general Bayesian network since
every node (X[i]) has (X[i+1]) and C - the category node - as parents.
In the above example, we can express the joint probability as such:
P (X[1], ..., X[N], E[1], .. ,E[N]) = P (X[1]) P (E[1]|X[1]) `prod_(t=2)^N`P (X[t]|X[t-1]) P (E[t]|X[t])
This can be easily checked by following the arcs of the corresponding network.
Same as with the ‘general’ Bayesian network, our goal is to classify the data by maximization of a function, usually the maximal Likelihood, MLE.
This task is simplified since we know the structure of the network and we do not need to have the classifier to ‘learn’ it.
In what follows we explain how to compute the Likelihood in case that the variable are not hidden first and when they are hidden.
MLE estimation with non-hidden data
Here we recall the principle of MLE - or log MLE - computation (estimation) in the case the variables are not hidden.
To compute the joint probability in a Bayesian network it is in fact quite simple, it is needed to get the products of all the CPDs, e.g., the product of the conditional probabilities of each Xi
regarding to its parents in the network, Par(i).
P (X[1], ..., X[n]) = i = `prod_(t=2)^N`P (Xi[|]P ar(i))
Therefore the log-MLE can be computed by:
Max[`theta`] `f(theta)` = Max log `prod_(i=1)^n` P (Xi|P ar(i), `theta`)
In the case of the Hidden Markov model as described in Example #1, we have the following results:
Max[`theta`] `f(theta)` = Max[`theta`] log P (X[1]| `theta`) P (E[1]|X[1], `theta` ) `prod_(t=2)^N` P (X[t]|X[t-1],`theta`) P (E[t]|X[t])
In a hidden Markov model, the variables (X[i]) are not known so it is not possible to find the max-likelihood (or the maximum a-posteriori) that way.
Expectation–maximization (EM) Algorithm
In the case where the variables are hidden, which is the case here, it is necessary to use a special algorithm to compute the MLE, namely the EM algorithm.
The starting point is to consider an arbitrary distribution Q so that we can compute a lower bound for the log-likelihood.
log p (E|`theta`) = log `sum_X` p (E, X|`theta`) = log `sum_X Q(X)` `(p(E,X|theta))/(Q(X))`.
Since log is concave we get:
`log p(E|theta) <= sum_X Q(X) log (p(E,X|theta))/(Q(X))`
`log p(E|theta) <= sum_X Q(X) log p(E,X|theta) - sum_X Q(X) log Q(X)`
If we put
`F(E,theta) = sum_X Q(X) log p(E,X|theta) - sum_X Q(X) log Q(X)`
then `F(E,theta)` is a lower bound for the log-likelihood.
`F(E,theta)` is the opposite of the Free Energy as defined in statistical physics.
This is the basic for the expectation–maximization (EM) method which is an iterative algorithm allowing to find the maximum likelihood in such case where the variables X are hidden.
The EM method alternates between an expectation (E) step, which creates a distribution `Q` for the expectation of the log-likelihood, and a maximization (M) step, which provides the parameters
maximizing the log-likelihood found on the E step.
Baum-Welch Algorithm
In the precise case of a Hidden Markov model the Baum–Welch algorithm uses the forward-backward algorithm to compute the data in the E-step.
Usage of Hidden Markov Models Classifiers
Hidden Markov Models (HMM) are used for example for :
• Speech recognition;
• Writing recognition;
• Object or face detection;
• Fault Diagnostic.
• Web page ranking
Usage for the classification of RADAR Objects
As an example of successful techniques for target recognition, Cepstral-analysis, as well as wavelet-based transforms, can be used to extract feature vectors. Such feature vectors are used as input
data for a Hidden Markov Model.
We do not wish to elaborate on the various concrete systems using Markov Model techniques for classification of RADAR data but we merely aim at providing a few bibliographic references for the reader
who would wish to go deeper into the details.
Read more about artificial intelligence in Radar Technology.
Contact us to learn about training radars for ATC and University Education. Just click on the image below:
Some concrete implementations of Hidden Markov Models classifiers for RADAR object can be found in the following references:
• [1] Robust Doppler classification Technique based on Hidden Markov models (2003), Jahangir, M. et al. , IEE Proc
• [2] Atemgeräuscherkennung mit Markov-Modellen und Neuronalen Netzen beim Patientenmonitoring (2000), by Kouemou, G. Dissertation an der Fakultät für Elektrotechnik und Informationstechnik der
Universität Karlsruhe
• [3] Hidden Markov Models in Radar Target Classification (2007a), Kouemou, G. & Opitz, F. International Conference on Radar Systems, Edinburgh UK
• [4] Automatic Radar Target Classification using Hidden Markov Models (2007b), Kouemou, G. & Opitz, F. . International Radar Symposium, Cologne Germany
• [5] Unsupervised Classification of Radar Images Using Hidden Markov Chains and Hidden Markov Random Fields, by Roger Fjørtoft, Member, IEEE, Yves Delignon, Member, IEEE, Wojciech Pieczynski, Marc
Sigelle, Member, IEEE, and Florence Tupin
• [6] Classification of sequenced SAR target images via hidden Markov models with decision fusion, by Timothy W. Albrecht; Kenneth W. Bauer Jr.
• [7] Naïve Bayesian radar micro-doppler recognition, by Graeme E. Smith ; Karl Woodbridge, by Chris J. Baker
• [8] Recursive Bayesian classification of surveillance radar tracks based on kinematic with temporal dynamics and static features, by Lars W. Jochumsen ; Morten Ø. Pedersen ; Kim Hansen ; Søren H.
Jensen; Jan Østergaard
Other Sources and References
• More articles on artificial intelligence (2019 - today), by Martin Rupp, Ulrich Scholten, Dawn Turner and more
• LEARNING ALGORITHMS FOR CLASSIFICATION: A COMPARISON ON HANDWRITTEN DIGIT RECOGNITION (2000), by Yann LeCun, L. D. Jackel & al, AT&T Bell Laboratories
• [9] This is often badly understood. This simply means that the future (state) will not depend on anything from the past (states) but only of the present (state). | {"url":"https://www.skyradar.com/blog/artificial-intelligence-hidden-markov-model-classifiers-and-radar-objects-classification-by-machine-learning-2-2","timestamp":"2024-11-14T17:44:42Z","content_type":"text/html","content_length":"135469","record_id":"<urn:uuid:f2e283bd-605e-4f95-bd75-f45e1d5e4785>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00772.warc.gz"} |
Statistical Process Control Tutorial Guide 010207 - ID:5ccb4c5dc783e
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Statistical Process Control Tutorial Guide 010207
Tutorial Guide Statistical Monitoring Quality in Healthcare Process Clinical Indicators Support Team Control 1 Contents Introduction History of SPC 1 Understanding Variation 1.1 1.2 1.3 1.4 Types of
Variation Sources of Variation Causes of Variation Tools for Identifying Process Variation 2 SPC Charts \u2013 Dynamic Processes 3 SPC Charts \u2013 Static Processes 4 Alternative SPC Charts 2.1 2.2
2.3 2.4 Constructing a Run Chart Interpreting a Run Chart Constructing a Control Interpreting a Control Chart 3.1 Constructing a Funnel Chart 3.2 Interpreting a Funnel Chart 4.1 CUSUM and EWMA Charts
4.2 g-Charts Contacts Useful References Appendix 2 Introduction NHSScotland routinely collects a vast array of data from healthcare processes. The analysis of these data can provide invaluable
insight into the behaviour of these healthcare processes. Statistical Process Control (SPC) techniques, when applied to measurement data, can be used to highlight areas that would benefit from further
investigation. These techniques enable the user to identify variation within their process. Understanding this variation is the first step towards quality improvement. There are many different SPC
techniques that can be applied to data. The simplest SPC techniques to implement are the run and control charts. The purpose of these techniques is to identify when the process is displaying unusual
behaviour. The purpose of this guide is to provide an introduction to the application of run charts and control charts for identifying unusual behaviour in healthcare processes. SPC techniques are a
tool for highlighting this unusual behaviour. However, these techniques do not necessarily indicate that the process is either right or wrong – they merely indicate areas of the process that could
merit further investigation. History of SPC 1928 saw the introduction of the first Statistical Process Control (SPC) Charts. Commissioned by Bell Laboratories to improve the quality of telephones
manufactured, Walter Shewhart developed a simple graphical method – the first of a growing range of SPC Charts. Understanding the causes of variation within an industrial process proved indispensable
as actions could be taken to improve process and output. In the 1950’s, with the effective use of SPC, Deming converted post war Japan into the world leader of manufacturing excellence. This approach
is increasingly being applied in healthcare by thinking of healthcare systems as processes. As well as providing a basis for quality improvement within healthcare, SPC Charts also offer alternative
methods of displaying data. 3 1. Understanding Variation 1.1 Types of Variation Variation exists in all processes around us. For example: • • • Every person is different No two snowflakes are identical
Each fingerprint is unique The two types of variation that we are interested in are ‘common cause’ and ‘special cause’ variation. Common Cause All processes have random variation - known as ‘common
cause variation’. A process is said to be ‘in control’ if it exhibits only common cause variation i.e. the process is completely stable and predictable. Special Cause Unexpected events/unplanned
situations can result in ‘special cause variation’. A process is said to be ‘out of control’ if it exhibits special cause variation i.e. the process is unstable. SPC charts are a good way to identify
between these types of variation, as we will see later. SPC charts can be applied to both dynamic processes and static processes. Dynamic Processes A process that is observed across time is known as
a dynamic process. An SPC chart for a dynamic process is often referred to as a ‘time-series’ or a ‘longitudinal’ SPC chart. Static Processes A process that is observed at a particular point in time
is known as a static process. An SPC chart for a static process is often referred to as a ‘crosssectional’ SPC chart. A cross-sectional SPC chart is a good way to compare different institutions. For
example, hospitals or health boards can be compared as an alternative to league tables as we will see later. 4 Example 1 Coloured beads pulled from a bag – a dynamic process A bag contains 100 beads
that are identical - except for colour. Twenty of the beads are red and 80 are blue. Scoopfuls of 20 are repeatedly drawn out, with replacement, and the number of red beads in each scoop is observed.
Figure 1 shows the result of 25 scoops. Figure 1 Number of red beads observed in 25 scoops Twenty of the 100 beads in the bag are red, which means that the proportion of red beads in the bag is 1/5.
Therefore, if a sample of 20 is drawn each time, we expect four of the beads in the sample to be red, on average. In figure 1 the plotted points oscillate around four. In general, every time a sample
of 20 is drawn you won’t necessarily observe four reds. The number that you observe will vary due to random variation. The random variation that you see in the graph above is common cause variation
as there is no unusual behaviour in this process. If a sample of 20 beads were drawn from the bag and 10 or more red beads were consistently being observed then this would indicate something unusual
in the process i.e. special cause variation which may require further investigation. The example above is a simplification of Deming’s red bead experiment where the red beads represent an undesired
outcome of the process. This process is not dissimilar to the many situations that often occur in healthcare processes. This is how data, which is collected over time, is typically presented and it
shows the behaviour and evolution of a dynamic process. 5 Example 2 Coloured beads pulled from a bag – a static process There are 10 groups in a room and each group has a bag that contains 20 beads –
four of these beads are red. Each group is required to draw out 10 beads and the number of red beads in each groups’ scoop is observed. Figure 2 shows the result from the 10 groups. Figure 2 Number
of red beads observed in each groups’ scoop The proportion of red beads in the bag is again 1/5. Therefore, if each group draws out a sample of 10, we expect two of the beads in the sample to be red,
on average. In figure 2 the plotted points oscillate around two. The variation in this sample is again random variation (common cause variation). This example illustrates how data is typically
presented at a single point in time and it is an example of a static process. This situation arises when data is analysed across units. For example, NHS boards, GP practices, surgical units etc and
is known as a cross-sectional chart. 1.2 Sources of Variation Variation in a process can occur through a number of different sources. For example: • • • • • 6 People Materials Methods Measurement
Environment - - Every person is different Each piece of material/item/tool is unique Signatures for example Samples from certain areas etc can bias results The effect of seasonality on hospital
admissions 1.3 Causes of Variation W. A. Shewhart recognised that a process can contain two types of variation. Variation contributable to random causes and/or to assignable causes. Variation in a
process due to Random causes (common causes) Assignable causes (special causes) W. E. Deming later derived the expressions ‘common cause variation’ (variation due to random causes) and ‘special cause
variation’ (variation due to assignable causes). Common cause variation is an inherent part of every process. Generally, the effect of this type of variation is minimal and results from the regular
rhythm of the process. Special cause variation is not an inherent part of the process. This type of variation highlights something unusual occurring within the process and is created by factors that
were not part of the process’ design. However, these causes are assignable and in most cases can be eliminated. If common cause is the only type of variation that exists in the process then the
process is said to be ‘in control’ and stable. It is also predictable within set limits i.e. the probability of any future outcome falling within the limits can be stated approximately. Conversely,
if special cause variation exists within the process then the process is described as being ‘out of control’ and unstable. Summary 1 2 3 Variation exists everywhere Processes displaying only common
cause variation are predictable within statistical limits Special cause variation should be eliminated if possible 1.4 Tools for Identifying Process Variation Now we know that variation exists in all
processes we can proceed to identify which type of variation is present. One method of identifying the type of variation present is by using SPC charts. Originally developed for use in manufacturing,
many applications are now involving healthcare processes for quality improvement purposes. The following section explains the fundamentals of SPC in more detail. 7 2. SPC Charts – Dynamic Processes
Statistical Process Control (SPC) Charts are essentially: • • • • Simple graphical tools that enable process performance monitoring Designed to identify which type of variation exists within the
process Designed to highlight areas that may require further investigation Easy to construct and interpret Two of the most popular SPC tools in use today are the run chart and the control chart. They
are easy to construct, as no specialist software is required. They are easy to interpret, as there are only a few basic rules to apply in order to identify the variation type without the need to
worry too much about the underlying statistical theory. The following sections step through the construction and interpretation of run charts and control charts. 2.1 Constructing a Run Chart Run
Chart A time ordered sequence of data, with a centreline drawn horizontally through the chart. A run chart enables the monitoring of the process level and identification of the type of variation in
the process over time. The centreline of a run chart consists of either the mean or median. The mean is used in most cases unless the data is discrete. Discrete Data Where the observations can only
take certain numerical values. Almost all counts of events e.g. number of patients, number of operations etc Continuous Data These data are usually obtained by a form of measurement where the
observations are not restricted to certain values. For example - height, age, weight, blood pressure etc. Steps to create a Run Chart 1 Ideally, there should be a minimum of 15 data points. 2 Draw a
horizontal line (the x-axis), and label it with the unit of time. Draw a vertical line (the y-axis), and scale it to cover the current data, 3 plus sufficient room to accommodate future data points.
Label it with the outcome. 4 Plot the data on the graph in time order and join adjacent points with a solid line. 5 Calculate the mean or median of the data (the centreline) and draw this on the
graph. 8 Example 1 (continued) Coloured beads pulled from a bag – a dynamic process The run chart for this data is shown in figure 3. Figure 3 Run chart for the number of red beads observed in 25
scoops Median = 4 It is a good idea to state which measure has been used for the centreline. As the data for the above example (number of red beads observed) is discrete, the median has been used to
construct the centreline. The following definitions are useful before proceeding onto the rules for detecting special variation within run charts and later, control charts. Useful Observations Those
observations that do not fall directly on the centreline are known as ‘useful observations’. The number of useful observations in a sample is equal to the total number of observations minus the
number of observations falling on the centreline. In the above example, four observations fall on the centreline. Therefore, there are 25 – 4 = 21 useful observations in the sample. N.B. If the mean
(=3.88) had been used for the calculation of the centreline, as no observations would have fallen on the centreline, the number of useful observations would have been 25 (the number of observations
in the sample). Run A sequence of one or more consecutive useful observations on the same side of the centreline. The observations falling directly on the centreline can be ignored. 9 Example 1
(continued) Coloured beads pulled from a bag – a dynamic process The run chart for this data is shown in figure 4. Figure 4 Run chart for the number of red beads observed in 25 scoops with runs
highlighted in red Median = 4 Trend A sequence of successive increases or decreases in your observations is known as a ‘trend’. An observation that falls directly on the centreline, or is the same as
the preceding value is not counted. From the run chart in example 1, the longest trend is of length 3. One of these trends occurs between observations 13 and 16 where there is an increasing sequence
of length 3 (observation 14 is not counted since it falls on the centreline). 2.2 Interpreting a Run Chart A run chart is a useful tool for identifying which type of variation exists within a
process. The following rules can be applied to the run chart for determining the type of variation in the process. 10 Run Chart Rules Identifying Special Cause Variation Number of Runs If there are
too few or too many runs in the process. The table below is a guide based on the number of useful observations in your sample. Shift If the number of successive useful observations, falling on the
same side of the centreline, is greater than 7. Trend If the number of successive useful observations, either increasing or decreasing, is greater than 7. Zig-Zag If the number of useful
observations, decreasing and increasing alternately (creating a zig-zag pattern), is greater than 14. Wildly different If a useful observation is deemed as wildly different from the other observations.
This rule is subjective and is easier to identify when interpreting control charts. Cyclical Pattern If a regular pattern is occurring over time – for example a seasonality effect. Number of useful
observations Too few runs Too many runs Number of useful observations 15 16 17 18 19 20 21 22 23 24 25 26 27 4 5 5 6 6 6 7 7 8 8 9 9 9 12 12 13 13 14 15 15 16 16 17 17 18 19 28 29 30 31 32 33 34 35
36 37 38 39 40 Too few runs 10 10 11 11 11 11 12 13 13 13 14 14 15 Too many runs 19 20 20 21 22 22 23 23 24 25 25 26 26 The rules listed above are purely guidelines. Some textbooks may quote different
sizes of trends, shifts and zig-zags. The above are standard to the work that is carried out within the Clinical Indicators Support Team (CIST) and ISD wide but their primary intention was for
applications in industry. Although SPC lends itself well to healthcare processes, healthcare processes deal with lives. With this in mind, common sense is often the best guideline - SPC charts will
illustrate the variation within your process but if fewer observations in a trend, shift or zig-zag etc is unusual behaviour in your process then this is just as good an indication of special cause
variation and is therefore worth investigating. 11 2.3 Constructing a Control Chart Control Chart A time ordered sequence of data, with a centreline calculated by the mean. Control charts bring the
addition of control limits (and warning limits – optional). A control chart enables the monitoring of the process level and identification of the type of variation in the process over time with
additional rules associated with the control (and warning) limits. Steps to create a Control Chart 1 First, select the most appropriate control chart for your data, which is dependent on the
properties of your data. See flow chart. 2 Proceed as for the run chart, using the mean as the centreline. 3 Calculate the standard deviation (sd) of the sample using the formula listed in the
appendix (for appropriate chosen control chart). 4 Calculate the control limit: centreline±(3*sd) 5 Calculate the warning limits (optional): centreline±(2*sd) Standard Deviation (sd) The spread of
the observations. For example, if there is a large amount of variation between observations then the sd will be bigger than the sd for observations more tightly packed together (i.e. with less
variation). Control Chart Types Selecting an appropriate control chart for your data X X-bar c u np p Continuous Poisson Binomial 1 observation 1+observation - Constant AoO Heterogeneous AoO -
Constant SoE Heterogeneous SoE Area of Opportunity (AoO) The parameter of each observation – see examples below: Constant AoO – Calculating weekly mortality rates for a surgeon that always works a
5-day week. Heterogeneous AoO - Calculating weekly mortality rates for a surgeon that works a variable amount of days each week. Sum of Events (SoE) The denominator of each observation – see examples
below: Constant SoE – Calculating the number of weekly admissions within a set population where the population does not change over a given time. Heterogeneous SoE - Calculating the number of weekly
admissions within a population that does change over time. 12 Poisson Data that has a Poisson distribution is discrete and is based on events occurring over time (or space) at a fixed rate on average,
but where each event occurs independently and at random. For example, the number of new hip fracture admissions. Binomial Data that has a Binomial distribution is discrete and is based on data with
only two possibilities e.g. the probability of being dead or alive, male or female etc. Data type? Data is continuous Yes Data is discrete Is there more No than one observation per subgroup? No Yes
Xbar-chart X-chart Are the number of observations countable? Is the AoO equal? c-chart No No u-chart Yes Is the SoE constant at each time point? p-chart Yes np-chart 13 Example 1 (continued) Coloured
beads pulled from a bag – a dynamic process The best type of control chart to use for the data below would be either an npchart (to measure the number of red beads in the 25 scoops) or a p-chart (to
measure the proportion of red beads in the 25 scoops). Below are the steps to creating a p-chart (see appendix). Where: n = number of beads drawn in each scoop p = number of red beads observed
(success) For this process: Mean (x) Rate (r) SD = Σ(p/n) = 97/500 = p/n = 3/20, 5/20, 2/20, 6/20 etc = sqrt((x*(1-x))/n) = sqrt((97/500*403/500)/20) Control Limits Warning Limits = x ± 3*SD = x ±
2*SD = 0.194 ± 3*0.088421 etc = 0.194 ± 2*0.088421 etc Figure 4 p chart for the proportion of red beads observed in 25 scoops In this case the lower control and warning limits have been set to the
maximum value of either the formula (given above) and zero, as the proportion of red beads cannot take a value below zero. When dealing with percentages, in most cases, the upper control and warning
limits can be dealt with in the same way (i.e. the minimum value of the formula and 100). 14 2.4 Interpreting a Control Chart The same rules for identifying special cause variation in run charts also
apply to control charts, with the addition of two extra rules. Control Chart Rules Additional rules for identifying Special Cause Variation Control Limits If there is one or more observation outwith
the control limits. Warning Limits If there are two successive observations outwith the same warning limits. The setting of control limits and warning limits are an attempt to balance the risk of
committing two possible types of error: • • Type I – False positives Type II – False negatives Type I Identifying special cause variation when there is none. As the limits are set at 3 standard
deviations from the centreline, only 99.7% of our observations are expected to fall within the limits (and 95% for warning limits) if the process is stable. This means that 3 in 1000 (and 50 in 1000)
observations are expected to fall outside the control limits even when the process is stable. Type II Not identifying special cause variation when there is. This occurs when action is not signalled
for an observation that falls within the limits when the process is actually out of control. As mentioned earlier, the rules for identifying special cause variation are guidelines and may be altered
in light of the process that is being investigated. Similarly, the boundaries of the control and warning limits can also be adjusted. The combined risk of committing Type I and Type II errors is
minimised when the control limits are set at 3 standard deviations from the centre line (Carey and Lloyd, 1995) however in some cases this may be deemed as too conservative. For example, if poor
surgical performance is the process that is being investigated, in order to increase the chances of identifying possible aberrant practice, it may be beneficial to choose tighter limits. 15 3. SPC
Charts – Static Processes SPC Charts are most typically plotted over time for a single process. However, it is also possible to construct SPC Charts at a static point in time for a process carried
out by multiple institutions (e.g. NHS Boards, Hospitals etc), which are often referred to as cross-sectional charts. The cross-sectional chart that we are going to cover is one of the most common
SPC charts for static processes and is known as a funnel chart due to the fact that the control limits take the shape of a ‘funnel’. 3.1 Constructing a Funnel Chart Funnel Chart SPC Chart for
cross-sectional data at a particular point in time. The rate of the process (e.g. mortality rate, survival rate etc) is plotted on the vertical axis and the denominator (i.e. population, number of
admissions etc) is plotted on the horizontal axis. The centreline is calculated by the mean. Generally, only control limits are calculated, as the rule for warning limits does not apply to
cross-sectional data. Steps to create a Funnel Chart 1 Order the data by the denominator (d) in ascending order. 2 Calculate proportions (p) for each individual institution and an overall proportion
(this will be the centreline (c)). 3 Calculate the standard deviation (sd): sqrt(d* c *(1- c)). 4 Calculate the control limits (ucl and lcl): c±(3*sd)/d Example 2 (continued) Coloured beads pulled
from a bag – a static process Using the data from example 2 we will now assume that each bag in fact contained a variable number of red beads (the total number of all beads in a bag is still 20). The
table below illustrates the data. Group A B C D E F G H I J 16 Number of red beads per bag (d) 4 6 5 4 3 2 4 7 8 5 Number of red beads observed (obs) 2 3 2 4 1 2 3 3 4 1 After sorting the data by the
denominator (d), in this case the numbers of red beads in each bag calculate the proportions (p) and limits (lcl and ucl). Group F E D G A C J B H I d 2 3 4 4 4 5 5 6 7 8 obs 2 1 4 3 2 2 1 3 3 4 p 1
0.33 1 0.75 0.5 0.4 0.2 0.5 0.43 0.5 c 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.52 sd 0.71 0.87 1.00 1.00 1.00 1.12 1.12 1.22 1.32 1.41 lcl -0.54 -0.35 -0.23 -0.23 -0.23 -0.15 -0.15 -0.09 -0.05
-0.01 ucl 1.58 1.39 1.27 1.27 1.27 1.19 1.19 1.13 1.09 1.05 We then create a chart using a ‘scatter plot’, where the x-axis values are always plotted as the denominator (d). The only other values to
plot, against the x-axis values, are the rate (r), centreline (c) and the lcl and ucl. Figure 5 Funnel Chart for the proportion of red beads observed in each groups’ scoop The graph above indicates
that all the groups are in control and do not display any cause for concern. 17 3.2 Interpreting a Funnel Chart The rules for identifying special cause variation in a static process are very simple
and are identified purely by an observation falling out with the control limits. Funnel Chart Rule If one of the observations falls outwith the control limits. If one of the observations does fall
outwith the control limits, it is often worth investigating that particular process more fully with a control chart over time (i.e. dynamic process). 4. Alternative SPC Charts 4.1 CUSUM and EWMA
Charts There are many other different types of SPC Charts that may be more appropriate for the type of investigation that your process requires. For example Cumulative Summation (CUSUM) Charts are
more sensitive to small shifts than the types of SPC Charts that have been discussed so far. Likewise, Exponentially Weighted Moving Average (EWMA) Charts, as well as being more sensitive to smaller
shifts, also have the advantage of taking into account past data, which avoids biasing the process variation to the current time period. These charts are not widely used in healthcare as they are
more complex to construct and more difficult to interpret. There are also very few cases in healthcare that would require additional time and resources directed towards investigating very small shifts
that are most likely an effect of common cause variation. However, when used appropriately they can provide useful, additional analysis. 4.2 g-charts The most widely used alternative to the ‘regular’
SPC Charts within the health service has been g-charts. These charts measure the number-between specific observations and are used for processes with low frequencies. In particular, g-charts have been
used extensively in monitoring Healthcare Acquired Infections (HAIs). Perhaps best described through the billboards in construction sites that state ‘x days since the last accident’, g-charts simply
look at the number of days, patients or catheters etc since the last event of interest e.g. infection. 18 g-charts are interpreted in the same way as the control charts that we have already seen i.e.
special cause variation is identified when an observation falls outwith the control limits and/or when two successive points fall out with the warning limits. These type of charts are only applicable
with dynamic processes but have already proved invaluable throughout hospital wards in Scotland that have implemented g-charts to monitor the success of specific care bundles. Contacts For further
information on SPC charts visit our website: www.indicators.scot.nhs.uk or alternatively you can contact: Rebecca Kaye email:
[email protected]
tel: (0131) 275 6434 Margaret MacLeod email:
[email protected]
tel: (0131) 275 6520 or subscribe to our bulletin (placing ‘add me’ in the subject field) email:
[email protected]
19 Useful References Adab Peymane, Rouse Andrew M, Mohammed Mohammed A, Marshall Tom. Performance league tables: the NHS deserves better. BMJ (2002) 324; 95-98. Benneyan J C. Use and interpretation
of statistical quality control charts. International Journal of Quality in Health Care (1998) 10; 69-73. Benneyan J C. Statistical quality control methods in infection control and hospital
epidemiology. Part I: Introduction and basic theory. Infection Control and Hospital Epidemiology (1998) 19, 3; pg 194 Part II: Chart use, statistical properties, and research issues. Infection
Control and Hospital Epidemiology; (1998); 19, 4; pg. 265 Berwick D M. Controlling variation in health care: A consultation from Walter Shewhart. Medical Care (1991) 29; 1212-1225. Carey Raymond G.
How do you know that your care is improving? Part I: Basic concepts in statistical thinking. Journal of Ambulatory Care Management (2002) 25. Part II: Using control charts to learn from your data.
Journal of Ambulatory Care Management (2002) 25; 78-88. Curran Evonne T, Benneyan James C, Hood John. Controlling methicillinresistant staphylococcus aureus: a feedback approach using annotated
statistical process control charts. Infection Control and Hospital Epidemiology (2002) 23; 13-18. Finison L J, Finison K S. Applying control charts to quality improvement. Journal of Healthcare
Quality (1996) 18; 32-41. Hanslik T, Boelle P and Flahault A. The control chart: an epidemiological tool for public health monitoring. Public Health (2001) 115; 277-281. Mohammed Mohammed A, Cheng K
K, Rouse Andrew, Marshall Tom. Bristol, Shipman, and clinical governance: Shewhart’s forgotten lessons. Lancet (2001) 357; 463-467. Montgomery, D. C. Introduction to statistical quality control
(second ed) (1991). New York: Wiley Paris P Tekkis, Peter McCulloch, Adrian C Steger, Irving S Benjamin, Jan D Poloniecki. Mortality control charts for comparing performance of surgical units:
validation study using hospital mortality data BMJ (2003) ;326:786. Plsek P. Tutorial: introduction to control charts. Quality Management in Health Care (1992) 1; 65-73. 20 Sellick J A. The use of
statistical process control charts in hospital epidemiology. Infection Control in Hospital Epidemiology (1993) 14; 649-656. Shahian D M, Williamson W A, Svensson L G, Restuccia J D, d’Agostino R S.
Applications of statistical quality control to cardiac surgery. Annals of Thoracic Surgery (1996) 62; 1351-1359. Spegielhalter, D. J. (2002). Funnel plots for institutional comparison. Quality &
Safety in Health Care 11. 390-391. VamderVeen L M. Statistical process control: a practical application for hospitals. Journal for Health Care Quality (1992) 14; 20-29. Wheeler, D. J. and D. S.
Chambers. Understanding statistical process control (1990). Wokingham, England: Addison-Wesley. 21 Appendix In this section you will find the formulae required for constructing a control chart for
your data, assuming that you have made the most appropriate choice of chart. X-chart Assume you have m observations, Xi, i =1, 2,…, m. Calculate the process average, m X =∑ X i . i =1 Calculate the
absolute moving ranges (MRs) between adjacent observations, where MR{i,i+1} = |Xi - Xi+1|, i =1, 2,…, m-1. Calculate the mean range, R , as 1 m−1 R = ∑ MR{i ,i +1}. m −1 i =1 Set the control limits
at X ±(2.66 ×R ). c-chart Assume that you have m observations from a Poisson(μ) distribution, i.e. X i ~Poisson( μ ), i =1,2,...,m, where: Xi is the number of occurrences for observation i, and μ is
the process average. Since the true process average, μ, is not known, we replace its value by the observed process average, which is given by m X =∑ X i . i=1 Since it is the number of occurrences
that is plotted, i.e. the sequence of values, {X1, X2, …, Xm}, we’re required to calculate that standard deviation, s, for each Xi, i=1,2,…m. Since the area of opportunity is constant for all i, s is
simply calculated as s = X. It is this s that is used for the calculation of the control (and warning) limits. 22 u-chart As with the c-chart, assume that you have m observations from a Poisson( μ)
distribution, i.e. X i ~Poisson( μ ), i =1,2,...,m. Since it is the proportion of occurrences that is plotted, i.e. the sequence of Y i = X i / ni ~ Poisson(μ/ni) and ni is simply a values, {Y1, Y2,
…, Ym}, where scaling constant that allows for the heterogeneity of the area of opportunity, we’re required to calculate that standard deviation, s, for each Yi, i=1,2,…m. This is given by s (Y i ) =
X ni . It is this s that is used for the calculation of the control (and warning) limits. Note that since a different s is required to be calculated for each proportion, Yi, i=1,2,…, m, different
control (and warning) limits are required to be calculated, too. np-chart Assume you have m observations from a Bin(ni ,p) distribution, i.e. Xi ~ Binomial(ni ,p), i =1, 2,…, m, where: Xi is the
number of non-conforming units for observation i, ni is the number of units for observation i, and p is the probability of “success”. In addition, let m N =∑ ni be the sum of the units, i =1 m X =∑ X
i be the sum of non - conformingunits, i =1 1 m X = ∑ X i be the mean numberof non - conformingunits,and m i =1 ˆ =X / N be the observedprobabilit p y of obtaininga non - conformingunit. Since it is
the number of non-conforming units that is plotted, i.e. the sequence of values, {X1, X2, …, Xm}, we’re required to calculate the standard deviation, s, for each Xi, i=1,2,…m. This is given by ˆ (1
−p ˆ ). s = ni p However, sincen =n1 =n2 = L nm , s is simply ˆ (1 −p ˆ ). s = np It is this s that is used for the calculation of the control (and warning) limits. 23 p-chart As with the np-chart,
assume that we have m observations from a Bin(ni ,p) distribution, i.e. Xi ~ Binomial(ni ,p), i =1, 2,…, m, Since it is the proportion of non-conforming units that is plotted, i.e. the Yi = X i / ni
, we’re required to sequence of values, {Y1, Y2, …, Ym}, where calculate the standard deviation, s, for each Yi, i=1,2,…, m. This is given by s (Y i ) =s ( X i / n) 1 = s( X i ) n 1 ˆ (1−p ˆ) ni p =
ni ˆ (1−p ˆ ) n i , i =1,2,...,m. = p It is this s that is used for the calculation of the control (and warning) limits. Note that since a different s is required to be calculated for each proportion,
Yi, i=1,2,…, m, different control (and warning) limits are required to be calculated, too. 24 | {"url":"https://xdocs.net/documents/statistical-process-control-tutorial-guide-010207-5ccb4c5dc783e","timestamp":"2024-11-11T05:06:45Z","content_type":"text/html","content_length":"105341","record_id":"<urn:uuid:6ec08b08-83bd-439a-b10f-86d3e7b2c7af>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00432.warc.gz"} |
Water Tank Volume Calculator - Easily Estimate Your Tank Capacity - Calculator Pack
Water Tank Volume Calculator
If you're planning on installing a water tank, it's important to know how much water it can hold. This is where our Water Tank Volume Calculator comes in handy! With just a few simple inputs, you can
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water, irrigation, or livestock, knowing its capacity is essential for proper maintenance and usage. So, let's dive in and learn how to use this tool to help you make informed decisions about your
water storage needs.
Water Tank Volume Calculator
Calculate the volume of a water tank based on specific dimensions.
Water Tank Volume Calculator Results
Length 0
Width 0
Height 0
Volume 0
Share results with your friends
determining the volume of aquariums is crucial for pet enthusiasts. Our aquarium volume calculator streamlines this calculation. To delve deeper into tank volume calculations and understand their
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How to Use the Water Tank Volume Calculator
The Water Tank Volume Calculator is a tool that allows users to determine the volume of water a tank can hold based on its specific dimensions. This calculator is an essential tool for homeowners,
builders, and anyone involved in water storage and management. In this article, we will explore how to utilize the Water Tank Volume Calculator effectively.
Instructions for Utilizing the Calculator
To use the Water Tank Volume Calculator, you will need to input the length, width, and height of the tank in the corresponding input fields. The length refers to the distance from one end of the tank
to the other, the width refers to the distance from one side of the tank to the other, and the height refers to the distance from the bottom of the tank to the top.
It is essential to input accurate data into the calculator to obtain reliable results. The input data is necessary to determine the volume of water the tank can hold. The more precise the data, the
more accurate the results.
The output fields of the calculator consist of the input values and the volume of water the tank can hold. The volume of water is expressed in cubic units, such as cubic meters or cubic feet. It is
important to note that the output of the calculator is only an estimate of the tank's volume.
Water Tank Volume Calculator Formula
The formula for calculating the volume of a tank is simple. Multiply the length, width, and height of the tank to get the volume. In mathematical terms:
Volume = Length x Width x Height
Illustrative Examples
Suppose we have a water tank with a length of 5 meters, a width of 4 meters, and a height of 3 meters. The volume of water the tank can hold is:
• Volume = Length x Width x Height
• Volume = 5 x 4 x 3
• Volume = 60 cubic meters
Therefore, the water tank can hold 60 cubic meters of water.
Illustrative Table Example
Length Width Height Volume
5m 4m 3m 60m³
3m 2m 2m 12m³
6m 5m 4m 120m³
The Water Tank Volume Calculator is a useful tool for determining the amount of water a tank can hold. By inputting the length, width, and height of the tank, the calculator can estimate the volume
of water the tank can hold. It is important to provide accurate input data to obtain reliable results. With the help of this calculator, homeowners, builders, and anyone involved in water storage and
management can effectively plan and manage their water resources. | {"url":"http://calculatorpack.com/water-tank-volume-calculator/","timestamp":"2024-11-05T15:37:39Z","content_type":"text/html","content_length":"32894","record_id":"<urn:uuid:3f5b4c2b-a572-485b-afd5-031b47e5189d>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00627.warc.gz"} |
C++ Programming/Code/Standard C Library/Functions/modf - Wikibooks, open books for an open world
Syntax #include <cmath>
double modf( double num, double *i );
The function modf() splits num into its integer and fraction parts. It returns the fractional part and loads the integer part into i.
Related topics
frexp - ldexp | {"url":"https://en.m.wikibooks.org/wiki/C%2B%2B_Programming/Code/Standard_C_Library/Functions/modf","timestamp":"2024-11-03T06:56:51Z","content_type":"text/html","content_length":"23287","record_id":"<urn:uuid:eb6fe0d4-327b-496e-b9c0-544fffbce066>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00660.warc.gz"} |
A gyrokinetic simulation model for 2D equilibrium potential in the scrape-off layer of a field-reversed configuration
The equilibrium potential structure in the scrape-off layer (SOL) of the field-reversed configuration (FRC) can be affected by the penetration of edge biasing applied at the divertor ends. The
primary focus of the paper is to establish a formulation that accurately captures both parallel and radial variations of the two-dimensional (2D) potential in SOL. The formulation mainly describes a
quasi-neutral plasma with a logical sheath boundary. A full-f gyrokinetic ion model and a massless electron model are implemented in the GTC-X code to solve for the self-consistent equilibrium
potential, given fixed radial potential profiles at the boundaries. The first essential point of this 2D model lies in its ability to couple radial and parallel dynamics stemming from resistive
currents and drag force on ions. The model successfully recovers the fluid force balance and continuity equations. These collisional effects on 2D potential mainly appear through the density profile
changes, modifying the potential through electron pressure gradient. This means an accurate prescription of electron density and temperature profiles is important in predicting the potential
structure in the FRC SOL. The Debye sheath potential and the potential profiles applied at the boundaries can be additional factors contributing to the 2D variations in SOL. This comprehensive full-f
scheme holds promise for future investigations into turbulent transport in the presence of the self-consistent 2D potential together with the non-Maxwellian distributions and open boundary conditions
in the FRC SOL.
Field-reversed configuration (FRC) is an alternative approach to realizing magnetic fusion, distinguishing itself from the more prevalent designs such as tokamaks and stellarators. Characterized as
an elongated prolate compact toroid (CT) with a solely poloidal magnetic field, the FRC is lauded for its engineering simplicity.^1 Due to its easy construction in engineering aspects, FRC continues
to draw research interest, especially in light of recent advancements at the TAE Technologies, Inc. The C2-W experiments at TAE successfully extend steady FRC plasmas for more than 30ms by using a
neutral beam injection (NBI) system and mirror plugs.^2–5 Notably, the performance is reported to be mainly limited by the NBI duration. Energetic ion population from the NBI helps to stabilize the
FRC plasmas because of large Larmor radius (FLR) effects^6 and strong radial electric field shear.
The C2-W experiments also use an edge-biasing system to produce a negative radial electric field, resulting in an E × B toroidal rotation opposite to the ion diamagnetic flow. The induced rotation
plays a crucial role in mitigating macroscopic wobble and rotational instabilities, enhancing the stability of the system.^7 Additionally, the $E×B$ shearing effects have been demonstrated to be
beneficial for reducing the turbulent transport in FRC.^8,9
Microscopic drift-wave turbulence on the scale of thermal ion gyroradius is typically suppressed in the FRC core, but in the scrape-off layer (SOL), ion-to-electron scale drift-wave turbulence has
been observed from the experiments and simulations.^10–13 This observation opens up the potential for utilizing the biasing system in the SOL as a means to reduce turbulent transport and improve the
FRC performance. Indeed, reduction of turbulence correlation length was reported in previous FRC measurements.^12 In tokamaks, similar equilibrium $E×B$ sheared flows have been conclusively
demonstrated to suppress the drift-wave turbulence by reducing linear growth rate, nonlinear eddy size, fluctuation intensity, and turbulent transport.^14–17 This motivates our research to
investigate the effects of equilibrium $E×B$ sheared flow under edge-biasing conditions on turbulent transport within the FRC SOL.
Using the gyrokinetic toroidal code (GTC),^18 we have conducted several studies on the turbulence behavior in the FRC. Lau et al.^13 demonstrated that both electron and ion-scale drift waves could
become unstable in the SOL, exhibiting critical pressure gradients in line with experimentally observed thresholds. Subsequent nonlinear simulations using a global particle code ANC^19 find that
linear drift-wave instabilities first grow in the SOL, then nonlinearly spreads from the SOL to core, which exhibits a toroidal wavenumber spectrum comparable to the experimental measurements.^20,21
A modified GTC version for the FRC geometry, GTC-X, is developed for the global simulation of nonlinear turbulent transport in the whole device.^22 It is reported that the ion temperature gradient
(ITG) mode is globally connected and axially varying across central FRC region, mirror throat area, and formation exit area. The self-generated zonal flows can suppress such ITG instability in the
nonlinear simulation of the FRC SOL.^23
In our preliminary effort to integrate an equilibrium potential into FRC turbulence simulations using GTC-X, we discovered that combining the additional $E×B$ flow with the diamagnetic flow yields a
total shearing rate. This total $E×B$ shearing rate can effectively suppress turbulent transport in the FRC SOL region.^8 However, it is important to note that the equilibrium potential introduced
was an unrealistic one-dimensional (1D) function of the flux surface by ignoring the presheath potentials that vary along the magnetic field-line.
When the biasing system is applied in the FRC SOL, it is important to understand how potential boundary profiles can modify the potential structure in the SOL. This understanding is crucial as the
$E×B$ shearing effects, introduced by the biased potential, have a direct impact on the behavior of turbulence within the region. Considering that current FRC experiments can well sustain for over
30ms, we can reasonably anticipate the establishment of a steady-state equilibrium in the presence of a biasing system. In this stable scenario, the potential structure within the SOL can be treated
as a time-invariant background, facilitating a more straightforward study of turbulence phenomena. Previous research has indicated that both the linear growth rate and the saturation timescale of
turbulent transport in the FRC SOL typically occur within a 1 ms timeframe.^8,22 This result supports the hypothesis that the background equilibrium and the turbulence phenomena can be studied
independently, with the former serving as an equilibrium to the latter.
To study the effects of biasing potential on turbulence, a self-consistent model to calculate the equilibrium potential is needed. Given that particles go from the core region to the SOL, eventually
exiting the device via the divertors, our sought-after equilibrium model inherently requires a two-dimensional (2D) approach, accompanied by open boundary conditions in the parallel direction. A
transport model, a Quasi-1D (Q1D) code with fluid ions and electrons, together with a kinetic neutral beam species,^24 has simulated the C2 plasma evolution with a self-consistent potential. Later,
this model was extended to a 2D version, incorporating parallel transport in the SOL region. The simulation captured the density profile evolution which can be compared to the experimental
To study the potential structure in SOL, a KSOL code with kinetic electrons has been developed by the TAE team to study the 1D parallel variation of presheath potential between the outer mirrors and
the divertors.^26 The simple Boltzmann response of electrons is extended by using the Vlasov–Fokker–Planck equation. This approach captures the kinetic effects on potential structure in an expanding
magnetic field with anisotropic electron pressures. By comparing with the Q2D fluid model and the Boltzmann relation, it was concluded that the 1D parallel potential profile was insensitive to the
different electron models, while the inclusion of ion acceleration in the model could be more important to the profile.^25,26 This finding motivates us to consider a fluid electron model and focus on
the 2D effects of poloidal flow in SOL.
In this paper, we take a different approach from the 1D KSOL simulations at TAE and focus on the 2D potential structure in the central SOL near the core region. By developing a fluid electron model
in GTC-X code, our model integrates a resistive current from electron–ion collisions or other enhanced transport, as well as a drag force accounting for ion interactions with impurities and neutrals.
These elements are crucial for establishing a connection between radial and parallel transport and creating a 2D potential structure in the FRC SOL. In addition, to provide the kinetic equilibrium
for turbulence study, gyrokinetic (GK) ions are incorporated to obtain the kinetic effects on ions. The primary purpose is to obtain a steady-state equilibrium, ensuring its applicability as a
reliable equilibrium for the upcoming turbulence simulations in the central vessel.
In the FRC SOL, electrostatic potential varies on both microscopic and macroscopic scales. On the microscopic scale, there exists a parallel potential drop over a Debye length (Debye sheath) in front
of the divertors where the magnetic field-lines intercept the conducting surfaces. There is also a perpendicular potential variation over a thin layer within a width of several ion gyroradii
(magnetic presheath)^27 at the outer radius in front of the cylindrical wall in the presence of an oblique magnetic field-line. These microscopic regions are far away from the FRC core and are not
simulated when we investigate the macroscopic transport of the FRC SOL in this paper. We discuss a simplified Debye sheath model at the end of the paper and highlight the need for a more accurate
sheath model in our simulations. Future upgrades to the simulation boundary conditions of our hybrid gyrokinetic-fluid model could benefit from implementations found in fluid transport codes like
SOLPS-ITER for tokamak SOLs.^28,29
The focus of the present work is on the parallel potential drop over the macroscopic region of the SOL where ions are accelerated to the ion sound speed before entering the Debye sheath. The
equilibrium electrostatic potential in this macroscopic presheath^30 depends on the magnetic structures, collisions, and atomic processes in the SOL, which affects the penetration of the divertor
biasing into the confinement vessel to improve the FRC confinement.^5,7 The gyrokinetic simulation model is valid for the quasi-neutral plasma in this macroscopic SOL region as documented in our
previous turbulence simulation papers.^8,19,22,23
Please note that there are inherent limitations in our proposed model described in this paper, which will necessitate further improvements in future research. The model addresses only the
quasi-neutral plasma before the sheath layer, employing a logical sheath^31 boundary condition to connect our simulations with the wall potential at the divertors. Appropriate sheath models are
required for accurate comparisons with experiments. Additionally, the radial coupling in the fluid equations is limited to resistive currents and a phenomenological drag force on ions. Other physics
which may cause the potential structure changes due to the modification of particle distributions, such as the secondary electron emission, ionization of neutrals, and other factors, are not covered
in this paper.
The rest of the paper is organized as follows. In Sec. II, we formulate the 2D equilibrium potential model with gyrokinetic ions and fluid electrons. The model describes a quasi-neutral plasma with a
logical sheath boundary in FRC SOL. Both the radial and parallel dynamics are considered by including the resistive currents and drag force on ions. We verify this simulation model in the GTC-X for a
simplified 1D case by neglecting the resistive and drag force in Sec. III. The ion force balance and the ion continuity equation are verified for this model. In Sec. IV, the 2D model is discussed in
detail for the GTC-X simulation. Parameter scans for the resistivity and the drag force are carried out to understand how the potential structure in the FRC SOL region is established. The current
density distribution in the FRC SOL is also discussed with this new model. Section V discussed the influence of the potential boundary profiles on the 2D structure of equilibrium potential. Section
VI discussed the Debye sheath potential model that was required to connect our simulations to the actual wall potential at the divertor ends. The demand for a better sheath potential model was
discussed in this section. Section VII is a summary of the simulation results and potential issues and future plan for this 2D FRC equilibrium model with edge biasing.
The primary objective of this 2D equilibrium potential model including the presheath region is to calculate a self-consistent potential, given a preassigned boundary. This equilibrium can
subsequently be linked to the edge-biasing applied at the divertor walls by incorporating an appropriate Debye sheath model. The 2D model described here lays the foundation for future transport
studies by providing a self-consistent gyrokinetic particle distribution function as an equilibrium. As illustrated in Fig. 1, the gyrokinetic (GK) ion simulation enables us to derive the ion guiding
center distribution. From this, we can calculate essential parameters such as particle density, ion parallel flow velocities, and pressures. With these quantities, we employ the Hall-MHD electron
model and a quasi-neutral condition to solve for the electrostatic potential, thereby advancing our understanding of the system's behavior and laying the groundwork for future explorations of
turbulence phenomena.
A. Gyrokinetic model for ion species in equilibrium simulation
The gyrokinetic formulation is applicable when dealing with physical phenomena characterized by a frequency much smaller than the cyclotron frequency and a gyroradius much smaller than the system
size. In regions like the FRC core and near magnetic null points in the FRC, where the gyroradius is relatively large, a fully kinetic approach is more appropriate. However, in the SOL, the
gyrokinetic assumptions remain valid due to the stronger magnetic field and significant temperature drop at the edge, which reduces the gyroradius. While high $β$ values (the ratio of kinetic
pressure to magnetic pressure) may be expected near the FRC core, our simulations typically yield a $β$ of around 0.01 in the SOL. As a result, we employ the electrostatic gyrokinetic equation to
calculate the ion distribution function, which can be later utilized in turbulence studies.
By averaging over the fast gyromotion, the long-timescale dynamics of the ion gyro-center distribution
is described in a reduced five-dimensional phase space
, with
. Here,
is the unit vector of the equilibrium magnetic field
, and
is the parallel velocity along magnetic field-line. Its parallel acceleration is calculated as
. The guiding center position,
, can be expressed in terms of the particle position,
, and its gyroradius,
, as
, where
represents the ion cyclotron frequency.
are the ion charge and mass, respectively. The perpendicular velocity is
with a magnitude of
, where
is the magnetic moment. The gyrophase
is averaged out when deriving Eq.
^33 $Cfi$
represents a general collision operator as discussed in Ref.
, and the electrostatic potential,
, is gyro-averaged through the operator
In order to reduce the particle noise, the traditional gyrokinetic simulation uses the perturbative
method as described in previous studies.
In this method, the distribution function
is divided into two components: an equilibrium part, denoted as
, and a time-varying perturbation part,
. The equilibrium component satisfies the zeroth-order equation
Here, we also separate the Lagrangian operator,
, into equilibrium and perturbation parts,
, where
$δL=vE⋅∇−B*miB∥*⋅ Zi∇ϕ∂∂v∥$
. The collision operator is also separated as
. For the specific case of FRC geometry, it can be demonstrated that,
serves as an exact solution for Eq.
being functions of magnetic flux
. In our previous studies, this local Maxwellian equilibrium has been used to investigate the microturbulence driven by ion temperature gradient (ITG) instability. A weight equation, based on this
, is derived to track the evolution of the perturbed
However, when computing a self-consistent equilibrium, the local Maxwellian solution becomes inappropriate. This is particularly evident in the SOL region, where particles can escape the machine
through open field-lines. To address this issue, we must assume open boundaries at the divertor ends, which can result in a non-Maxwellian distribution of particles with a macroscopic flow velocity.
Consequently, constructing an analytical form of $fi0$ for Eq. (2) in this open boundary scenario becomes challenging.
To construct the complicated equilibrium, we now pursue a full-f simulation approach. In this method, we no longer separate the $fi0$ from $δfi$. Instead, we evolve the complete gyrokinetic equation
with the full-f weight, $wf=fig$, where $g$ is the marker distribution. The full-f weight should satisfy $Lwf=Cfi$, given the marker distribution $g$ as also a solution for the Lagrangian operator.
In this case, the full-f weight remains constant throughout the simulation if we ignore the effects of collision on $g$.^40
From the gyrokinetic formulation, we can compute the ion particle distribution,
, from the guiding center distribution as
$Fi=fi+ZiB∂fi∂μ ϕ−ϕ$
To obtain the particle density,
$Ni=∫dx′dv Fiδ(x′−x)$
, the first term gives us the guiding center density
$ni¯=∫Bmidv∥dμ∫dα∫dR fiR,v∥,μ,tδR+ρ−x,$
while the second term corresponds to the polarization density
. A simplified form can be derived for the second term by approximating the guiding center density as a Maxwellian distribution,
. Notably,
, the polarization density becomes
as a result. The double gyrophase-averaged potential,
, is defined as
$ϕ̃x,t=1ni0∫Bmidv∥dμ∫dα∫dR fiR,v∥,μ,tϕδR+ρ−x.$
Similarly, we can also calculate the parallel ion flow velocity
and the anisotropic ion pressure
, based on their guiding center counterparts
$uı∥¯=1Ni∫Bmidv∥dμ∫dα∫dR v∥fiδR+ρ−x,Pi∥=pı∥¯+pi∥,pol, Pi⊥=pı⊥¯+pi⊥,pol,$
$pı∥¯=∫Bmidv∥dμ∫dα∫dRmiv∥−Ui02fiδR+ρ−x, pı∥,pol=∫Bmidv∥dμ∫dα∫dRmiv∥−Ui02qiB∂fi∂μ ϕ−ϕδR+ρ−x,pı⊥¯=∫Bmidv∥dμ∫dα∫dR12miv⊥2fiδR+ρ−x,pi⊥,pol=∫Bmidv∥dμ∫dα∫dR12miv⊥2qiB∂fi∂μ ϕ−ϕδR+ρ−x.$
When calculating the pressure, it is important to consider that the pressure is defined by the averaged kinetic energy in the reference frame moving with the average flow velocity of each species.
Particularly in the FRC SOL, a notable parallel flow is directed toward the divertors. Therefore, when computing pressure in Eq. (6), it is essential to subtract the fluid flow velocity $Ui0$. The
contribution of perpendicular flow velocity, being 1–2 orders of magnitude smaller than the parallel flow, is omitted in the formulation. This observation is confirmed through simulations discussed
in subsequent sections.
To compute polarization quantities, obtaining the gyro-averaged potential $ϕ$ is crucial. Presently, the GTC-X code performs gyro-averaging for a specific toroidal mode number only,^8 so we here use
a long-wavelength approximation for the gyro-average.^41 Future developments may focus on implementing a more generalized gyro-averaging algorithm within the GTC-X framework.
Another challenge in analytically integrating the polarization quantities in Eqs. (3)–(6) is to get the guiding center density distribution $fi$. Since the actual density distribution can deviate
from the local Maxwellian, we adopt a shifted Maxwellian as a simplified model, $fi≈fSM=ni0mi2πTi03/2exp−miv∥−Ui02+2μB2Ti0$. This heuristic model offers valuable insights into the approximation of
polarization terms in Eqs. (3)–(6) in the long-wavelength limit. Evaluating integrals with the shifted Maxwellian is straightforward, which yields the guiding center contributions: $nı¯=ni0$,
$ui∥¯≈Uı0$, and $pı¯=ni0Ti0$.
To compute the polarization terms analytically, we first compute the gyro-averaged potential as $ϕR=∑kϕkeik⋅RJ0k⊥2ρi2$, where $ρi=1ΩiTi0mi=cmiTi0ZiB$ and $J0$ is the Bessel function.^41 Employing
the long-wavelength approximation ( $k⊥ρi≪1$), we can express $ϕ̃$ using the Padé approximation, $ϕ̃x=11−ρi2∇⊥2ϕx$. Subsequently, the polarization density can be approximated as
$ni,pol≈Zini0Ti0ρi2∇⊥2ϕ$. Intriguingly, in the long-wavelength limit, the particle parallel flow velocity is the same as the guiding center flow velocity, $Ui∥=Ui0$, as long as we correctly account
for the polarization density. Following a similar approach, we can obtain $pi∥,pol≈ni,polTi0$ and $pi⊥,pol≈32ni,polTi0$. The particle density and pressure, incorporating polarization corrections,
now become $Ni=ni01+ZiTi0ρi2∇⊥2ϕ$, $Pi∥=ni0Ti01+ZiTi0ρi2∇⊥2ϕ$, and $Pi⊥=ni0Ti01+32ZiTi0ρi2∇⊥2ϕ$.^42 Although a heuristic shifted Maxwellian distribution is used in our derivation, these
expressions serve as a useful lowest-order approximation for polarization effects without loss of generality.
B. Fluid model for electron species in equilibrium simulation
Due to the faster timescale of electron motion compared to ions, directly evolving electron particles can be time-consuming. To circumvent this difficulty, a reduced model for electrons is often
preferred. Our proposed fluid model incorporates momentum equations for both electrons and ions. The resistive current allows for coupling between radial and parallel physics, enabling the derivation
of a 2D structure. The electrostatic potential is computed using the resistive Ohm's law
represents the self-consistent equilibrium potential. The
denotes the resistive current with
. The
are, respectively, the resistivity in parallel and perpendicular directions. Additionally,
represents the ion flow velocity,
stands for the magnetic field, and
is the electron density. Notice that the electron inertia term in this equation is dropped considering the smallness of electron mass.
For classical transport induced by electron–ion collisions, the distortion of the electron distribution function results in different parallel and perpendicular resistivity with $η∥=0.51η⊥$ for an
ion charge $Zi=e$.^43 However, this ratio might differ when considering anomalous transport, such as turbulence-induced transport and other resistive effects. When scanning a wide range of
resistivity values in the later sections of this paper, we use the same parallel and perpendicular resistivity ( $η=η∥=η⊥$) for simplicity.
In the current work, only isotropic pressure, $Pe=Pe∥≈Pe⊥$, is used in the fluid equations. When accounting for temperature variations in different direction, it becomes necessary to substitute the
pressure gradient term $∇Pe$ with the divergence $∇⋅Pe$, where $Pe=Pe∥−Pe⊥bb+Pe⊥I$ is the pressure tensor with distinct parallel and perpendicular pressure components. This modification will
introduce an additional mirror force term related to the magnetic field changes, which will be discussed in Sec. IV. For derivations in this section and Appendix A, we will assume an isotropic
pressure in all the equations.
To calculate the potential and address the other unknown variables in Eq. (7), we introduce simplifications in our model. We treat the magnetic field as static throughout the simulation, assuming
that the magnetic field's evolution and dissipation occur over a longer timescale. We also neglect the change of the magnetic field due to change of the plasmas current, i.e., we assume that the
divertor biasing does not significantly change the magnetic field. In this way, ions and electrons reach equilibrium with the electrostatic potential before any significant magnetic field alterations
take place. Thus, our simulation results can be interpreted as an equilibrium state given a specific magnetic field structure.
In the parallel direction, the electric field is balanced by the electron pressure gradient and the resistive force arising from the parallel current. To compute the current term, additional
equations are required, we use the steady-state quasi-neutral condition
Because we are interested in the steady-state equilibrium, all the time derivatives can be dropped in our fluid model and the system evolves toward the equilibrium through the nonlinear ion
gyrokinetic equation, Eq.
. The electron density is also obtained by quasi-neutral condition,
. Since we focused on the presheath region, any charge separation is omitted in the fluid equation. The ion density comes from the guiding center density, Eq.
, and the polarization density in the gyrokinetic ion formulation, i.e.,
To calculate the electron pressure, we consider the electron temperature $Te$, with the equation of state $Pe=NeTe$. Because electron parallel transit time is much shorter than the collisional
time, we assume that electron is isothermal, i.e., electron temperature is a function of the flux surface only, $Te=Teψ$. In the current simulations, we use a predefined radial profile for electron
temperature to avoid the need for the electron energy equation. Neglecting the parallel electron temperature variation in our model results in the exclusion of the thermal force in Eq. (7) from the
electron–ion collision, which potentially alter the electrostatic potential.
Next, we need to determine the ion flow velocity in Eq.
. We have already obtained the parallel component using the ion gyrokinetic formulation with the pushforward transformation, i.e., Eq.
. However, calculating the perpendicular ion flow velocity is more complicated due to the expression of velocity in guiding center coordinates, involving magnetic moment
and the gyrophase
. A direct integral approach is not suitable, necessitating the inclusion of additional fluid equations for ion species to determine these quantities
Here, we introduce the total time derivative,
. Once again, we can safely neglect the partial derivative with respect to time for equilibrium simulation. The friction force,
, accounts for the damping effects arising from sources other than ion or electron collisions, such as turbulence fluctuations or collisions with neutrals or impurities. The
is the effective coefficient for the resulted drag force, and
stands for the effective diffusion coefficient. The resistive current term is the counterpart of the transport that also appeared in Eq.
. By adding Eqs.
, we arrive at the total force balance
where the total pressure
To maintain consistency with the fluid electron model, we also need to modify the gyrokinetic equation for ions. This includes incorporating the collisions with electrons and the friction force from
into the guiding center motion, Eq.
. As a straightforward solution, we can treat the drag force as an external force acting on the ion particles directly, similar to the electromagnetic forces. In this way, the friction force in the
fluid picture will be equivalent to a drift velocity,
, and a parallel acceleration,
, where
encompasses all collisional effects from other species, such as
. This modification transforms the guiding center motion as follows:
C. Implementation of the fluid equations in field-line coordinates for GTC-X code
In GTC-X, we use a field-aligned mesh for the field solver.^22 The coordinate system $ψ,ζ,S$ includes the poloidal magnetic flux $ψ$, the angular coordinate $ζ$, which is consistent with the
cylindrical coordinates, and the normalized field-line distance coordinate $S$, which relates to the distance along each magnetic field-line. $S∈0,1$ can be treated as a substitute for the parallel
coordinate $Z∈−Z0,+Z0$ when we constructed the GTC-X code.
To solve the electrostatic potential, we require six component equations from the vector equation set, Eqs.
. These equations correspond to six unknown variables: the potential
, three components of the current
and two components of the perpendicular ion flow velocity
. We express these equations in the magnetic coordinates
$JψBSg=−miNiUi⋅∇Ui⋅eζ−σUiζ+ν∇2Ui⋅eζ total toroidal force balance,$
$∂∂SgJS=−∂∂ψgJψ (quasi-neutrality),$
$∂∂Sϕ=−η∥JS+1Nee∂∂SPe (electron parallel force balance),$
$JζBSgNee=1Nee∂∂ψPe−∂∂ψϕ−η∥−η⊥gψSgSSJS+η⊥Jψ+UiζBSg (electron radial force balance),$
$UiψBSg=−η⊥Jζ+JψBSgNee (electron toroidal force balance, classical transport),$
$UiζBSg=η∥−η⊥gψSgSSJS+η⊥Jψ+∂∂ψϕ+1NiZi∂∂ψPi+miNiUi⋅∇Ui⋅eψ+σUiψ−ν∇2Ui⋅eψ (ion radial force balance),$
are the contravariant and covariant components based on the corresponding basis vectors of the field-line coordinates.
are components of geometric tensor. The
is the Jacobian, and its inverse can be calculated as
Here, we have already simplified the equations by assuming symmetry in the toroidal direction, i.e.,
To numerically solve Eqs. (12)–(17), we convert the covariant components to the contravariant components, using the geometric tensor $gαβ=eα⋅eβ$, such as $Jα=gαβJβ$. This ensures that all the
equations involve only six unknown contravariant variables. For simplicity, we have not included the detailed forms of the convection term, $Ui⋅∇Ui$, and the diffusion term, $ν∇2Ui$, in the
field-line coordinates. The detailed computation of the geometric tensor and the convection/diffusion terms will be presented in Appendix A. Throughout the simulation in this paper, we have not
included the diffusion term $ν∇2Ui$ due to its complex form in the field-line coordinates, which may be implemented in the future work. Nevertheless, the effects of the convection term $Ui⋅∇Ui$ are
discussed in detail in Sec. IV.
To improve clarity, we have organized our equations in such a way that a specific variable is listed on the left-hand side of each equation. This indicates which variable that each equation
addresses. Equation (12) reveals that the radial current in our model arises from flow convection, drag force, and diffusion. Due to the presence of radial current, the quasi-neutral condition, Eq.
(13), necessitates a parallel current. The parallel Ohm's law then determines the 2D potential structure based on the resistive parallel current and pressure gradient. Eqs. (15)–(17) give the value
of $Jζ$, $Uiψ$, and $Uiζ$ that these flow velocities again serve as important sources to generate the radial current. These six equations form a closed equation set that can be simultaneously
solved using the HYPRE solver implemented in the GTC-X code.^44
To verify the model in GTC-X simulations, we initiate our analysis from a simplified model by setting the resistivity
) and the drag force coefficient
. In this scenario, the parallel electron force balance, Eq.
, implies that the electrostatic potential is solely determined by the electron pressure gradient. Consequently, this simplification decouples Eq.
from other equations, and a simple solution of
on each flux surface can be obtained:
Here, we used the quasi-neutral condition,
, and the equation of state,
. Assuming a constant electron temperature along the flux surface, the potential can be calculated by the ion density. The
, which corresponds to a zero potential, can be arbitrarily selected along a given field-line.
This simplified model effectively behaves as a 1D model, with the potential on each flux surface being independent from each other. In this case, the radial force balance with the electrostatic
potential is inherently satisfied due to the lack of interaction between the parallel and radial directions. To implement this 1D model in the 2D simulation, a boundary potential profile
can be manually set on each flux surface,
$S=0, 1$
correspond to the parallel boundary locations,
, and
. This implies that
directly corresponds to the potential profiles at the edge of the quasi-neutral plasma. When a biasing profile is applied in the FRC SOL, a passive toroidal ion flow is induced, represented by
. The solution discussed here, similar to Steinhauer, can be used to explain the plasma rotation with end-shorting condition.
The potential boundary $ϕ0(ψ)$ here is right at the edge of the quasi-neutral plasmas, implying that we need to add the additional potential drop across the Debye sheath layer when considering the
external potential set by the biasing system on the divertor walls. The true potential on the wall $ϕw$ should be the sum of both the boundary potential $ϕ0$ in this paper and the Debye sheath
potential drop $ϕD$, $ϕwψ=ϕ0ψ+ϕDψ$. Incorporating this Debye sheath potential with a proper sheath boundary model is needed for calculating the structures of the parallel current and flow near the
In our simulation, the potential drop across the sheath, $ϕD$, can be either calculated using the logical sheath boundary^31 or estimated through simple Debye sheath models. For example, $ϕD$ can be
modeled as a function of electron temperature, $ϕDψ=−Teψelnmi4πme$. This formulation assumes zero current through the Debye sheath, cold ions moving at a speed of $2Temi$, and electrons with a
half-Maxwellian distribution flowing out of the region.^46 In our present model, with both $ϕw$ and $Te$ as preassigned values, $ϕDψ$ will be a fixed offset to the wall potential $ϕw$ on each flux
surface. Consequently, we can obtain a fixed boundary condition, $ϕ0=ϕw+Teelnmi4πme$ for the quasi-neutral plasma.
In future simulations, if the electron temperature is modeled self-consistently or a more comprehensive Debye sheath model is employed, $ϕD$ may depends on variables such as the current density $J$.
In such cases, the boundary condition for the quasi-neutral plasma $ϕ0$ must align with these simulated quantities rather than being a fixed value as currently used. In Sec. VI, we will discuss the
selection of appropriate Debye sheath models for our simulations and will examine the implications of assuming zero current through the Debye sheath. Before that, $ϕ0$ will remain a fixed boundary
condition, based on the simplified Debye sheath model, to facilitate discussions on the effects of other parameters.
To establish an equilibrium, it is necessary to include a particle source in the simulation, compensating for particles leaving the FRC SOL region in the parallel direction. A straightforward
approach is to refill the central region with the same number of particles lost at the boundaries, which conserves the total number of particles in simulations. A practical implementation of this
concept could involve a local Maxwellian velocity distribution $fM$, with similar profiles as the initial density $ni0ψ$ and temperature $Ti0ψ$. Such model maintains similar thermal properties as
the initial state, which can be regarded as an external particle injection with zero flow velocity and fixed density and temperature. In this paper, we use such a simple source model for verifying
the 2D equilibrium model in the FRC SOL. In the future study, we will use more realistic particle and energy sources from experimental measurements or modeling that include the particle transport
from the core to the SOL and external injections such as the neutral beam injection and the particle fueling.
Following these considerations, the density source rate is chosen as $ṅsrcψ,SΔt=AsrcSni0ψ$. $AsrcS$ is a normalization factor that ensures the total refilled particle number $Nsrc$ equals to the
lost particle number $Nloss$ within a time interval $Δt$, i.e., $Nsrc=∫dRAsrcSni0ψ=Nloss$. In the simulation, $AsrcS$ is held constant within the region $Z/R0∈−4.0, 4.0$ and set to zero outside the
region. This will result in a uniform density source rate within the region, as the plateau shown in Fig. 3(b). It is worth noting that this is a simplified model, and more realistic source density
profiles may be explored in future studies, particularly as more experimental data becomes available.
By including the source term in the ion equation, we derived the continuity equation and the parallel force balance with drift-kinetic ions by neglecting the finite Larmor effects in Eq.
Here, we use the symmetry in toroidal direction, so
. The parallel gradient is defined as
. Notice that Eqs.
are the fluid equations for the guiding center quantities, which can be directly calculated from ion guiding center distribution
When refilling the lost particles back to the center with a local Maxwellian, part of them will be trapped by the mirror throat of FRC. When the system reaches equilibrium, the particle source rate
is balanced with the collisional loss rate toward the divertors. To simulate the portion of ions constantly escaping the magnetic potential well, we use an ion–ion pitch-angle scattering operator to
randomize the ion velocities, corresponding to $Cfi$ in Eq. (1). In this way, some of ion particles can be scattered into the loss cone and escape the region in the parallel direction. This
scattering operator is crucial in this 1D simulation to establish such dynamic balance with the source $ṅsrc$.
Finally, the ion flow boundary we used was consistent with the so-called logical sheath boundary^31 to satisfy the quasi-neutral condition. A simplified estimation of the Debye sheath potential drop
is to assume a certain form of the electron distribution function and compute the flux as $Γevce=∫vce∞dv∥∫−∞∞dv⊥ Fev$, where $vce$ is the cutoff velocity for the slowest electron. Since the electron
flux is the same as the ion flux, $Γe=Γi$, we can obtain the critical velocity of reflected electron $vce$ from the ion flux and then estimate the sheath potential drop as $ϕD=−m2evce2$.^31 This
sheath potential should be added to the potential boundary $ϕ0$ in our model to calculate the potential drop between the divertor wall and the plasma. Future research will focus on simulating the
actual electron distribution to refine the sheath potential drop calculation in order to compare with experiments.
To verify Eqs. (20) and (21), the GTC-X simulation uses the 2D axisymmetric FRC magnetic geometry provided by the MHD equilibrium code LR_eqMI,^47 with the FRC magnetic field structure described in
the original GTC-X formulation paper.^22 The poloidal flux surfaces and magnetic field geometry within the simulation domain in this paper is plotted in Fig. 2. The radial simulation domain spans
from $ψR=1.43,Z=0$ to $ψR=2.40,Z=0$, with $R$, $Z$ value normalized to the radius of the equilibrium magnetic axis $R0=26.8 cm$.
The parallel domain encompasses $Z=±20.0$. Notably, the inner boundary approaches the separatrix $ψsepR=1.423,Z=0$, and the parallel boundary includes the inner mirror throat location around $Z=
±9.37$. For the C2-W device, a second mirror throat is located near $Z=±30.0$. The 1D KSOL simulations by TAE team focus specifically on the second mirror location, where the magnetic field-line
expands rapidly after the mirror throat, and a kinetic electron model is used to explore the deviation from the simple Boltzmann electron response.^26 Due to the drastic density drop after the second
mirror throat, including this region in GTC-X simulations up to the divertor would require substantially more computational resources. For this reason, the area near the second magnetic mirror
location was not included in our current simulations. However, the formulation presented in this paper remains applicable for possible extensions into the magnetic expansion region toward the
The ion temperature is a function of $ψ$, exhibiting a sharp decrease from $861$ to 8eV toward the outer radius, and the electron temperature is a constant $Te=80 eV$ in this equilibrium. Although
$Ti$ near the separatrix is approximately ten times that of $Te$, $Ti$ quickly drops to a comparatively low value, resulting in a mean ion temperature of $206 eV$. The density profile also
undergoes a significant drop from $0.857n0$ to 0.081 $n0$, where $n0=2.44×1013 cm−3$ represents the peak density at the magnetic axis.
In our simulations, we use an explicit time integration with a 2nd-order Runge–Kutta method to evolve the dynamical system including the gyrokinetic and fluid equations. The time step, $4.3×10−5 ms$
, is chosen based on the convergence study in our previous paper^8 to accurately resolve ion guiding center motion, which sets the shortest timescale of the simulated system. The simulation time for
the system to evolve toward an equilibrium solution is on the order of 1 ms, which is much longer than that ion pitch-angle scattering time.
Figures 3(b) and 3(c) present the parallel variation of each term in Eqs. (20) and (21), evaluated at a mid-radius flux surface where the initial $Ti$ is comparable to $Te$. The amplitude of
magnetic field, in Fig. 3(a), is normalized to $B0=530.65 G$ at the magnetic axis. A pronounced peak in magnetic field strength is observable at the mirror throat locations $Z=±9.37$. The large
gradient leads to significant variation of the $∇∥B$ terms in these equations.
As we can see in Fig. 3, the correct calculation of these mirror force effects can be important when achieving the force balance and the particle conservation. The source term $ṅsrc$ manifests in
Fig. 3(b) as a plateau within $Z=±4$, marking the region where particles lost in the parallel direction are uniformly replenished.
Figure 4(a) shows the 2D potential structure, and Fig. 5(a) shows the density distribution in the simulation. Due to the particle loss at $Z=±20$, a low-density region can be observed outside the
mirror throat. Despite a density peak near the inner boundary around $Z=0$, the highest potential values are found near the outer boundary. This is because we choose a boundary condition of $ϕW=ϕD$
( $ϕ0=0$) at $Z=±20$. In this case, the potential change only depends on the relative density drop $Niψ,SNiψ,1$ along each field-line, instead of the absolute value $Niψ,S$. Larger potential
value at the outer boundary only indicates a larger relative ratio of density on those flux surfaces.
By verifying Eqs. (20) and (21) through our simulation, GTC-X's ability to accurately represent the parallel physics of ion species is confirmed. The next step is to extend the electron model,
specifically Eq. (18), to couple the radial and parallel physics for a more comprehensive understanding of electrostatic potential structure in SOL. This will be addressed through an exploration of
Eqs. (12)–(17) in Secs. IV–VI.
IV. EFFECTS OF RESISTIVITY AND DRAG FORCE IN THE 2D FRC SOL MODEL
In this section, we will focus on the 2D model of Eqs.
and start our investigation by estimating the physical parameters in these equations. As we mentioned in Sec.
, our discussion does not differentiate between parallel and perpendicular resistivity in this paper. Instead, we approach the resistivity as a general representation of frictional effects arising
from anomalous transport. For the resistivity
, we use the classical resistivity from electron–ion collisions as a reference value
$ηclassical=meνeiNee2, with νei=4.206×10−6Nicm−3Zi2lnΛTeeV3/2.$
With typical parameters of
$Ne=2.444×1013 cm−3$
, and
$Te=80 eV$
, we calculate the electron–ion collision frequency as
$νei=1.86×106 s−1$
. The corresponding resistivity, denote value as
, is termed the classical resistivity in all our simulations. In subsequent analyses, we assess the impact of the resistive current by scaling the resistivity
up by
orders of magnitude. This amplification accounts for enhanced transport phenomena, possibly stemming from a range of turbulent and collisional interactions. This approach aligns with experimental
observations, which often indicate resistivity levels surpassing the classical predictions.
The drag force can originate from various collisional processes or turbulent fluctuations. We use the ion–ion collisional frequency, $νii=4.8×10−8nilnΛTi3/2mi/mp$, as a reference for this drag force
in our paper. With a typical ion temperature of $Ti=200 eV$ and $mi/mp=2$, we have $νii=4.45×103 s−1$. The corresponding value for the drag force coefficient is $σ0=miNiνii$.
As we mentioned in Sec. IIB, the fluid model Eqs. (12)–(17) were derived based on the assumption of isotropic pressure. However, when taking into account the anisotropy in pressure, a mirror force
term will appear in the fluid equation. This term is important when we explain the force balance shown in Fig. 3(c). Following the algebra shown in Appendix B, the divergence of pressure tensor will
give us an additional term associated with $∇⋅bb$. When the parallel ( $P∥$) and perpendicular ( $P⊥$) pressures are different, we need to modify the pressure gradients, $∂P∂S$ by
$∂P∥∂S+P⊥−P∥1B∂B∂S$, and $∂P∂ψ$ by $∂P⊥∂ψ+P∥−P⊥∇×bζRB$ for both electrons and ions. In this paper, as $Te$ is assumed to be a constant, the electron pressure is always isotropic, $Pe∥=Pe⊥$. The
only modification under this assumption is in Eq. (17), the ion radial force balance. Furthermore, modifications would be necessary to accommodate anisotropic electron pressure in future studies.
In the 1D simulation, the parallel equation is decoupled from other equations and only the potential $ϕ0$ at the plasma edges on each flux surface is needed to solve the electrostatic potential.
However, more boundary conditions are needed in the 2D simulation with resistivity and drag forces. In the parallel direction, we still provide a fixed $ϕ0ψ$ at the plasma edges. The parallel current
at the midpoint is assumed to be zero, i.e., $J∥=J⋅b=0$ at $Z=0$ due to the symmetry between positive and negative Z regions. In the radial direction, we apply additional radial boundary conditions
for potential and current by assuming, $∂∂ψ2=0$, i.e., $ϕ$ and $Jψ$ changes linearly at the radial boundaries when calculating the radial derivatives in Eqs. (13), (15), and (17).
After determining all the necessary parameters in the fluid electron model, we conduct a simulation with $η=η0$ and $σ=σ0$. Similar to the 1D case, several ion collisional times are simulated to
obtain the steady-state equilibrium. Figures 4(b) and 5(b) display the resulting electrostatic potential and the density structure. Comparatively, the density profile from this simulation appears
less peaked in the center region than what was observed in the 1D parallel simulation, which can be attributed to the influences of resistive and collisional effects. Especially, the drift velocity
$vF$ due to the drag force diffuse ion particles away from the core region. This phenomenon is more distinctly visible in the potential plot of Fig. 4(b), where more contour lines extend toward the
outer radial boundary and the parallel divertor boundary, beyond $Z>9.37$. These results underscore the significance of incorporating 2D effects into the simulation to capture the complex behavior
of the plasma, especially in the presence of resistive and collisional interactions.
Figure 6(a) illustrates the current density flow within the FRC SOL, revealing a pronounced current density near the axis. This enhancement at smaller radii is partly attributed to the FRC's
cylindrical geometry, where the volume increases with radius, leading to a lower current density at a larger radius. The current density is presented in GTC units, $JGTC=en0R0Ωcp=5.33×106 A/m2$,
where $Ωcp=5.08×106 s−1$ is the gyrofrequency of a proton at the magnetic axis $B0$. Maximum amplitude for poloidal current is $2.4×10−3$, while the toroidal current ranges from $−3.1×10−2$ to
The inclusion of the convection term, represented by $Ui⋅∇Ui⋅eζ$ in Eq. (12), can play a crucial role in predicting the current distribution in the SOL. This term can be a main source for radial
current $Jψ$, when ion flow velocity is significant. Including this term ensures the model accounts for the influence of ion flow inertia, particularly in regions beyond the mirror throat where the
parallel flow can reach values comparable to the sound speed. Similar level of parallel ion flow was also reported in previous Q2D simulations.^25
When the convection term is omitted in the simulation, the results show a significant change in the current density distribution, even though the potential and ion density remain similar to the
previous case with the convection term included. In Fig. 6(b) (without the convection term for $η=η0$ and $σ=σ0$), the current direction is reversed near the axis when the convection term is not
included. Maximum amplitude for poloidal current in this case is $1.3×10−4$, while the toroidal current ranges from $−3.0×10−2$ to $8.9×10−3$.
The convection term contributes an additional radial current $Jψ=miNiBSωiψUiS$, as derived in Appendix A. Due to a relatively large ion toroidal flow (shown by the contours in Fig. 6) and parallel
flow, a stronger radial current is generated and modifies the current structure in FRC SOL. Comparing Figs. 6(a) and 6(b), the poloidal current structure can be correlated with the toroidal flow when
the convective effects are considered in the simulation. Therefore, it can be essential to include the convection term when predicting the current distribution.
In GTC-X, the convection term is approximated by using the flow velocity from previous time step. Although such treatment can be used to achieve a steady-state solution, a more rigorous nonlinear
solver for these terms should be developed in the future works. The inertial forces effects associated with strong perpendicular flows are neglected in the current gyrokinetic model in the GTC-X.
Systematic derivation of such effects exists for tokamaks with strong toroidal rotation.^48–50 These effects will be incorporated in our future work for a more consistent simulation of the convection
effects in the FRC.
Next, to investigate how the resistivity and drag force can modify the 2D potential structure, we vary the resistivity $η$ and the drag force $σ$ by applying a multiplicative factor. In Fig. 7, the
radial profile of the potential $ϕ$ at $Z=0$ is presented for changing drag force coefficient $σ$ from $1σ0$ to $50σ0$, while fixing the resistivity at $η0$. The dashed line serves as a reference
line from the 1D simulation in Sec. III.
With an increase in $σ$, ions are subjected to stronger drag force, leading to their redistribution toward the outer radius and the parallel boundaries. This redistribution moderates the density
gradient observed in the 1D simulation, resulting in a less pronounced potential drop in the parallel direction. When we fix the potential as zero (end-shorting boundary) at $Z=±20$, lower potential
is observed at the center of the simulation domain, as shown in Fig. 7(a).
Since the $E×B$ shearing rate is more relevant in the turbulence suppression, we also plot the shearing rate, $ωs=R2Bd2ϕdψ2$, in Fig. 7(b). When the resistivity and drag force coefficient are set to
their baseline values, $η=η0$ and $σ=σ0$, the inclusion of the drag force and the resistivity smooth out the potential profile which slightly reduces the shearing rate. However, as $σ$ continues to
increase, the impact of these collisional factors on the shearing rate becomes more and more obvious, due to particle density redistribution. This means, under strong collisional effects, an accurate
estimation of particle density in SOL can be important for predicting the $E×B$ shearing rate.
In Fig. 8, we carry out a similar parameter scan for the resistivity $η$ from $1η0$ to $50η0$, while fixing the drag force coefficient at $σ0$. The overall potential drop between the 1D simulation
and the $η=η0$, $σ=σ0$ case is mainly due to the inclusion of a drag force $σ=σ0$ as we have discussed in Fig. 7. For the case “^* $η=50η0, σ=σ0$,” we do not include the convection term in Eqs.
(12) and (17) due to the numerical stability when including convection term with such a large resistivity.
The inclusion of $η$ near the classical resistivity value has relatively minor effects on the $E×B$ shearing rate, as compared to the inclusion of $σ$. As we will see in the following discussion,
this is attributed to the fact that the resistive current term $ηJ$ plays a minimal role in balancing the parallel forces. The predominant factor influencing the potential change is the alternation
of the density profile, which is mainly affected by the drag force in the current parameter range.
Based on the parameter scan results, we choose a resistivity value that is five times the classical value, $η=5η0$, for subsequent discussions. Past research has indicated that such a resistivity
value is more pertinent for accurately forecasting the C2 FRC plasma's experimental evolution.^25 The drag force coefficient is equivalent to the ion–ion collisional timescale, $σ=σ0$, allowing us
to observe the impact of this added drag force within reasonable simulation timeframe. In future studies, we anticipate adopting more realistic resistivity and drag force coefficients once we obtain
additional experimental measurements.
To better understand the electron fluid model, we plot out the electron force balance in Fig. 9 with $η=5η0$, $σ=σ0$. Along the parallel direction at the middle flux surface $R=1.89$ in Fig. 9(c),
the electrostatic potential, labeled as “ $−∂Sϕ$,” is mainly balanced by the electron pressure, labeled as “ $∂SPe$.” This observation is similar to the 1D case, where the Boltzmann-like response,
Eq. (19), can be effectively employed to estimate $ϕ$ based on the ion density. The resistive current effects “ $−ηJS$,” on the other hand, can be negligible when $η$ is around classical value.
The electron radial force balance at $Z=0$ in Fig. 9(a) reveals significant contributions from $Ui×B$ and $J×B$, which together give the electron flow $Ue×B$. This toroidal electron flow is then
comparable to the $∇Pe$ and the $∇ϕ$ term in Eq. (7). In the toroidal force balance at $R=1.89$ in Fig. 9(b), the absence of toroidal variations due to the symmetry condition $∂∂ζ=0$ results in a
balance between the radial current $Jψ$, the ion radial flow $Uψ$ and the resistive current $Jζ$. There is an interesting separation of two regions for the toroidal force balance: within the mirror
throat ( $Z=9.37$), the radial ion flow and the toroidal resistivity are the main contributions; outside the mirror throat, the radial ion flow and the radial resistive current dominates the
toroidal force balance. This could be attributed to the increase in ion flow toward the divertors, which in turn generates a higher plasma current as the particles exit the SOL.
The force balance can be different when a much higher resistivity is considered. With
, the force balance is shown in
Fig. 10
. All the ion flow velocities are reduced due to the higher collision frequency. Instead of a dominant toroidal flow and current in
Fig. 10(a)
, the radial force is now in a complex balance with all terms except for the resistivity at
. In the toroidal force balance [
Fig. 10(b)
], though the radial current is still mainly carried by the ion flow outside the mirror throat, the radial transport within the mirror throat are now in a complex balance between,
, and
. Based on these considerations, a simplified electron model from Eqs.
can be proposed:
where the resistive current effect is mainly considered in the toroidal electron force balance. The drag force, on the other hand, plays a vital role in shaping the overall plasma dynamics and enters
the electron dynamics through the radial current
and the density distribution in
After the detailed analysis of the 2D potential model, the focus now shifts to understanding how the edge biasing voltage impacts the potential structure in the confining vessel of the FRC SOL. In
experiments, the divertor biasing is achieved by an array of concentric annular electrodes to achieve discrete voltages at different radial locations, and these electrodes are insulated from each
other.^4,5,7 To achieve smooth potential structures in simulations, we manually select a continuous radial profile $ϕw$ instead of discrete potential steps to qualitatively resemble the potential
drops created by electrodes.
Assuming the Debye sheath drop $ϕD$ as a function of $Te$ and the current $J$, we have the boundary condition $ϕ0=ϕw−ϕDTe,J$. In parctice, the wall potential can be controlled by the electrodes,
allowing $ϕ0$ to be determined self-consistently based on temperature and current data. For the simple Debye sheath we discussed, $ϕ0=ϕw+Teelnmi4πme$ can be a fixed potential boundary in the
simulations. However, this model assumes zero current through the Debye sheath. In Sec. VI, we will explore the validity of this assumption and discuss under what conditions it may hold.
In this section, we explore how changes to the simplified boundary potential affect the 2D equilibrium potential. Notice that the Debye sheath potential $ϕD$ in this form is no different than
providing an additional shift of $ϕ0$ profile. We can produce the desired potential boundary profiles by selecting appropriate wall potentials. However, one should always keep in mind that this
relationship may not hold if $ϕD$ also depends on the current, in which case its effects on $ϕ0$ would be more complex.
To implement the fixed potential boundary, we first use an analytic profile $ϕ0ψ=ϕ0,max21−tanh0.34−ψ¯−ψ¯00.29$ as the potential boundary condition at the quasi-neutral plasma edge $Z=±20$, where $ψ¯
=ψψ1−ψ0$ is the poloidal flux normalized by the simulation domain size of $ψ0,ψ1$, and $ϕ0,max$ is the maximum potential of the radial profile. For simplification, we describe our simulations by
$ϕ0$ in most cases since it is directly applied at the boundaries of our simulation domain. A discussion on the difference between $ϕ0$ and $ϕw$ is provided in Sec. VI.
When there is no resistivity and drag force ( $η=0$, $σ=0$), the potential at the boundary is simply a constant shift of potential value along each flux surface. Same amount of $E×B$ shear will be
added in the parallel direction, saying that the boundary $ϕ0$ can be accurately transfer to the center region of FRC. When the resistivity and drag force are included, there can be 2D variations due
to the radial coupling, so the introduced $E×B$ shear from the potential boundary may change when penetrating toward the middle region of FRC.
Figure 11 illustrates how the radial potential profile at the plasma center of $Z=0$ can change when we apply different potentials at the plasma edge of $Z=±20$. In general, the potential profile we
applied at the boundary introduces a potential shift throughout FRC SOL. With $η=5η0$, $σ=σ0$, the resistive $ηJ$ term has little effects in the parallel electron force balance, as we discussed in
Sec. IV. Thus, a solution similar to Eq. (19) can be obtained, meaning that the total potential can be separated into two parts: the boundary condition $ϕ0ψ$ and the Boltzmann-like response due to
In Fig. 11(a), the potential difference between the quasi-neutral plasma edge ( $Z=20$) and the center profile ( $Z=0$) already exists when a zero potential boundary $eϕ0,max=0$ is used. A
density structure with $eϕ0,max=200 eV$ is also provided in Fig. 12(b), which can be qualitatively compared to Fig. 5(b). The similarity in density structures, whether or not a non-zero potential
profile is applied, leads to a similar potential drop along the parallel direction in both scenarios.
However, if we take a closer look at the $E×B$ shearing rate in Fig. 12(b), the situation is more complex than a straightforward superposition of these effects. Specifically, the $E×B$ shearing at $Z
=0$ with $eϕ0,max=200 eV$, denoted as $ωSZ=0, 200 eV$, cannot be simply expressed as the sum of the shearing due to the potential $ϕ0ψ$, i.e., $ωSZ=20, 200 eV$, and the shearing caused by
original density variation with zero edge profile, $ωSZ=0, 0 eV$. The density structure is also modified when an edge potential profile is introduced into the FRC SOL, and more subtle changes exist
in the radial potential profiles.
To see a stronger effects of the resistive current, we also run a non-zero $ϕ0$ case with $η=100η0$, $σ=10σ0$ in Fig. 13. Due to stronger effects of the resistivity and the drag force, the radial
profile of the electrostatic potential at $Z=0$ can differ more from the applied $ϕ0$ boundary. The toroidal electron force balance and the drift velocity in ion dynamics, $vF$, can both modify the
density distribution in FRC SOL and finally changes the radial potential profile.
To conclude the discussion, we want to show that Debye sheath potential drop can possibly affect the 2D equilibrium potential structure. As discussed in Secs. III–V, the boundary condition $ϕ0ψ$ is
the potential profile at the edge of the quasi-neutral plasma, so an additional Debye sheath potential drop should be included if we want to compare the simulation with the actual potential value
$ϕw$ at the divertor walls. As defined in Sec. III, $ϕ0=ϕw−ϕD$, the zero boundary conditions $ϕ0=0$ in Secs. III–V correspond to $ϕw=ϕD$. That is, an externally applied potential on the divertor
wall is set to be identical to the Debye sheath potential drop.
In Secs. III–V, a simple Debye sheath model was assumed, $ϕDψ=−Teψelnmi4πme$, based on a textbook derivation.^46 Given that the electron temperature is predetermined in our model, the required
additional potential drop $ϕD$ is also a fixed profile throughout the simulation. Essentially, this approach acts as an offset correction to the profile $ϕW$ and leads to a fixed boundary $ϕ0$ at the
quasi-neutral plasma.
For comparison between the simulations with $ϕw=ϕD$ ( $ϕ0=0$) and with $ϕw=0$ ( $ϕ0=−ϕD$), we use an electron temperature profile $Teψ=Te01.0+0.25tanh0.34−ψ¯−ψ¯00.29−1$, with $Te0=80 eV$.
Figure 14(b) illustrates the potential structure under the $ϕw=0$ conditions. The potential $ϕ0$ at $Z/R0=±20$ is larger at the inner radius, due to the higher electron temperature from the specified
profile. This leads to a higher Debye sheath potential drop $ϕD$.
When analyzing the differences in the 2D potential structure with $ϕw=ϕD$ and $ϕw=0$, the variations can be elucidated in a manner akin to what is depicted in Fig. 12(a), but now with a $ϕ0$ profile
decreasing in the radial direction. Figure 15(b) illustrates the radial force balance incorporating the sheath potential drop as a boundary condition. Here, an extra $E×B$ flow can be observed if we
look at the radial profile labeled as “ $−∂ψϕ$” in Figs. 15(a) and 15(b). This flow emerges predominantly due to the Debye sheath potential boundary, where $∂ϕ0∂ψ=−∂ϕD∂ψ$.
The Debye sheath potential drop in our simulation, as a simple function of electron temperature, essentially acts as an additional adjustment to the $ϕ0$ profile at the boundary. This model assumes
zero net current through the Debye sheath, $J∥=0$, by balancing the ion and electron contributions. Specifically, at the boundary of the plasma, the parallel current $J∥$ can be expressed as the sum
of the contributions from ions and electrons, $J∥=ZiΓi∥−eΓe∥$, where $Γi∥$ and $Γe∥$ are parallel particle fluxes of ions and electrons, respectively.
Given our maintenance of the quasi-neutral condition, the total ion flow from all boundaries must equal the total electron flow. This means that any local current entering the region at one boundary
must exit at another. Illustratively, if we examine the radial profile of the parallel current at $Z=20$ shown in Fig. 6(a), we observe current entering the region from the larger radius and exiting
at the smaller radius, as further detailed in Fig. 16(a).
Figure 16(b) presents the current flow when a boundary potential profile with $eϕ0,max=200 eV$ was applied. Although a lower potential at the smaller radius tends to direct currents toward that area,
this trend was not pronounced in our simulations. The current pattern remained largely unchanged after altering the potential boundary profile, consistent with our earlier findings that most
potential variations at the boundary primarily influence $E×B$ flow. Additionally, the effects of resistive current terms are relatively minor in the overall force balance.
This observation suggests that assuming zero current through the Debye sheath and applying a simplified Deba sheath model at the larger radius may be a reasonable approximation. However, this should
not be considered a definitive conclusion, as our fluid model includes several simplifications which limits our ability to compare with actual experiments. In the future, more comprehensive Debye
sheath models will be necessary to accurately match the current and potential drop near the boundary.
For example, considering a Debye sheath model that includes non-zero current through the Debye sheath boundary. This necessitates accounting for the current flow balance $J∥=ZiΓi∥−eΓe∥Te,ϕD$ at the
boundary. Assuming electrons flow out of the boundary with a half-Maxwellian distribution, we obtain an electron flux, $Γe∥Te,ϕD=ne0Te2πmeexpeϕDTe$, from which the Debye sheath potential can be
estimated with the simulated $J∥$ and $Γi∥$. However, this still requires several assumptions, such as a definite flow direction of electrons, i.e., $J∥<ZiΓi∥$. Also, special attention is meeded
when $Γe∥$ approaches zero, since $ϕD→−∞$ given finite $ne0$ and $Te$. These potential issues underline the need for future tests and implementations of a self-consistent Debye sheath boundary in
the model.
Future investigations will utilize current flow data at the boundary to explore more sophisticated Debye sheath models and validate them against experimental data, improving the model's fidelity.
Potential useful references for developing such sheath boundaries could be the Chapter 3 in a recent textbook by Rozhansky, 2023^51 and the related treatment in the SOLPS-ITER code.^28,29 The simple
Debye sheath model used in this study mainly acts as a preliminary tool, highlighting the importance of incorporating a precise sheath model for more accurate simulations.
In this paper, an axisymmetric 2D equilibrium potential in the FRC SOL is simulated by assuming the timescale separation between the turbulence evolution and the steady-state equilibrium with a
preassigned potential boundaries. The formulation to calculate the presheath equilibrium includes a full-f gyrokinetic ion model and a massless electron model for a quasi-neutral plasma before
entering the microscopic Debye sheath layer or thin magnetic presheath layer. Due to the particle flow through the SOL toward the divertors, the equilibrium presheath potential is intrinsically 2D,
which involves the balance between radial and parallel transport.
The model was first verified in a simplified simulation when the resistivity and the drag force on ions are neglected, which reduces the equilibrium to 1D. The simplified model uses a Boltzmann-like
electron response and can restore the ion parallel force balance and continuity equation on each flux surface. The potential boundary in this 1D equilibrium can perfectly transfer to the center
region of the FRC SOL from the divertors.
After verifying the 1D physics in the parallel direction, parameter scans were performed to analyze the influences of resistivity and drag force on the 2D equilibrium. When considering classical
resistivity corresponding to electron–ion collisions, the resistive current exhibited minimal effects in electron parallel force balance. The drag force, initially approximated using the ion–ion
collision frequency, induced an outward transport of particles directed toward the divertors. This results in a slightly lower potential profile in the center region of the SOL. A more pronounced
impact on electron toroidal force balance and subsequent radial transport behavior was observed when the resistivity was increased to two orders of magnitude above the classical value, combined with
a drag force equivalent to ten times the ion–ion collision frequency.
The parallel electron force balance is always Boltzmann-like within the parameter ranges in this paper, which provides a simple solution to estimate the potential structure in the FRC SOL. The
resistive current can be important in explaining the electron toroidal force balance, but not in the electron parallel and radial force balance. The collisional effects mainly appear through the
density profile change, and then modify the potential structure through electron pressure gradient.
The effects of a changing radial profile of electrostatic potential at the quasi-neutral plasma boundary were investigated. However, an additional potential drop, associated with the Debye sheath
layer, should be added for accurate comparisons with the actual potential at the divertor walls. In this paper, we only explored a simple Debye sheath model by including a Debye sheath potential drop
proportional to electron temperature. This simplified Debye sheath model makes the Debye sheath potential drop an additional correction to the potential boundary applied at the edge of quasi-neutral
plasma. This can be a rough estimation since the present simulation settings predicted a relatively small current flow. However, more accurate Debye sheath model should be considered to
self-consistently match the sheath drop and the current flow, in order to compare with experimental measurements in the future.
Eventually, the interplay between the potential edge profiles and the parallel variation of potential due to the density structure within the SOL will determine the radial electric field at the
center of the FRC SOL. The $E×B$ shearing rates can also be drastically changed under different resistivity and drag force. An accurate estimation of density profiles can be a key point in order to
correctly predict the electrostatic equilibrium potential.
Although we have scanned a certain range of parameter space of our 2D model, simulations with a more realistic setting are still required. In this paper, a uniform density sourcing rate within a
fixed simulation region is used, but the real particle source should be particle transported from the core region into the SOL region, together with other particle injections, such as NBI systems and
fueling system. More accurate resistivity and drag force coefficients are also needed, which requires more experimental data such as the neutrals density. Additionally, current information at the
boundary should also be considered for a self-consistent calculation of Debye sheath potential drop as we just mentioned. A well-defined Debye sheath model could significantly enhance the capability
to accurately estimate the potential changes near the divertor walls.
Though not discussed in this paper, our full-f scheme has successfully replicated the ion temperature gradient (ITG) instability observed in previous delta-f simulations. This comprehensive full-f
scheme holds promise for future investigations into turbulent transport in FRC, incorporating the dynamic 2D equilibrium obtained in this paper.
The authors would like to thank Dr. Laura Galeotti for providing the FRC equilibrium data, Dr. Robert D. Falgout for setting up the HYPRE linear solver in the GTC-X code, and Dr. Peter Yushmanov for
helpful discussions. This work was supported by TAE Grant No. TAE-200441, DOE SciDAC, and INCITE programs and (T.T.) U.S. DoE ECP (Exascale Computing Project). Simulations used resources on the Oak
Ridge Leadership Computing Facility at Oak Ridge National Laboratory (DOE Contract No. DEAC05-00OR22725) and the National Energy Research Scientific Computing Center (DOE Contract No.
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
W. H. Wang: Formal analysis (equal); Writing – original draft (equal). X. S. Wei: Validation (equal); Writing – review & editing (supporting). Z. Lin: Conceptualization (equal); Methodology (equal);
Supervision (equal); Writing – review & editing (equal). C. Lau: Visualization (supporting); Writing – review & editing (supporting). S. Dettrick: Visualization (supporting); Writing – review &
editing (supporting). T. Tajima: Resources (supporting); Writing – review & editing (supporting).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
To implement the fluid model for axisymmetric 2D equilibrium potential in GTC-X, we must express Eqs.
in the field-line coordinates,
. This involves the calculation of geometric tensor
are labels for the three covariant basis vectors, which are defined as
, and
. In GTC-X, we also use cylindrical coordinates
as the base coordinates system.
Using transformation functions in cylindrical coordinates, we can calculate the values of geometric tensor in the physical space
$eψ=∂r∂ψ=∂R∂ψR̂+∂Z∂ψẐ, eζ=Rζ̂, eS=∂R∂SR̂+∂Z∂SẐ,gψψ=eψ⋅eψ=∂R∂ψ2+∂Z∂ψ2, gζζ=eζ⋅eζ=R2,gSS=eS⋅eS=∂R∂S2+∂Z∂S2,gψS=eψ⋅eS=∂R∂ψ∂R∂S+∂Z∂ψ∂Z∂S.$
The calculation of contravariant components
of the geometric tensor and the Jacobian
are shown in previous work by Bao
et al.^22
Note that the parallel components in the $b$ direction differ from the projection onto the field-line basis $eS$. In the FRC geometry, the magnetic field without toroidal component is represented as
$B=∇ψ×∇ζ=BRR̂+BZẐ$, with $eS=g∇ψ×∇ζ$. This leads to the relationship, $b=BB=1BgeS$. Therefore, the parallel flow can be related to the covariant component as $Ui,∥=Ui⋅b=1BgUS$ in the FRC geometry.
By expressing $UiS=gSψUiψ+gSSUiS$, we can substitute $UiS=gSS−1UiS−gSψUiψ$ into the equation. The equation set can be further simplified by recognizing $B=Bb=1geS$. Thus, $BS=1g$ and $BS=B2g$.
Up to now, we have successfully derived a set of linear expressions for our equations, except for the nonlinear convection term, $Ui⋅∇Ui$. To address this challenge, one approach is to utilize the
flow velocity from the previous time step and calculate the complicated convection term as a known source term in the current time step. Because we are interested in the steady-state solution, this
approach can be reasonable, provided that the simulation can numerically converge. In the future, we may explore the development of a nonlinear algorithm to rigorously solve these nonlinear terms.
By arranging the known terms on the right-hand side and the unknown variables on the left-hand side, express the equations in a matrix form
Here, both the convection term and the diffusion term are calculated using the values from previous time step. To clarify this, we label the quantities from previous time step with the superscript, “
To compute the convection term in general coordinates, we employ the vector identity,
to avoid the need for direct derivation on the basis vector. By introducing the ion flow vorticity,
, the convection term we used in Eq.
can then be expressed as
Again, we exploit the symmetry in the toroidal direction,
, and
sums over all the coordinate components.
Similarly, we can use the identity,
, to reformulate the diffusion term in the general coordinates
$∇2Ui⋅eζ=∇∇⋅Ui−∇×∇×Ui⋅eζ=1g ∂∂Sωi,ψ−∂∂ψωi,S,∇2Ui⋅eψ=∇∇⋅Ui−∇×∇×Ui⋅eψ=−∂∂ψ1g∂∂ψgUiψ+1g∂∂SgUiS−1g ∂∂Sωi,ζ,$
$ωiψ=gψψωiψ+gψSωiS, ωiζ=gζζωiζ, ωiS=gSψωiψ+gSSωiS$
For a general pressure tensor,
, with different pressure in parallel and perpendicular direction, we can take the divergence as
The first two terms can reduce to the
term in Eqs.
, when we equate
. The
will only appear when the pressures are distinct in different directions. To calculate this additional term, we use the identity
and notice
For FRC geometry,
is only in
so we can write
$∇×bζ=1B∂BR∂Z−∂BZ∂R+∂B∂RbZ−∂B∂ZbR in R,ζ,Z coordinates,$
where we have used
. Now, the pressure tensor in FRC geometry becomes
We simply need to replace $∂P∂S$ by $∂P∥∂S+P⊥−P∥1B∂B∂S$, and $∂P∂ψ$ by $∂P⊥∂ψ+P∥−P⊥∇×bζRB$ for any species in Eqs. (12)–(17), to account for the additional effects from $∇⋅bb$ term in the pressure
, “
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Interesting Research on Lessons – What You Didn’t Know
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Computer Oriented Stastistical Methods for JNTU-H 18 Course (II - I -
Computer Oriented Stastistical Methods for JNTU-H 18 Course (II - I - CSE - MA303BS) (Decode)
UNIT - I Probability : Sample Space, Events, Counting Sample Points, Probability of an Event, Additive Rules, Conditional Probability, Independence, and the Product Rule, Bayes’ Rule. Random
Variables and Probability Distributions : Concept of a Random Variable, Discrete Probability Distributions, Continuous Probability Distributions, Statistical Independence. (Chapter - 1) UNIT - II
Mathematical Expectation : Mean of a Random Variable, Variance and Covariance of Random Variables, Means and Variances of Linear Combinations of Random Variables, Chebyshev’s Theorem. Discrete
Probability Distributions : Introduction and Motivation, Binomial Distribution, Geometric Distributions and Poisson distribution. (Chapter - 2) UNIT - III Continuous Probability Distributions :
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Maximum Likelihood Estimation. Statistical Hypotheses : General Concepts, Testing a Statistical Hypothesis, Tests Concerning a Single Mean, Tests on Two Means, Test on a Single Proportion, Two
Samples : Tests on Two Proportions. (Chapter - 4) | {"url":"https://technicalpublications.in/products/9789389420401-3","timestamp":"2024-11-09T11:04:38Z","content_type":"text/html","content_length":"97378","record_id":"<urn:uuid:5100b0d2-936f-4cd0-87e2-28220ac32eb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00659.warc.gz"} |
Mean age from observations in the lowermost stratosphere: an improved method and interhemispheric differences
Articles | Volume 23, issue 7
© Author(s) 2023. This work is distributed under the Creative Commons Attribution 4.0 License.
Mean age from observations in the lowermost stratosphere: an improved method and interhemispheric differences
The age of stratospheric air is a concept commonly used to evaluate transport timescales in atmospheric models. The mean age can be derived from observations of a single long-lived trace gas species
with a known tropospheric trend. Commonly, deriving mean age is based on the assumption that all air enters the stratosphere through the tropical (TR) tropopause. However, in the lowermost
stratosphere (LMS) close to the extra-tropical (exTR) tropopause, cross-tropopause transport needs to be taken into account. We introduce the new exTR–TR method, which considers exTR input into the
stratosphere in addition to TR input. We apply the exTR–TR method to in situ SF[6] measurements from three aircraft campaigns (PGS, WISE and SouthTRAC) and compare results to those from the
conventional TR-only method. Using the TR-only method, negative mean age values are derived in the LMS close to the tropopause during the WISE campaign in Northern Hemispheric (NH) fall 2017. Using
the new exTR–TR method instead, the number and extent of negative mean age values is reduced. With our new exTR–TR method, we are thus able to derive more realistic values of typical transport times
in the LMS from in situ SF[6] measurements. Absolute differences between both methods range from 0.3 to 0.4 years among the three campaigns. Interhemispheric differences in mean age are found when
comparing seasonally overlapping campaign phases from the PGS and the SouthTRAC campaigns. On average, within the lowest 65K potential temperature above the tropopause, the NH LMS is 0.5±0.3 years
older around March 2016 than the Southern Hemispheric (SH) LMS around September 2019. The derived differences between results from the exTR–TR method and the TR-only method, as well as
interhemispheric differences, are higher than the sensitivities of the exTR–TR method to parameter uncertainties, which are estimated to be below 0.22 years for all three campaigns.
Received: 03 Nov 2022 – Discussion started: 11 Nov 2022 – Revised: 17 Feb 2023 – Accepted: 01 Mar 2023 – Published: 03 Apr 2023
The lowermost stratosphere (LMS) is the lowest part of the extra-tropical (exTR) stratosphere. Its upper boundary is usually defined as the 380K isentrope and approximates the lower boundary of the
stratosphere in the tropics. The chemical composition of the LMS plays an important role in the climate system. Different transport paths and timescales determine the chemical composition of the LMS
for a wide range of trace gases. The most prominent transport mechanism in the stratosphere is the Brewer–Dobson circulation (BDC), which transports air from the tropical (TR) tropopause to the exTR
and polar stratosphere (Butchart, 2014). The residual circulation part of the BDC is characterized by two branches (Birner and Bönisch, 2011; Plumb, 2002): one branch extends deep into the middle
atmosphere and slowly transports air to high latitudes where it eventually descends to lower altitudes. The shallow branch in the lower part of the stratosphere transports air poleward below the
subtropical transport barrier and is characterized by comparably fast transport timescales. In addition to residual transport, air is transported within the stratosphere by bidirectional mixing. Both
residual transport and mixing are induced by wave activity on different scales and are part of the BDC. In addition to the BDC, exTR cross-tropopause transport strongly affects the chemical
composition of the LMS. This exTR transport mechanism is modulated by the subtropical jet (Gettelman et al., 2011).
The age of air is a widely used concept to describe tracer transport in the stratosphere (Waugh and Hall, 2002). In principle, infinitesimal fluid elements enter the stratosphere across a source
region. The transit time (or “age”) of each individual fluid element is the elapsed time since it last made contact with a source region. A macroscopic air parcel in the stratosphere consists of an
infinite number of such fluid elements, each with its own transit time. The transit time distribution for the air parcel is called the “age spectrum” (Kida, 1983). Past studies were able to obtain
information on age spectra from observations of multiple trace gases (Andrews et al., 1999, 2001; Bönisch et al., 2009; Hauck et al., 2020; Ray et al., 2022). The first moment of the age spectrum is
the mean age of air. It can be derived from measurements of a single inert trace gas species with a monotonic trend in the troposphere. CO[2], SF[6] and a variety of “new” age tracers have been used
in past studies to derive the mean age of air from observations (e.g., Engel et al., 2017; Leedham Elvidge et al., 2018).
The age of air from observations provides a stringent test for numerical models. The number of available trace gas observations that are suited to derive mean age is vastly higher than that to derive
age spectra. In addition, deriving mean age relies on making fewer assumptions compared to deriving age spectra. This makes mean age a valuable measure to compare models to observations. Still,
observational estimates of mean age rely on several simplified assumptions, depending on the trace gas used, which significantly add to the uncertainty in mean age across large areas of the
stratosphere. For example, in order to derive mean age from SF[6] measurements, an infinite lifetime is commonly assumed. In contrast, recent studies showed that a mesospheric sink of SF[6] leads to
a significant bias towards higher ages, especially on old mean age values derived from SF[6] observations (Leedham Elvidge et al., 2018; Loeffel et al., 2022). Another common assumption is that all
air enters the stratosphere through the TR tropopause. However, Hauck et al. (2019, 2020) showed that in the vicinity of the tropopause, transport across the exTR tropopause is also important to
adequately describe age spectra and mean age in the LMS. While the assumption of a single entry point is a good approximation for the stratosphere above about 380K potential temperature θ, this is
thus not the case for the LMS. Together with the interhemispheric gradient in tropospheric trace gas mixing ratios, this limits the ability to derive the mean age of air in the LMS. Further
improvements of the methods to derive the mean age of air from observations are thus desirable in order to provide robust real world estimates of transport timescales in sensitive regions of the
atmosphere and be able to compare them to model results.
With this work we focus on the mean age of air in the LMS, where the old bias of the SF[6] mean age is presumably low. We introduce an extended method that considers exTR input into the stratosphere
in addition to TR input (hereafter the exTR–TR method). In Sect. 2, we describe the concept and implementation of our new exTR–TR method. In Sect. 3, we firstly compare results from the exTR–TR
method to the conventional method, which only considers TR input (hereafter referred to as the TR-only method). These results are based on in situ measurements taken during three aircraft campaigns.
Secondly, we compare Northern Hemispheric (NH) and Southern Hemispheric (SH) mean age in the LMS based on these results. Thirdly, we present a sensitivity study on the exTR–TR method. We summarize
our findings in Sect. 4.
2Calculating mean age in the LMS considering multiple entry regions
2.1General concept
A common approach to describe the mixing ratio χ(x) of a suitable age tracer at an arbitrary location x in the stratosphere is
$\begin{array}{}\text{(1)}& \mathit{\chi }\left(\mathbit{x}\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}\mathit{\chi }\left({\mathbit{x}}_{\mathrm{0}},{t}^{\prime }\right)\cdot G\
left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)\mathrm{d}{t}^{\prime },\end{array}$
where $\mathit{\chi }\left({\mathbit{x}}_{\mathrm{0}},{t}^{\prime }\right)$ is the tracer mixing ratio time series in the source region x[0] as a function of transit time t^′ and the age spectrum $G\
left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)$. The approach expressed in Eq. (1) is based on the assumption that all fluid elements that enter the stratosphere at the same time
have the same tracer mixing ratio. Albeit, in the real world, there is no suitable age tracer with the same mixing ratio time series throughout the troposphere. Hence, the mixing ratio time series is
likely to be different in different entry regions. By using Eq. (1), so far studies that derived the mean age of stratospheric air from measurements of one inert trace gas commonly relied on the
assumption that all air enters the stratosphere through the tropical (TR) tropopause (TR-only method), which appears valid for large parts of the stratosphere. In the LMS however, exTR input needs to
be considered (Hauck et al., 2019, 2020).
We introduce the new exTR–TR method, which builds on an extended approach to derive mean age in the LMS from an inert monotonic tracer that considers exTR input into the stratosphere in addition to
TR input. In a generalized way, our extended approach accounts for input into the stratosphere from N individual source regions x[i] with individual mixing ratio time series $\mathit{\chi }\left({\
mathbit{x}}_{i},{t}^{\prime }\right)$ by calculating a weighted mixing ratio time series. The relative importance of individual source regions can be described by so-called origin fractions (e.g.,
Orbe et al., 2013, 2015). We use the origin fractions f[i](x) as derived by Hauck et al. (2020) as weights for each χ(x[i]); f[i](x) is the fraction of air at x that entered the stratosphere through
x[i]. By applying this assumption, Eq. (1) translates into Eq. (2):
$\begin{array}{}\text{(2)}& \begin{array}{r}\mathit{\chi }\left(\mathbit{x}\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}{\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left({f}_{i}\left(\
mathbit{x}\right)\cdot \mathit{\chi }\left({\mathbit{x}}_{i},{t}^{\prime }\right)\right)\cdot G\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)\mathrm{d}{t}^{\prime }\\ ={\sum }_{i
=\mathrm{0}}^{N-\mathrm{1}}\left({f}_{i}\left(\mathbit{x}\right)\cdot \underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}\mathit{\chi }\left({\mathbit{x}}_{i},{t}^{\prime }\right)\cdot G\left(\
mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)\mathrm{d}{t}^{\prime }\right).\end{array}\end{array}$
Note that Eq. (2) is only valid if the sum of all origin fractions equals 1:
$\begin{array}{}\text{(3)}& {\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}{f}_{i}\left(\mathbit{x}\right):=\mathrm{1}.\end{array}$
There are currently no long-term time series from measurements at the tropopause that are suited for mean age calculations. For this reason, we assume that each long-term time series in each entry
region i can be described by the tropical ground time series shifted by an individual constant t[xi]:
$\begin{array}{}\text{(4)}& \mathit{\chi }\left({\mathbit{x}}_{i}\right)=\mathit{\chi }\left({\mathbit{x}}_{\mathrm{TR}\phantom{\rule{0.125em}{0ex}}\mathrm{ground}},\phantom{\rule{0.125em}{0ex}}{t}^
{\prime }-{t}_{xi}\right).\end{array}$
The negative sign points out that looking at increasing transit times mean looking backwards in time. In the case of an ideal inert linear tracer with the y intercept a and slope b and by applying
Eq. (4), Eq. (2) can be transferred to Eq. (5) in order to calculate the mean age Γ(x):
$\begin{array}{}\text{(5)}& \mathrm{\Gamma }\left(\mathbit{x}\right)=\frac{a-\mathit{\chi }\left(\mathbit{x}\right)}{b}+{t}_{\mathrm{m}}\left(\mathbit{x}\right),\end{array}$
where the weighted mean time shift is ${t}_{\mathrm{m}}\left(\mathbit{x}\right)={\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left({f}_{i}\left(\mathbit{x}\right)\cdot {t}_{xi}\right)$.
In the case of an ideal inert quadratic tracer with curvature c and a known ratio of moments $\mathit{\lambda }=\frac{{\mathrm{\Delta }}^{\mathrm{2}}}{\mathrm{\Gamma }}$ with the width of the age
spectrum Δ and again by applying Eq. (4), Eq. (2) can be transferred to Eq. (6) in order to calculate the mean age:
$\begin{array}{}\text{(6)}& \begin{array}{rl}& \mathrm{\Gamma }{\left(\mathbit{x}\right)}_{\mathrm{1},\mathrm{2}}=-\mathit{\lambda }+{t}_{\mathrm{m}}\left(\mathbit{x}\right)+\frac{b}{\mathrm{2}c}\\ &
\phantom{\rule{0.25em}{0ex}}±\sqrt{{\left(-\mathit{\lambda }+{t}_{\mathrm{m}}\left(\mathbit{x}\right)+\frac{b}{\mathrm{2}c}\right)}^{\mathrm{2}}-\frac{a+b{t}_{\mathrm{m}}\left(\mathbit{x}\right)-\
mathit{\chi }\left(\mathbit{x}\right)}{c}-{\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot {t}_{xi}^{\mathrm{2}}\right]}\end{array}.\end{array}$
Details on deriving Eqs. (5) and (6) are given in Appendix A. Obviously, Eqs. (5) and (6) can also be applied to the single-entry region case, i.e., in context of the conventional TR-only method.
This is equivalent to deriving mean age from an ideal inert linear tracer following Hall and Plumb (1994), respectively in the quadratic case following Volk et al. (1997).
Alternatively, instead of assuming ideal linear or ideal quadratic evolving tracer mixing ratios, G can be approximated by a mathematical function, e.g., an inverse Gaussian following (Hall and
Plumb, 1994). Information on the width of the age spectrum needs to be included (as in the ideal quadratic tracer case). This way, Eq. (1) (TR-only) or Eq. (2) (exTR–TR) can be used directly to
create a lookup table for Γ from a range of age spectra G as described in several studies (e.g., Fritsch et al., 2020; Leedham Elvidge et al., 2018; Ray et al., 2017). Mean age then is inferred from
the best match between measured χ(x) and mixing ratios given in the lookup table. We refer to this approach as the G-match approach in the following paragraph.
Our exTR–TR method will only work for inert monotonic tracers, e.g., SF[6]-like tracers. Tracers that are characterized by seasonally varying trends in their mixing ratios, which propagate into the
stratosphere, e.g., CO[2]-like tracers, will lead to ambiguous mean age results in the LMS using the exTR–TR method. We tested calculating mean age from SF[6] measurements using Eq. (6) versus
following the G-match approach and found only negligible differences for mean ages greater than 1 year. For lower mean ages, the G-match approach leads to numerical issues that cause larger
deviations. Therefore, we decided to use Eq. (6) for all mean age calculations in the context of this study.
The new exTR–TR method requires additional information compared to the conventional TR-only method. In order to account for input from different entry regions, information on the fraction of air that
originated from each entry region is essential in the first place. Secondly, the age tracer's mixing ratio time series at each entry region needs to be known. In the following section, we introduce a
parameterization of the origin fractions published in Hauck et al. (2020). Further, we derive entry mixing ratio time series by shifting the tropical ground mixing ratio time series by a constant
amount of time. The software implementation of the exTR–TR method is described in the supplementary information.
2.2.1Parameterizations of origin fractions from CLaMS
Information on the fraction of air originating from different source regions is an essential input to our new exTR–TR method. We use the seasonally averaged origin fractions from the Chemical
Lagrangian Model of the Stratosphere (CLaMS), e.g., Pommrich et al. (2014) published in Hauck et al. (2020). Hauck et al. (2020) derived such fractions based on origin tracers initiated at three
tropopause sections in the model for extra-tropical input from the Southern Hemisphere (30 to 90^∘S, hereafter referred to as SH input), tropical input (30^∘S to 30^∘N, hereafter referred to as TR
input) and extra-tropical input from the Northern Hemisphere (30 to 90^∘N, hereafter referred to as NH input). In total, there are 15 seasonal distributions of origin fractions f[i,seas](x)
published in Hauck et al. (2020) (see also their Fig. 2): five seasonal sets (annual mean (ANN), December–January–February (DJF), March–April–May (MAM), June–July–August (JJA) and
September–October–November (SON)) for each entry region (SH, TR, NH). Hauck et al. (2020) found that cross-hemispheric transport is negligible, with origin fractions below 10% from the extra-tropics
of the respective other hemisphere. Hence, in order to calculate the mean age at a given location in the stratosphere, we only consider the exTR origin fraction of the respective hemisphere and
assume that the rest originates from the TR tropopause (i.e., ${f}_{\mathrm{TR}}=\mathrm{1}-{f}_{\mathrm{exTR}}$). In doing so, the number of seasonal distributions of origin fractions reduces from
15 to 10.
In order to facilitate accessing the origin fractions from Hauck et al. (2020) and to reduce computational effort, we designed a general mathematical parameterization function φ[i,seas] with 12
parameters to derive 2-D parameterizations for exTR origin fractions. The process of designing φ[i,seas] was guided by a non-physical but entirely geometrical approach. We chose the potential
temperature difference to the local 2PVU tropopause (ΔΘ) as the vertical coordinate and equivalent latitude (eq. lat.; i.e., latitudes sorted by potential vorticity) as the horizontal coordinate.
Details on the parameterizations and how we derived them are given in the Appendix B. Figure 1 shows φ[i,seas] (top row), f[i,seas](x) (middle row) and the absolute difference between f[i,seas](x)
and φ[i,seas] (bottom row) exemplarily for NH spring (March, April, May: MAM; left column), NH fall (September, October, November: SON; middle column) and SH spring (SON; right column). The remaining
seven distributions are presented in the same way as in Fig. 1 in the Supplement (Fig. S1). The absolute differences between φ[i,seas] and f[i,seas](x) shown in Fig. 1 are less than 10% for NH MAM
(panel g) and SH SON (panel i) and only exceed 10% in a small region at the Equator around 25K above the tropopause for NH SON (panel h). The root mean squared difference (RMSD) is less than 3%
for all distributions shown in Fig. 1 and less than 4% for all 10 distributions (including the 7 distributions shown in Fig. S1).
2.2.2Entry region mixing ratio time series
Our new exTR–TR method uses a TR ground-reference time series $\mathit{\chi }\left({\mathbit{x}}_{\mathrm{TR}\phantom{\rule{0.125em}{0ex}}\mathrm{ground}},{t}^{\prime }\right)$ together with constant
time shift values t[xi] in order to simulate reference time series in the three entry regions as defined by the origin fractions from Hauck et al. (2020) (see Eq. 4). This will work for inert
monotonic tracers like SF[6]. In contrast, the entry region mixing ratio time series of tracers like CO[2], which are characterized by a pronounced seasonality in the troposphere, most likely cannot
be approximated satisfactorily with this approach. These tracers are thus not suited for deriving mean age with the exTR–TR method in the LMS. Here, we first describe which TR ground-reference time
series we use and secondly, how we derived constant time shift values t[xi].
In this study, we use SF[6] as an age tracer. Simmonds et al. (2020) used ground measurements from the AGAGE (Advanced Global Atmospheric Gases Experiment) network (Prinn et al., 2018) together with
measurements of archived air samples and the 2-D AGAGE 12-box model (Cunnold et al., 1978, 1983; Rigby et al., 2013) to derive a monthly resolved time series of SF[6] mixing ratios from the 1970s to
2018. We use the TR ground SF[6] mixing ratios of an updated version of this dataset (Laube et al., 2022), which has been extended until the end of 2019, as a reference time series $\mathit{\chi }\
left({\mathbit{x}}_{\mathrm{TR}\phantom{\rule{0.125em}{0ex}}\mathrm{ground}},{t}^{\prime }$) for calculating the mean age of air.
In order to derive t[xi] for each of the three entry regions, we use the annual mean optimized 3-D SF[6] mixing ratios output from the Model for Ozone and Related Tracers (MOZART v4.5) for 1970 to
2008, published by Rigby et al. (2010, Supplement). Rigby et al. (2010) derived a new estimate of SF[6] emissions using the Emissions Database for Global Atmospheric Research (EDGAR v4) as a prior
and optimizing the emissions using SF[6] ground measurements from the AGAGE network including monitoring site data and archived sample measurements together with MOZART and meteorological data from
the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis project. The annually averaged, 3-D optimized SF[6] mixing ratio fields that we use
to derive t[xi] are part of their result. In our approach, we only considered the data from 1973 to 2008, since the data from 1970 to 1972 may be influenced by the start conditions of the model
(Rigby et al., 2010). We calculated t[xi] for each of the three entry regions following three steps:
• i.
Calculate a mean TR ground SF[6] time series by using MOZART data between −30 to 30^∘N weighted by latitude.
• ii.
For each grid cell and each year of the 3-D SF[6] field time series, interpolate SF[6] mixing ratios to TR ground time using (i) and calculate time shift to TR ground.
• iii.
For each entry region, calculate mean and standard deviations weighted by latitude and pressure for time shifts from (ii) for 1973 to 2008 altogether to eventually obtain t[xi] and information on
associated uncertainty.
The latitudinal extents of the entry regions that we calculated t[xi] for are the same as for the origin fractions by Hauck et al. (2020). For the exTR entry regions, we included data between 500 and
200hPa. For the TR entry region, we included data between 300 and 100hPa. Table 1 lists t[xi] and standard deviations for the three entry regions SH exTR, TR and NH exTR. Positive values of t[xi]
indicate that the corresponding region lags behind TR ground SF[6] mixing ratios. Note that for NH exTR, t[xi] is negative. This means that this region precedes TR ground SF[6] mixing ratios. This
finding is consistent with SF[6] source regions being located primarily in the Northern Hemisphere (Rigby et al., 2010).
We performed a Monte Carlo simulation in order to test whether t[xi] can be considered constant over time for each entry region. Firstly, for each entry region, we calculated weighted means and
standard deviations for each year (instead of for the whole time period as in iii). Figure 2 shows the resulting time shift time series from 1973 to 2008. Secondly, for each year, we took 10000
samples from a Gaussian distribution using those weighted means and standard deviations in order to create 10000 time series for each entry region. Thirdly, we applied a linear fit to each of the
10000 time series and calculated the mean and the standard deviation of the slope for each entry region. The resulting mean slopes, standard deviations and the ratio of mean slope and standard
deviation are listed in Table S1 in the Supplement. For NH exTR and TR, the mean slopes deviate less than 1 standard deviation from 0. For SH exTR, the mean slope deviates less than 1.2 standard
deviations from 0. Hence, we do not detect a significant trend. These findings strengthen our confidence into our assumption that we can use the constant time shifts t[xi] listed in Table 1 together
with $\mathit{\chi }\left({\mathbit{x}}_{\mathrm{TR}\phantom{\rule{0.125em}{0ex}}\mathrm{ground}},{t}^{\prime }$) to describe the SF[6] entry mixing ratio time series reasonably well. In the context
of our exTR–TR method, we assume that this also holds true for the subsequent decade from 2008 on. This decade is not covered by the model from Rigby et al. (2010) that we used to derive t[xi],
however, it is covered by $\mathit{\chi }\left({\mathbit{x}}_{\mathrm{TR}\phantom{\rule{0.125em}{0ex}}\mathrm{ground}},{t}^{\prime }$) (Laube et al., 2022, updated from Simmonds et al., 2020).
Previous studies used a similar procedure as outlined above (steps i–iii) to estimate transport timescales while referencing the NH midlatitude ground (Orbe et al., 2021; Waugh et al., 2013). We
found that t[xi] varies less over the time period 1973–2008 when referencing the tropical ground in the MOZART dataset. In order to derive more robust entry mixing ratio time series for our exTR–TR
method, we thus decided to use the tropical ground as a reference. We emphasize that each t[xi] as defined here is an integrated empirical measure. No useful information pertaining to transport paths
or transit times from the TR ground to the entry regions is contained in t[xi]. We only use t[xi] to derive entry mixing ratio time series at locations where suitable long-term time series are not
available from measurements.
2.3Stratospheric observations of age tracer SF[6]
We apply our new exTR–TR method to in situ measurements of SF[6] that were obtained during three HALO research campaigns. The first campaign, PGS (Oelhaf et al., 2019), is a combination of three
missions: POLSTRACC (Polar Stratosphere in a Changing Climate), GW-LCYCLE (Investigation of the Life Cycle of Gravity Waves) and SALSA (Seasonality of Air mass transport and origin in the Lowermost
Stratosphere). The PGS campaign was split into two campaign phases that were conducted in NH winter 2015–2016 between 13 December and 2 February and NH early spring 2016 between 26 February and
18 March. The second campaign, WISE (Wave-driven Isentropic Exchange, https://www.wise2017.de, last access: 5 May 2022), took place mainly in NH fall 2017 between September and October. The flight
tracks of HALO during the PGS and WISE campaigns are shown in Figs. 2 and 1b of Keber et al. (2020). They cover large parts of midlatitudes and high latitudes in the NH. Thirdly, we consider data
from the SouthTRAC (Southern Hemisphere Transport, Dynamics and Chemistry) campaign which took place in SH spring 2019. The SouthTRAC campaign was also split into two campaign phases, conducted
between 6 September and 9 October (Rapp et al., 2021) and between 2 and 15 November. The flight tracks for SouthTRAC, which covered a wide geographical area of the SH, are shown in Jesswein et al.
Measurements of SF[6] and CFC-12 were obtained in-flight in the context of all three campaigns with a time resolution of 1min using the ECD channel of the two-channel Gas chromatograph for
Observational Studies using Tracers (GhOST) instrument in a similar setup as used in the SPURT campaign (Bönisch et al., 2009; Engel et al., 2006). SF[6] has been measured with a precision of 0.6%
(0.56%) during SouthTRAC (PGS, WISE). CFC-12 has been measured with a precision of 0.23% (0.2%) during SouthTRAC (PGS, WISE). All measurements are reported relative to the AGAGE SIO-05 scale
(Miller et al., 2008; Prinn et al., 2018; Rigby et al., 2010; Simmonds et al., 2020). Due to the better precision of CFC-12 measurements, the original SF[6] data were smoothed using a local SF[6]
–CFC-12 correlation 10min before and after each measurement following Krause et al. (2018), prior to calculating mean age values. The height of the dynamical 2PVU tropopause (e.g., Gettelman et
al., 2011) as well as eq. lat. coordinates were obtained via CLaMS driven by ERA-5 reanalysis along the flight tracks. With this study, we exclusively focus on the LMS. Therefore, only tracer
measurements at or above the dynamical tropopause were considered (i.e., with Δθ≥0K). Tracer measurements and flight coordinates can be downloaded from Wagenhäuser et al. (2022). Additional data
associated to the HALO aircraft campaigns that are beyond the scope of this study are accessible via the HALO database at https://halo-db.pa.op.dlr.de/ (last access: 30 March 2023; DLR, 2023).
We derive mean age in the LMS using in situ SF[6] measurements from three aircraft campaigns (see Sect. 2.3). Results are presented in a 2-D tropopause-relative coordinate system. The potential
temperature relative to the local dynamical tropopause (defined by the value of 2PVU) Δθ is used as the vertical coordinate. Horizontally, data are sorted by eq. lat. In order to visualize and
compare our results, datasets were processed in three steps:
1. Mean age was calculated for each data point that was measured above the local 2PVU tropopause.
2. For each campaign dataset, mean ages were averaged in Δθ–eq. lat. bins (5K – 5^∘). Only bins that contained at least five data points were considered.
3. The averaged mean ages have been corrected for mesospheric loss using a linear correction function by Leedham Elvidge et al. (2018), given in their Fig. 4:
$\begin{array}{}\text{(7)}& {\mathrm{\Gamma }}_{\mathrm{corr}}=\mathrm{0.85}\cdot \mathrm{\Gamma }-\mathrm{0.02}\phantom{\rule{0.25em}{0ex}}\text{years}.\end{array}$
3.1Method comparison using campaign-averaged results
We applied our new exTR–TR method for deriving mean age in the LMS considering exTR and TR input into the stratosphere to all three campaign datasets. Further, we applied the conventional TR-only
method, which considers only TR input into the stratosphere, in order to compare the results from both methods. The results were averaged and corrected for mesospheric loss.
Figure 3 shows the resulting Δθ–eq. lat. distributions of averaged mean age mA for PGS (left column), WISE (middle column) and SouthTRAC (right column), derived using the conventional TR-only method
mA[TR-only] (top row), using our new exTR–TR method mA[exTR−TR] (middle row), and the difference between the two methods ΔmA[methods] (bottom row). There are negative values down to −0.54 years close
to the tropopause below Δθ=10K in the WISE dataset using the TR-only method (panel b). In the same region, mean ages between −0.23 and 0.35 years are found using the new exTR–TR method (panel e).
Mean ages below 0 as derived from the TR-only method do not allow for a reasonable interpretation regarding transport timescales in the LMS. In contrast, mean ages derived using our new exTR–TR
method appear physically reasonable, even close to the tropopause. During the WISE campaign, low gradients in mA[exTR−TR] values reveal a well-mixed LMS (panel e), while during PGS and ST, stronger
gradients in mA[exTR−TR] are found (panels d and f).
The maximum absolute difference between the average exTR–TR and TR-only methods' derived mean ages |ΔmA[methods]| is 0.31 years (WISE and PGS) and 0.42 years (SouthTRAC) on the same order of
magnitude for all three campaigns, but in different direction for the SH (see also Fig. 3, bottom row). For all three campaigns |ΔmA[methods]| is largest close to the tropopause at midlatitudes and
high latitudes and approaches 0 years further up and closer to the Equator. This distribution is similar to the distribution of exTR origin fractions from CLaMS. In fact, |ΔmA[methods]| and the exTR
origin fractions are highly correlated (r>0.99 for all three campaigns). This results from the design of the exTR–TR method, which explicitly considers exTR input into the stratosphere.
Note that in the SH data from the SouthTRAC campaign, mean ages are generally lower when derived using the exTR–TR method than when using the TR-only method. For NH data (WISE, PGS), the opposite is
the case. This is a direct consequence from the TR ground mixing ratio time series being lagged by a positive (in the SH) and a negative (in the NH) empirical time shift to obtain the respective
entry mixing ratio time series (see Table 1).
3.2SouthTRAC and PGS campaign: SH vs. NH late winter/early spring
The SouthTRAC and PGS campaigns both involved flights during the respective hemisphere's late winter or early spring. We compare results from the SouthTRAC campaign's phase 1 dataset (ST1) to results
from the PGS campaign's phase 2 dataset (PGS2), derived with the exTR–TR method. This selection is a compromise between including a high number of trace gas measurements and having a large seasonal
overlap between both datasets. Again, the results were averaged and corrected for mesospheric loss.
Figure 4 shows mA[exTR−TR] for ST1 (panel a), for PGS2 (panel b) and the difference between the two mA[PGS2−ST1] (panel c). In order to calculate interhemispheric differences, the mA[exTR−TR]
distribution from ST1 was converted from eq. lat. degrees North to eq. lat. degrees South by flipping it horizontally. Hence, the respective pole corresponds to 90^∘ eq. lat. for both datasets in
panel (c). Mean ages mA[exTR−TR] between −0.2 and 0.1 years are found within the lowest 10K above the tropopause during ST1 (panel a). During PGS2, mA[exTR−TR] between 0.2 and 1.1 years are found in
the equivalent region in the NH (panel b). The differences in mean age mA[PGS2−ST1] (panel c) reveal higher mean ages during PGS2 than during ST1 from the tropopause up to 65K above the tropopause
throughout midlatitudes and high latitudes, with a few exceptions. On average, below Δθ=65K, the LMS is 0.5±0.3 years older during PGS2 than during ST1. Above Δθ=65K, a more complex picture is
observed: at Δθ levels between 65 and 85K at latitudes between 40 and 55^∘, the LMS is even older during PGS2 than during ST1, with mA${}_{\mathrm{PGS}\mathrm{2}-\mathrm{ST}\mathrm{1}}=\mathrm{0.7}±
\mathrm{0.4}$ years on average. In contrast, the opposite is the case at the same Δθ levels but at poleward latitudes higher than 55^∘: mean ages during ST1 are older than during PGS2, with mA
[PGS2−ST1] reaching values down to −2.1 years. A less clear picture emerges when comparing mean ages derived with the TR-only method (see Appendix C: Fig. C1). This could be explained by the fact
that the TR-only method disregards the interhemispheric gradient in SF[6] mixing ratios. In the LMS, the resulting mean age values are thus low biased in the NH, while they are old biased in the SH
using the TR-only method. These biases happen to obscure interhemispheric differences in mean age in the LMS which have been detected using the new exTR–TR method on the same dataset.
Our findings indicate that on the one hand, during ST1, old air from higher altitudes descends in a confined way at high latitudes. There is a sharp vortex edge with a strong gradient in the SH. On
the other hand, during PGS2, descending old air is mixed vertically and horizontally with young air in the LMS. The vortex edge is less sharp than during ST1, resulting in younger air at high
latitudes and altitudes and older air outside the PGS2 vortex region compared to ST1.
These results cover only isolated time periods of less than 2 months for each campaign. In addition, as discussed by Jesswein et al. (2021), the extent of the respective polar vortices and therefore
also the location of the respective vortex edge are likely to be different for both hemispheres. Hence, different vortex characteristics contribute to the differences observed in Fig. 4.
Nevertheless, our findings are in agreement with multiannual simulation results from Konopka et al. (2015), who found a pronounced minimum in wave forcing driving the shallow branch of the BDC in the
midlatitudes of the lower stratosphere in the SH between June and October, as opposed to a maximum in boreal spring in the NH.
3.3Sensitivity study
Our new exTR–TR method requires input of several parameters which all have individual uncertainties. In the following discussion, the sensitivity of the exTR–TR method to these uncertainties is
investigated in the context of three aircraft campaigns.
We identified seven uncertain parameters:
• i.
extra-tropical origin fraction (f[exTR,seas](x));
• ii.
time shift to extra-tropical entry region (t[exTR]);
• iii.
time shift to tropical entry region (t[TR]);
• iv.
measurement precision of age tracer mixing ratios (χ(x));
• v.
ratio of moments (λ);
• vi.
chemical depletion of age tracer SF[6];
• vii.
reference time series calibration-scale uncertainty.
Parameters (i) to (vi) may vary for each individual observational sample, making them eligible for a sensitivity analysis. In contrast, the uncertainty of the calibration scale of the reference time
series (vii) affects all derived absolute mean ages, not in an individual but in a consistent way. Therefore, we excluded it from the sensitivity analysis, albeit knowing that it contributes to the
overall uncertainty in deriving mean age from tracers. Furthermore, we excluded the chemical depletion of SF[6] (vi) from the sensitivity analysis since it is not yet well understood and the subject
of current comprehensive research (e.g., Loeffel et al., 2022). Leedham Elvidge et al. (2018) showed that younger mean age values derived from SF[6] measurements, which point to shorter transport
paths, are less affected by the mesospheric sink than older mean age values. Adequately addressing uncertainties in mean age due to the chemical depletion of SF[6] is beyond the scope of this paper,
which focuses primarily on young air in the LMS.
Parameters (i) to (v) are suited for a sensitivity analysis within the scope of this study using a Monte Carlo simulation. Since typical mean ages and origin fractions vary across different locations
and seasons in the LMS, the sensitivity of the exTR–TR method is investigated and results are shown in the same 2-D tropopause-relative coordinate system that is used for the results shown in
Sect. 3.1 and 3.2: Δθ is used as vertical coordinate, while horizontally, data are sorted by eq. lat. We conduct the sensitivity analysis by applying the following procedure to each of the three
aircraft campaign datasets individually.
In order to obtain a reduced set of representative data and therefore reduce computational effort of the subsequent steps, SF[6] mixing ratios and dates of observation are averaged into 5K Δθ and 5^
∘ eq. lat. bins. For each bin, the following three steps are applied:
Step 1. For each uncertain parameter (i)–(v), a random number is drawn based on the parameter's best estimate and its uncertainty for that specific bin (see below for details). This is done 1000
times to create 1000 sets of parameters.
Step 2. These 1000 sets of parameters are used to calculate 1000 mean age values.
Step 3. The standard deviation of those 1000 mean age values is calculated to obtain an overall sensitivity value for this bin.
In this way, we derive the overall sensitivity. Further, we investigate the relative importance of the uncertain parameters (i)–(v). For this purpose, additional sensitivity calculations are done
where only one uncertain parameter is varied while leaving the others at their best estimate.
3.3.2Parameter uncertainties
Here, we describe how uncertainties associated with parameters (i)–(v) are implemented in step 1 of the sensitivity analysis.
• i.
The exTR origin fraction f[exTR](x) varies spatially and over time. For each of the three aircraft campaigns, the spatial distribution of uncertainties in f[exTR](x) is derived from the
parameterized origin fraction φ[i,seas](x) (see Sect. 2.2.1) individually. Therefore, for each bin, the mean absolute half difference (MAHD[seas]) between φ[i,seas](x) and φ[i,nextseas](x),
respectively φ[i,previousseas](x), is calculated. In addition, the root mean squared difference (RMSD[space]) between each bin and its eight surrounding bins is calculated. Both measures, MAHD
[seas] and RMSD[space], are combined in the root sum squared to finally derive the spatial distribution of uncertainties in the exTR origin fraction for each campaign. Random values are drawn
from a Gaussian distribution using this root sum squared.
• ii., iii.
Obtaining t[exTR] for the NH and for the SH tropopause and t[TR] for the tropical tropopause regarding SF[6] from the annually averaged 3-D model output is described in Sect. 2.2.2. We use the
weighted mean values and standard deviations given in Table 1 as input for a Gaussian distribution from which random values are drawn.
• iv.
The measurement precision of age tracer mixing ratios χ(x) is given campaign-wise in Sect. 2.3. Since we use smoothed SF[6] mixing ratios by considering local CFC-12–SF[6] correlations for mean
age calculations, we here apply the better measurement precision for CFC-12 mixing ratios to draw samples from a Gaussian distribution.
• v.
Regarding the ratio of moments λ, random values are drawn from a triangular distribution with a minimum of λ=0.7 years, a center of λ=1.2 years and a maximum of λ=2 years.
3.3.3exTR–TR method sensitivities during PGS, WISE and SouthTRAC
The sensitivities of the exTR–TR method to uncertainties in input parameters have been calculated following the procedure outlined above. The resulting distributions of sensitivity values are shown
in Fig. 5. The most sensitive regions are found between 20–40^∘ poleward of the Equator below Δθ=20K during all three aircraft campaigns with maximum values of 0.22 years (PGS, panel a), 0.19 years
(WISE, panel b) and 0.16 years (SouthTRAC, panel c). Above Δθ=20K, the sensitivity values are distributed evenly (standard deviation <0.02 years), with average values of 0.15 years (PGS) and
0.14 years (WISE and SouthTRAC). These sensitivities are lower than the differences between mean ages derived using the exTR–TR method and the TR-only method, which are found to be larger than
0.3 years close to the tropopause.
The contribution of the individual parameters (i)–(v) is shown in Fig. 6. Each row depicts isolated sensitivities to uncertainties in a single parameter with all other parameters being held at their
best estimate. This allows us to test the relative importance of the individual parameters to the exTR–TR method's overall sensitivity. Most strikingly, uncertainties in the ratio of moments
(parameter v) seem to contribute only negligibly to the overall sensitivity (panels m–o). Measurement uncertainties in the stratospheric mixing ratio χ(x) contribute to the overall sensitivity
spatially evenly distributed to a moderate extent (panels j–l). Due to the slightly worse measurement precision during SouthTRAC and in addition due to the decelerating relative growth rate of SF[6]
mixing ratios, the uncertainties in χ(x) have a stronger impact on the overall sensitivity during SouthTRAC than during the other two campaigns. In the upper part of the LMS (above Δθ=50K),
uncertainties in t[TR] dominate the overall sensitivity (panels g–i). Below, uncertainties in t[exTR] and in f[exTR](x) gain importance (panels a–f). Note that the SH uncertainties in t[exTR] are low
(see Table 1), which is reflected by contributing to the overall sensitivity during the SouthTRAC campaign (panel f) only to a minor to moderate extent.
4Summary and conclusions
In this work, the new exTR–TR method to derive stratospheric mean age of air in the LMS from observational tracer mixing ratio data is presented. In order to take exTR input into the stratosphere
into account, our implementation of the exTR–TR method uses seasonally averaged exTR origin fractions from CLaMS (Hauck et al., 2020), for which we provide a parameterization, and a long-term tracer
mixing ratio time series for each entry region $\mathit{\chi }\left({\mathbit{x}}_{i},{t}^{\prime }\right)$. Following Hauck et al. (2020), the entry regions are defined as a northern (90–30^∘N), a
TR (30^∘N–30^∘S) and a southern (30–90^∘S) tropopause section. Owing to the lack of continuous long-term measurements of age tracers at the tropopause, we approximated $\mathit{\chi }\left({\
mathbit{x}}_{i},{t}^{\prime }\right)$ by applying a constant empirical time shift t[xi] to the available TR ground tracer mixing ratio time series for each entry region. For the age tracer SF[6],
individual t[xi] were obtained in this study by averaging optimized 3-D model output between 1973–2008 that was published by Rigby et al. (2010). We emphasize that the resulting t[xi] are exclusively
used to approximate $\mathit{\chi }\left({\mathbit{x}}_{i},{t}^{\prime }\right)$ and that they do not represent real-world transport times between TR ground and the entry regions.
We applied the exTR–TR method to in situ SF[6] measurements taken during three aircraft campaigns in different geographical regions and at different times: PGS in NH late winter or early spring 2016,
WISE in NH fall 2017 and SouthTRAC in SH early spring 2019. The resulting mean age values were averaged into bins over multiple flights using tropopause-relative altitude and eq. lat. coordinates (Δθ
bins of size 5K, eq. lat. bins of size 5^∘). These averaged mean age values were corrected for mesospheric loss by applying a linear function published by Leedham Elvidge et al. (2018). In addition,
the conventional TR-only method, which assumes that all air enters the stratosphere through the TR tropopause, was applied to the same data and the results were post-processed in the same way in
order to compare the results. Using the conventional TR-only method, negative mean age values are derived in the LMS close to the tropopause during the WISE campaign. Using the new exTR–TR method
instead, the number and extent of negative mean age values are reduced. Maximum absolute differences between the resulting averaged mean age values from the two methods range from 0.31 to 0.42 years
among the three campaigns and go in different directions for the two hemispheres. With our new exTR–TR method, we are thus able to derive more realistic values of typical transport times in the LMS
from measurements. This allows a comparison of the two hemispheres based on campaign data. We compared results derived using the exTR–TR method from the PGS campaign phase 2 (PGS2) to SouthTRAC
campaign phase 1 (ST1) in order to investigate hemispheric differences with a maximal seasonal overlap of the campaigns. On average, below Δθ=65K, the LMS was 0.5±0.3 years older during PGS2 than
during ST1 across all eq. lats that are covered by both datasets. We attribute this older LMS to mixing with old vortex air during PGS2, as opposed to a more confined vortex edge with higher age
gradients during ST1. Although these findings only cover an isolated time period of less than 2 months for each campaign and do not account for different polar vortex characteristics, they are in
agreement with multiannual simulation results from Konopka et al. (2015), who found a pronounced minimum in wave forcing driving the shallow branch of the BDC in the midlatitudes of the lower
stratosphere in the SH between June and October, as opposed to a maximum in boreal spring in the NH.
The sensitivity of the exTR–TR method to uncertainties of six input parameters was investigated at different locations using a Monte Carlo approach. The mesospheric loss of SF[6] was excluded from
this sensitivity analysis since it is currently not well understood and beyond the scope of this work. The combined sensitivity was found to be less than 0.22 years for all locations for all three
campaigns. The most sensitive region for each hemisphere was located between 20–40^∘ poleward of the Equator below Δθ=20K. This is related to the setup of the experiment with a boundary at 30^∘ in
each hemisphere. Uncertainties in the origin fractions and in t[xi] have the largest isolated impact on the sensitivity of the exTR–TR method. Overall, these sensitivities are lower than the
differences between mean ages derived using the exTR–TR method and the TR-only method. Hence, our new exTR–TR method yields mean age values that differ considerably from results obtained using the
conventional TR-only method in the LMS. In future studies, the exTR–TR method could be used to improve deriving estimates of total and inorganic chlorine from observations of organic chlorine in the
LMS as in Jesswein et al. (2021).
Appendix A:Calculating mean age in the LMS considering multiple entry regions and an ideal tracer
In the case of an ideal inert linear evolving tracer, the tropical ground time series as a function of transit time t is given by
$\begin{array}{}\text{(A1)}& \mathit{\chi }\left({\mathbit{x}}_{\mathrm{TR}\phantom{\rule{0.125em}{0ex}}\mathrm{ground}},{t}^{\prime }\right)=a-b{t}^{\prime }.\end{array}$
The negative sign indicates that looking at increasing transit times means looking backwards in time.
Assuming a constant time shift t[xi] for each entry region i, the tracer time series at x[i] is
$\begin{array}{}\text{(A2)}& \mathit{\chi }\left({\mathbit{x}}_{i},{t}^{\prime }\right)=a-b\cdot \left({t}^{\prime }-{t}_{xi}\right).\end{array}$
Considering individual transit time distributions ${G}_{i}\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)$ for each origin fraction f[i](x), the stratospheric mixing ratio χ(x) of
a suitable age tracer at an arbitrary location x in the stratosphere is
$\begin{array}{}\text{(A3)}& \mathit{\chi }\left(\mathbit{x}\right)={\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot \underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\
int }}\mathit{\chi }\left({\mathbit{x}}_{i},{t}^{\prime }\right)\cdot \phantom{\rule{0.125em}{0ex}}{G}_{i}\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)\mathrm{d}{t}^{\prime }\
Hence, by inserting Eq. (A2) into Eq. (A3), the stratospheric mixing ratio can be expressed as
$\begin{array}{}\text{(A4)}& \begin{array}{rl}\mathit{\chi }\left(\mathbit{x}\right)& ={\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot \underset{\mathrm{0}}{\overset{\
mathrm{\infty }}{\int }}\left(a-b{t}^{\prime }+b{t}_{xi}\right)\cdot {G}_{i}\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)\mathrm{d}{t}^{\prime }\right]\\ & ={\sum }_{i=\mathrm
{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot \left(a+b{t}_{xi}\right)\right]\\ & -\sum _{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot b\cdot \underset{\
mathrm{0}}{\overset{\mathrm{\infty }}{\int }}{t}^{\prime }\cdot {G}_{i}\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)\mathrm{d}{t}^{\prime }\right].\end{array}\end{array}$
The mean age Γ is the first moment of the age spectrum, given by
$\begin{array}{}\text{(A5)}& \mathrm{\Gamma }\left(\mathbit{x}\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}{t}^{\prime }\cdot G\left(\mathbit{x},{t}^{\prime }\right)\mathrm{d}{t}^
{\prime }.\end{array}$
In the case of ${G}_{i}\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)$, Eq. (A5) translates into the mean age of air originating from source region i (Γ[i](x)):
$\begin{array}{}\text{(A6)}& {\mathrm{\Gamma }}_{i}\left(\mathbit{x}\right)=\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}{t}^{\prime }\cdot {G}_{i}\left(\mathbit{x},{t}^{\prime }\right)\
mathrm{d}{t}^{\prime }.\end{array}$
Inserting Eq. (A6) into Eq. (A4) yields the following:
$\begin{array}{}\text{(A7)}& \begin{array}{rl}\mathit{\chi }\left(\mathbit{x}\right)& ={\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot \left(a+b{t}_{xi}\right)\right]\
\ & -\sum _{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot b\cdot {\mathrm{\Gamma }}_{i}\left(\mathbit{x}\right)\right].\end{array}\end{array}$
Since the sum of all origin fractions equals 1, Eq. (A7) can also be written as
$\begin{array}{}\text{(A8)}& \begin{array}{rl}\mathit{\chi }\left(\mathbit{x}\right)& =a+{\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot b{t}_{xi}\right]\\ & -b\cdot \
sum _{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot {\mathrm{\Gamma }}_{i}\left(\mathbit{x}\right)\right].\end{array}\end{array}$
The mean age Γ(x) equals the sum of individual Γ[i](x), weighted by their respective origin fraction f[i](x):
$\begin{array}{}\text{(A9)}& \mathrm{\Gamma }\left(\mathbit{x}\right)={\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot {\mathrm{\Gamma }}_{i}\left(\mathbit{x}\right)\
By inserting Eq. (A9) into Eq. (A8), we can thus reduce the number of unknown parameters:
$\begin{array}{}\text{(A10)}& \mathit{\chi }\left(\mathbit{x}\right)=a+{\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\right)\cdot b{t}_{xi}\right]-b\cdot \mathrm{\Gamma }\left(\
Equation (A10) can be solved for Γ, which yields
$\begin{array}{}\text{(A11)}& \mathrm{\Gamma }\left(\mathbit{x}\right)=\frac{a-\mathit{\chi }\left(\mathbit{x}\right)}{b}+{\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left({f}_{i}\left(\mathbit{x}\right)\
cdot {t}_{xi}\right),\end{array}$
which is equivalent to Eq. (5). The same result can be obtained mathematically when we use the origin fractions as weights only for the mixing ratio time series and neglect the concept of ${G}_{i}\
left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)$ (starting with Eq. 2 instead of Eq. A3). Differences across individual ${G}_{i}\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^
{\prime }\right)$ thus have no influence on calculating the mean age from an ideal inert linear evolving tracer. In contrast, in the case of an ideal inert quadratic evolving tracer, the Ansatz
expressed in Eq. (A3) cannot be solved for Γ(x) without knowledge of individual Γ[i](x). However, if the quadratic term of the tracer mixing ratio time series is sufficiently low, then the concept of
${G}_{i}\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)$ can be neglected by using the Ansatz expressed in Eq. (2).
In order to derive mean age from an ideal inert quadratic evolving tracer with multiple entry regions, we extended the equations given by Volk et al. (1997). In this case, the TR ground mixing ratio
time series is given as a function of transit time by
$\begin{array}{}\text{(A12)}& \mathit{\chi }\left({\mathbit{x}}_{\mathrm{TR}\phantom{\rule{0.125em}{0ex}}\mathrm{ground}},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)=a-b{t}^{\prime }+c{t}^{\
prime \mathrm{2}}.\end{array}$
Assuming a constant time shift t[xi] for each entry region i, the tracer time series at x[i] is
$\begin{array}{}\text{(A13)}& \mathit{\chi }\left({\mathbit{x}}_{i},{t}^{\prime }\right)=a-b\cdot \left({t}^{\prime }-{t}_{xi}\right)+c\cdot {\left({t}^{\prime }-{t}_{xi}\right)}^{\mathrm{2}}.\end
Hence, by inserting Eq. (A13) into Eq. (2), the stratospheric mixing ratio can be expressed as
which is equivalent to
Note that for better readability, f[i](x) is written as f[i].
By extracting constant factors from the integral and applying Eq. (A5), Eq. (A15) can also be written as
The width of the age spectrum Δ is the square root of the second centered moment of the age spectrum, which is given by
$\begin{array}{}\text{(A17)}& {\mathrm{\Delta }}^{\mathrm{2}}\left(\mathbit{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}{\left({t}^{\prime }-\mathrm
{\Gamma }\left(\mathbit{x}\right)\right)}^{\mathrm{2}}\cdot G\left(\mathbit{x},{t}^{\prime }\right)\cdot \mathrm{d}{t}^{\prime }.\end{array}$
Equation (A17) can be transformed to
$\begin{array}{}\text{(A18)}& \underset{\mathrm{0}}{\overset{\mathrm{\infty }}{\int }}{t}^{\prime \mathrm{2}}\cdot G\left(\mathbit{x},\phantom{\rule{0.125em}{0ex}}{t}^{\prime }\right)\mathrm{d}{t}^{\
prime }=\mathrm{2}\mathrm{\Delta }{\left(\mathbit{x}\right)}^{\mathrm{2}}+\mathrm{\Gamma }{\left(\mathbit{x}\right)}^{\mathrm{2}}.\end{array}$
Inserting Eq. (A18) into Eq. (A16) yields
Since the sum of all origin fractions equals 1 and with the weighted mean time shift ${t}_{\mathrm{m}}\left(\mathbit{x}\right)={\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}_{i}\left(\mathbit{x}\
right)\cdot {t}_{xi}\right]$, Eq. (A19) can also be written as:
Inserting the ratio of moments $\mathit{\lambda }={\mathrm{\Delta }}^{\mathrm{2}}/\mathrm{\Gamma }$ into Eq. (A20) yields the quadratic equation Eq. (A21):
which can be rearranged to
$\begin{array}{}\text{(A22)}& \begin{array}{rl}\mathrm{0}& =\frac{a+b{t}_{\mathrm{m}}\left(\mathbit{x}\right)-\mathit{\chi }\left(\mathbit{x}\right)}{c}+{\sum }_{i=\mathrm{0}}^{N-\mathrm{1}}\left[{f}
_{i}{t}_{xi}^{\mathrm{2}}\right]+\mathrm{\Gamma }\left(\mathbit{x}\right)\\ & \cdot \left(\mathrm{2}\mathit{\lambda }-\frac{b}{c}-\mathrm{2}{t}_{\mathrm{m}}\left(\mathbit{x}\right)\right)+\mathrm{\
Gamma }{\left(\mathbit{x}\right)}^{\mathrm{2}},\end{array}\end{array}$
and finally solved for Γ:
$\begin{array}{}\text{(A23)}& \begin{array}{rl}& \mathrm{\Gamma }{\left(\mathbit{x}\right)}_{\mathrm{1},\mathrm{2}}=-\mathit{\lambda }+{t}_{\mathrm{m}}+\frac{b}{\mathrm{2}c}\\ & \phantom{\rule
{0.25em}{0ex}}±\sqrt{{\left(-\mathit{\lambda }+{t}_{\mathrm{m}}+\frac{b}{\mathrm{2}c}\right)}^{\mathrm{2}}-\frac{a+b{t}_{\mathrm{m}}-\mathit{\chi }\left(\mathbit{x}\right)}{c}-{\sum }_{i=\mathrm{0}}^
{N-\mathrm{1}}\left[{f}_{i}\cdot {t}_{xi}^{\mathrm{2}}\right]}.\end{array}\end{array}$
Appendix B:CLaMS origin fraction parameterizations
We designed a general mathematical parameterization function φ[i,seas] with 12 parameters to derive 2-D parameterizations for exTR origin fractions in ΔΘ−eq. lat. space. The process of designing φ[i
,seas] was guided by a non-physical but entirely geometrical approach pursuing three priorities for each of the 10 considered f[i,seas](x) at once:
• i.
φ[i,seas] should be able to reproduce major geometrical features of the distributions.
• ii.
The maximum difference between φ[i,seas](x) and f[i,seas](x) should be as low as possible.
• iii.
The mean deviation between φ[i,seas](x) and f[i,seas](x) should be as low as possible.
In addition to the three priorities, the number of parameters needed to achieve (i), (ii) and (iii) should preferably be low. The resulting general mathematical parameterization function is a
combination of Gaussian distributions and cumulative Gumbel distributions with 12 parameters in total:
$\begin{array}{}\text{(B1)}& \mathrm{peak}\mathrm{1}\left(\text{eq. lat},\mathrm{\Delta }\mathrm{\Theta }\right)={e}^{-{e}^{-\frac{\mathrm{|}\text{eq. lat}\mathrm{|}-{x}_{\mathrm{0}}}{{x}_{\mathrm
{1}}}}}\cdot {e}^{-{\left(\frac{\mathrm{\Delta }\mathrm{\Theta }-{y}_{\mathrm{1}}}{{y}_{\mathrm{0}}}\right)}^{\mathrm{2}}},\text{(B2)}& \mathrm{peak}\mathrm{2}\left(\text{eq. lat},\mathrm{\Delta }\
mathrm{\Theta }\right)={g}_{a}\cdot {e}^{-{\left(\frac{\text{eq. lat}-{g}_{x\mathrm{1}}}{{g}_{x\mathrm{0}}}\right)}^{\mathrm{2}}}\cdot {e}^{-{\left(\frac{\mathrm{\Delta }\mathrm{\Theta }-{g}_{y\
mathrm{1}}}{{g}_{y\mathrm{0}}}\right)}^{\mathrm{2}}},\text{(B3)}& \mathrm{offset}\mathrm{_}\mathrm{gumbel}\left(\mathrm{\Delta }\mathrm{\Theta }\right)={b}_{y}\cdot {e}^{-{e}^{-\frac{\mathrm{\Delta }
\mathrm{\Theta }-{e}_{\mathrm{0}}}{{e}_{\mathrm{1}}}}},\text{(B4)}& \begin{array}{rl}{\mathit{\phi }}_{i,\mathrm{seas}}\left(\text{eq. lat},\phantom{\rule{0.125em}{0ex}}\mathrm{\Delta }\mathrm{\Theta
}\right)& =\mathrm{peak}\mathrm{1}\left(\text{eq. lat},\mathrm{\Delta }\mathrm{\Theta }\right)\\ & +\mathrm{peak}\mathrm{2}\left(\text{eq. lat},\mathrm{\Delta }\mathrm{\Theta }\right)\\ & +\mathrm
{offset}\mathrm{_}\mathrm{gumbel}\left(\mathrm{\Delta }\mathrm{\Theta }\right).\end{array}\end{array}$
The seasonally averaged f[i,seas](x) data published by Hauck et al. (2020) are gridded in 2^∘ latitude and 37 vertical potential temperature levels between 280 and 3000K. Additionally, the
difference in potential temperature to the local tropopause (ΔΘ) is provided for each data point. In order to find optimal fitting parameters using a least-square fit, for each of the 10 considered f
[i,seas](x) we only considered data from the respective hemisphere and for the lower 20 vertical levels (i.e., 280 to 480K). The resulting parameters for each of the 10 considered f[i,seas](x) are
listed in Table B1. The Python code for applying φ[i,seas] as given in Eq. (B4) and automatically including the information given in Table B1 is available from Wagenhäuser (2022a).
Appendix C:SouthTRAC phase 1 and PGS phase 2 campaign differences using the TR-only method
Code and data availability
The Python software implementation of the exTR–TR method is available at https://doi.org/10.5281/zenodo.7267203 (Wagenhäuser and Engel, 2022).
The Python software code repository “f_exTR” for deploying our parameterizations of the CLaMS origin fractions is available at https://doi.org/10.5281/zenodo.7267114 (Wagenhäuser, 2022a).
The Python software code repository “sf6-timeshifts-from-rigby2010” for deriving SF[6] time shifts to tropical ground using model data from Rigby et al. (2010) (Sect. 2.2.2) is available at https://
doi.org/10.5281/zenodo.7267089 (Wagenhäuser, 2022b).
Tracer measurements, flight coordinates and mean age values derived using both the exTR–TR method and the TR-only method can be downloaded at https://doi.org/10.5281/zenodo.7275822 (Wagenhäuser et
al., 2022).
TW developed the mathematical framework and Python software code for the exTR–TR method in close collaboration with AE. AE initiated this study. TW, MJ, TK, TS, and AE operated the GhOST instrument
during the SouthTRAC campaign. TW wrote the paper in collaboration with AE. All authors contributed to the final version of the paper.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We would like to thank the DLR staff for the operation of the HALO and the support during all three campaigns. Many thanks also to all former master students at the Goethe University of Frankfurt,
who helped to carry out measurements during the campaigns. Moreover, we thank Jens-Uwe Grooß for facilitating access to model-based Δθ and eq. lat. data along the flight tracks. We thank
Matthew Rigby and Luke Western for providing updated AGAGE 12-box model output of SF[6] mixing ratios. We further thank Ronald Prinn, Ray Weiss, Paul Krummel, Dickon Young, Simon O'Doherty, and
Jens Mühle for facilitating access to the AGAGE data (http://agage.mit.edu, last access: 30 March 2023). The AGAGE stations used in this paper are supported by the National Aeronautics and Space
Administration (NASA). Support also comes from the UK Department for Business, Energy & Industrial Strategy (BEIS) for MHD, the National Oceanic and Atmospheric Administration (NOAA) for RPB, and
the Commonwealth Scientific and Industrial Research Organization (CSIRO) and the Bureau of Meteorology (Australia) for CGO.
This research was supported under the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Priority Program SPP 1294 “Atmospheric and Earth System Research with HALO” – “High Altitude
and Long Range Research Aircraft” (project nos. EN367/5, EN367/8, EN367/11, EN367/13, EN367/14, and EN367/16). Financial support also came from the DFG – TRR 301 (project ID 428312742) and the
National Aeronautics and Space Administration (NASA) (grant nos. NNX16AC98G to MIT, and NNX16AC97G and NNX16AC96G to SIO).
This open-access publication was funded by the Goethe University Frankfurt.
This paper was edited by Rolf Müller and reviewed by three anonymous referees.
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Waugh, D. W., Crotwell, A. M., Dlugokencky, E. J., Dutton, G. S., Elkins, J. W., Hall, B. D., Hintsa, E. J., Hurst, D. F., Montzka, S. A., Mondeel, D. J., Moore, F. L., Nance, J. D., Ray, E. A.,
Steenrod, S. D., Strahan, S. E., and Sweeney, C.: Tropospheric SF[6]: Age of air from the Northern Hemisphere midlatitude surface, J. Geophys. Res.-Atmos., 118, 11429–11441, https://doi.org/10.1002/
jgrd.50848, 2013. | {"url":"https://acp.copernicus.org/articles/23/3887/2023/","timestamp":"2024-11-01T23:38:01Z","content_type":"text/html","content_length":"416406","record_id":"<urn:uuid:834a9e24-c671-4ec5-8e84-d33b3e695f44>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00273.warc.gz"} |
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Module std.range.primitives
This module is a submodule of std.range.
It defines the bidirectional and forward range primitives for arrays: empty, front, back, popFront, popBack and save.
It provides basic range functionality by defining several templates for testing whether a given object is a range, and what kind of range it is:
isInputRange Tests if something is an input range, defined to be something from which one can sequentially read data using the primitives front, popFront, and empty.
isOutputRange Tests if something is an output range, defined to be something to which one can sequentially write data using the put primitive.
isForwardRange Tests if something is a forward range, defined to be an input range with the additional capability that one can save one's current position with the save primitive, thus allowing
one to iterate over the same range multiple times.
isBidirectionalRange Tests if something is a bidirectional range, that is, a forward range that allows reverse traversal using the primitives back and popBack.
isRandomAccessRange Tests if something is a random access range, which is a bidirectional range that also supports the array subscripting operation via the primitive opIndex.
It also provides number of templates that test for various range capabilities:
hasMobileElements Tests if a given range's elements can be moved around using the primitives moveFront, moveBack, or moveAt.
ElementType Returns the element type of a given range.
ElementEncodingType Returns the encoding element type of a given range.
hasSwappableElements Tests if a range is a forward range with swappable elements.
hasAssignableElements Tests if a range is a forward range with mutable elements.
hasLvalueElements Tests if a range is a forward range with elements that can be passed by reference and have their address taken.
hasLength Tests if a given range has the length attribute.
isInfinite Tests if a given range is an infinite range.
hasSlicing Tests if a given range supports the array slicing operation R[x .. y].
Finally, it includes some convenience functions for manipulating ranges:
popFrontN Advances a given range by up to n elements.
popBackN Advances a given bidirectional range from the right by up to n elements.
popFrontExactly Advances a given range by up exactly n elements.
popBackExactly Advances a given bidirectional range from the right by exactly n elements.
moveFront Removes the front element of a range.
moveBack Removes the back element of a bidirectional range.
moveAt Removes the i'th element of a random-access range.
walkLength Computes the length of any range in O(n) time.
put Outputs element e to a range.
Name Description
back(a) Implements the range interface primitive back for built-in arrays. Due to the fact that nonmember functions can be called with the first argument using the dot notation, array.back is
equivalent to back(array). For narrow strings, back automatically returns the last code point as a dchar.
empty(a) Implements the range interface primitive empty for types that obey hasLength property and for narrow strings. Due to the fact that nonmember functions can be called with the first
argument using the dot notation, a.empty is equivalent to empty( a).
front(a) Implements the range interface primitive front for built-in arrays. Due to the fact that nonmember functions can be called with the first argument using the dot notation, array.front
is equivalent to front(array). For narrow strings, front automatically returns the first code point as a dchar.
moveAt(r, i) Moves element at index i of r out and returns it. Leaves r[ i] in a destroyable state that does not allocate any resources (usually equal to its .init value).
moveBack(r) Moves the back of r out and returns it. Leaves r.back in a destroyable state that does not allocate any resources (usually equal to its .init value).
moveFront(r) Moves the front of r out and returns it.
popBack(a) Implements the range interface primitive popBack for built-in arrays. Due to the fact that nonmember functions can be called with the first argument using the dot notation, array.
popBack is equivalent to popBack(array). For narrow strings, popFront automatically eliminates the last code point.
popBackExactly Eagerly advances r itself (not a copy) exactly n times (by calling r.popFront). popFrontExactly takes r by ref, so it mutates the original range. Completes in Ο(1) steps for ranges
(r, n) that support slicing, and have either length or are infinite. Completes in Ο(n) time for all other ranges.
popBackN(r, n) popFrontN eagerly advances r itself (not a copy) up to n times (by calling r.popFront). popFrontN takes r by ref, so it mutates the original range. Completes in Ο(1) steps for ranges
that support slicing and have length. Completes in Ο(n) time for all other ranges.
popFront(a) Implements the range interface primitive popFront for built-in arrays. Due to the fact that nonmember functions can be called with the first argument using the dot notation, array.
popFront is equivalent to popFront(array). For narrow strings, popFront automatically advances to the next code point.
popFrontExactly Eagerly advances r itself (not a copy) exactly n times (by calling r.popFront). popFrontExactly takes r by ref, so it mutates the original range. Completes in Ο(1) steps for ranges
(r, n) that support slicing, and have either length or are infinite. Completes in Ο(n) time for all other ranges.
popFrontN(r, n) popFrontN eagerly advances r itself (not a copy) up to n times (by calling r.popFront). popFrontN takes r by ref, so it mutates the original range. Completes in Ο(1) steps for ranges
that support slicing and have length. Completes in Ο(n) time for all other ranges.
put(r, e) Outputs e to r. The exact effect is dependent upon the two types. Several cases are accepted, as described below. The code snippets are attempted in order, and the first to compile
"wins" and gets evaluated.
save(a) Implements the range interface primitive save for built-in arrays. Due to the fact that nonmember functions can be called with the first argument using the dot notation, array.save is
equivalent to save(array). The function does not duplicate the content of the array, it simply returns its argument.
walkLength This is a best-effort implementation of length for any kind of range.
Manifest constants
Name Type Description
hasAssignableElements Returns true if R is an input range and has mutable elements. The following code should compile for any range with assignable elements.
hasLength Yields true if R has a length member that returns a value of size_t type. R does not have to be a range. If R is a range, algorithms in the standard library are only
guaranteed to support length with type size_t.
hasLvalueElements Tests whether the range R has lvalue elements. These are defined as elements that can be passed by reference and have their address taken. The following code should compile
for any range with lvalue elements.
Returns true iff R is an input range that supports the moveFront primitive, as well as moveBack and moveAt if it's a bidirectional or random access range. These may be
hasMobileElements explicitly implemented, or may work via the default behavior of the module level functions moveFront and friends. The following code should compile for any range with
mobile elements.
hasSlicing Returns true if R offers a slicing operator with integral boundaries that returns a forward range type.
hasSwappableElements Returns true if R is an input range and has swappable elements. The following code should compile for any range with swappable elements.
isBidirectionalRange Returns true if R is a bidirectional range. A bidirectional range is a forward range that also offers the primitives back and popBack. The following code should compile for
any bidirectional range.
Returns true if R is a forward range. A forward range is an input range r that can save "checkpoints" by saving r.save to another value of type R. Notable examples of input
isForwardRange ranges that are not forward ranges are file/socket ranges; copying such a range will not save the position in the stream, and they most likely reuse an internal buffer as
the entire stream does not sit in memory. Subsequently, advancing either the original or the copy will advance the stream, so the copies are not independent.
isInfinite Returns true if R is an infinite input range. An infinite input range is an input range that has a statically-defined enumerated member called empty that is always false,
for example:
isInputRange Returns true if R is an input range. An input range must define the primitives empty, popFront, and front. The following code should compile for any input range.
isOutputRange Returns true if R is an output range for elements of type E. An output range is defined functionally as a range that supports the operation put(r, e) as defined above.
isRandomAccessRange Returns true if R is a random-access range. A random-access range is a bidirectional range that also offers the primitive opIndex, OR an infinite forward range that offers
opIndex. In either case, the range must either offer length or be infinite. The following code should compile for any random-access range.
Name Type Description
ElementEncodingType E The encoding element type of R. For narrow strings (char[], wchar[] and their qualified variants including string and wstring), ElementEncodingType is the character type of
the string. For all other types, ElementEncodingType is the same as ElementType.
ElementType T The element type of R. R does not have to be a range. The element type is determined as the type yielded by r.front for an object r of type R. For example, ElementType!(T[])
is T if T[] isn't a narrow string; if it is, the element type is dchar. If R doesn't have front, ElementType!R is void. | {"url":"https://docarchives.dlang.io/v2.101.0/library/std/range/primitives.html","timestamp":"2024-11-03T13:09:59Z","content_type":"text/html","content_length":"111792","record_id":"<urn:uuid:1af5666e-e35c-41fa-9d8c-a80399266408>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00786.warc.gz"} |
Optimal induced universal graphs for bounded-degree graphs
We show that for any constant δ ≥ 2, there exists a graph T with O(nδ=2) vertices which contains every n-vertex graph with maximum degree Δ as an induced subgraph. For odd Δ this significantly
improves the best-known earlier bound of Esperet et al. and is optimal up to a constant factor, as it is known that any such graph must have at least (n δ =2) vertices. Our proof builds on the
approach of Alon and Capalbo (SODA 2008) together with several additional ingredients. The construction of T is explicit and is based on an appropriately defined composition of high-girth expander
graphs. The proof also provides an efficient deterministic procedure for finding, for any given input graph H on n vertices with maximum degree at most Δ, an induced subgraph of T isomorphic to H.
Publication series
Name Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume 0
Other 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
Country/Territory Spain
City Barcelona
Period 1/16/17 → 1/19/17
All Science Journal Classification (ASJC) codes
• Software
• General Mathematics
Dive into the research topics of 'Optimal induced universal graphs for bounded-degree graphs'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/optimal-induced-universal-graphs-for-bounded-degree-graphs-2","timestamp":"2024-11-11T11:36:47Z","content_type":"text/html","content_length":"49866","record_id":"<urn:uuid:255fbecd-53ec-4b07-8775-a6dc3a204915>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00188.warc.gz"} |
Time Comparisons In Power BI: This Year vs Last Year
I want to go over how you can easily do time comparisons in Power BI and specifically calculate this year versus last year. I’m going to show you a couple of techniques that you can utilize for time
comparisons like this. You can watch the full video of this tutorial at the bottom of this blog.
These are some of the easiest things that you can do in Power BI. They are also some of the first examples that you should work through as you’re learning how to operate DAX effectively, especially
on the Power BI desktop.
First, just let’s just do a quick recap of how you should set up your models.
This is the waterfall technique that I talk a lot about. Others call it a snowflake, but I like this concept of the waterfall and its filters flowing down.
Your Date table is absolutely key here. You do not want to do time intelligence time comparisons or time comparisons without a Date table. You have to have a good Date table. If you want to learn how
to set up one really effectively and quickly, definitely check out a lot of Enterprise DNA content around Date tables.
Time Comparisons Using SAMEPERIODLASTYEAR
To begin with, I’m going to a table here so that we can review the numbers and I’ve got my slicer selected in 2016 here as well.
Instead of sales, I’m going to create another core measure, which is Total Quantity. I’m going to sum up the quantity column, which is in my Sales table.
I just think of this as a core measure, and then I can branch out into all of these other calculations, like time intelligence, moving averages, dynamic grouping, and other different patterns or
formula combinations. It’s just reusing the patterns over and over again.
I’m going to drag my Total Quantity measure into the table so we can see the results. Now that I have this, I can quickly calculate my last year’s quantity.
I’ll go new measure and I’m going to call this Quantity LY (last year). This is where I can use a function called CALCULATE. This is the most important function in the DAX formula language. It
enables you to change the context of a calculation. In this measure, I still want to calculate that total quantity, but I want to do it in a previous timeframe.
In this first example, I’m going to show you this simple function called SAMEPERIODLASTYEAR. It does exactly what it says. It returns a set of dates and the current selection from the previous year.
So basically by putting this inside CALCULATE, I’m able to bring my quantity from one timeframe into another timeframe. And the SAMEPERIODLASTYEAR function allows me to do it exactly with one year
difference. I’m going to show you a better combination to use, but I just showed you this one because I don’t want you to get too confused.
Now, I’ll drag Quantity LY and you see that we’re basically comparing the quantity sold this year on the 1^st of January 2016 to what I sold last year, 1^st of January 2015.
So, if I click on 2015 on the slicer, you’ll see that this first number should be 115. And from there, we can now run time comparisons.
Using Measure Branching Technique
We can actually work out the difference of this year versus last year. And so from that, I can say Quantity Diff YoY (difference year on year). And then all I need to do is subtract Quantity LY from
Total Quantity. I can just reference my measures within a measure. This is called measure branching.
And now when I drag this measure in, you can see that this calculation has been dynamically generated from these two measures, Total Quantity and Quantity LY. There’s nothing hardcoded because when I
click on 2017, I’m going to see the difference. There was nothing done in 2014.
So in this data set, which is a generic old data set, I can see the difference. And we can turn that into a visualization quite easily – now I can get the quantity difference on a daily basis.
We might want to filter this down a bit more. Maybe we’ll create another slicer here that enables us to select a specific month, so we can see on a monthly basis.
And remember, you can also change the context within here. If I wanted to not select anything there, I can actually see the monthly difference very easily without having to change any of my
Time Comparisons Using DATEADD
So now I’ve showed you how to use the SAMEPERIODLASTYEAR function. There is also a function called DATEADD, which enables you to do this as well. I prefer DATEADD because it is more versatile.
And for this, I want to jump quickly to the analyst topic case. So the Analyst Hub is Enterprise DNA’s web-based application that supports your Power BI development. Inside there is a range of
different apps and I’ve already embedded my DATEADD formula pattern in here.
Instead of me writing it out, I’m just going to come in here, search for my formula (sales last year), and then copy it.
Then, I’ll bring it into my model, go new measure and create another name, as we can’t use the same. I’m going to call this Quantity Last Year. All I need to do is change the parameters here.
Instead of Total Sales, I’m going to place Total Quantity.
And then just like that, I have this new Quantity Last Year calculation, which is basically going to return exactly the same number.
It’s going to return exactly the same number here, there is literally no difference between this calculation here and the SAMEPERIODLASTYEAR function. But the benefit of using DATEADD is the
***** Related Links *****
Time Comparison For Non Standard Date Tables In Power BI
Common Time Intelligence Patterns Used In Power BI
Comparing Any Sale Versus The Last Sale (No Time Intelligence) – Advanced DAX In Power BI
In this blog tutorial, I showed you a couple of ways to calculate this year versus last year. You can use the SAMEPERIODLASTYEAR, but I highly recommend the DATEADD function for time comparisons. We
have a lot of content about this function on Enterprise DNA, so definitely check them out.
I also recommend that you use the Analyst Hub. You can save all your patterns there and bring them into your model.
Start using these techniques, including the waterfall model set up. These are the first things that you should be exploring within Power BI from a calculation point of view.
Once you learn how to do this, you can quickly do interesting analyses, especially with all the additional filters that you can place on your data when you build an optimized data model in the
Good luck with this one. All the best!
This project aims to implement a full data analysis pipeline using Power BI with a focus on DAX formulas to derive actionable insights from the order data.
A comprehensive guide to mastering the CALCULATETABLE function in DAX, focusing on practical implementation within Power BI for advanced data analysis.
An interactive web-based application to explore and understand various data model examples across multiple industries and business functions.
A comprehensive project aimed at enhancing oil well performance through advanced data analysis using Power BI’s DAX formulas.
Learn how to leverage key DAX table functions to manipulate and analyze data efficiently in Power BI.
Deep dive into the CALCULATETABLE function in DAX to elevate your data analysis skills.
One of the main reasons why businesses all over the world have fallen in love with Power BI is because...
A hands-on project focused on using the TREATAS function to manipulate and analyze data in DAX.
A hands-on guide to implementing data analysis projects using DAX, focused on the MAXX function and its combinations with other essential DAX functions.
Learn how to leverage the COUNTX function in DAX for in-depth data analysis. This guide provides step-by-step instructions and practical examples.
A comprehensive guide to understanding and implementing the FILTER function in DAX, complete with examples and combinations with other functions.
Learn how to implement and utilize DAX functions effectively, with a focus on the DATESINPERIOD function.
Comprehensive Data Analysis using Power BI and DAX
Exploring CALCULATETABLE Function in DAX for Data Analysis in Power BI
Data Model Discovery Library
Optimizing Oil Well Performance Using Power BI and DAX
Mastering DAX Table Functions for Data Analysis
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Debugging DAX: Tips and Tools for Troubleshooting Your Formulas
Practical Application of TREATAS Function in DAX
MAXX in Power BI – A Detailed Guide
Leveraging the COUNTX Function In Power BI
Using the FILTER Function in DAX – A Detailed Guide With Examples
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Printable Calendars AT A GLANCE
Segment Lengths In Circles Worksheet Answers
Segment Lengths In Circles Worksheet Answers - Web if two segments intersect outside a circle, the following theorems are true. 9 · x = 3 · 6. Web segment lengths in circles. Web segment lengths in
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12.8 x 15.1 7.1 13) 14.1 x 30 9.9 14) 15.7 x 18.9 8.4. If 2 intersect inside a circle, then the product of the lengths of the segments of one chord is. In this video we're going to start the cuda
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Web segment lengths in circles worksheet. Assume that lines whic name date period 1/(1+4) =. A tangent of length \(6 \text{. Section 10.6 segment lengths in circles theorem: Using segments of
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Segment Lengths In Circles Worksheet Printable Word Searches
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Segment Lengths In Circles Worksheet Printable Word Searches
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Unit 10 Circles Homework 8 Answer Key Interleaved Practice Enhances
A tangent of length \(6 \text{. You will receive your score and. Web two intersecting chords in a circle have lengths of segments as \(2 \text{ cm}\), \(8 \text{ cm}\), \(3 \text{ cm}\) and \(x\).
The product of the lengths of one secant segment and its external. Web quickly learn how to find segment lengths are circles (chords, tangents, &.
PPT 10.6 Segment Lengths in Circles PowerPoint Presentation, free
0 w kppear tangent are tangent. In the diagram shown below, prove the following. The and more includes today's geometry lesson. A tangent of length \(6 \text{. Web segment lengths in circles (chords,
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Unit 10 Test Circles Answer Key Gina Wilson Unit 12 Trigonometry
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Segment Lengths In Circles Worksheets
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Segment Lengths In Circles Worksheet Answers Arc Length Worksheet
9 · x = 3 · 6. Web segment lengths in circles. Web if two segments intersect outside a circle, the following theorems are true. Using the above theorem, we have. You will receive your score and.
arc length area of sector worksheet
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Segment Lengths In Circles Worksheet Answers - Rq · rp = rs · rt. The one page interactive worksheet. A tangent of length \(6 \text{. Web segment lengths in circles (chords, secants, and tangents)
mazestudents will practice finding segment lengths in circles created by intersecting chords, intersecting secants,. 0 w kppear tangent are tangent. Using segments of tangents and. Web segment
lengths in circles worksheet. Web in this geometry worksheet, 10th graders use the theorems regarding circles and tangents to find the indicated missing segment length. Web two intersecting chords in
a circle have lengths of segments as \(2 \text{ cm}\), \(8 \text{ cm}\), \(3 \text{ cm}\) and \(x\). Web segment lengths in circles.
Web a worksheet by kuta software llc 7) 10.6 4.3 x 11.4 8) 8.3 4.4 x 9.4 9) 8.4 x 10.4 6.1 10) 10.8 x 12.5 6.3 11) 15 x 16.7 9.4 12) 12.8 x 15.1 7.1 13) 14.1 x 30 9.9 14) 15.7 x 18.9 8.4. Web quickly
learn how to find segment lengths are circles (chords, tangents, & secants) using 3 popular theorems. In the diagram shown below, prove the following. Using the above theorem, we have. Web segment
lengths in circles.
Divide each side by 9. Using segments of tangents and. Web segment lengths in circles. 9 · x = 3 · 6.
Rq · Rp = Rs · Rt.
Choose an answer and hit 'next'. In this video we're going to start the cuda software infinite geometry free worksheet. Web quickly learn how to find segment lengths are circles (chords, tangents, &
secants) using 3 popular theorems. The product of the lengths of one secant segment and its external.
Web Segment Lengths In Circles (Chords, Secants, And Tangents) Mazestudents Will Practice Finding Segment Lengths In Circles Created By Intersecting Chords, Intersecting Secants,.
If 2 intersect inside a circle, then the product of the lengths of the segments of one chord is. Web a worksheet by kuta software llc 7) 10.6 4.3 x 11.4 8) 8.3 4.4 x 9.4 9) 8.4 x 10.4 6.1 10) 10.8 x
12.5 6.3 11) 15 x 16.7 9.4 12) 12.8 x 15.1 7.1 13) 14.1 x 30 9.9 14) 15.7 x 18.9 8.4. You will receive your score and. Web segment lengths in circles.
Web 9.5 Segment Lengths In Circles Solve For X.
Web in this geometry worksheet, 10th graders use the theorems regarding circles and tangents to find the indicated missing segment length. Divide each side by 9. Assume that lines which appear
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9 · X = 3 · 6.
Web segment lengths in circles worksheet. Section 10.6 segment lengths in circles theorem: Ea ⋅ eb = ec ⋅ ed. In the diagram shown below, prove the following.
Related Post: | {"url":"https://ataglance.randstad.com/viewer/segment-lengths-in-circles-worksheet-answers.html","timestamp":"2024-11-02T23:42:01Z","content_type":"text/html","content_length":"37194","record_id":"<urn:uuid:0f7dc772-6852-4385-bb36-9b6d04d5a5b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00529.warc.gz"} |
Bret's Amortization Calculator FAQ (2024)
Hi. From the e-mail I have received over the years, thecalculator gets a lot of use by all kinds of people, even somefolks in the financial industry. In this document, I will tryto respond to the
most frequently asked questions.
• Go to the calculator
• Visit Bret's Blog
• Go to Bret's Home Page
1. Can you help me figure out this financial problem...?
2. What is amortization?
3. Can you describe the data entry fields?
4. How do I use the calculator?
5. Why is one line in the schedule highlighted?
6. How do I prevent the calculator from changing the Payment Amount when I am trying to compute the Number of Regular Payments?
7. How do I calculate an amortization if I make extra principal payments?
8. How do I get the current principal balance?
9. Can the calculator help me with refinancing?
10. Can I calculate a negative amortization?
11. Does the calculator work on a 30/360 basis, or actual/365, or actual/actual?
12. Can you make the calculator show me payment dates?
13. Can you have the calculator figure out the interest paid per year for tax purposes?
14. Can I download this calculator so I can use it on my computer or server?
15. Do you know of another calculator that has feature X?
16. Do you have a spreadsheet template, or any other finance software, that I can download?
Can you help me figure out this financial problem...?
Probably not. I am a computer programmer, not a finance person.I can answer questions about my amortization calculator, how it works,and the (possibly naive) assumptions I've made regarding
amortization.And though I can balance my own checkbook,I have no experience in the banking or finance industries.
What is amortization?
Amortization is a means of paying out a predetermined sum (theprincipal) plus interest over a fixed period of time, so that theprincipal is completely eliminated by the end of the term. This wouldbe
trivial if interest weren't involved, since one could simply dividethe principal amount into a certain number of payments and be donewith it. The trick is to find the right payment amount,
whichincludes some principal and some interest. The math isn't celestialmechanics, but it probably doesn't come standard on the basicpocket calculator.
For the curious, there's a mathematical presentation (PDF) of the problem and its solution.I've done some additional work which shows how to calculatethe principal remaining after a given number of
payments, and how toamortize with an initial payment moratorium in this document.If you're trying to find some original loan parameters for an amortizationschedule in the process of repayment,
there's some additional mathhere which mayhelp. Finally, there's also a document showingequations for calculating the total accumulated interest paid outafter a certain number of payments.
This calculator assumes that each payment should be the same amount,and that a payment consists of some amount for principal reduction andthe interest calculated on the principal balance (including
the principal part of the current payment). I have been told that some Canadian mortgages are not calculated using this method.(Thanks to Andrew Bell, who sent along a link describing Canadian
mortgage compounding.)
Amortization is used most often in mortgages (at least in theUnited States) and short-term loans, but the technique can also beapplied to figure out how long it would take to pay off a givencredit
card debt (for example). In fact, this latter applicationwas why I wrote the calculator in the first place.
Can you describe the data entry fields?
For loans, this is the amount that's borrowed, the amount which will be paid off by the end of the amortization period.
Annual Interest Rate
Typically, this rate would be the APR (Annual Percentage Rate) without additional fees. This rate is divided by the Payments per Year to obtain a periodic interest rate which is actually used by
the calculator. What may be called the "annual rate" can be quite confusing. See the Wikipedia entry for the Annual Percentage Rate and related articles for more detailed information.
Payments per Year
This should be self-explanatory. Monthly payments would by indicated by 12 Payments per Year, twice monthly payments by 24, etc. (This also determines the number of compounding periods in the
year.) For payments every 2 weeks, enter 26, but beware that this is an approximation, since every 10 years you'll actually make 27 payments in the course of one calendar year. A similar caveat
applies to any schedule based on any multiple of weekly payments because a calendar year contains just slightly more than 52 weeks, not to mention the additional complications of leap years.
Number of Regular Payments
The number of payments, combined with Payments per Year defines the term of the loan. If you're looking at a 30-year mortgage with monthly payments, you'd enter 360 into this field (12 × 30). I
call these regular payments to distinguish them from the optional Balloon Payment.
Payment Amount
This is the amount one would pay every regular payment period.
Balloon Payment
This is an optional field. Some loans are set up so that there's a lump sum paid at the end of the term, most of it principal, but typically some interest component as well. If you are running a
balloon scenario, just enter the amount of the balloon payment in this field. If the field is blank, the calculator will assume that there is no balloon amount involved (unless it is the only
blank field). The calculator will treat the balloon payment as if it occurs one payment period after the last regular payment, so this value includes one additional cycle's interest payment.
How do I use the calculator?
When you click “Calculate”, the calculator figures outwhich field is blank (or zero), and then determines what that valueshould be, given the other numbers you've filled in. However, the calculator
will not figure out thePayments per Year if this is left blank: you will begiven an error message.The Balloon Payment field is treated specially: itnormally remains blank. If it is the only blank
field, however,the calculator assumes that you want to calculate what the balloon payment should be, given the other values.
IMPORTANT NOTE: You are welcome to use thiscalculator as a guide in your decision-making or to explorealternatives, but please consult your lending institution or financialadvisor before making your
final decision, since I am not a finance person. (See #1 above.)
As you see, the calculator can be used to calculate a numberof different things. For example, if you are car-shopping and youwant to know how much you can reasonably borrow given that youcan safely
make a $325 monthly payment over 5years, put 325 in the Payment Amount field, 60 in the Number of Regular Payments box, and fill in an APR that yourlending institution will give you. When you click
“Calculate”,the Principal field will be filled in with the amount you canborrow under these conditions. Each time you try a different set ofnumbers, be sure to delete or clear out the field that you
want thecalculator to figure for you.
Since I originally wrote the calculator to figure out how long itwould take me to eliminate some credit card debt, I'll provide an example of that calculation. Assume you've got a $5000 balance on a
credit card at 18%. Put 5000 in the Principalfield, and 18.0 in the Annual Interest Rate field.Clear the Number of Regular Payments field, and set thePayment Amount box to the amount you think you
can afford topay each month. When you click “Calculate”, the number ofpayments will be filled in for you. The paymentamount may be adjusted so that the entire $5000 (plus interest) willbe paid off in
the given term.
As a final example of how to use the calculator, assume that you'vegot a $45,000 principal balance remaining on a house at 7.5%, andyou'd like to pay that amount off in 5years so that you canretire
without a having a house payment. To figure out how muchyou should be paying now to pay off the loan, put 45000 inthe Principal field, 7.5 in the Annual Interest Rate field, and 60 in the Number of
Regular Payments field (12months × 5years). Clear thePayment Amount field and click “calculate”. By paying thecalculated Payment Amount, you should be able to retiremortgage-free.
If the “Show Amortization Schedule” option has been activated,when you click “Calculate”, you will be shown a table of all thepayments, their principal and interest components, and running totals
ofthe principal and interest components.
In addition to finding the value that belongs in the emptyfield, the calculator shows you an estimate of the total amountof interest you'll be paying. A more accurate estimate of interestwill be
found on the last line of the amortization schedule. Theactual interest you will pay depends on how your financialinstitution rounds its numbers, but the estimate should be close.
Why is one line in the amortization schedule highlighted?
This line indicates the cross-over point for the payment schedule,the point in the amortization when the principal part of the paymentexceeds the interest part of a payment for the first time. Not
allschedules will have a cross-over point, though most typical mortgageamortizations probably will.
How do I prevent the calculator fromchanging the Payment Amount when I am trying to computethe Number of Regular Payments?
The calculator only wants to work with an integer number ofpayments. Sometimes, the Payment Amount you enter may resultin a non-integer number of payments. In such a case, the calculatorrounds the
Number of Regular Payments to the nearest integer,and then re-calculates what the Payment Amount should be underthese altered conditions. If this is not the behavior you want, thenthe calculator can
be coerced into doing it your way.
Let's assume that you borrow $5000 from your great-grandmother at8% (you want to be fair, afterall). You want to make payments of $250monthly. How long will it take to pay off? You enter the
parametersinto the calculator, and it tells you that the loan can be paid off in22months, but that you will only pay $245.10 per month. Forsome reason, this is unacceptable (perhaps because it does
not quitecover great-grandma's $250 monthly bingo habit), so you invoke theBalloon Payment field to handle the left-overs: subtract onefrom the Number of Regular Payments (resulting in 21 in
thiscase), and reset the Payment Amount to 250.00. Whenyou click "Calculate" this time, the Balloon Payment field isfilled in with your final payment. In this scenario, you'll make 21regular payments
of $250.00, and one "balloon" payment of$134.36.
How do I calculate an amortization if I make extra principalpayments?
The calculator can only handle extra payments under the followingconditions:
1. Extra payments are the same amount each time
2. Extra payments are made at the same time as regular payments
3. Extra payments are made every regular payment period
If any of these conditions don't apply to your situation, thenyou probably need some sort of a spreadsheet to help you generatean amortization table. I've written an Excel loan amortization
spreadsheetto assist folks in analyzing these cases.
If the above conditions are met, then you may add the extra amountto the Payment Amount field and re-calculate. For example, ifyou have $100,000 loan at 8% over 30 years, the calculator
determinesthat the Payment Amount is $733.76. If you want to make anadditional payment of $100 per month, set Payment Amount to833.76, clear the Number of Regular Payments field,and click
“Calculate”. Under these conditions, the loan term hasbeen reduced to 242 payments, or just over 20 years.
How do I get the current principal balance?
Plug in the loan parameters, and set the “Show Amortization Schedule”option before clicking “Calculate”. Determine how many payments you have made so far, and look up the Principal Balance in the
finalcolumn of the amortization table.
Can the calculator help me with refinancing?
A little. First, find out what your principal balance is (seethe previous question).Now enter the principal balance in the Principalfield. If you're planning a zero cash out-of-pocket re-fi,you
should add closing costs/points to the Principal amount.Then you can play with the numbers in the other fields.
Can I calculate a negative amortization?
Yes, the calculator can perform negative amortizations. A negativeamortization loan is a scenario where the periodic payment is lessthan the interest that is due for that period. In this case,the
unpaid interest is added into the principal amount, and sothe debt grows over time rather than being reduced. Since theseloans are never paid off, they are usually temporary or
short-termarrangements, after which the loan is "recast" into an actual payoffscenario.
As an example,assume that we would like to know how much we will owe after anegative amortization period of 5years, given that we have borrowed$100K at 8%, and we'll be making monthly payments of
$600.00.We enter Principal at 100000.00, Annual Interest Rateat 8.0, Payments per Year at 12, set Number of Regular Payments to 60 (12×5), and set thePayment Amount to 600.00. Click on "Show
AmortizationSchedule", and then click "Calculate". The Balloon Paymentfield will be filled in with the amount of money owed at the end of thenegative amortization period ($105,597.78, in this
example).If you look at the amortization schedule,you will see that the "Principal" column contains negative values, whichmakes sense because this column is intended to show the principalreduction.
You will also note that the principal balanceis increasing over time.
NOTE: I've recently noticed that some of the loan summaryinformation that the calculator produces in negative amortizationscenarios is inaccurate and misleading. The final line of the paymentschedule
is also inconsistent. I will fix these things when I havethe opportunity, but for now, please be aware of these "gotchas".
Does the calculator work on a 30/360 basis,or actual/365, or actual/actual?
Strictly speaking, none of the above.For all payment schedules, the calculator treats the "Payments per Year"as equally-distributed. E.g., a "bi-weekly" payment schedule,26payments per year, is
treated as having 365/26=14.038...daysper payment period by the calculator. A true bi-weekly schedule wouldinstead use 14days exactly, and every 10years, there wouldbe an extra (27th) payment during
the year.
A 30/360 basis treatsthe year as having 360days, each month having 30days, resulting in12equally-spaced payments per year. For monthly payments,the calculator appears to be on a 30/360 basis since
the math endsup the same: (APR×30)/360 is the same as APR/12.
My understanding of the "actual" bases is that the APR is dividedby the number of days in the year to provide a daily interest rate,and the interest is calculated on the true number days between
scheduled payments. If one were to plot the interest paid over time,the actual basis methods will produce slightly jittery curves relativeto the curves this calculator will produce.
Can you make the calculator show me payment dates?
As convenient as this feature may be, it's not gonna happen anytimesoon for several reasons. The calculator is rather simple right now,and it would need to have a whole lot more intelligence to
handle dates properly. I like to program, so it's notthat I'm averse to writing more code, but in addition, the input formwould require more fields to be filled in by the user. I made aconscious
effort to keep the input required to a minimum while maintaining the computational flexibility I wanted.
More profound issues are raisedif dates are added: for one, I would need to know how to calculate anamortization based on a bi-weekly payment schedule. I really don'tknow how financial institutions
handle this case in real life. [If youwork for a lending institution and have specific info on how theperiodic interest rate is calculated for a true bi-weekly paymentschedule, please fill me in!] In
short, the calculator program is just a “quickie” (it took longer to figure out the math than to write the actual program), and I prefer its current simplicity.
Can you have the calculator figure out the interestpaid per year for tax purposes?
To handle this properly requires that the calculator know when afiscal year begins and ends, and therefore requires some knowledge ofdates (see previous question). However, you can sidestep the
issueand do the calculation manually. Suppose that payment 31 is the finalpayment of the previous fiscal year, and payment 57 is the finalpayment of the current fiscal year. To get the interest paid
duringthe current fiscal year, subtract Cum Int for payment 31 fromCum Int for payment 57.
Can I download this calculator so I can use it on my computer or server?
Sorry, but this calculator and its source code are not available.It is old and idiosyncratic. Since I cannot offer support for it, Ido not license it or make it available in any other form, even
formoney. (Well, OK, if you wanna put me through grad school at MIT, Imight consider it. ;-)
Do you know of another calculator that has feature X?
Sorry, I'm not trying to keep up with the Jones's calculators, and Idon't keep track of what other people may offer. Frankly, I'msurprised that this calculator still seems to be so popular after
allthese years. I would have thought that someone else would haveoutclassed this puppy long ago.
Do you have a spreadsheet template, or anyother finance software, that I can download?
Funny that you should ask. In June 2010 I wrote an amortization spreadsheet in Excelwhich may be useful with irregular extra-payment or late feesituations which the online calculator isn't
well-equipped to handle.And since it's something you download onto your own computer, it mayalso be handy as a means of tracking your loan or mortgage payments.Use at your own risk! | {"url":"https://floridaexecutivevilla.com/article/bret-s-amortization-calculator-faq","timestamp":"2024-11-08T22:54:31Z","content_type":"text/html","content_length":"130180","record_id":"<urn:uuid:b1a9eefb-f4eb-4613-ad1f-3c27c3c5fd1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00455.warc.gz"} |
greedy_tsp(G, weight='weight', source=None)[source]#
Return a low cost cycle starting at source and its cost.
This approximates a solution to the traveling salesman problem. It finds a cycle of all the nodes that a salesman can visit in order to visit many nodes while minimizing total distance. It uses a
simple greedy algorithm. In essence, this function returns a large cycle given a source point for which the total cost of the cycle is minimized.
The Graph should be a complete weighted undirected graph. The distance between all pairs of nodes should be included.
weightstring, optional (default=”weight”)
Edge data key corresponding to the edge weight. If any edge does not have this attribute the weight is set to 1.
sourcenode, optional (default: first node in list(G))
Starting node. If None, defaults to next(iter(G))
cyclelist of nodes
Returns the cycle (list of nodes) that a salesman can follow to minimize total weight of the trip.
If G is not complete, the algorithm raises an exception.
This implementation of a greedy algorithm is based on the following:
□ The algorithm adds a node to the solution at every iteration.
□ The algorithm selects a node not already in the cycle whose connection to the previous node adds the least cost to the cycle.
A greedy algorithm does not always give the best solution. However, it can construct a first feasible solution which can be passed as a parameter to an iterative improvement algorithm such as
Simulated Annealing, or Threshold Accepting.
Time complexity: It has a running time \(O(|V|^2)\)
>>> from networkx.algorithms import approximation as approx
>>> G = nx.DiGraph()
>>> G.add_weighted_edges_from(
... {
... ("A", "B", 3),
... ("A", "C", 17),
... ("A", "D", 14),
... ("B", "A", 3),
... ("B", "C", 12),
... ("B", "D", 16),
... ("C", "A", 13),
... ("C", "B", 12),
... ("C", "D", 4),
... ("D", "A", 14),
... ("D", "B", 15),
... ("D", "C", 2),
... }
... )
>>> cycle = approx.greedy_tsp(G, source="D")
>>> cost = sum(G[n][nbr]["weight"] for n, nbr in nx.utils.pairwise(cycle))
>>> cycle
['D', 'C', 'B', 'A', 'D']
>>> cost | {"url":"https://networkx.org/documentation/latest/reference/algorithms/generated/networkx.algorithms.approximation.traveling_salesman.greedy_tsp.html","timestamp":"2024-11-06T20:31:07Z","content_type":"text/html","content_length":"38095","record_id":"<urn:uuid:669d6171-27e8-4b7f-bf5c-3f1e9b69ca0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00426.warc.gz"} |
Research in presence of multiple theories
Consider a hypothetical question, a researcher was given with a research question; compare the mathematical ability of male and female students of grade 5. The researcher collected data of 300 female
students and 300 male students of grade 5 and administered a test of mathematical questions. The average score for female students was 80% and average score of male student was 50%, the difference
was statistically significant and therefore, the researcher concluded that the female students have better mathematical aptitude.
The findings seem strong and impressive, but let me add into the information that the male students were chosen from a far-off village with untrained educational staff and lack of educational
facilities. The female students were chosen from an elite school of a metropolitan city, where the best teachers of the city actually serve. What should be the conclusion now? It can be argued that
actually difference doesn’t come from the gender, the difference is coming from the school type.
The researcher carrying out the project says ‘look, my research assignment was only to investigate the difference due to gender, the school type is not the question I am interested in, therefore, I
have nothing to do with the school type’.
Do you think that the argument of researcher is valid and the findings should be considered reliable? The answer is obvious, the findings are not reliable, and the school type creates a serious bias.
The researcher must compare students from the same school type. This implies you have to take care of the variables not having any mention in your research question if they are determinants of your
dependent variable.
Now let’s apply the same logic to econometric modeling, suppose we have the task to analyze the impact of financial development on economic growth. We are running a regression of GDP growth on a
proxy of financial development; we are getting a regression output and presenting the output as impact of financial development on economic growth. Is it a reliable research?
This research is also deficient just like our example of gender and mathematical ability. The research is not reliable if ceteris paribus doesn’t hold. The other variables which may affect the output
variable should remain same.
But in real life, it is often very difficult to keep all other variables same. The economy continuously evolves and so are the economic variables. The other solution to overcome the problem is to
take the other variables into account while running regression. This means other variables that determine your dependent variable should be taken as control variables in the regression. This means
suppose you want to check the effect of X1 on Y using model Y=a+bX1+e. Some other research studies indicate that another model exist for Y which is Y=c+dX2+e. Then I cannot run the first model
ignoring the second model. If I am running only model 1 ignoring the other models, the results would be biased in a similar way as we have seen in our example of mathematical ability. We have to use
the variables of model 2 as control variable, even if we are not interested in coefficients of model 2. Therefore, the estimated model would be like Y=a+bX1+cX2+e
Taking the control variables is possible when there are a few models. The seminal study of Davidson, Hendry, Sarba and Yeo titled ‘Econometric modelling of the aggregate time-series relationship
between …. (often referred as DHSY)’ summarizes the way to build a model in such a situation. But it often happens that there exists very large number of models for one variable. For example, there
is very large number of models for growth. A book titled ‘Growth Econometrics’ by Darlauf lists hundreds of models for growth used by researchers in their studies. Life becomes very complicated when
you have so many models. Estimating a model with all determinants of growth would be literally impossible for most of the countries using the classical methodology. This is because growth data is
usually available at annual or quarterly frequency and the number of predictors taken from all models collectively would exceed number of observations. The time series data also have dynamic
structure and taking lags of variables makes things more complicated. Therefore, classical techniques of econometrics often fail to work for such high dimensional data.
Some experts have invented sophisticated techniques for the modeling in a scenario where number of predictor becomes very large. These techniques include Extreme Bound Analysis, Weighted Average
Least Squares, and Autometrix etc. The high dimensional econometric techniques are also very interesting field of econometric investigation. However, DHSY is extremely useful for the situations where
there are more than one models for a variable based on different theories. The DHSY methodology is also called LSE methodology, General to Specific Methodology or simply G2S methodology.
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Analogies between the geodetic number and the Steiner number of some classes of graphs
A set of vertices $S$ of a graph $G$ is a geodetic set of $G$ if
every vertex $v\not\in S$ lies on a shortest path between two
vertices of $S$.
The minimum cardinality of a geodetic set of $G$ is the geodetic
number of $G$ and it is denoted by $g(G)$. A Steiner set of $G$ is a set of vertices $W$ of
$G$ such that every vertex of $G$ belongs to the set of vertices of
a connected subgraph of minimum size containing the vertices of $W$. The minimum cardinality of a Steiner set of $G$ is the Steiner
number of $G$ and it is denoted by $s(G)$. Let $G$ and $H$ be two graphs and let $n$ be the order of $G$. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking
one copy of $G$ and $n$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ to the $i^{th}$-vertex of $G$. We study the geodetic number and the Steiner number of corona
product graphs.
We show that if $G$ is a connected graph of order $n\ge 2$ and $H$ is a
non complete graph, then $g(G\odot H)\le s(G\odot H)$, which partially solve the open problem presented in [\emph{Discrete Mathematics} \textbf{280} (2004) 259--263] related to characterize families
of graphs $G$ satisfying that $g(G)\le s(G)$.
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How can I get the right notation?(e^3x)
How can I get the right notation?(e^3x)
I'd like to match the order between constant and variable. For example, generally we write a tangent function as e^3x, but in sage It is reversed lik xe^3 and the code is var('x') print(e ^ 3 * x)
How can I get the right notation?
1 Answer
Sort by » oldest newest most voted
You may try :
sage: foo=e^3*x
sage: foo
sage: foo.coefficient(x)
sage: print("%s%s"%(foo.coefficient(x),x))
but this is, IMNSHO, chemically pure (analytical quality) foolishness : you're aiming to something (printing æsthetics) that has no algorithmic definition.
A ((very) slightly) little less silly :
sage: R1.<t>=PolynomialRing(SR)
sage: sum([var("p{}".format(u))*t^u for u in range(5)])
p4*t^4 + p3*t^3 + p2*t^2 + p1*t + p0
edit flag offensive delete link more
wow! it worked! thank you!!!!!
wnghks2516 ( 2021-02-08 06:38:26 +0100 )edit | {"url":"https://ask.sagemath.org/question/55585/how-can-i-get-the-right-notatione3x/","timestamp":"2024-11-02T15:48:36Z","content_type":"application/xhtml+xml","content_length":"53222","record_id":"<urn:uuid:9232fddc-cde9-454d-8e56-9605db5eae52>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00595.warc.gz"} |
Step-By-Step Multiplication
You don't know how this keeps happening. You try to keep your room clean and tidy, but it always ends up a mess! And now it has become so messy that it is overwhelming.
Tackling a big mess isn’t so bad if you break it down.
The best thing to do is start small and take it one step at a time. Maybe you start by picking up one corner of the room, or maybe put all the clothes in the hamper or all the books in the bookshelf.
Completing small tasks in the right order can make the job less scary. The same is also true of multiplying 2-digit numbers!
• Do you ever look at a huge mess and wonder where to start, like cleaning your room or solving a tough math problem?
That’s how multiplying two-digit numbers can feel at first—it seems impossible to do in your head!
But don’t worry, just like cleaning, you can break it down into steps, and it won’t feel overwhelming at all!
Look at the problem 32 x 14.
Step One
First, set the problem up vertically — one number on top of the other. You can use a place value chart like this one.
Set up your problem so that the 32 is on top with the 3 in the tens place and the 2 in the ones place.
Then, write 14 underneath so that the 1 is in the tens column and the 4 is in the ones column.
Now you can start multiplying.
Step Two
Start with the bottom number in the ones place. In this case it is 4.
Multiply it by the top number in the ones place. In this case, it is 2.
You got it, 8! Write that down in the ones column.
Next, take the same bottom number (4) and multiply it by the top number in the tens place, In this case, it is 3.
You go it. 12.
Because there is no number in the hundreds place, you can write the whole number 12.
Step Three
Now is the trickiest and most forgettable part of multiplying 2-digit numbers: the zero placeholder!
Multiplying by the bottom number in the tens place makes your answer ten times bigger, so you must start writing in the tens place. To do that, place a zero in the ones place as a placeholder.
Step Four
Repeat the multiplication process, but start with the bottom number in the tens place. In this case, 1.
Multiply it by the top number in the ones place (2).
2, of course! But remember that the 1 in this case is actually a ten (10 x 2), so you write the 2 in the tens column.
Multiply the same bottom number (1) by the top number in the tens place (3).
Yep! 3.
Since the 1 and 3 are in the tens column, it is actually like multiplying 10 times 30. This would give you 300, so it makes sense that the 3 will go in the hundreds column.
Phew! So far so good.
Step Five
Now it is time to add it all together. Start on the right side with the ones column and work your way across.
• What is 8 + 0?
• What is 2 + 2?
• What is 1 + 3?
Your final answer is 448!
You just solved a big multiplication problem by breaking it into small steps!
Now, try this one: 35 x 28. Start just like before—write the numbers one on top of the other.
Remember, begin with the bottom number in the ones place (the 8), and multiply it by the top number ones place (the 5).
That's right! 40.
• But how do you write 40 in the ones place?
Write the zero in the ones place, but then carry the 4 over to the tens place where it belongs. You will work with it in a moment when you multiply by the tens place.
Now multiply 8 by 3, which equals 24.
• But what do you do about the 4 you carried to the tens place?
Simply add the 4 to your answer of 24. 24 + 4 equals 28, so write 28 below.
Great job! Now, cross off the 4 you carried so it doesn’t get in the way as you continue.
Here comes an important step! Since you're moving on to the tens place, don’t forget to put a zero as a placeholder in the ones place.
Next, multiply the bottom tens place number (2) by the top ones place number (5).
That’s right! It’s 10.
Again, you can't put the whole ten in the tens place because it is really saying 20 x 5, so place the 0 in the tens place and carry the 1 to the tens place
Now, multiply 2 times 3, which equals 6. Don't forget to add the 1 you carried over!
6 = 1 equals 7, so write 7 in the hundred place.
Awesome! Now it’s time to add everything together.
• What is 0 + 0?
• What si 8 + 0?
• What is 2 + 7?
Well done! The answer is 980.
You did an amazing job multiplying those big numbers!
Now, you’re ready for the Got It? section where you can practice your skills! | {"url":"https://www.elephango.com/index.cfm/pg/k12learning/lcid/14154/Step-By-Step_Multiplication","timestamp":"2024-11-14T12:06:28Z","content_type":"text/html","content_length":"74983","record_id":"<urn:uuid:3a86c4a7-9dbc-4b21-8580-33648c37d1f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00633.warc.gz"} |
Retaining Wall Calculator
How to Calculate the Amount of Materials Needed to Build a Retaining Wall
Retaining Wall Calculator
When building a retaining wall, you need to estimate the number of stone blocks, capstones, and cubic yards of gravel needed to complete the project. Getting an accurate landscaping estimate will
help you plan for the cost of materials and labor to build a retaining wall. You can figure the amount of materials following the steps below, or by using the convenient retaining wall calculator on
the left.
This article assumes that you build a retaining wall following this method: First dig a trench that is twice as wide as the wall, as long as the finished wall, and 8 inches deep. Then fill the base
of the trench with coarse gravel, and build the wall so that about 6 inches (0.5 feet) of the wall rests below ground. Leave about 4-6 inches of space between the back of the retaining wall and the
soil, then use more coarse gravel for backfill behind the wall.
Calculating Number of Stone Blocks
The number of stone blocks depends on the finished height and length of the retaining wall. Multiply the wall's length by its height plus 1/2 a foot to find the total square footage of the wall. That
is, calculate
Wall Area = L(H + 0.5) sq. ft,
where L is the length of the wall in feet and H is the finished height of the wall in feet. Next find the face area of each stone block by multiplying its length and width in inches. That is,
Stone Block Area = AB sq. inches,
where A is the length of the block in inches and B is the width of the block in inches. Now, compute the total number of stone blocks needed with the formula
Number of Blocks = 144L(H + 0.5)/(AB).
The factor of 144 is necessary to convert feet to inches. Example: Suppose a wall is to be 20 feet long and 3 feet height, built with stones that are 6 inches by 10 inches. Then the number of stones
needed is
144(20)(3 + 0.5)/[(6)(10)] = 10080/60 = 168.
You should purchase stone blocks to spare in case of breakage.
Calculating the Number of Capstones
To figure the number of capstones needed, dividing the length of the wall in feet by the length of the capstones in inches, then multiply by 12. For example, if the wall is 20 feet long and the
capstones are 8 inches wide, then the number of capstones needed is [20/8](12) = 30. You may need to buy a few more in case some of the capstones break.
Calculating the Amount of Gravel
First estimate the amount of gravel needed to pack the trench at the base of the retaining wall. If the wall is L feet long and T inches thick, then the trench needs to be L feet long, about 8 inches
deep, and 2T inches wide. Assuming that about 5/8 of the trench's volume will be filled with gravel, the cubic yards of gravel need for the trench is given by the formula
Trench Gravel = (5/1944)LT cubic yards,
where L is in feet, T is in inches, and the number 5/1944 accounts for the 8 inch depth and conversion factors for feet to yards and inches to yards. Next, estimate the cubic yards of gravel needed
for backfill behind the retaining wall. Assuming about 4 inches of space between the stones and the earth, the volume of gravel needed is given by the formula
Backfill Gravel = LH/81 cubic yards,
where L and H are the length and finished height of the wall in feet. Example: Suppose a retaining wall that is 20 feet long, 3 feet high, and 6 inches thick. Thus, L = 20, H = 3, and T = 6. The
total amount of gravel needed is
(5/1944)LT + LH/81
= (5/1944)(20)(6) + (20)(3)/81
= 0.30864 + 0.74074
= 1.05 cubic yards of gravel.
© Had2Know 2010 | {"url":"https://www.had2know.org/garden/retaining-wall-materials-calculator.html","timestamp":"2024-11-13T02:29:10Z","content_type":"text/html","content_length":"38828","record_id":"<urn:uuid:eb256685-9db5-48c3-bb83-a10519c64ddb>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00401.warc.gz"} |
How it works
⚡ Lightning
Lightning fees are calculated using the following method:
• Get the complete list of all known public channels from our Lightning node
• Filter channels: keep only the ones that are currently active and had recent updates
• Compute the average of all fees (fixed and proportional), weighted by channel capacity, excluding 5% of outliers.
The result is the average fee to travel through a route on the Lightning Network.
Fees on Lightning are computed differently from on-chain fees.
One of the primary differences is that the transaction size (in bytes transmitted through the network) is irrelevant.
They are composed of two components:
• Fixed fee: a constant amount in satoshis that is charged no matter the payment amount.
• Proportional fee: a percentage that is applied to the payment amount.
In addition, these fees are charged once per hop through a route, so the more hops a payment goes through, the more fees will be charged. Each intermediate node is free to choose its own fee policy.
Question: will I be charged the exact amount that is displayed on this website?
The answer is: it depends. The fees displayed on this website are averages of all public routes, for 1 hop. If your payment goes through just 1 hop, on a very average and typical route, it should be
approximately the same amount. Although we can't predict how far you are from each destination you want to pay, so the more hops, the more fees.
Also, if you are connected to an expensive routing node, you might pay above-than-average rates.
Question: I have a direct channel to the payment recipient, what fees am I going to pay?
One word: ZERO :)
Direct payments do not incur any fee.
🔗 On-chain
Here is a quick summary of the method used to compute estimates. The algorithm uses a simplified model taking into account 3 factors:
Current weight of transactions in the mempool
Speed at which new weight is entering the mempool
Randomness in block production intervals
Unlike some other fee estimation algorithms, it doesn't look at the previous mined blocks at all. Instead, it looks at the factors that are going to drive the production of the next blocks: the
mempool, the speed of increase of the mempool and the probability at which it is being drained.
Its goal is to give reasonable estimates given the presently known mempool dynamics, while avoiding overestimation.
The mempool is categorized into "fee buckets". A bucket represents data about all transactions with a fee greater than or equal to some amount (in sat/vbyte).
Each bucket contains 2 numeric values:
• current_weight, in WU (Weight-Units), represents the transactions currently sitting in the mempool.
• flow, in WU/min (Weight-Units per minute), represents the speed at which new transactions are entering the mempool. Currently that is sampled by observing the flow of transactions during twice
the timespan of each target interval (ex: last 60 minutes of transactions for the 30 minutes target interval)
For simplicity, transactions are not looked at individually. Focus is on the weight, like a fluid flowing from bucket to bucket.
For each target interval (30 mins, 1 hour, 2 hours etc...), we're trying to find the cheapest fee rate that is likely to become fully cleared (0 WU) with a given probability.
The probability is defined by the "confidence" setting on the website. Current values are:
• Optimistic – I'm feeling lucky – : 50%
This is to be used if your primary objective is fee minimization and if you do not care if there's some chance to get delayed. Might work if you get the next blocks fast enough with no unlucky
• Standard: 80%
This profile seems to give reasonable balanced estimates while avoiding overestimation or underestimation most of the time.
• Cautious: 90%
This one has a tendency to overestimate, to compensate for potential unlucky rounds of blocks.
Now let's simulate what's going to happen during each timespan lasting minutes:
• New transactions entering the mempool. While it's impossible to predict sudden changes to the speed at which new weight is added to the mempool, for simplicty's sake we're going to assume the
flow we measured remains constant.
added_weight = flow * minutes
• Transactions leaving the mempool due to mined blocks. Each block removes up to 4,000,000 WU from a bucket, however the exact number of blocks that are going to occur during the interval is
uncertain. So what we'd like to do is to find out the mimimum number of blocks we should expect (with our chosen probability).
The occurence of blocks follows a Poisson distribution, so what we can do is calculate the inverted Poisson CDF (in Python: 1 - scipy.stats.poisson(λ).cdf(k)), with λ = minutes / 10 (expected
average number of blocks), then iteratively increase the k parameter (number of blocks) until the output probability is < to our chosen probability and then we return the previous k value.
Once we know the minimum expected number of blocks we can compute how that would affect the bucket's weight:
removed_weight = 4000000 * blocks
• Finally we can compute the expected final weight of the bucket:
final_weight = current_weight + added_weight - removed_weight
The cheapest bucket whose final_weight is ≤ 0 is going to be the one selected as the estimate.
Small correction
Because the window used to sample the flow of transactions increases proportionally to each target interval, it sometimes gives incoherent results with estimates that decrease then increase as the
window gets larger (if there was significant variations in the flow of transactions during this time).
Since this makes no sense (if a low fee gets you confirmed faster, then there is no need to increase the fee to target a longer window), so for each estimate we take the minimum value of all
estimates at windows shorter or equal.
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Herbert Kociemba's optimal Rubik's Cube solver - Cube Explorer
Herbert Kociemba’s Optimal Cube Solver
– Cube Explorer
Ever since the Rubik's Cube’s debut many decades ago, it has posed many questions far beyond the solving of it. One of these puzzles has been the optimal solution for a given scrambled state. Once we
had solved the puzzle, we wanted to push it further. How can we solve any given scrambled cube in the least number of moves possible? Mathematics and several decades of research has proved that any
scramble can be solved in under 20 moves. But how do we find the smallest combination of moves? The answer resides in Herbert Kociemba's Cube Explorer program.
The desktop version of Cube Explorer
Start the optimal online solver >>
Let’s first take a look at the long winded approach – a cube solver that would take a scrambled cube and find the shortest sequence of moves required to solve it. Since the value must be <20 and
there is no collection of the exact number of moves for each of the 43 quintillion possibilities, let’s use results from the Cube Explorer itself – Out of 100,000 unique scrambles, 93,772 of these
had minimums of 17 or 18 moves (26,673 and 67,099 respectively). Therefore we shall use an approximation of 17.75 moves per solve. There are 18 different possible moves, and if we assume that,
following the completion of a move, the following move must be performed on a different layer (reducing the possible move number to 15), then there are 18 x 15^16.75 possible combinations of moves,
of which the shortest one would have to be found. For every single scramble, approximately 901,158,314,000,000,100,000 (901 quintillion) solutions would have to be checked. This would take an immense
amount of hours for even some of the best machinery; keeping in mind that even the 43 quintillion possible scrambles is too big a number to process, hence the Cube Explorer’s test with 100,000
So how exactly do we find the shortest number of moves to solve a scramble without checking an immensely large number of possible moves? It all starts with the Two-Phase-Algorithm.
The Two-Phase-Algorithm
Phase 1 of the Two-Phase-Algorithm uses a set of the 18 possible moves that, regardless of order or number of moves, cannot change the orientation of edge and corner pieces – G1. This set includes <U
D R2 F2 L2 B2>, and it also means that pieces within the U and D layers cannot move out of the two layers (only 2 moves can be performed which would simply move pieces from U to D and vice-versa).
Since this is a subset of all possible scrambles (a subset which has a large number of acceptable outputs), it is much easier to get to than going from a scrambled cube to a solved one in one step.
An algorithm is performed which increases the count of allowed moves used to find a solution that takes the cube to a G1 state. Once it finds the shortest number of moves required to do this, it
continues until it has a large enough supply of possible solutions ranging from the lowest number of moves to a much larger number.
In Phase 2, the permutations of the edges and corners are restored. However, G1 is still an incredibly large subset; it just wouldn’t be possible to do all of this accurately. Therefore, an estimate
is made on the number of moves required for Phase 2. The algorithm continues to find shorter and shorter solutions by using some not-so-optimal Phase 1 values that produce more optimal Phase 2
values. For example, an 8 move Phase 1 followed by a 15 move Phase 2 is less optimal than a 10 move Phase 1 followed by a 5 move Phase 2. As the Phase 1 value increases, the Phase 2 value decreases.
Eventually, a solution will be found where the value of Phase 2 is 0 – This is the optimal solution.
Of course, there could be other solutions where Phase 2 is needed, however the implementation above is much more efficient and provides very accurate results.
Symmetry is also another concept used in the Cube Explorer. A cube can be rotated and produce a different case. Therefore geometrical transformation can be applied to cubes to drastically reduce the
number of possible cases and enhancing the solver.
For picture cubes, the centre piece’s orientation is vital otherwise the cube is not “solved”. There are 2048 different centre combinations, and Kociemba proved that any picture cube could be solved
in 21 moves or less.
For more information about the solver, please visit Kociemba’s website (kociemba.org) which explains in more detail the Cube Explorer software and has links to download and donate. | {"url":"https://ruwix.com/the-rubiks-cube/herbert-kociemba-optimal-cube-solver-cube-explorer/","timestamp":"2024-11-03T21:29:47Z","content_type":"text/html","content_length":"11769","record_id":"<urn:uuid:c2c32a2f-ee6b-4c1f-87a0-1e50a6894e12>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00854.warc.gz"} |
Heriot-Watt University
Conjugacy growth in (some Artin) groups
Geometry and Topology Seminar
30th April 2024, 2:00 pm – 3:00 pm
Fry Building, 2.04
In this talk I will give an overview of what is known about conjugacy growth and the formal series associated with it in infinite discrete groups. I will highlight how the rationality (or rather lack
thereof) of these series is connected to both the algebraic and the geometric nature of groups such as (relatively) hyperbolic or nilpotent, and how tools from analytic combinatorics can be employed
in this context.
I will then present recent work on the conjugacy growth of dihedral Artin groups, joint with Gemma Crowe. | {"url":"https://www.bristolmathsresearch.org/seminar/tba-72/","timestamp":"2024-11-05T18:30:21Z","content_type":"text/html","content_length":"54028","record_id":"<urn:uuid:81847117-c2d1-476b-b7d9-ec9825ef1bad>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00465.warc.gz"} |
Question #cb6b8 | Socratic
Question #cb6b8
1 Answer
Via Quantum Mechanics with lots and lots of math.
In Quantum Mechanics, there is the Schrodinger Equation . The equation is capable determining many properties of a particle in a system based on time and position of the particle.
In a steady state Schrodinger Equation (we are not interested in the time variable);
The equation is $E = - {\overline{h}}^{2} / \left(2 m\right) {\nabla}^{2} \psi + U \psi$.
$\psi$ is the wave function of a particle; a function that describes the nature of your particle.
${\nabla}^{2} = {\partial}^{2} / \left(\partial {x}^{2}\right) + {\partial}^{2} / \left(\partial {y}^{2}\right) + {\partial}^{2} / \left(\partial {z}^{2}\right)$ in Cartesian coordinate system.
${\nabla}^{2} = \frac{1}{r} \frac{\partial}{\partial r} \left({r}^{2} \partial \frac{\psi}{\partial r}\right) + S \in \left(\theta\right) \frac{\partial}{\partial \theta} \left(S \in \left(\theta\
right) \partial \frac{\psi}{\partial \theta}\right) + {\partial}^{2} \frac{\psi}{\partial {\phi}^{2}}$ in Polar Coordinate system.
Lets see the hydrogen atom, the easiest one.
$E = {E}_{1} / {n}^{2}$. $n$ is the principle quantum number which corresponds to the orbital shell (also energy state).
What we want to do is to solve the Schrodinger equation to find $\psi$.
Solving the equation is very tedious and requires arbitrary constants $l$ and ${m}_{l}$. Refer to Separation of Variables method .
Solving this implies that $\psi$ has non-zero value if the values of $l$ has integer values and does not exceed $n - 1$ .
!! $l$ will determine your orbitals. $n$ determines the values of $l$.!!
In other words, if $n = 3$ you must have $l = 0 , 1 \mathmr{and} 2$. Energy level 3 has 3 orbitals.
$s , p \mathmr{and} d$ orbitals. Other values of $l$ in this case will cause $\psi$ to breakdown.
The electron will cease to exist.
Therefore, if $n = 3$, you will have 3 solutions for $\psi$ based on $l = 0 , 1 , 2$.
Sorry if I can't express this any simpler. The history of the atomic model from Neils Bohr onwards is very theoretical.
Impact of this question
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Topics in projective algebraic optimization | KTH
Topics in projective algebraic optimization
Time: Fri 2023-12-08 14.00
Location: F3 (Flodis) Lindstedsvägen 26 & 28
Video link: https://kth-se.zoom.us/j/63466474196
Language: English
Doctoral student: Lukas Gustafsson , Matematik (Inst.)
Opponent: Cordian Riener, UiT The Arctic University of Norway
Supervisor: Sandra Di Rocco, Matematik (Avd.); Kathlén Kohn, Matematik (Inst.)
QC 2023-11-17
This thesis explores optimization challenges within algebraic statistics, employing both topological and geometrical methodologies to derive new insights. The main focus is the optimization degree of
nearest point and Gaussian maximum likelihood estimation problems with algebraic constraints. The optimization degree counts the number of complex critical points for an optimization problem. It is
interesting as it can aid numerical solvers by providing an upper bound on the number of solutions to a set of equations, without computing them explicitly. The study extends to a parallel research
trajectory, complementing and expanding the primary themes by studying relative tangency for critical point loci and characterizing the ideal of the line-multiview variety, inspiring further study of
reconstructing 3D objects from 2D images in computer vision.
Paper A focuses on linear concentration models and critical point counts for the Gaussian log-likelihood function when restricted to a linear space. The paper unveils new Gaussian maximum likelihood
degree formulae from line geometry and Segre classes. We also study codimension one models and scenarios with zero maximum likelihood degree in particular.
In Paper B, we extend the inquiry from Paper A by exploring Gaussian likelihood geometry of arbitrary projective varieties. We introduce the maximum likelihood degree of a homogeneous polynomial on a
projective variety, delving into quantifying critical points for a rational function. We find geometric characterizations of the maximum likelihood degree in terms of Euler characteristics, dual
varieties, and Chern classes.
Paper C advances the investigation into multivariate Gaussian statistical models with rational maximum likelihood estimator (MLE). A correspondence is established between such models and solutions to
a nonlinear first-order partial differential equation (PDE). This link sheds light on the problem of classifying Gaussian models with rational MLE, relating it to the open problem of classification
of homaloidal polynomials in birational geometry.
Paper D computes the generic, or expected, maximum likelihood degree of a variety as an analog to the known polar class formula for the Euclidean distance degree. Additionally, as a follow-up to
paper C, the complex projective curves of maximum likelihood degree 1 are classified in paper D. This allows further work into when a complex curve can be realized as a real statistical models. Both
paper C and D connect the maximum likelihood degree as a possible generalization to the Euclidean distance degree for projective varieties.
Paper E intersects algebraic geometry and computer vision, focusing on projected lines from multiple pinhole cameras. The line multiview variety captures these projections as an algebraic variety.
The main result establishes the ideal of this variety, generated by 3x3-minors of a matrix derived from projected line equations. The predecessor of the line-multiview variety is the point-multiview
variety, with image correction being a driving motivation for introducing the Euclidean distance degree. Notably, Paper E opens the door for studying the Euclidean distance degree of the
line-multiview variety and its uses in 3D reconstruction.
Paper F delves into the concept of Euclidean distance estimates within the context of a specific subset of the available data. To contruct a robust foundational theory, this paper introduces the
concepts of relative duality and relative characteristic classes. It demonstrates that classical formulas can be equivalently expressed in the relative setting, thereby shedding light on the
geometric intricacies inherent to this relative analysis. | {"url":"https://www.kth.se/en/om/2.266/topics-in-projective-algebraic-optimization-1.1296930?date=2023-12-08&orgdate=2023-03-30&length=1&orglength=277","timestamp":"2024-11-05T06:13:01Z","content_type":"text/html","content_length":"54134","record_id":"<urn:uuid:2a05dc15-7c44-4e1a-a7db-82b81af6b676>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00095.warc.gz"} |
Integrated Knowledge Solutions
On July 17, a new family of AI models, LLaMA 2 was announced by Meta. LLaMA 2 is trained on a mix of publicly available data. According to Meta LLaMA 2 performs significantly better than the previous
generation of LLaMA models. Two flavors of the model: LLaMA 2 and LLaMA 2-Chat, a model fine tuned for two-way conversations, were released. Each flavor further has three versions with the parameters
ranging from 7 billions to 70 billions. Meta is also freely releasing the code and data behind the model for researchers to build upon and improve the technology.
There are several ways to access LLaMA 2 for development work; you can download it from HuggingFace or access it via Microsoft Azure or Amazon SageMaker. For those interested in interacting with the
LLaMA 2-Chat version, you can do so by visiting llama2.ai, a chatbot model demo hosted by the venture capitalist Andreessen Horowitz. This is the route I took to interact with LLaMA 2-Chat.
Since I was reading an excellent paper on symbolic regression, I decided to query LLaMA 2-Chat about this topic. Before I show my chat with the model, let me explain symbolic regression if you are
not familiar with it. In the traditional linear regression, the model form, linear or polynomial etc., is assumed and the coefficients/parameters of the model are determined to get the best possible
accuracy. In contrast, the symbolic regression involves searching a space of analytical expressions with the corresponding parameter values to best model a given dataset.
I started off by asking if LLaMA-2 Chat is better than GPT-4. I followed it up by asking about symbolic regression as shown below.
The answer provided was not specific. So I asked LLaMA 2 for a concrete example. This resulted in the conversation shown below.
Clearly, the example provided is that of linear regression and not of symbolic regression. Pointing this out to LLaMA 2 resulted in the following conversation, where again I had to point out that
symbolic regression searching for different functions.
As you can see, LLaMA 2 had difficulty explaining symbolic regression and needed to be prompted for making mistakes. Next, I decided to go to ChatGPT to see what kind of response it would produce.
Below is the ChatGPT output.
As you can see, ChatGPT was clear in explaining symbolic regression and even mentioned about the use of genetic algorithms and genetic programming that are key to symbolic regression.
So my take is to stick with Chat-GPT for getting help on topics of interest. LLaMA 2 is lacking in providing clear explanations. Of course, my take is based only on conversation about one topic only.
Pre-trained large language models (LLMs) are being used for numerous natural language processing applications. These models perform well out of the box and are fine-tuned for any desired down-stream
application. However, fine-tuning these models to adapt to specific tasks often poses challenges due to their large parameter sizes. To address this, a technique called Low Rank Adaptation (LoRA) has
emerged, enabling efficient fine-tuning of LLMs. In this post, we will try to understand LoRA, and delve into its importance and application in fine-tuning LLMs. We will begin our journey by first
looking at the concept of rank of a matrix, followed by a look at matrix factorization, and then to LoRA.
Rank of a Matrix
The rank of a matrix indicates the number of independent rows or column in the matrix. As an example, consider the following 4x4 matrix A:
A = [[2, 4, 6, 8], [1, 3, 5, 7], [4, 8, 12, 16], [3, 9, 15, 21]]
Looking at the first and third row of this matrix, we see that the third row is just a scale up version of the first row by a factor of 2. The same is true for the second and fourth rows. Thus, the
rank of matrix A is 2 as there are only two independent rows.
The rank of a matrix of size mxn cannot be greater than min{m,n}. In other words, the rank of a matrix cannot be greater than the smallest dimension of the matrix. We say a matrix is a full rank
matrix if its rank equals the largest possible rank for that matrix.
When a matrix is not a full rank matrix, it tells us that the underlying matrix has some redundancy in it that can be exploited for data compression or dimensionality reduction. This is done by
obtaining a low-rank approximation of the matrix. The process of obtaining a low-rank approximation of a matrix is involves matrix factorization. Some of these factorization methods are briefly
described below.
Matrix Factorization
Matrix factorization is the process of decomposing a matrix into multiple factors. Some of the matrix factorization are:
1. Singular Value Decomposition (SVD)
In SVD, a real-valued matrix A of size m x n is factorized as $ A = UDV^t$, where 𝐔 is an orthogonal matrix of size m x m of left singular vectors and 𝐕 is an orthogonal matrix of size n x n of right
singular vectors. The matrix 𝐃 is a diagonal matrix of size m x n of singular values. A low rank approximation to matrix A of rank r is obtained by using only a subset of singular values and the
corresponding left and right singular vectors as given by the following expression. In other words, the approximation is obtained by the weighted sum of rank one matrices.
$ \hat{ \bf A} = \sum\limits_{j=1}\limits^{k} d_{jj}\bf U_j\bf V^t,\text{ }k\leq r$
SVD is a popular matrix factorization method that is commonly used for data compression and dimensionality reduction. It has also been used for compressing convolutional neural networks. You can read
more about SVD and its use for compression at this blog post.
2. Principal Component Analysis (PCA)
PCA aims to find the principal components that capture the most significant variance in the data. It works with data matrices that have been normalized to have zero mean. Let's say $X$ of m rows and
n columns is one such data matrix where each row represents an observation vector of n features. PCA computes the eigenvalues and eigenvectors of the covariance matrix $C = \frac{1}{(1-n)}XX^t$ by
factorizing it as $\frac{1}{(1-n)}WD^tW$, where $W$ is an orthogonal matrix of eigenvectors and $D$ is the diagonal matrix of eigenvalues. PCA is a popular technique for dimensionality reduction.
3. Non-Negative Matrix Factorization (NMF)
NMF is another technique for obtaining low rank representation of matrices with non-negative or positive elements. Given a data matrix $A$ of m rows and n columns with each and every element $a_{ij}
≥ 0$, NMF seeks matrices $W$ and $H$ of size m rows and k columns, and k rows and n columns, respectively, such that $A≈WH$, and every element of matrices $W$ and $H$ is either zero or positive. The
value of k is set by the user and is required to be equal or less than the smallest of m and n. The matrix $W$ is generally called the basis matrix, and $H$ is known as expansion or coefficient
matrix. The underlying idea of this terminology is that a given data matrix $A$ can be expressed in terms of summation of k basis vectors (columns of $W$) multiplied by the corresponding coefficients
(columns of $H$). Compared to SVD, the NMF based factorization offers a better interpretation of the original data matrix as it is represented/approximated as a sum of positive matrices/vectors. NMF
has been used to perform document clustering, making recommendations, visual pattern recognition such as face recognition, gene expression analysis, feature extraction, source separation etc.
Basically, it can be used in any application where data matrix $A$ has no negative elements. You can read more about NMF at this blog post.
Low Rank Adaptation (LoRA) of Large Language Models
The first thing to note is that LoRA doesn't perform a low rank approximation of the weight or parameter matrix; it rather modifies it by generating a new low rank matrix that captures the needed
parameter changes as a result of the fine tuning the LLM. The pre-trained matrix $W$ is frozen while fine tuning and the weight changes are captured in a delta weight matrix $\Delta W$ through
gradient learning. The delta weight change matrix is a low rank matrix which is set as a product of two small matrices, i.e. $\Delta W = AB$. The $A$ matrix is initialized with values coming from a
gaussian distribution while $B$ matrix is initialized with elements all equal to zero. This ensures that the pre-trained weights matrix is the only contributing matrix at the start of fine tuning.
The figure below illustrates this setup for LoRA.
LoRA Scheme: Matrix W is kept fixed and only A and B are trained.
Let's now try to understand the reasoning behind LoRA and its advantages. The main motivation is that the pretrained models are over-parameterized with low intrinsic dimensionality. Further, the
authors of LoRA hypothesize that change in weights during model fine tuning also has a low intrinsic rank. Thus, it is suffice to use a low rank matrix to capture the weight changes during fine
tuning. LoRA offers several advantages. First, it is possible to share the pretrained model for several downstream tasks with each task having its own LoRA model. This obviously saves storage needs
as well as makes task switching easier. Second, LoRA makes the LLMs adaptation for different tasks easier and efficient. Third, it is easy to combine with other fine tuning methods, if desired. As an
example of the parameter efficiency of LoRA, consider the pretrained matrix of size 200x400. To perform adaptation, let matrix $A$ be of size 200x8 and matrix $B$ be of size 8x400 giving rise to the
delta weight change matrix of the desired size of 200x400. The number of parameters thus needed by LoRA is only 200*8+8*400 = 4800 as compared to the number of parameter, 200*400 = 80000, needed to
adjust without LoRA.
An important consideration in using LoRA is the choice of the rank of the $\Delta W$ matrix. Choosing a smaller rank leads to a simpler low-rank matrix, which results in fewer parameters to learn
during adaptation. However, the adaptation with a smaller rank $\Delta W$ may not lead to the desired performance. Thus, the rank choice offers a tradeoff that typically requires experimentation to
get the best adaptation.
LoRA in PEFT
PEFT stands for a general parameter-efficient fine-tuning library from Huggins Face that includes LoRA as one of its techniques. The few lines of codes below illustrate its basic use.
from transformers import AutoModelForSeq2SeqLM
from peft import get_peft_config, get_peft_model, LoraConfig, TaskType
model_name_or_path = "bigscience/mt0-large"
tokenizer_name_or_path = "bigscience/mt0-large"
peft_config = LoraConfig(
task_type=TaskType.SEQ_2_SEQ_LM, inference_mode=False, r=8, lora_alpha=32, lora_dropout=0.1
model = AutoModelForSeq2SeqLM.from_pretrained(model_name_or_path)
model = get_peft_model(model, peft_config)
# output: trainable params: 2359296 || all params: 1231940608 || trainable%: 0.19151053100118282
In the above example, mt0-large model is being fine tuned for a sequence to sequence conversion task. The rank of the delta weight change is specified as 8. The model has 1.2 B parameters but LoRA
needs only 2.36M parameters, 19% of the total parameters, to train. If we are to change the rank to 12, the number of trainable parameters increases to 3538944, 28.7% of the total parameters.
Clearly, the choice of rank is an important consideration when using LoRA.
LoRA's performance has been evaluated against full fine tuning and other efficient techniques for parameter computation. LoRA has been found to generally outperforms other efficient fine tuning
techniques by a significant margin while yielding comparable or better performance than full fine tuning.
To wrap up, LoRA is an efficient technique for fine tuning large pretrained models. It is poised to play an important role in fine tuning and customizing LLMs for numerous applications.
It would be my pleasure to hear your comments/suggestions to make this site more interesting. | {"url":"https://www.iksinc.tech/2023/07/","timestamp":"2024-11-05T04:23:00Z","content_type":"application/xhtml+xml","content_length":"85056","record_id":"<urn:uuid:8d84768e-1671-4aa6-a26d-2a2ffdd60468>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00663.warc.gz"} |
Digital Math Resources
Display Title
Math Example--Measures of Central Tendency--Mode: Example 37
Math Example--Measures of Central Tendency--Mode: Example 37
Measures of Central Tendency
This example showcases a situation of measures of central tendency, where the goal is to identify a key summary measure in a set of data. An example of finding the mode in a set of numbers is
presented. The numbers are listed, sorted, and analyzed to determine if there is a mode. The result is "No Mode." This example reinforces the concept that not all data sets have a mode, particularly
when each number in the set appears only once.
Measures of Central Tendency lessons are instrumental in providing students with a better understanding of how to interpret data through these examples. Each example highlights distinct scenarios
which reinforce the concept of determining frequency of occurrences within given sets, enhancing students' analytical skills.
Seeing multiple worked-out examples is crucial in solidifying a student's grasp on a concept. Each example contributes unique perspectives and challenges that can arise when thinking about data sets.
This varied approach not only caters to diverse learning styles but also ensures that all students can see the relevance of these concepts in their learning journey.
Teacher's Script
Let's examine this intriguing example. We have the following set of numbers: 49, 50, 18, 47, 29, 33, 27, 23, 34, 43, 17, 10, 15, and 22. Our task is to find the mode. Remember, the mode is the value
that appears most frequently in a data set. Let's start by arranging these numbers from least to greatest. Now, look carefully at our sorted list. Do you notice any numbers that appear more than
once? That's right, each number appears only once. What does this mean for our mode? Exactly, we have no mode in this data set. This is an important lesson because it shows us that not every data set
will have a mode. In real-world data, this could indicate a wide variety of values with no clear 'most common' value. For instance, if these numbers represented the ages of people in a small group,
having no mode would suggest a diverse age range with no particular age being more common than others.
For a complete collection of math examples related to Measures of Central Tendency click on this link: Math Examples: Measures of Central Tendency: Mode Collection.
Common Core Standards CCSS.MATH.CONTENT.6.SP.B.4, CCSS.MATH.CONTENT.6.SP.A.3, CCSS.MATH.CONTENT.HSS.ID.A.2, CCSS.MATH.CONTENT.HSS.ID.A.3
Grade Range 6 - 12
Curriculum Nodes • Probability and Data Analysis
• Data Analysis
Copyright Year 2014
Keywords data analysis, tutorials, measures of central tendency, mode, average | {"url":"https://www.media4math.com/library/math-example-measures-central-tendency-mode-example-37","timestamp":"2024-11-07T12:02:38Z","content_type":"text/html","content_length":"52578","record_id":"<urn:uuid:7bcbda98-314e-45ca-9c6b-4e5b85a401fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00643.warc.gz"} |
Videos - How Far? - OCR (A) Chemistry A-level - PMT
How Far? Videos
Joel F.
University of Nottingham - PhD Pharmacy
Expert PhD Educator and Researcher Dedicated to Student Achievement
£70 / hour
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January mocks on the horizon? Kick-start your revision with our 2-day online Mock Preparation Courses. Chemistry AQA and OCR (A) - 22-23rd December. Check them out now!
The videos below are from the YouTube channels MaChemGuy and Mr C Dunkley. Subscribe to keep up to date with the latest videos.
Overview: How Far?
This is an overview of Topic 22: “How Far?”. It includes typical Kc and Kp questions and units.
The Equilibrium Constant Kc
This video introduces the equilibrium constant Kc and uses worked examples to aid explanations.
Kc Calculations 1
This video demonstrates how to approach a typical exam style question using Kc.
Kc Calculations 2
This video demonstrates how to approach a typical exam style question using Kc.
Kc for Heterogeneous Equilibria
This video shows how to calculate Kc for heterogeneous equilibria and provides a typical question.
Explain in terms of Kc
This video explains how to answer typical questions asking about the composition of an equilibrium mixture when a factor is changed in terms of Kc.
The Equilibrium Constant Kp
Video introducing Kp equilibrium constant and explains how the pressures of components of gaseous equilibria can be used to calculate an equilibrium constant.
Kp Calculations involving Total Pressure
This video demonstrates how to calculate partial pressures from total pressure using examples.
Kp Calculations involving Mole Fractions
This video demonstrates how to calculate mole fractions and partial pressure and provides worked calculations for Kp.
Exam Paper Kp Calculation
Video looking at a typical exam question of Kp calculation. | {"url":"https://www.physicsandmathstutor.com/chemistry-revision/a-level-ocr-a/module-5/how-far-videos/","timestamp":"2024-11-05T10:33:27Z","content_type":"text/html","content_length":"96037","record_id":"<urn:uuid:819ff7a8-2cbe-495e-9010-94fa02010540>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00740.warc.gz"} |
How mathematicians connected with physicists
ProfessorDavid R. Morrison just posted
Geometry and Physics: An Overview
We present some episodes from the history of interactions between geometry and physics over the past century.
He says "we", but he is the sole author.
In 1954, during the era of minimal communication between mathematics and theoretical physics, C. N. Yang and R. L. Mills [YM54] introduced gauge transformations consisting of locally varying
symmetries taking values in a compact Lie group3 G, and studied physical theories which are invariant under such gauge transformations. These generalized the already-familiar abelian gauge
transformations from electromagnetism - the same ones we encountered in Section 1 - for which G = U(1). These gauge theories (or "Yang-Mills theories") eventually became the basis of the Standard
Model of particle physics, the formulation of which was finalized in the mid 1970s using the group4 G = (SU(3) x SU(2) x U(1))/Z6.
In the late 1960s and early 1970s, Yang got acquainted with James Simons, then the mathematics department chair at SUNY Stony Brook where Yang was a professor of physics. In the course of their
conversations,5 Yang and Simons came to recognize that there were important similarities between formulas which were showing up in Yang's work, and formulas which appeared in parts of mathematics
which Simons was familiar with. Simons identified the relevant mathematics as the mathematical theory of connections on fiber bundles, and recommended that Yang consult Steenrod's foundational
book on the subject [Ste51] (which coincidentally was published just a few years prior to the work of Yang and Mills). Yang found the book difficult to read, but through further discussions with
Simons and other mathematicians (including S.-S. Chern) he came to appreciate the power of the mathematical tools which fiber bundle theory offered. ...
Simons communicated these newly uncovered connections with physics to Isadore Singer at MIT who in turn discussed them with Michael Atiyah of Cambridge University. It is likely that similar
observations were made independently by others.
I have heard this story directly from Singer, Chern, and others, but I find it hard to believe.
I believe Chern said that Yang took a differential geometry course from him in China, before that 1954 paper was written. So Yang did not really re-invent gauge theories. Yang was also an ego-maniac,
so maybe he pretended to.
Indeed, the names "
gauge theory
" and "gauge transformation" date back to some mathematical physics by
Hermann Weyl
in 1918. Wikipedia says that Pauli popularized the first widely accepted gauge theory in 1941. Some of the main ideas seem to have been published as early as 1914.
At some point it must have been obvious that special relativity could be elegantly described as spacetime with the metric dx
+ dy
+ dz
- dt
, with electromagnetism being a
on a circle (S
) bundle. But I cannot find who explicitly said this first.
Weyl was very close to saying this in 1919, and so was
, also in 1919. So the idea that this formulation was only discovered in the 1970s is absurd.
One possible explanation is that mathematicians did not realize the importance of bundles, not derived from tangent spaces, until the 1950s. Also, mathematicians quit talking to physicists around
1950. So mathematicians had an intuitive understanding in the 1920s, but would not have expressed it in terms of bundles until the 1950s, and physicists never learned it. I am not sure it ever made
it into physics textbooks until recently.
At any rate, the
standard model
of particle physics is based on replacing the circle with SU(3)xSU(2)xU(1), and using the same geometric formalism. So you would think that physicists would think that the geometric interpretation of
electromagnetism would be fundamental and important enough for elementary textbooks.
While Morrison's paper has many examples of mathematical advances related to geometry and physics, the gauge theory of the standard model is the only one that involves genuine physics. The others
could be more accurately described as mathematics that was partially inspired by physics, but which does not actually apply to any physical situation.
1 comment:
1. Call me arrogant but isn't it rather easy to reinvent such gauge theories? See this class where the intersection of analysis, algebra and geometry (differential geometry) looks like the third eye
of mickey mouse. Ahhh physics... when my brain needs some easy thoughts. | {"url":"https://blog.darkbuzz.com/2018/06/how-mathematicians-connected-with.html","timestamp":"2024-11-09T20:31:32Z","content_type":"text/html","content_length":"111953","record_id":"<urn:uuid:73c9aeb2-759f-4378-8fab-6f0f65791165>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00195.warc.gz"} |
The Worst Prediction in the Whole of Physics » Vlatko Vedral
The Worst Prediction in the Whole of Physics
Published: October 18, 2023
One of the most amazing consequences of quantization of the electromagnetic field is the appearance of the so called zero-point energy or the quantum vacuum. This is a direct consequence
(mathematically speaking) of the fact that – in quantum mechanics – the electric and the magnetic fields become q-numbers and, unlike in classical physics, they can no longer be specified
simultaneously (they do not commute). The energy of the field equals the sum of the squares of the electric and the magnetic fields, but – because they are now q-numbers – even in the lowest energy
state we cannot make this sum vanish (if we could, we would be able to specify the values of both electric and the magnetic field, which would violate the fact that they are q-numbers; in other
words, if the energy of the quantum vacuum was zero, this fact alone would violate the Heisenberg Uncertainty Relations since it would mean that both the electric and the magnetic field are
simultaneously zero!).
I’ve written about the reality of the quantum vacuum before (for which we have indirect evidence in the spontaneous emission of radiation by matter, in the Lamb shift, the Casimir effect and so on),
however, I would now like to talk about one of its paradoxical consequences. We don’t understand this one, and it could signal some kind of trouble in the foundations of quantum physics (or field
theory or both).
To a good approximation, we can think of our universe as a box filled with the quantum vacuum. Each frequency that can occur within the universe (the lowest frequency being inversely proportional to
the size of the universe and the highest being inversely proportional to the smallest possible length in the universe – let’s assume this is the so-called Planck length) gets one half of Planck’s
constant times the frequency worth of energy in the vacuum.
Now in quantum electrodynamics, this enormous amount of energy is not a problem because only the energy differences are relevant. In fact, in many applications we can ignore the vacuum energy and set
it equal to zero without any consequences (though the vacuum state will still play a role).
But there is one force of nature that unfortunately ought to see this vacuum energy in all its totality, absolutely speaking, and not just the differences between energies. Enter gravity.
According to General Relativity, our best theory of gravity, the total energy of the universe does actually affect gravity. Even an otherwise empty universe would, according to quantum mechanics,
produce an enormous amount of energy that would gravitate strongly. In the equations of general relativity, this vacuum energy sometimes goes by the name of the cosmological constant.
Our best astronomical observations (to do with the rate of the expansion of the universe) are telling us that the cosmological constant is tiny. It is the energy equivalent of having a single
Hydrogen atom on average in every cubic meter of the universe! However, if we calculate the quantum vacuum energy of the universe it turns out to be 120 orders of magnitude bigger. Hence: the worst
prediction in the whole of physics.
Now, of course, there are a number of ways out of this problem.
One is simply to say that our quantum calculation is not correct. We only added up the electromagnetic contribution (i.e. due to the bosonic fields), however, when we add up all fields, various
contribution might just cancel out (or almost cancel out). The logic here could be that the fermionic fields (e.g., electrons) produce the same vacuum energy but of the opposite sign to the bosonic
Another way out is to deny the connection between the cosmological constant and the vacuum energy. Perhaps the cosmological constant is due to something else, say the total mass in the universe.
Yet another way of resolving the apparent paradox is to deny the reality of this quantum vacuum energy. Maybe this energy does not exist, or, maybe, even if it did, it has no gravitational influence
whatsoever. Some physics heavyweights like Pauli and Schwinger thought this way.
The most interesting possibility, I think, would be if this discrepancy really signalled that there was a fundamental problem with quantum mechanics itself. Perhaps just as classical physics
predicted an infinite amount of energy radiated by a black body (clearly not true in practice), the huge amount of energy that quantum physics predicts in the vacuum (which, incidentally, would also
be infinite if there was no smallest length, a question that we have no answer to at present) could also mean the breakdown of quantum physics.
So, could the “worst” prediction in physics give us a clue about how to “fix” quantum physics? Perhaps, but given the success that quantum physics has enjoyed in the microscopic world of atoms and
photons and subatomic particles and molecules, it is difficult to see how one ought to modify it while preserving all the good explanations it has provided us with so far.
The key to fixing classical physics was the introduction of q-numbers, namely the fact that quantities such as position, momentum, energy and so on, are no longer just represented by the ordinary
(real) c-number, but ought to be represented by more complex objects (matrices). Briefly, this resolved the black body issues since first emitting light and then absorbing it, now, in quantum
physics, has a different probability to first absorbing light and then emitting it (in the technical jargon, the two processes no longer commute, just as the product of two matrices generally
doesn’t). Writing a detailed balance between matter and light then gives us the famous quantum Planck distribution (which is the correct formula for the black body).
So, should we be upgrading q-numbers to some other entities, say the w(eird)-numbers? My personal feeling is that it might not be that quantum physics is at fault here. It could be that we should be
looking into modifying General Relativity instead. But I don’t mean quantizing it (yes, we have to do that too, however, I don’t think this will solve the problem of the cosmological constant). I
mean that perhaps gravity is not what we understand it at present to be (curvature of spacetime).
One can only speculate on how gravity ought to be modified. Wheeler succinctly described general relativity by saying that “matter tells spacetime how to curve, and spacetime tells matter how to
move”. Maybe there is not such thing as spacetime (it is all about relative positions of masses). Maybe there is no such thing as matter either…I mean at the fundamental level, these are not the
right “elements of reality”.
In this spirit, there is an interesting idea, called induced gravity, which is the idea that gravity is not really a fundamental force, but is the result of all the quantum vacuum fluctuations. So,
it’s not that the vacuum contributes on top of gravity, but it is, in fact, gravity in toto. This is very interesting; however, it has not really been developed properly. For instance, suppose we
have a mass in a superposition of two different places at the same time, what kind of an (induced) gravitational field does this produce?
Also, and this is related to something I’ve written about before (the BMV effect), what kind of dynamics do we have if two masses are each superposed and then coupled by induced gravity. Can such
gravity entangle the two masses? I hope the answer is yes, since – otherwise, the whole idea of gravity induced by quantum mechanics might be suspicious. But luckily, our experiments are getting
closer and closer to being able to test this and, as I frequently say in my blogs, now is certainly a great time to be a physicist!
Sign up to my substack if you'd like to have my articles delivered straight to your inbox
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Find Nearby Stores For Copper Sulfate
Find Nearby Stores for Copper Sulfate
Welcome to Warren Institute! In our latest article, we will be exploring the availability of copper sulfate near you. Whether you're a teacher looking for a reliable source for your classroom
experiments or a student needing it for a science project, finding the right supplier is crucial. We will provide you with valuable insights and tips on where to buy copper sulfate locally. From
agricultural supply stores to online retailers, we'll guide you through the different options and help you make an informed decision. Stay tuned for our comprehensive guide on purchasing copper
The Importance of Copper Sulfate in Mathematics Education
In this section, we will discuss why copper sulfate is a valuable tool for mathematics education.
Copper sulfate is commonly used in educational settings to demonstrate various mathematical concepts and principles. Its vibrant blue color makes it visually appealing and helps engage students'
interest. Additionally, copper sulfate can be used to create crystal structures, which can aid in teaching geometry and spatial reasoning.
• Enhances visual learning: The distinct color of copper sulfate allows students to easily observe and analyze mathematical patterns or relationships.
• Promotes hands-on learning: Working with copper sulfate crystals provides students with a tangible and interactive experience, promoting a deeper understanding of mathematical concepts.
• Reinforces geometry skills: The crystal structures formed by copper sulfate can be used to explore geometrical shapes, symmetry, and three-dimensional figures.
Where to Find Copper Sulfate for Mathematics Education
This section will guide you on where to buy copper sulfate for mathematics education purposes.
Local scientific supply stores: Many cities have scientific supply stores that cater to educational institutions and individuals interested in conducting experiments. These stores often carry copper
sulfate in various forms, such as crystals or powders.
Online marketplaces: Websites like Amazon, eBay, or educational resource platforms offer a wide range of products, including copper sulfate specifically intended for educational use. It is essential
to verify the credibility of the seller and read customer reviews before making a purchase.
Chemical suppliers: Some chemical suppliers specialize in providing laboratory-grade materials to educational institutions. Contact local chemical suppliers or check their websites to inquire about
the availability of copper sulfate.
Safety Considerations when Using Copper Sulfate in Mathematics Education
In this section, we will discuss important safety guidelines to follow when using copper sulfate for mathematics education.
Proper handling: When working with copper sulfate, it is crucial to wear appropriate personal protective equipment (PPE), such as gloves and safety goggles. Avoid direct contact with the skin, eyes,
or mouth and wash hands thoroughly after handling.
Storage: Store copper sulfate in a secure container, away from children and pets. Keep it in a cool, dry place to prevent moisture absorption or accidental spills.
Disposal: Dispose of copper sulfate according to local regulations and guidelines. Do not pour it down the drain or dispose of it in regular trash.
Alternatives to Copper Sulfate in Mathematics Education
This section will explore alternative materials that can be used in mathematics education if copper sulfate is not readily available.
Food coloring or watercolor paints: These can be used to create visually appealing representations of mathematical concepts. For example, different colors can be used to show patterns or highlight
specific areas in a graph.
Colored construction paper or cardstock: These materials can be cut into shapes and used for hands-on geometry activities or creating visual aids for mathematical concepts.
Virtual simulations and software: Various educational software or online platforms provide virtual simulations that allow students to explore mathematical concepts in a digital environment. These can
be used as an alternative when physical materials are not accessible.
Remember, while copper sulfate has its advantages, there are always alternative methods and materials available to facilitate mathematics education effectively.
frequently asked questions
How can I incorporate the concept of copper sulfate into my math lesson plan?
One way to incorporate the concept of copper sulfate into a math lesson plan is by using it as a real-life example during a discussion on percentages and ratios. For example, you can introduce the
idea of copper sulfate as a common ingredient in fertilizers and ask students to calculate the percentage of copper sulfate in a given fertilizer mixture. This not only reinforces their understanding
of percentages and ratios but also connects math to practical applications in agriculture or other relevant fields.
Are there any math activities or experiments involving copper sulfate that I can do with my students?
Yes, there are several math activities or experiments involving copper sulfate that you can do with your students. One example is to have them measure the mass of a given amount of copper sulfate and
then calculate its molar mass using the periodic table. Another activity could involve determining the concentration of a copper sulfate solution by performing titrations. These hands-on experiments
not only reinforce mathematical concepts such as measurement and calculation, but also allow students to observe chemical reactions and apply their knowledge in a practical setting.
Can you recommend any educational resources or websites where I can find information on purchasing copper sulfate for math education purposes?
Yes, you can find information on purchasing copper sulfate for math education purposes on websites such as Amazon, eBay, or educational supply stores like Carolina Biological Supply Company.
Is copper sulfate commonly used in mathematics education, and if so, how can I obtain it for my classroom?
No, copper sulfate is not commonly used in mathematics education. It is primarily used in chemistry experiments and as a pesticide. If you are looking for materials to enhance your mathematics
classroom, there are many other resources and manipulatives available that are specifically designed for teaching mathematical concepts.
Where can I find a local supplier or store that sells copper sulfate for math teaching?
You can find a local supplier or store that sells copper sulfate for math teaching at agricultural supply stores or online chemical suppliers.
In conclusion, finding a reliable source to purchase copper sulfate near you is crucial for implementing hands-on activities in mathematics education. By incorporating this versatile compound into
experiments and demonstrations, students can deepen their understanding of key mathematical concepts while also developing critical thinking and analytical skills. Whether it be exploring geometric
shapes or investigating the properties of different materials, copper sulfate offers educators a valuable tool to engage students in interactive learning experiences. So, don't hesitate to research
local hardware stores, agricultural supply centers, or online retailers to find the best options available. Remember, access to quality resources is key to fostering a stimulating and enriching
mathematics education environment.
If you want to know other articles similar to Find Nearby Stores for Copper Sulfate you can visit the category General Education. | {"url":"https://warreninstitute.org/where-can-i-buy-copper-sulfate-near-me/","timestamp":"2024-11-05T23:05:56Z","content_type":"text/html","content_length":"105567","record_id":"<urn:uuid:1d1f07fa-812d-4237-a461-6c9aa0fbd5c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00064.warc.gz"} |
Mathematical methods in physics
Code: 201726
ECTS: 5.0
Lecturers in charge: izv. prof. dr. sc. Marko Erceg
Lecturers: izv. prof. dr. sc. Marko Erceg - Exercises
1. komponenta
Lecture type Total
Lectures 45
Exercises 30
* Load is given in academic hour
(1 academic hour = 45
COURSE AIMS AND OBJECTIVES:
Course goals are to acquire
theoretical and practical
knowledge in the theory of
ordinary and partial
differential equations.
COURSE DESCRIPTION AND
1. Ordinary differential
2. Linear differential
equations. Linear
differential equations of the
first order
3. Existence and uniqueness
theorems for the Cauchy
problem for the homogeneous
linear equation of n-th order
4. Linear independence of
functions and the Wronskian
5. Linear differential
equation with constant
6. Nonhomogeneous
differential equations. The
method of undetermined
coefficients. The method of
Variation of Parameters.
7. Solving differential
equations by power series.
8. Second order linear
differential equation with
regular singularities
9. Legendre polynomials and
Legendre differential
equation. A Generating
Function for Legendre
10. The associated Legendre
functions. Spherical
11. Laplace's equation. The
method of separation of
12. Wave equation
13. Bessel functions and
Bessel differential equation
14. Schrodinger equation.
Laguerre polynomials
Prerequisit for:
Enrollment :
Attended : Differential and
integral calculus 2
Examination :
Passed : Differential and
integral calculus 2
1. komponenta Lecture typeTotal Lectures 45 Exercises 30 * Load is given in academic hour (1 academic hour = 45 minutes)
COURSE AIMS AND OBJECTIVES: Course goals are to acquire theoretical and practical knowledge in the theory of ordinary and partial differential equations. COURSE DESCRIPTION AND SYLLABUS: 1. Ordinary
differential equations 2. Linear differential equations. Linear differential equations of the first order 3. Existence and uniqueness theorems for the Cauchy problem for the homogeneous linear
equation of n-th order 4. Linear independence of functions and the Wronskian 5. Linear differential equation with constant coefficients 6. Nonhomogeneous differential equations. The method of
undetermined coefficients. The method of Variation of Parameters. 7. Solving differential equations by power series. 8. Second order linear differential equation with regular singularities 9.
Legendre polynomials and Legendre differential equation. A Generating Function for Legendre Polynomials. 10. The associated Legendre functions. Spherical harmonics 11. Laplace's equation. The method
of separation of variables 12. Wave equation 13. Bessel functions and Bessel differential equation 14. Schrodinger equation. Laguerre polynomials
Enrollment : Attended : Differential and integral calculus 2 Examination : Passed : Differential and integral calculus 2
4. semester
Izborni matematički predmet 2 - Regular study - Mathematics and Physics Education
Izborni matematički predmet 2 - Regular study - Mathematics and Physics Education
All notices and course related materials will be available on course web page. | {"url":"https://www.chem.pmf.hr/math/en/course/mmip","timestamp":"2024-11-06T01:58:20Z","content_type":"text/html","content_length":"81429","record_id":"<urn:uuid:3f50020b-4c1c-426b-bbe4-d784d30f17e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00206.warc.gz"} |
17 research outputs found
We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized
maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the optimal strategy can be calculated without knowing the time horizon in advance.
They put forward the question what families have exchangeable SNML strategies. This paper fully answers this open problem for one-dimensional exponential families. The exchangeability can happen only
for three classes of natural exponential family distributions, namely the Gaussian, Gamma, and the Tweedie exponential family of order 3/2. Keywords: SNML Exchangeability, Exponential Family, Online
Learning, Logarithmic Loss, Bayesian Strategy, Jeffreys Prior, Fisher Information1Comment: 23 page
Web crawling is the problem of keeping a cache of webpages fresh, i.e., having the most recent copy available when a page is requested. This problem is usually coupled with the natural restriction
that the bandwidth available to the web crawler is limited. The corresponding optimization problem was solved optimally by Azar et al. [2018] under the assumption that, for each webpage, both the
elapsed time between two changes and the elapsed time between two requests follow a Poisson distribution with known parameters. In this paper, we study the same control problem but under the
assumption that the change rates are unknown a priori, and thus we need to estimate them in an online fashion using only partial observations (i.e., single-bit signals indicating whether the page has
changed since the last refresh). As a point of departure, we characterise the conditions under which one can solve the problem with such partial observability. Next, we propose a practical estimator
and compute confidence intervals for it in terms of the elapsed time between the observations. Finally, we show that the explore-and-commit algorithm achieves an $\mathcal{O}(\sqrt{T})$ regret with a
carefully chosen exploration horizon. Our simulation study shows that our online policy scales well and achieves close to optimal performance for a wide range of the parameters.Comment: Published at
AAAI 202
Online portfolio selection has received much attention in the COLT community since its introduction by Cover, but all state-of-the-art methods fall short in at least one of the following ways: they
are either i) computationally infeasible; or ii) they do not guarantee optimal regret; or iii) they assume the gradients are bounded, which is unnecessary and cannot be guaranteed. We are interested
in a natural follow-the-regularized-leader (FTRL) approach based on the log barrier regularizer, which is computationally feasible. The open problem we put before the community is to formally prove
whether this approach achieves the optimal regret. Resolving this question will likely lead to new techniques to analyse FTRL algorithms. There are also interesting technical connections to
self-concordance, which has previously been used in the context of bandit convex optimization
Pattern recognition is a central topic in learning theory, with numerous applications such as voice and text recognition, image analysis and computer diagnosis. The statistical setup in
classification is the following: we are given an i.i.d. training set (X1, Y1),...,(Xn, Yn), where X i represents a feature and Y in {0, 1} is a label attached to that feature. The underlying joint
distribution of (X, Y) is unknown, but we can learn about it from the training set, and we aim at devising low error classifiers f: X→Y used to predict the label of new incoming features. In this
paper, we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown mixed qubit states. Given a number of 'training' copies from each of the states, we would like
to 'learn' about them by performing a measurement on the training set. The outcome is then used to design measurements for the classification of future systems with unknown labels. We found the
asymptotically optimal classification strategy and show that typically it performs strictly better than a plug-in strategy, which consists of estimating the states separately and then discriminating
between them using the Helstrom measurement. The figure of merit is given by the excess risk equal to the difference between the probability of error and the probability of error of the optimal
measurement for known states. We show that the excess risk scales as n^{–1} and compute the exact constant of the rate
We analyze the regret, measured in terms of log loss, of the maximum likelihood (ML) sequential prediction strategy. This “follow the leader ” strategy also defines one of the main versions of
Minimum Description Length model selection. We proved in prior work for single parameter exponential family models that (a) in the misspecified case, the redundancy of follow-the-leader is not 1 2
log n+O(1), as it is for other universal prediction strategies; as such, the strategy also yields suboptimal individual sequence regret and inferior model selection performance; and (b) that in
general it is not possible to achieve the optimal redundancy when predictions are constrained to the distributions in the considered model. Here we describe a simple “flattening” of the
sequential ML and related predictors, that does achieve the optimal worst case individual sequence regret of (k/2)log n + O(1) for k parameter exponential family models for bounded outcome spaces;
for unbounded spaces, we provide almost-sure results. Simulations show a major improvement of the resulting model selection criterion
The paper considers sequential prediction of individual sequences with log loss (online density estimation) using an exponential family of distributions. We first analyze the regret of the maximum
likelihood ("follow the leader") strategy. We find that this strategy is (1) suboptimal and (2) requires an additional assumption about boundedness of the data sequence. We then show that both
problems can be be addressed by adding the currently predicted outcome to the calculation of the maximum likelihood, followed by normalization of the distribution. The strategy obtained in this way
is known in the literature as the sequential normalized maximum likelihood or last-step minimax strategy. We show for the first time that for general exponential families, the regret is bounded by
the familiar (k/2) log(n) and thus optimal up to O(1). We also show the relationship to the Bayes strategy with Jeffreys' prior
Minimization of the rank loss or, equivalently, maximization of the AUC in bipartite ranking calls for minimizing the number of disagreements between pairs of instances. Since the complexity of this
problem is inherently quadratic in the number of training examples, it is tempting to ask how much is actually lost by minimizing a simple univariate loss function, as done by standard classification
methods, as a surrogate. In this paper, we first note that minimization of 0/1 loss is not an option, as it may yield an arbitrarily high rank loss. We show, however, that better results can be
achieved by means of a weighted (cost-sensitive) version of 0/1 loss. Yet, the real gain is obtained through margin-based loss functions, for which we are able to derive proper bounds, not only for
rank risk but, more importantly, also for rank regret. The paper is completed with an experimental study in which we address specific questions raised by our theoretical analysis
We extend the classical problem of predicting a sequence of outcomes from a finite alphabet to the matrix domain. In this extension, the alphabet of n outcomes is replaced by the set of all dyads,
i.e. outer products uu^T where u is a vector in R^n of unit length. Whereas in the classical case the goal is to learn (i.e. sequentially predict as well as) the best multinomial distribution, in the
matrix case we desire to learn the density matrix that best explains the observed sequence of dyads. We show how popular online algorithms for learning a multinomial distribution can be extended to
learn density matrices. Intuitively, learning the n^2 parameters of a density matrix is much harder than learning the n parameters of a multinomial distribution. Completely surprisingly, we prove
that the worst-case regrets of certain classical algorithms and their matrix generalizations are identical. The reason is that the worstcase sequence of dyads share a common eigensystem, i.e. the
worst case regret is achieved in the classical case. So these matrix algorithms learn the eigenvectors without any regret | {"url":"https://core.ac.uk/search/?q=author%3A(Kotlowski%2C%20Wojciech)","timestamp":"2024-11-03T17:25:15Z","content_type":"text/html","content_length":"121615","record_id":"<urn:uuid:7b4d3378-69bc-4e26-a577-15c2ef4378bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00849.warc.gz"} |
Machine Learning Notes: Convolutional Neural Networks
Convolutional Neural Networks
Say we have a image that is composed of multiple atomic elements, and we want to be able to identifty specific groupings of those elements within a specific image? How could we achieve this through
Machine Learning?
It turns out there is one approach that can achieve this - Convolutional Neural Networks (CNN). These allow us to take a input image, generate feature maps and given a set of parameters build up
combinations of these features to (similarlly to LRs and MLPs), to determine a confidence rate of a specific combination being present in a given image.
Math Model
The below demonstrates how the model is built up, where @M_n, L_n, G_n@ are the feature maps generated by each layer, @\phi_K, \psi_K, \omega_K@ are the parameters for each layer, and @\ell_n@ is the
prediction strength. @@ \displaylines{ M_n = f(I_n; \phi_1, …, \phi_K) \\ L_n = f(M_n; \Psi_1, …,\Psi_K ) \\ G_n = f(L_n; \omega_1, …,\omega_K ) \\ \ell_n = \ell(G_n; W) } @@
How this Model learns?
Assuming we have labelled data: @{I_n, y_n}(n=1, N)@, and those labels are binary @\in \{ +1, -1 \}@.
We can use the energy function of the model parameters: @@ E(\phi, \Psi, \omega, W) = 1/N \sum loss(y_n, \ell_n) @@
Which will generate the loss between the prediction label and the generated prediction for a given image.
The parameters will then need to be estimated by finding @\hat{\phi}, \hat{\Psi}, \hat{\omega}, \hat{W}@ that minimises @E(\phi, \Psi, \omega, W)@.
This is the difficult part, estimation of the parameters can be difficult depending on the amount of parameters, and the dataset can have multiple Local Optimal Solutions (areas in the loss value vs
parameter value dataset where the loss is minimised most).
#ai #course #machine learning #math #notes | {"url":"https://sylvanb.dev/machine-learning-notes-convolutional-neural-networks/","timestamp":"2024-11-03T03:45:58Z","content_type":"text/html","content_length":"10973","record_id":"<urn:uuid:75828f0b-376f-4a2b-978f-aad1cc47c597>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00864.warc.gz"} |
Summation based benchmark for calculators
01-02-2018, 01:39 PM
Post: #61
grsbanks Posts: 1,219
Senior Member Joined: Jan 2017
RE: Summation based test for calculators
(01-02-2018 01:36 PM)pier4r Wrote: Oh. I know that the 12C was a business model but I thought that the basic trig/exp was there. My bad. Removing.
\( e^x \) and \( y^x \) are there, it's just the trig functions that are not
01-03-2018, 02:46 PM
Post: #62
xerxes Posts: 178
Member Joined: Jun 2014
RE: Summation based test for calculators
A record breaking result for your list:
max=10 using a loop
63.5 seconds
Calculator Benchmark
01-03-2018, 04:00 PM
(This post was last modified: 01-03-2018 04:02 PM by pier4r.)
Post: #63
pier4r Posts: 2,248
Senior Member Joined: Nov 2014
RE: Summation based test for calculators
(01-03-2018 02:46 PM)xerxes Wrote: A record breaking result for your list:
max=10 using a loop
63.5 seconds
Thanks for the info that is even slower than me manually (although with a newer calc, the 506w).
And I have even a couple of ideas that I can be, without much stress, even faster with the 506w.
In the previous test I put the value in X, I recalled the formula in F4 and I added the result to M with M+.
I could just use M as increment, The formula in F4 as A+function(M) and then save the result in A. This may be clearly faster than my 47 seconds. Maybe I can break the 40s barrier.
I will try when I have time.
edit: anyone with a 71B ?
Wikis are great, Contribute :)
01-03-2018, 04:05 PM
(This post was last modified: 01-03-2018 10:16 PM by grsbanks.)
Post: #64
grsbanks Posts: 1,219
Senior Member Joined: Jan 2017
RE: Summation based test for calculators
(01-03-2018 04:00 PM)pier4r Wrote: Thanks for the info that is even slower than me manually (although with a newer calc, the 506w).
The TI-53 might be even slower than that. I'll give it a go this evening.
Edit: Nope. The TI-62 is slower. Kind of...
With only 32 (non-merged) steps to play with I couldn't do a loop a fixed number of times so I let it run for about 15 minutes and then stopped it to see how many iterations had been completed. It
was completing them at a rate of one iteration every 5.64 seconds.
01-09-2018, 10:20 PM
Post: #65
pier4r Posts: 2,248
Senior Member Joined: Nov 2014
RE: Summation based test for calculators
Little bump:
still missing
- 71B
- Some sharp PC
- other capable (programmable or with sum function) calculators
Really no one with a 71B? Pretty please! Virtual hugs!
Wikis are great, Contribute :)
01-09-2018, 10:53 PM
Post: #66
TheKaneB Posts: 175
Member Joined: Jul 2014
RE: Summation based benchmark for calculators
Sharp EL-5120 Solver
With 100 iteration it runs for approx. 36 seconds, giving a result of 139.297187 (all digits correct according to my HP Prime).
It lacks proper loop instructions, so I used a manual counter with a IF ... GOTO instruction
X1 = 1
A1 = 0
A1 = A1 + 3 <x-root-of>(e^(sin(atan(X1))))
X1 = X1 + 1
IF X1 <= 100 GOTO A
PRINT A1
Software Failure: Guru Meditation
01-10-2018, 07:52 PM
(This post was last modified: 01-10-2018 08:30 PM by Michael de Estrada.)
Post: #67
Michael de Estrada Posts: 373
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
HP-33C (Spice): N=10, 44 sec, Result=13.71183501
HP-25C (Woodstock): N=10, 29 sec, Result=13.71183501
Again, the older Woodstock is significantly faster than the newer Spice.
01-10-2018, 08:13 PM
Post: #68
pier4r Posts: 2,248
Senior Member Joined: Nov 2014
RE: Summation based benchmark for calculators
I am impressed by the 33C, 34C and the TI equivalents.
A 71B? Anyone?
Wikis are great, Contribute :)
01-10-2018, 08:50 PM
Post: #69
Michael de Estrada Posts: 373
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
I edited my post above to add the HP-25C.
01-11-2018, 08:11 PM
Post: #70
Michael de Estrada Posts: 373
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
N=10, Time= 32 sec, Result= 13.71183501
N=100, Time= 329 sec, Result= 139.2971873
which is 11 sec faster than the HP-67.
01-11-2018, 08:19 PM
(This post was last modified: 01-11-2018 08:20 PM by pier4r.)
Post: #71
pier4r Posts: 2,248
Senior Member Joined: Nov 2014
RE: Summation based benchmark for calculators
Added. Much appreciated!
Someone with a 71B! For the glory!
maybe is it not possible? If I remember correctly the 71B should be basic programmable but maybe I am confused.
Wikis are great, Contribute :)
01-11-2018, 08:30 PM
Post: #72
Maximilian Hohmann Posts: 1,413
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
(01-11-2018 08:19 PM)pier4r Wrote: Someone with a 71B! For the glory!
Which program or formula? There are so many on this tread that I lost count... A 71B would be ready here on my desk!
01-11-2018, 08:42 PM
Post: #73
pier4r Posts: 2,248
Senior Member Joined: Nov 2014
RE: Summation based benchmark for calculators
(01-11-2018 08:30 PM)Maximilian Hohmann Wrote: Which program or formula? There are so many on this tread that I lost count... A 71B would be ready here on my desk!
see the frmula in the middle of the post. It is written in bold "formula to use".
If you want to use a program that is equivalent, you can do it too. I cannot provide you a program though, I do not posses a 71B.
Und Tausend Dank!
Wikis are great, Contribute :)
01-11-2018, 09:16 PM
(This post was last modified: 01-11-2018 09:39 PM by Maximilian Hohmann.)
Post: #74
Maximilian Hohmann Posts: 1,413
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
Allora ... col HP-71B, usando un programma semplicissimo:
5 X=0
10 FOR I=1 TO 1000
20 X=X+(EXP(SIN(ATAN(I))))^(0.333)
30 NEXT I
40 PRINT X
ottengo un risultato in 2 minuti e 58 secondi.
Adesso proverò la stessa cosa con un HP-75D (se riesco a ricordormi come si fa a programmarlo).
Lo stesso programma eseguito dal mio HP-75D ottiene un risultato (uguale!) in 2 primi e 28 secondi.
01-11-2018, 10:03 PM
(This post was last modified: 01-11-2018 10:13 PM by pier4r.)
Post: #75
pier4r Posts: 2,248
Senior Member Joined: Nov 2014
RE: Summation based benchmark for calculators
Woah. Kannst du Italienisch? (Kann ich Sie duzen?)
Auf jedem fall, wenn es möglich ist, lieber auf Deutsch schreiben, dass ich üben muss.
Translated: Woah, can you speak Italian? (Can I use the informal you with You?)
Anyway, if possible, I'd like to write in German, since I must train.
Ergebnisse hinzugefügt. Fast 3 Minuten, oder 2 un halb sind nicht schlecht.
Results added. 3 minutes (almost) or 2 and half are not that bad.
Wikis are great, Contribute :)
01-11-2018, 10:17 PM
(This post was last modified: 01-11-2018 10:50 PM by Maximilian Hohmann.)
Post: #76
Maximilian Hohmann Posts: 1,413
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
(01-11-2018 10:03 PM)pier4r Wrote: Kann ich Sie duzen?
Sure, "
You can say you to me
" as former German chancellor Kohl (allegedly) once said to Margaret Thatcher :-)
And yes, I (still) speak some Italian because I had the big luck to grow up in that beautiful country (all in all I lived there for 20 years and will almost certainly return there when I have to
retire from work).
One of my proud possessions is an unused HP-97 with Italian box and manuals which they threw away in the research facility where my father worked - it still has it's engraved "Euratom" (
) inventory plaque on it's back :-) Unfortunately I don't have it here so I can't run the benchmark on it now.
Edit: In the meantime I ran the exact same BASIC program on a Casio FX-880P. It takes 2 minutes and 38 seconds (for 1000 loops), so almost exactly in the middle of the HP-71 and 75. Mind you, the
Casio came nearly 10 years after the HP-75 and yet it is slower!
01-11-2018, 10:50 PM
Post: #77
Michael de Estrada Posts: 373
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
Los resultados deben estar identicos a los mismos en el HP 67.
01-12-2018, 08:44 AM
(This post was last modified: 01-12-2018 08:51 AM by pier4r.)
Post: #78
pier4r Posts: 2,248
Senior Member Joined: Nov 2014
RE: Summation based benchmark for calculators
Is the hp 97 not the big brother of the hp 67? Shouldn't it be faster?
edit: added the 880P (now it has 2 results)
plus, yes Italy is nice (its landscapes) as mostly every place in the world. I mean even the tundra in Russia is peculiar in some way.
Still I don't like the average mindset there, at least from the regions under Florence/Rome. In the North they were/are a little bit more organized (nonetheless there are some groups in the North
that, well, are a bit backwards. Like people following separatists parties in the North-East or populists).
Wikis are great, Contribute :)
01-12-2018, 08:47 AM
Post: #79
grsbanks Posts: 1,219
Senior Member Joined: Jan 2017
RE: Summation based benchmark for calculators
(01-11-2018 10:50 PM)Michael de Estrada Wrote: Los resultados deben estar identicos a los mismos en el HP 67.
Si. El HP-97 es simplemente un HP-67 con una impresora.
01-18-2018, 09:47 PM
Post: #80
Michael de Estrada Posts: 373
Senior Member Joined: Dec 2013
RE: Summation based benchmark for calculators
HP-32S (Pioneer)
1000 loops —> 206 seconds, Result = 1395.34628770
100 loops —> 23 seconds, Result = 139.29718705
10 loops —> 3 seconds, Result = 13.71183502
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Development of Operational Limit Diagrams For Offshore Lifting Procedures
34th OMAE – ASME International Conference on Ocean, Offshore and Artic Engineering – Canada – Junho 2015
Lifting operations with offshore cranes are fundamental for proper functioning of a platform. Despite the great technological development, offshore cranes load charts only consider the significant
wave height as parameter of environmental load, neglecting wave period, which may lead to unsafe or overestimated lifting operations. This paper aims to develop a method to design offshore crane
operational limit diagrams for lifting of personnel and usual loads, in function of significant wave height and wave peak period, using time domain dynamic analysis, for a crane installed on a
floating unit. The lifting of personnel with crane to transfer between a floating unit and a support vessel is a very used option in offshore operations, and this is in many cases, the only
alternative beyond the helicopter. Due to recent fatal accidents with lifting operations in offshore platforms, it is essential the study about this subject, contributing to the increase of safety.
The sea states for analysis were chosen covering usual significant wave heights and peak periods limits for lifting operations. The methodology used the SITUA / Prosim software to obtain the dynamic
responses of the personnel transfer basket lifting and container loads on a typical FPSO. Through program developed by the author, it was implemented the automatic generation of diagrams as a
function of operational limits. It is concluded that using this methodology, it is possible to achieve greater efficiency in the design and execution of personnel and routine load lifting, increasing
safety and a wider weather window available.
Offshore cranes are present in most of the platforms nowadays, no matter they are fixed or a floating unit, and also in a great variety of construction and supply boats. They are used to lift a wide
range of loads, including food, pipes, containers and personnel.
Offshore cranes are also used to perform lift operations for construction and maintenance aboard the offshore unit.
Because these operations involve high risks, all lifts should be designed and planned to mitigate the danger and increase reliability, preserving lives, loads and equipment involved.
In the design, planning and execution of the lift, one should ensure that equipment and accessories will not be overloaded; there will be no load collisions with any obstacles, including the crane
itself; the load will not be subjected to excessive speeds and accelerations among other undesirable situations.
The information generally available to the owner of the crane or who will do the design of the lifting is limited to the load charts and some operational limits as significant wave height (Hs),
maximum wind speed, maximum angles of trim and band. So, it is not taken into account a number of other factors that can directly influence the safety of lifting, as wave period, platform heading,
minimum and maximum length of the crane cable, speed of lifted load, among other factors.
It is proposed in this paper the “offshore crane operational limit diagram” (OCOLD), that has the goal to take into account
all factors that affects the lift operation, considering these effects in only one diagram, where an envelope that satisfies all safety criteria is established, and will be used in decisionmaking for
planning, design and execution of the lift operation.
The OCOLD can be used in design and planning of the lift, showing the values of Hs and Tp that meet the operational requirements previously established, and once defined Hs and Tp, one can define a
weather window, based on metocean data available for the location of operation. It is possible to reverse the procedure, that is, based on metocean data and expected Hs and Tp for the location, it is
possible to establish the operational parameters that may limit the lift procedure, like radius of operation, weight of load, speed of the lifted load, among others.
The OCOLD can also be used before routine lifts to ensure that an operation can be conducted within the safety envelope for a specific environmental and operational condition.
This methodology can be applied to any floating unit like FPSO, semisubmersibles, spars, construction vessels and also in fixed platforms.
Preparation of the OCOLD
The necessary information of the floating unit for the preparation of OCOLD are: platform’s response amplitude operator (RAO) for movements at correspondent draft, crane position in relation to the
movement center of platform, minimum and maximum cable length during operation, minimum radius and maximum radius of operation as well as the load weight to be lifted.
For environmental loads, one must provide the wave spectrum of the geographic region considered. The wind effect, directly on the load, is not considered in this paper and its effect on the unit, is
taken into account indirectly in the heading of platform.
It is also necessary the input of the operating limits and limits established by the technical standards and manufacturers of lifting equipment. The operating limits can be various, as established by
the operator of the platform and may limit for the lifted load: inclination, speed and load acceleration, proximity to obstacles such as ship’s side or superstructure (offset limitation). Different
values for limits may be established: nominal values, which exclude additional factors of safety, values for safe operation, which include factors of safety, and values for emergency operation, which
reduces the nominal values for an accepted level in emergency situation.
For example, codes, guides and standards for offshore lifting of personnel can define maximum significant height (Hs) for operation [1], and crane manufacturer, also based on international standards,
may also define maximum trim and heel angle for the offshore unit, as well as safe working load (SWL) and others. Figure 1 shows the flowchart for OCOLD preparation.
Regarding lifting of personnel using baskets, since 56% of accidents occur by collision [2], limits may be established with multiple safety volumes, negative or positive, where the basket cannot
achieve at if it is negative, or cannot get out of it if the volume is positive. This concept was developed by Roncetti [3] for lifting and rigging simulation applied in shipbuilding and offshore
The consideration of this limit in preparing the OCOLD may be made as follows: in the non-linear dynamic analysis stage, for each load case, the violation of each safety volume is checked and if it
is violated, this case is not represented in the diagram. The contour is formed with the combinations of maximum Hs that meet the geometric limits. This outline does not have scalar value.
From the input data, it is elaborated the structural model for dynamic simulation in SITUA / Prosim [4]. The load cases are variations of Hs and Tp pairs which, together with the selected spectrum,
generate the wave load that will act on the floating platform.
It is possible in a single structural model, to include all parameters combinations as headings, crane cables with different lengths, different radius of operation, different loads, among others,
allowing to obtain in single execution of the program, all results necessary to generate de diagrams.
After the dynamic simulation, the results, called intermediaries, are separated and processed by the postprocessing module developed by the authors, whose source code is found in [5]. For each set of
data that defines a diagram, it is performed the calculation of extreme values of the predefined limits, such as speed, acceleration, displacement, axial force in cable, etc. These calculations are
done for each load, radius of operation and cable length.
Next, it is calculated by linear or quadratic interpolation, a contour diagram as function of Hs and Tp for determining each operating limit. Figure 2 illustrates an example of diagram with contours
in function of Hs and Tp, highlighting the contour of the basket speed for safety operation limits, nominal operation limits and emergency operation limits. All values of this diagram are the
extremes values found for each Hs and Tp pair, varying the length of the crane cable.
Once obtained the contours for each operating limit, it is calculated the safe envelope that corresponds to each Tp, the highest value of Hs that is below of all thresholds simultaneously. It may be
generated how many diagrams that are necessary to cover load weights ranges, radius of operation ranges, most probable platform or ship headings and other parameters.
Once prepared the OCOLD, it can be used indefinitely, provided there is no change in initial parameters. Decision making for routine lifts can follow the flowchart shown in Figure 3.
Once known parameters of the lifting operation, such as operating radius, load weight, heading and other previously established, one choose the corresponding diagram prepared based on this lifting
configuration. Then, known Hs and TP, one check if this pair is within the safety area of the envelope. If it is, the lift operation can be executed. Otherwise, if it is possible to change any
parameter, such as operating radius, load weight or other, a new check is done. If it is not possible to change operational parameters, it is necessary to wait for more favorable sea condition or
abort the operation.
To demonstrate the application in lifting operations of the offshore crane operational limit diagram (OCOLD), a real FPSO data is used, considering a personnel lift operation and a routine lift of an
offshore container.
FPSO and Crane Parameters
The characteristics of the FPSO are listed on Table 1 and Figure 4 illustrates the FPSO dimensions and parameters used on this paper.
The structural model for dynamic analysis considered that the crane has a rigid structure. This consideration do not affect the final results significantly due to low resonant periods of the real
crane, ranging from 0,04 seconds to 0,33 seconds for the first six modes of vibration, not considering the pendulum modes. The cable of crane was modeled using truss elements. A detailed analysis can
be found in [5].
To determine the environmental loads, it was considered only the effect of waves, ignoring the wind and the current. JONSWAP wave spectrum adapted to the Campos Basin [4] was used and the combination
of Hs and Tp pairs based on values shown in Table 2 resulted in 90 load cases. These values of Hs and Tp are within a range of interest for the lifting activities, whose values are framed in periods
of 1 year recurrence, suitable for operating situations according to ISO 19901-6 [7].
Operational Limits
For the generation of OCOLD, it is necessary the establishment of operational upper limits as speed, acceleration, displacement, forces, floating unit movements, among others, defined by the platform
operator and based on technical standards.
The operational limits considered in this paper are shown in Table 3. Note that for the offshore container lifting, there is no need to establish different operational situations (nominal, safety or
emergency), as it is recommended when lifting personnel.
The limit of the basket speed is determined using [8] and the speed limit for the container is arbitrated. The roll threshold and dynamic amplification factor (FAD) were taken as the API
Specification 2C (API, 2004) and the threshold value for pitch is arbitrated.
The limits of the safe working load (SWL) are typical values for real offshore cranes, configured for lifting of personel or cargo as each case.
Other operating limits can be included depending on the type of lifting operation to be performed. In case of lifting of personnel, it may be included as a limiting factor, not only velocity and
acceleration but also the angle of inclination to the vertical of the transport basket. Also, for any case, one can consider if the basket or load are submerged, the maximum offset of load, maximum
offlead and sidelead of the crane cable and other limits.
After the dynamic analysis, the results were joined and processed by software developed by the authors. For each load case and limiting parameter, an extreme value analysis was conducted to establish
the envelope for each parameter.
Movement of the FPSO
In OCOLD composition, the motion limits for surge, sway, heave and yaw of the FPSO are not considered, although their effects are considered in the dynamic analysis. Figure 5 shows the contour plot
of extreme values, highlighting safety, nominal and emergency operation.
For pitch movement, all contours are above the value for Hs equal to 6,0 meters, so no plot is needed.
Although it is not considered in this paper the verification of collision or submersion of the load, an example of the path of the load is show in Figure 6. The study of trajectory of load is useful
to know how far it will be from obstacles or water.
One note that based on load path shown in Figure 6, amplitude for load case 47 is greater than for load case 48, even with lower Hs. That is explained by the movement of load that tends to whip more
Force in crane cable
For calculating the maximum extreme value for axial force acting on the cable during lift operation, an analysis of 90 time series for each cable length was conducted, determining the extreme maximum
value for each combination. For lifting one person, the extreme maximum force in function of Hs, Tp and cable length is shown in Figure 7. Radial values are the length of crane cable, and contours
are the axial force for each combination of Hs and Tp.
In this case, the maximum static force to the cable length of 15 meters, including the load is 3.95 kN and a cable length of 40 meters, the maximum static force is 4.91 kN.
It is noted that for Hs up to 4.0 meters there is no significant amplification of tensile force in cable comparing to static force, with a maximum of 7.0 kN, occurring in cable length of 40 meters,
load case 39, Hs 3.0 meters and Tp 12 seconds, which is the heave resonant period. The corresponding dynamic amplification factor is 1.43.
It is also shown that, in some situations the force decreases with increasing length of the cable, for example Hs exceeding 6.0 m and Tp equal to 11.0 seconds (load case 36), with the axial force for
32.5 meters long larger than the axial force for 35.0 meters long. The natural period of pendulum of the shorter cable is 11.4 seconds almost equal to the resonant period of heave, which is 11.5
seconds justifying the results found.
To calculate the contour diagram for axial force in crane cable, a similarly procedure to create the plot shown in Figure 7 is conducted, but with a rectangular layout, and adopting as a force value
for each pair of Tp and Hs the maximum value among all cable lengths. The graph thus has one dimension less (cable length) than the polar plot.
Figure 8 shows the diagram with contours of the operational limit for axial force in cable.
Dynamic amplification factor (DAF)
Based on axial force in cable and based on the static load for different cable lengths, one can draw the contour plot of the dynamic amplification factor (DAF) and establish operating limits. For
each point of the diagram, the static reference value corresponds to the cable that had the highest tensile force.
Figure 9 shows the contour plot of the DAF and the operational limits.
Once calculated the operational safety contours, nominal and emergency for each threshold considered, it is performed the calculation of the envelope for each operating situation. The envelope is
formed by calculating for each value of Tp, the largest value of Hs that is lower than all values in contours of operational situation analyzed. The envelope is automatically calculated by
post-processing program.
It is noted that, currently, technical standards define Hs limits for the lifting of personnel [1], and these limits are considered in the calculation of the envelope. The limitation of Hs value by
the crane safe working load, from load chart, is taken into account indirectly by limiting the force in the cable.
Figure 10 shows the contours of the limits adopted for the nominal operation, which is obtained without using the factor of safety. The yellow region indicates the permitted operational area, which
meets all of the nominal limits, in this case, limited by Hs and the transport basket speed.
Figure 11 shows the contours of the limits used for the emergency operation, which is constructed using less conservative values, but still within a controlled risk situation at the discretion of the
FPSO operator. The orange color region indicates the permitted operational area, which meets all the limits of the emergency operation, in this case, limited by Hs and the transport basket speed.
Figure 12 shows the contours of the limits used for safety operation situation or safe operation, which is established using more conservative values than the nominal ones, to compensate process
variation and uncertainty in the determination of natural periods, wave heights, RAO of the vessel, among others, at the discretion of the FPSO operator. The green area is the permitted operating
area, which meets all the limits of safety operation, in this case limited by Hs and transport basket speed.
It is noted that in all cases, operating constraints were Hs and speed of transport basket and thus should have their threshold well evaluated, in order to not allow false unsafe operations or not to
allow false safe operations that, in fact, are unsafe. It was also noted that in no condition occurred immersion of load, which is clearly identified in time series by the force in cable or the
position of the basket in relation to the wave height.
Final OCOLD
Once established the envelopes for each operational situation, one can assemble them into a single diagram called offshore crane operational limit diagram or OCOLD. In this diagram, the
representation of the operational limits contours is no longer necessary.
An OCOLD diagram should be developed for each condition and parameters of the platform, crane and load. In this case, this paper took into account the RAO at 21 meters draft, the radius of operation
of 20 meters, the load weight (1 person more equipment), maximum Hs, maximum and minimum crane cable lengths and heading. Figure 13 shows the resulting OCOLD for the presented case.
For any pair of Hs and Tp that intersects below the chosen operational condition (safety, nominal or emergency), the operation for lifting of personnel can be held.
The presented OCOLD considers Hs limitations for safety, nominal and emergency operations, using values 2.0, 2.5 and 3.0 meters respectively, as shown in Table 3.
It is observed in the above OCOLD that the rules limiting Hs equal to 2.0 meters are violated for Tp equal to 12 seconds and Hs of 1.8 meters for nominal operation and Hs of 1.5 meters for safe
The purpose of this paper is that the Hs parameter be not considered directly as a contour for operational limitation but, along with Tp, has its effects considered in dynamic response of load, cable
and crane. By doing so, all operating regions expand, increasing the weather window available. Figure 14 shows the OCOLD without limitation of Hs.
The allowed value of Hs expands significantly compared with OCOLD considering limitation of Hs (Figure 13), mainly below 9 seconds and values above 15 seconds, showing that when the wave period is
outside the range of resonant periods of the FPSO and crane cable, it is possible to use larger wave heights without compromising the safety of the operation for the analyzed case.
Offshore Container Lifting Operation
For lifting the offshore container, the procedure is analogous to lifting personnel, making the adoption of appropriate operational limitations. In this case it is not necessary or usual the
establishment of emergency or safety operations, since nominal values of limits already includes the factor of safety for each component of lift.
It is shown in Figure 15 intermediate operating limit diagram with contours for offshore container lifting, for nominal operation considering the limitation of Hs. One note that the limitation, for
periods for less than 11 seconds is given by the speed, up to 15 seconds it is given by DAF, up to 18.4 seconds, by roll movement and above this valve DAF is limiting again.
Figure 16 shows the OCOLD without limitation Hs, suggesting high operating clearance with Hs extension beyond the crane load chart.
Next are presented two practical applications of OCOLD developed using the methodology described above for typical situations in offshore platforms.
Application 1
It is assumed that an offshore platform needs to perform the lifting one person. The peak wave period (Tp) measured by the radar installed on the FPSO ranges from 8 to 9 seconds and Hs ranging from
2.0 to 3.1 meters, depending on the wind speed. It should be checked whether it is possible to perform this operation for safety situation.
Application 2
The FPSO is subjected to incidence of swell and sea with Tp ranging from 16 to 17 seconds and Hs ranging from 3.5 to 4.1 meters. Due to a medical emergency, because of an accident in supply boat that
serves the unit, a crew member needs to be carried on board the FPSO. It should be checked whether it is possible to perform this lifting operation in emergency situation.
Figure 17 of shows the OCOLD with limitation Hs and operational areas (Hs and Tp ranges values) for the proposed applications.
It can be seen in Figure 17 that in both applications one can’t do the lifting. In Application 1, the minimum measured value Hs is equal to the Hs of the safety operating limit, prohibiting the
operation. In the second application, due to Hs restriction, the operational area is completely in the unsafe region, also impeding the operation.
Figure 18 shows the OCOLD without Hs limitation, expanding the permissible values, making the two operations possible.
As in most of cases, lifting operations do not consider the effect of wave period and other important factors, it is concluded that OCOLD can improve the safety of offshore lifting operations showing
that an operation would be risky if not considering all parameters involved in lifting. By the other hand, OCOLD can optimize the use of equipment and provide a wider weather window, allowing
operations that a simple analysis would not permit.
To LAMCSO team, for the support using SITUA/Prosim, TechCon Engenharia e Consultoria for computation support.
[1] STANDARDS NORWAY. NORSOK Standard R-002: Lifting equipment. Lysaker, Norway, 2012.
[2] CATHERALL, Roger. Incidents with marine personnel transfer. Aberdeen 2013, available: http://www.reflexmarine.com/industry-expertise/crewtransfer-safety/ accessed 14-May-2014.
[3] RONCETTI, Leonardo. A software do design and management of lifting operations for shipbuilding and offshore construction. In: Pan American Conference of Naval Engineering, Maritime Transportation
and Ports Engineering, 2011, Buenos Aires.
[4] JACOB BP, BAHIENSE, R.A., CORREA, F.N., JACOVAZZO, B.M., 2012a. Parallel Implementations of Coupled Formulations for the Analysis of Floating Production Systems, Part I: Coupling Formulations.
Ocean Engineering 55 206-218, doi: 10.1016/j.oceaneng.2012.06.019.
[5] RONCETTI, Leonardo. 2014. Development of operational limit diagrams for offshore lifting procedures. (In Portuguese). Dissertation (Master’s degree in Civil Engineering) of COPPE/UFRJ, Rio de
Janeiro, Brazil.
[6] LANGEN, I., THAN, T. K., BIRKELAND, O., RØLVÅG, T. Simulation of dynamic behavior of a FPSO crane. Paper. 2003.
[7] International Organization for Standardization ISO 19901: petroleum and natural gas industries – Specificrequirements for offshore structures, part 6: Marine operations. Geneva, 2009.
[8] National Aeronautics and Space Administration (NASA). Issues on human acceleration tolerance after long-duration space flights. Technical memorandum. Houston: NASA, 1992.
[9] International Marine Contractors Association (IMCA). Guidance on the transfer of personnel to and from offshore vessels: IMCA SEL 025, IMCA M 202. United Kingdom, 2010.
Our offshore lifting projects: Click here
Our lifting courses: Clique aqui | {"url":"https://techcon.eng.br/artigos-tecnicos/development-of-operational-limit-diagrams-for-offshore-lifting-procedures/","timestamp":"2024-11-02T20:45:28Z","content_type":"text/html","content_length":"149881","record_id":"<urn:uuid:85b90efe-854c-4984-a1c9-1fc7ac0d721f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00418.warc.gz"} |
RBF as MFS approximations in higher dimension
Alves, Carlos J. S.
8th World Congress on Computational Mechanics (WCCM8), ECCOMAS, Barcelona (2008), (CD-Rom)
Some of the most common scalar PDEs have fundamental solutions that have a radial feature. There-fore, the method of fundamental solutions (MFS) is usually associated to radial basis functions (RBF)
as a particular case, when the basis functions have radial property. Here we will present a missing counterpart – some of the most common RBF approximation methods are just a particular case of the
MFS boundary approximation when applied in a higher dimension. This idea was presented in [1]. In thispresentation we will consider the relation between MFS boundary interpolation in dimension d+1
and RBF domain interpolation in dimension d, for the most commonly used RBF basis functions. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=4&member_id=78&doc_id=3154","timestamp":"2024-11-02T15:23:34Z","content_type":"text/html","content_length":"8813","record_id":"<urn:uuid:55cbbee1-0fb9-4664-b157-b2ccfaba6258>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00775.warc.gz"} |
A $17,900 truck is depreciated by 15.5% over 5 years. What's the truck worth at the end of 5 years? | HIX Tutor
A $17,900 truck is depreciated by 15.5% over 5 years. What's the truck worth at the end of 5 years?
Answer 1
If taken as written (depreciating 15.5% in total over the course of 5 years) = $15,125.50. If straight-line depreciation 15.5% each year, = $4,027.50. If declining balance, = $7,711.46.
If the van depreciates by 15.5% in total over 5 years, then we'd have:
#color(white)(00)17,900# - Original Cost #ul(xx 15.5%# - Depreciation rate #2,774.50# - Depreciation
I'll note we can get there by "going the other way" - by multiplying the cost of the van by #1-"depreciation rate"#:
However, depreciation rates are usually expressed as a per year measure and so I suspect the question is asking about a van that is depreciating by 15.5% per year over 5 years.
There are a couple of ways depreciation works. One way is straight-line depreciation, where we divide the original price and divide it into 15.5% chunks. If we take away 5 of those chunks, we get:
Another way to do depreciation is with declining balance depreciation (which means that we apply the depreciation percentage to the value of the van each year - and so as the value decreases, so does
the amount of depreciation). This gives:
Year 1: #17,900xx84.5%=15,125.50#
Year 2: #15,125.50xx84.5% = 12,781.05# (we round to the closest cent)
Year 3: #12,781.05xx84.5%=10,799.99#
Year 4: #10,799.99xx84.5%=9,125.99#
Year 5: #9,125.99xx84.5%=$7,711.46#
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Answer 2
Depreciation is usually regarded annually, because during the year the value drops from what it was at the start of the year.
Use the compound formula, but because the value is decreasing by #15.5%# we subtract. This means that at the end of a year, the van is only worth #84.5%# of the value at the start of the year.
#"Value" = P(1-r)^n#
#= 17,900(1-0.155)^5" "larr (15.5/100 = 0.155)#
Note that this is the same as:
#17,900 xx84.5/100 xx84.5/100 xx84.5/100 xx84.5/100xx84.5/100#
(Each year the value drops by #15.5%#)
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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/5a232650b72cff55c8eb1105-84670bd610","timestamp":"2024-11-07T17:08:20Z","content_type":"text/html","content_length":"586929","record_id":"<urn:uuid:f8f9fb0d-3150-4829-8ec4-9cb69454261e>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00003.warc.gz"} |
Packing without some pieces
Erdos and Hanani proved that for every fixed integer k >= 2, the complete graph K-n can be almost completely packed with copies of K-k; that is, K-n contains pairwise edge-disjoint copies of K-k that
cover all but an o(n) (1) fraction of its edges. Equivalently, elements of the set C(k) of all red-blue edge colorings of K-k can be used to almost completely pack every red-blue edge coloring of
K-n. The following strengthening of the result of Erdos and Hanani is considered. Suppose C' subset of C(k). Is it true that we can use elements only from C' and almost completely pack every red-blue
edge coloring of K-n ? An element C is an element of C(k) is avoidable if C' = C(k) \ C has this property and a subset F subset of C(k) is avoidable if C' = C(k) \ F has this property. It seems
difficult to determine all avoidable graphs as well as all avoidable families. We prove some nontrivial sufficient conditions for avoidability. Our proofs imply, in particular, that (i) almost all
elements of C(k) are avoidable (ii) all Eulerian elements of C(k) are avoidable and, in fact, the set of all Eulerian elements of C(k) is avoidable.
• Packing
• avoidable graph
• edge-coloring
• INTEGER
Dive into the research topics of 'Packing without some pieces'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/packing-without-some-pieces","timestamp":"2024-11-03T07:37:09Z","content_type":"text/html","content_length":"46897","record_id":"<urn:uuid:7ef5f8b7-977b-4b8d-b5d7-e6683b3ec9c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00750.warc.gz"} |
SciPy Tutorial {Comprehensive Guide for Beginners} | phoenixNAP KB
Computing is an essential part of scientific research. Mathematical, engineering, scientific and other technical problems are complex and require computing power and speed. Python provides the SciPy
library for solving technical problems computationally.
This article presents a SciPy tutorial and how to implement the code in Python with examples.
• Installed Python 2.7 or Python 3
• A Python environment for running the code.
• SciPy library installed.
• NumPy library installed (Follow our guide: How to Install NumPy).
• Matplotlib library installed for plotting (optional).
What is SciPy?
SciPy (Scientific Python) is an open-source scientific computing module for Python. Based on NumPy, SciPy includes tools to solve scientific problems. Scientists created this library to address their
growing needs for solving complex issues.
SciPy vs NumPy
The NumPy library (Numerical Python) does numerical computation. Scientists use this library for working with arrays since NumPy covers elementary uses in data science, statistics, and mathematics.
SciPy covers advanced calculations and functions. This library adds more data science features, all linear algebra functions, and standard scientific algorithms.
Why Use SciPy?
The SciPy library builds on top of NumPy and operates on arrays. The computational power is fast because NumPy uses C for evaluation.
The Python scientific stack is similar to MATLAB, Octave, Scilab, and Fortran. The main difference is Python is easy to learn and write.
SciPy Subpackages
The SciPy library has different groups of subpackages. There are two ways to import subpackages from the SciPy module:
import scipy.<sub package name> as <alias>
Or alternatively:
from scipy import <sub package name> as <alias>
In both importing methods, the alias is optional.
SciPy Functions
SciPy includes many of the primary array functions available in NumPy and some of the commonly used modules from the SciPy subpackages.
To import a function from a subpackage, use:
from scipy.<subpackage> import <function>
Note: Some of the NumPy functions available in SciPy show deprecation warnings.
Basic Functions
To get help and information for any SciPy function, use the help() command:
help(<name of function>)
The help() command does not need parameters. After executing without parameters, a prompt appears where you input the function name.
Another quick way to get help with any command in Python is to write the command name, put a question mark at the end, and run the code.
Special Functions
Special functions in the SciPy module include commonly used computations and algorithms. All special functions accept NumPy arrays as input. The calculations are elementwise.
To import the special subpackage, use:
import scipy.special as special
Or alternatively:
from scipy import special
To import a specific function from the special subpackage, use:
from scipy.special import <function name>
Evaluate the factorial of any number by running:
special.factorial(<integer or array>)
For example, to find the factorial of ten, use:
Permutations and Combinations
To find the number of permutations, use:
special.perm(<number of elements>, <number of elements taken>)
For example, to see the number of permutations of three elements taken two at a time:
Similarly, find the number of combinations with:
special.comb(<number of elements>, <number of elements taken>, repetition=<True or False>)
To find the number of combinations of three elements taken two at a time with repetition, enter:
special.comb(6,2, repetition=True)
Permutations and combinations are used in computer science sorting algorithms.
Exponential Functions
Exponential functions evaluate the exponents for different bases.
Calculate the exponents of base ten with:
special.exp10(<integer or array>)
For example:
Computer science often uses exponential functions of base two:
special.exp2(<integer or array>)
Calculate the tenth power of base two with:
Logarithmic Sum of Exponentials
The Logarithmic Sum of Exponentials (LSE or LogSumExp) is an approximation used by machine learning algorithms. Calculate the LSE with:
special.logsumexp(<integer or array>)
Bessel Function
Bessel functions appear in wave propagation, signal processing, and static potential problems. Find the Bessel function of the first kind with:
special.jn(<integer order>, <integer or array>)
Take advantage of the full stack to visualize the Bessel function. To find the second-order Bessel function of the first kind, use:
#import stack
import scipy.special as special
import matplotlib.pyplot as plt
import numpy as np
#The X-axis
x = np.linspace(1,50,100)
#Bessel function of the first kind order two
jn1 = special.jn(2,x)
Plot the results:
plt.title('Bessel function first kind order two')
plt.plot(x, jn1)
Integration and ODE Functions
SciPy provides a subpackage for calculations with definite integrals. To import the integrate subpackage, use:
import scipy.integrate as integrate
Or alternatively:
from scipy import integrate
Import a specific function from the subpackage integrate with:
from scipy.integrate import <function name>
General Integration
Calculate a single variable integral with the quad function from the integrate subpackage:
integrate.quad(<function>, <lower limit>, <upper limit>)
The function input is defined using a lambda function.
For example, to calculate the definite integral of the function x+1 between zero and one:
from scipy import integrate
f = lambda x: x+1
integrate.quad(f, 0, 1)
The output shows two values. The first value is the evaluated integral, and the second is the error of estimation.
Optimization Functions
SciPy has an optimization subpackage for finding the minimum or maximum of a function. The optimize subpackage includes solvers and algorithms for finding local and global optimal values.
To import the optimize subpackage:
from scipy import optimize
Or use:
import scipy.optimize as optimize
To import a specific function from the subpackage optimize, run:
from scipy.optimize import <function name>
Minimize a Function
Finding a minimum of a function is used in machine learning to lower an algorithm’s loss (or error).
For example, you can create a function and find the minimum. To do so, use the fmin function from the optimize subpackage in SciPy:
#Import stack
import numpy as np
from scipy import optimize
#Defining inverse sine function
def f(x):
return -np.sin(x)
x = np.linspace(0,5,100)
#Starting point
start = 3
#Simplex algorithm for optimization
optimized = optimize.fmin(f,start)
To plot the result, run:
import matplotlib.pyplot as plt
plt.plot(x, f(x))
plt.scatter(optimized, f(optimized))
plt.legend(['Function -sin(x)', 'Starting point', 'Optimized minimum'])
Fourier Transformation Functions
SciPy includes a subpackage for Fourier transformation functions called fftpack. The transformations are Discrete Fourier Transformations (DFT). All transforms are applied using the Fast Fourier
Transformation (FFT) algorithm.
To import the fftpack subpackage, use:
import scipy.fftpack as fftpack
from scipy import fftpack
Fast Fourier Transform
As an example, create a periodic function as a sum of three sine waves:
import numpy as np
freq_samp = 100
t = np.linspace(0, 1, freq_samp*2, endpoint = False )
f1, f2, f3 = 1, 5, 20
A1, A2, A3 = 3, 2, 1
x1 = A1*np.sin(f1*2*np.pi*t)
x2 = A2*np.sin(f2*2*np.pi*t)
x3 = A3*np.sin(f3*2*np.pi*t)
#Sum of waves
x = x1+x2+x3
Plot the waves using matplotlib:
import matplotlib.pyplot as plt
plt.xlabel('Time (s)')
Next, apply the fft and fftfreq functions from the fftpack to do a Fourier transform of the signal.
from scipy import fftpack
A = fftpack.fft(x)
freq = fftpack.fftfreq(len(x))*freq_samp*2
Plot the results to see the frequency domain:
plt.xlabel('Frequency (Hz)')
Signal Processing Functions
The subpackage signal includes functions used in signal processing. To import signal, run:
import scipy.signal as signal
Or alternatively:
from scipy import signal
A common task in signal processing is convolution. The SciPy subpackage signal has the function convolve to perform this task. For example, create two signals with different frequencies:
import numpy as np
t = np.linspace(0,1,100)
f1, f2 = 5, 2
#Two signals of different frequencies
first_signal = np.sin(f1*2*np.pi*t)
second_signal = np.sin(f2*2*np.pi*t)
Plot the signals:
import matplotlib.pyplot as plt
#Plotting both signals
plt.plot(t, first_signal)
plt.plot(t, second_signal)
plt.xlabel('Time (s)')
Import the signal subpackage from scipy. Use the convolve function from the signal subpackage to convolve the two signals:
#Importing the signal subpackage
from scipy import signal
#Convolving two signals
convolution = signal.convolve(first_signal, second_signal, mode='same')
Plot the results:
#Plotting the result
plt.plot(t, convolution)
plt.xlabel('Time (s)')
Interpolation Functions
Interpolation is used in the numerical analysis field to generalize values between two points. SciPy has the interpolate subpackage with interpolation functions and algorithms.
Import the interpolate subpackage with:
import scipy.interpolate as interpolate
from scipy import interpolate
One Dimensional Interpolation
The SciPy interpolate subpackage has the interp1d function for one dimensional interpolation of data. As an example, create toy data using numpy:
import numpy as np
#Create toy data
x = np.arange(0,10,0.5)
y = np.sin(x)
Interpolate the data with interp1d from the interpolate subpackage:
from scipy import interpolate
f = interpolate.interp1d(x, y)
#Create interpolation function
x_i = np.arange(0,10,3)
y_i = f(x_i)
Plot the results:
#Plot results
plt.plot(x_i, y_i)
plt.legend(['Interpolation', 'Data points'])
Linear Algebra
SciPy has a fully-featured linear algebra subpackage. The SciPy linear algebra subpackage is optimized with the ATLAS LAPACK and BLAS libraries for faster computation.
To import the linear algebra package from SciPy, run:
import scipy.linalg as linalg
Or use:
from scipy import linalg
All the linear algebra functions expect a NumPy array for input.
Calculate the determinant of a matrix with det from the linalg subpackage:
linalg.det(<numpy array>)
For example:
import numpy as np
#Generate a 2D array
A = np.array([[1,2],[3, 4]])
from scipy import linalg
#Calculate the determinant
Inverse Matrix
Determine the inverse matrix by using inv:
linalg.inv(<numpy array>)
For example:
import numpy as np
#Generate a 2D array
A = np.array([[1,2],[3,4]])
from scipy import linalg
#Calculate the inverse matrix
Eigenvectors and Eigenvalues
Eigenvectors and eigenvalues are a matrix decomposition method. The eigenvalue-eigenvector problem is a commonly implemented linear algebra problem.
The eig function finds the eigenvalues and eigenvectors of a matrix:
linalg.eig(<numpy array>)
The output returns two arrays. The first contains eigenvalues, and the second has eigenvectors for the given matrix. For example:
import numpy as np
#Generate a 2D array
A = np.array([[1,2],[3, 4]])
from scipy import linalg
#Calculate the eigenvalues and eigenvectors
Spatial Data Structures and Algorithms
Spatial data structures are objects made of points, lines, and surfaces. SciPy has algorithms for spatial data structures since they apply to many scientific disciplines.
Import the spatial subpackage from SciPy with:
import scipy.spatial as spatial
from scipy import spatial
A notable example of a spatial algorithm is the Voronoi diagram. For a given set of points, Voronoi maps divide a plane into regions. If a new point falls into a region, the point in the region is
the nearest neighbor.
Note: Voronoi diagrams relate to the k-Nearest Neighbor algorithm in machine learning.
As an example, create a Voronoi diagram from twenty random points:
from scipy.spatial import Voronoi
import numpy as np
points = np.random.rand(20,2)
voronoi = Voronoi(points)
from scipy.spatial import voronoi_plot_2d
fig = voronoi_plot_2d(voronoi,show_vertices=False)
Image Processing
SciPy has a subpackage for various n-dimensional image processing. To import the ndimage subpackage, run:
import scipy.ndimage as ndimage
Or use:
from scipy import ndimage
The SciPy misc subpackage contains a sample image for demonstration purposes. To import the misc subpackage and show the image:
from scipy import misc
from matplotlib import pyplot as plt
raccoon = misc.face()
#show image
Import the ndimage subpackage and apply a uniform_filter to the image. Show the image to see the results:
from scipy import ndimage
filtered = ndimage.uniform_filter(raccoon)
File IO (File Input / Output Package)
SciPy has a file input and output subpackage called io. The io subpackage is used for reading and writing data formats from different scientific computing programs and languages, such as Fortran,
MATLAB, IDL, etc.
Import the io subpackage from SciPy with:
import scipy.io as sio
Or use:
from scipy import io as sio
This tutorial provided the necessary ScyPy examples to get started. Python is easy to learn for beginners and scripts are simple to write and test. Combining SciPy with other Python libraries, such
as NumPy and Matplotlib, Python becomes a powerful scientific tool. The SciPy subpackages are well documented and developed continuously.
For further reading, check out our tutorial on the Pandas library: Introduction to Python Pandas.
Was this article helpful? | {"url":"https://phoenixnap.nl/kb/pittige-tutorial","timestamp":"2024-11-01T22:16:52Z","content_type":"text/html","content_length":"340150","record_id":"<urn:uuid:21946855-22d5-4632-bb89-1912577e0243>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00704.warc.gz"} |
Appliaction Of CAPM Model: A Case Study On Utility Industry
1. Theoretical assumptions and implications of Capital Asset Pricing Model (CAPM) derived from Sharpe (1964) and Lintner (1965)
The Capital Asset Pricing Model established by Sharpe (1964) and Lintner (1965) specifies that expected return on a particular stock is developed by considering the risk premium and risk free
interest rate. Early practical analysis of different models have supported the main prediction that beta is the sole explanatory factor, which tends to explain the cross-sectional variation in stock.
It is observed in recent empirical tests that the asset pricing model comprise a number of variables, which helps in explaining cross-sectional variations in stock returns, apart from the market
risk. The model had initiated the idea of asset pricing theory (Roll, 1997). Prior to that, there was no asset pricing models and the same was generated by considering the nature and taste of
investment opportunities available in the stock market. The model clearly tested predictions that are made with respect to return and risk of the stock. The models are used presently for estimating
cost of equity and examining performance of portfolios that are developed in the stock market. It states that there is always a linear relation between expected return of a stock, risk associated
with the stock and beta. Here, beta is defined as the variable, which focuses on explaining the cross-sectional return of stocks. The model was initially devised by Harry Markowitz. However, it was
later modified by Sharpe and Lintner. Initially, the model explained that if an investor is selecting a particular portfolio for a particular period of time and receiving a return, he is risk averse.
The model is based on algebraic statement, which predicts the relation between risk and return of the portfolio. The modified version of Sharpe (1964) and Lintner (1965) has the following
1. The security markets are expected to be perfectly competitive.
a) There are many small investors.
b) The investors are regarded as the price takers.
2. The markets are not expected to encounter any friction or disturbance.
a) The markets transactions do not bear any taxes or other costs.
3. Investors are very narrow-minded as well as risk averse.
a) The investors count only one and the same holding period (Lakonishok, Shleifer and Vishny, 1994)
4. The investments are restricted to publicly traded assets. The assets bear unlimited lending and borrowing at risk-free rates.
a) The assets, like, human capital, does not form the part of opportunity set of the investment plan.
5. All investors are regarded as rational beings (Mitchell and Stafford, 2000). They optimize the mean-variance before investing.
a) Investors are using the Markowitz portfolio method for selecting their investment plan.
6. Perfect information is collected by investors in order to get a clear idea regarding the security market and risk associated with it (U.S. Department of Treasury, 2013)
a) The investors have good access to the information that is provided to them.
b) The investors are expected to analyse the information in the same way (Fama and French, 2003).
The equation for the model can be written as the following:
E(Ri) = Rf + ?(E(Rm) - Rf)
Rm = Return from the market
E(Ri) = Expected return on the asset in which investment is made
Rf = risk-free rate of interest i.e. interest arising from government bonds
?= sensitivity of the expected excess asset returns to the expected excess market returns
The regression version of the model can be written as:
Rpt- Rft = ?+ ?*(Rmt - Rft) + ?t
Rpt-Rft = excess return of the portfolio (taken as dependent variable for solving the problem)
Rmt?Rft = excess return of the market (taken as independent variable for solving the problem)
? = estimated vales of the parameters
? = estimated vales of the parameters
?t = error at time t (Roll, 1997)
The value of beta depends on the type of assets and it gauges volatility of same, in terms of market risk. The following are the interpretation of beta with respect to different assets:
Beta Implication
Greater than 1 Performance of the shares is aggressive. They outperform the market, which implies that the stocks with ?> 1 provide higher return than the market.
Equals to 1 Performance of the share is neutral. The performance of the stocks is in line with that of the average return of the market.
Less than 1 Performance of the shares is noticed to be conservative and is less risky than the market returns.
Apart from the above explanation, it is observed that every market has a beta factor of 1. For example, if the beta factor is 2, then return of the stock varies twice as much as the market return. If
market return (Rm) is 5% more the risk free return, then the expected return of the company stocks with beat factor 2 is 10% above risk free rate of return.
2. Validity of the assumptions
The model was considered by taking time period as one year. The validity of model is approved only by expanding the time period to multiple years. It, thus, measures whether the return of the market
is stable or not. The changing market conditions and company’s cost structure indicate the fact that beta of the security market will not remain the same over years. Beta is regarded as risk
associated with the stock returns. The estimation of these betas is subject to statistical variability. The betas related to the industry are more reliable than individual betas of the company. | {"url":"https://penmypaper.com/free-paper/capm-model-utility-industry","timestamp":"2024-11-06T04:30:55Z","content_type":"text/html","content_length":"41580","record_id":"<urn:uuid:b57829e5-b9cd-4c9a-9995-303fe7eb5def>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00762.warc.gz"} |
Counterfactuals and Causal Reasoning | Adventures in Why
Counterfactuals and Causal Reasoning
A/B Testing Series
1. Counterfactuals and Causal Reasoning
So far in this series we have only considered the possibility that our actions have no effect on an observed outcome. This disheartening possibility is called the Null Hypothesis. Whenever we are
using random segmentation to investigate the causal relationship between an experience we are providing and the response of an audience, the observed treatment effect must be large relative to what
is plausibly attributable to random chance. To use the standard terminology, the effect must be statistically significant.
When either the audience size, or the causal effect is small, it is unlikely we will achieve statistical significance. Just because a result is not statistically significant, does not mean there is
no treatment effect. In order to understand what conclusions we may rightfully draw from such “null” results, we need a better understanding of what sort of outcomes are possible when our actions do
indeed affect the behavior of an audience.
A popular approach to causal inference is based on counterfactuals. The Stanford Encyclopedia of Philosophy provides an excellent discussion of the history and development of this approach.^1 The
basic idea is to consider what would have happened if a specific event had not occurred, or a specific agent had not been present. We compare this counterfactual reality with what was actually
observed following said event. As discussed in the Random Sampling article, this is easier said than done.
We can make use of the crystal ball we introduced in that post to provide an example of precisely what we mean by this. Recall from our subject line example that we have an audience of 1000 email
recipients, and we are investigating the impact of two candidate subject lines, $A$ and $B$, on open rates. We send the first subject line to everyone, and three people open the email. We peer into
our crystal ball, which conveniently provides a window into a counterfactual reality in which we had sent the second subject line to everyone. In that reality, five people opened the email. With a
crystal ball, not only can we determine which subject line leads to more email opens, we can actually list which recipients opened it in which reality.
Recipient $A$ $B$
Alice x x
Brian x
Charlotte x
David x x
Emily x
Frank x
Totals 3 5
In the table above, we see that Alice and David opened the email regardless of the subject line they received (an “x” denotes the person opened the email when receiving a particular treatment).
George—and the remaining audience members not listed—did not open the email regardless of the subject line they received. None of these recipients’ behaviors were affected by the subject line.
In contrast; Brian’s, Charlotte’s, Emily’s, and Frank’s behaviors were indeed influenced by the subject line they received. Brian only opened the email in the reality where he received subject line
$A$; whereas, Charlotte, Emily, and Frank only opened the email in the reality where they received subject line $B$.
Looking at the table, it is clear sending subject line $B$ to everyone is preferable to sending subject line $A$ to everyone. (Actually, sending subject line $B$ to everyone except Brian, and sending
subject line $A$ to him, is the best option of all. For now, we assume we only care about determining which subject line is the best overall. Providing tailored experiences to each individual is much
more challenging, and is outside the scope of the present discussion.) Unfortunately, the only way to generate the table is with a crystal ball, which doesn’t actually exist. We can only observe the
response to the treatment we actually provide, and we can only speculate about the response to treatments we do not provide. That is what makes causal inference so difficult.
With random segmentation, we randomly assign subject lines to audience members and observe the results. This allows us to fill in part of the table. For example, suppose we randomly select Alice,
Charlotte, David, and Frank (and 496 others) to receive subject line $A$, and the remaining people to receive subject line $B$. This leads to the following table.
Receives $A$ $A$ $B$ Receives $B$ $A$ $B$
Alice x ? Brian ?
Charlotte ? Emily ? x
David x ? George ?
Frank ? (497 others) ?
(496 others) ?
Totals 2 ? ? 1
Consistent with the previous table, Alice and David open the email; however, it is important to note they do not open it because they received subject line $A$. They would have opened it even if they
had received subject line $B$. But we only know this because of our crystal ball. In reality, we have no idea what any of the first group would have done had they received subject line $B$. That’s
why there are question marks in that column. Indeed, we can only speculate about why any of these individuals opened or did not open the email.
Similarly, for the group that receives subject line $B$, only Emily opened the email. In light of the original table, Emily did indeed open the email because she received subject line $B$. We know
this because in a parallel universe where she received subject line $A$, she did not open the email. But again, we can never know this in any realistic scenario.
What we do know is how people reacted to the subject lines they received. Out of the 500 people randomly selected to receive subject line $A$, 2 opened the email, for an open rate of $0.4\%$;
whereas, the open rate in the second group was only 1 out of 500 or $0.2\%$. Taking the results at face value, we would conclude that subject line $A$ is twice as good as subject line $B$, when in
fact it is worse.
Random segmentation does not always enable us to determine which treatment gives the best result, as this example shows, but neither does any other method. What random segmentation does provide is:
1. A method that gives more reliable answers the larger the audience.
2. Extremely precise measures of how reliable the method itself is, for audiences large or small.
I am unaware of any other method that does the same, which is why random segmentation is considered the gold standard of causal inference.
What does “Why?” mean anyway?
In the counterfactual approach, we are interpreting “why” in a specific way. We are fundamentally asking whether the occurrence of a specific event, or the presence of a specific agent was necessary
and sufficient for a particular outcome. If not necessary, the outcome would have happened even without the event or agent; if not sufficient, the event or agent is an incomplete explanation.
When we randomly selected Alice to receive subject line $A$, the latter was not necessary for Alice to open the email; she would still have opened the email had she received subject line $B$. On the
other hand, it would appear that receiving subject line $B$ was both necessary and sufficient for Emily to open the email. In more simple language, we say that Emily opened the email because she
received subject line $B$.
This logic is only applicable in a particular context. If we speculate about a third subject line, $C$, and if we believe Emily would have opened the email had she received $C$, then in that context,
$B$ was not necessary. If we additionally know that subject lines $B$ and $C$ use informal language in contrast to $A$, and that Emily does not appreciate formality, we might say that the subject
line alone is not a complete causal description. Rather, the tone of the subject line and Emily’s preferences, together, form a more complete explanation for the outcome. Then in that case, $B$ is
not sufficient. Conclusions about the causal relationship between an event or agent and an outcome depend on context and indeed on the goals of the inquiry.
While counterfactuals and random segmentation form a powerful and practically useful framework for causal inference, the approach has limitations. When we ask, “Why did Emily open the email?”, the
answer according to this approach is, “Because she received subject line $B$.” The approach offers no insight into what it was about subject line $B$ that appealed to Emily. Neither does it offer any
insight into what sort of subject line would appeal to George. Because of this, we cannot extrapolate what the response to other, untested subject lines might be.
A suitably rich collection of subject lines may enable us to investigate these issues. The theory of Experiment Design—and, presumably, theories of marketing and human psychology—have more to say on
these issues, which are outside the scope of the present discussion. Nonetheless, in many situations, we are merely attempting to determine the best option from a particular set of alternatives. The
counterfactual framework provides a logically compelling approach for considering what it actually means for one option to be best. Random segmentation provides a method not only for determining what
that best option is, but also for quantifying the reliability of our conclusions. While there are many important questions that cannot be addressed within this framework, random segmentation is both
practical and valuable. ^2 | {"url":"https://adventuresinwhy.com/post/counterfactuals/","timestamp":"2024-11-07T19:28:36Z","content_type":"text/html","content_length":"32025","record_id":"<urn:uuid:d324467b-954e-413b-a04d-7eb4c3a3ee7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00067.warc.gz"} |
How Do You Solve a Decimal Equation Using Multiplication?
Want to see how to solve an equation containing decimals? Then check out this tutorial! You'll see how to multiply decimals in order to solve an equation for a variable. Then, see how to check your
answer so you can be certain it's correct! | {"url":"https://virtualnerd.com/common-core/grade-8/8_EE-expressions-equations/C/7/7b/equation-decimal-solve-by-multiplication","timestamp":"2024-11-10T03:22:20Z","content_type":"text/html","content_length":"31301","record_id":"<urn:uuid:2835157b-9d7d-4172-92c9-1cb970899898>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00356.warc.gz"} |
Large transformer models often possess loss functions with heterogeneous curvature properties across different parameter dimensions. Consequently, the incorporation of per-coordinate curvature
information into the descent direction can potentially enhance convergence. However, calculating second-order information can be computationally expensive.
In response to the enormous costs associated with large language model pre-training, the authors of the paper [Liu23S] introduce a second-order optimizer designed to diminish the number of iterations
required for convergence, as compared to the prevailing first-order methods. Simultaneously, it aims to maintain a similar computational cost per iteration.
The researchers contrast their novel optimization method, Sophia, against AdamW—the prevalent solver used in training large language models. Sophia incorporates gradient smoothing components akin to
those in Adam, and combines these with smoothed second-order information. In short, the descent update becomes the moving average of the gradients divided by the moving average of the estimated
Hessian, all followed by an element-wise clipping procedure.
Adopting the notation used in the original paper, we let $\theta_t$ represent the solution at iteration $t$, $L_t(\theta_t)$ denote the mini-batch loss, and $\eta_t$ be the step size. The method
implemented by Sophia can be summarized as follows:
1. Exponential smoothing of minibatch gradients at each iteration:
$$m_t = \beta_1 m_{t−1} + (1 − \beta_1)abla L_t(\theta_t)$$
2. Exponential smoothing of the Hessian information every $k=10$ iterations:
$$h_t = \beta_2 h_{t−k} +(1−\beta_2)\hat{h}_t$$
and $h_t = h_{t−1}$ if $t \operatorname{mod} k \neq 1$. Here, $\hat{h}_t$ stands for a lightweight estimator of the Hessian’s diagonal at iteration $t$.
3. Per-coordinate clipping:
$$\theta_{t+1} \leftarrow \theta_t − \eta_t \cdot \operatorname{clip}(m_t/ \max(h_t, \varepsilon), \rho),$$
where $\operatorname{clip}(z, ρ) = \max(\min(z, ρ), −ρ)$.
The authors highlight two critical points. First, the stochastic estimator for the Hessian diagonal should not introduce substantial overhead per step. It should be computationally on par with simple
gradient computation (the original article proposes two options to achieve this). Second, the smoothing of the Hessian information and the clipping procedure offer stability to the optimization
process by mitigating the effects of inaccurate Hessian estimates, rapidly changing curvature, and challenges arising from non-convexity (i.e., when the algorithm moves uphill instead of following a
descent direction).
Experimental Results
The experimental evaluation conducted by the authors focused on training the GPT-2 model (with varying number of parameters) on the OpenWebText corpus.
The authors noted that using Sophia led to a significant performance improvement. Specifically, the number of iterations needed to achieve a certain level of validation loss was halved when compared
to using AdamW. As the Hessian computations contribute less than a 5% overhead, this reduction in iterations considerably decreases the total compute requirements. The authors encapsulate this
finding as follows:
“Sophia is 2x faster in terms of number of steps, total compute and wall-clock time.”
Furthermore, the authors observed that models optimized by Sophia exhibit validation losses comparable to significantly larger models trained with AdamW. Interestingly, this performance differential
increases as the size of the model grows.
“The scaling law is in favor of Sophia-H over AdamW”
At first glance, one might underestimate the potential of Sophia given its “sparse” use of second-order information. Yet, the results presented in the original article are nothing short of
impressive. Sophia applies an effective combination of stochastic estimation of curvature, smoothing and clipping to achieve a very well-designed balance between computational overhead and improved
convergence behavior.
Given the current trend towards larger and more complex language models, coupled with the substantial computational resources required for their training, the improvements Sophia brings to the table
are significant. While the experiments to date have focused on large language models trained on text, it would be exciting to investigate the potential of Sophia across a wider range of applications,
including various domains of Natural Language Processing, Computer Vision, and more.
The authors have conveniently provided an open-source implementation, which leverages PyTorch’s Optimizer base class. This allows it to be used directly as a drop-in replacement. | {"url":"https://transferlab.ai/pills/2023/sophia/","timestamp":"2024-11-13T14:20:58Z","content_type":"text/html","content_length":"24957","record_id":"<urn:uuid:964b3ff7-a357-47fd-9501-52450d19e88f>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00251.warc.gz"} |
Derivative-Calculator.org – Rules and Explanations
Let’s dive into the world of calculus! Differentiation is a fundamental concept that allows us to study the rates of change and the slopes of curves. To make the process easier, mathematicians have
developed a set of rules and techniques that simplify the process of calculating derivatives. In this section, we’ll explore the common differentiation rules together, providing clear explanations,
illustrative examples, and proofs to help you understand and apply these rules effectively. | {"url":"https://derivative-calculator.org/rules/","timestamp":"2024-11-04T11:12:18Z","content_type":"text/html","content_length":"13918","record_id":"<urn:uuid:216cc068-07ca-4086-93d0-0f33ae8a0026>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00001.warc.gz"} |
160 Centimeters to Feet
160 cm to ft conversion result above is displayed in three different forms: as a decimal (which could be rounded), in scientific notation (scientific form, standard index form or standard form in the
United Kingdom) and as a fraction (exact result). Every display form has its own advantages and in different situations particular form is more convenient than another. For example usage of
scientific notation when working with big numbers is recommended due to easier reading and comprehension. Usage of fractions is recommended when more precision is needed.
If we want to calculate how many Feet are 160 Centimeters we have to multiply 160 by 25 and divide the product by 762. So for 160 we have: (160 × 25) ÷ 762 = 4000 ÷ 762 = 5.249343832021 Feet
So finally 160 cm = 5.249343832021 ft | {"url":"https://unitchefs.com/centimeters/feet/160/","timestamp":"2024-11-07T05:53:01Z","content_type":"text/html","content_length":"22998","record_id":"<urn:uuid:74f098d5-01f6-4198-ae0d-6c3fa11bdb00>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00802.warc.gz"} |
Python Lists: Copy ListsPython Lists: Copy Lists - MakerTechLab.com | Arduino Python Projects Tutorials DIY
Python Lists: Copy Lists
Looping Through Python Lists
In Python, looping through a list allows you to perform actions on each item within the list. This can be achieved using various types of loops, such as the for loop, while loop, or even list
comprehension for more concise syntax.
Using a For Loop
One of the most common ways to iterate over list elements is by utilizing a for loop. You simply iterate through each item in the list, performing the desired operation.
thislist = ["apple", "banana", "cherry"]
for x in thislist:
This code will print each item in the list, one by one. If you’d like to explore more about for loops, check out our dedicated Python For Loops Chapter.
Iterating Using Index Numbers
Another method to loop through a list involves referencing each item’s index number. By combining the range() and len() functions, you can create an iterable that spans the indices of the list.
thislist = ["apple", "banana", "cherry"]
for i in range(len(thislist)):
In this case, the iterable generated by range(len(thislist)) is [0, 1, 2], allowing you to access each item through its index.
Using a While Loop
You can also loop through list items by employing a while loop. This approach requires you to initialize an index variable, check its value against the list length using len(), and increment the
index with each iteration.
thislist = ["apple", "banana", "cherry"]
i = 0
while i < len(thislist):
i += 1
This example demonstrates how to print all list items by looping through their index numbers with a while loop. To dive deeper into while loops, see our Python While Loops Chapter.
Efficient Looping with List Comprehension
For a more concise syntax, you can use list comprehension to loop through lists. This approach is ideal for those who prefer a more compact and efficient way to write loops.
thislist = ["apple", "banana", "cherry"]
[print(x) for x in thislist]
List comprehension provides a shorthand method for iterating over all items in a list, making your code more streamlined.
Leave a Comment | {"url":"https://www.makertechlab.com/python-lists-copy-lists/","timestamp":"2024-11-03T01:07:27Z","content_type":"text/html","content_length":"67522","record_id":"<urn:uuid:e226ab6c-3fdd-49b4-99b6-49e7895f4398>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00039.warc.gz"} |
Basis Point: Understanding its Impact on Financial Movements - Inspired Economist
Basis Point: Understanding its Impact on Financial Movements
✅ All InspiredEconomist articles and guides have been fact-checked and reviewed for accuracy. Please refer to our editorial policy for additional information.
Basis Point Definition
A basis point is a financial unit of measurement used to describe the percentage change in rates or the value of a financial product, where one basis point is equivalent to 0.01%, or 0.0001 in
decimal terms. It is commonly used in discussions around interest rates, credit spreads, and other percentage-based indications.
Use of Basis Points in Finance
Understanding Basis Points in Different Financial Contexts
Basis points are an integral part of financial discussions and transactions. They function as a common unit of measure in various aspects of finance.
Basis Points and Interest Rates
In the context of interest rates, one might often hear phrases like "the Fed has raised rates by 25 basis points". This means the Federal Reserve Bank, the central bank of the United States, has
increased the interest rate by 0.25%. It signifies how even slight adjustments in rates can have significant impacts on the economy.
Basis Points and Bond Yields
Similarly, bond yields are also often discussed in terms of basis points. For example, if a Treasury bond yield increases from 2.00% to 2.10%, it can be stated as a 10 basis point increase. This
allows compact representation and easy comparison of shifts in bond market conditions over time.
Basis Points and Mutual Funds
Within the realm of mutual funds, basis points play an essential role in determining expense ratios. The expense ratio, which is the operation cost of a fund, is often expressed in basis points. If a
fund has a 1.50% expense ratio, it's described as having a 150 basis point expense ratio. This perspective helps investors evaluate the cost-effectiveness of different funds before making an
Basis Points and Exchange Rates
In the world of foreign exchange markets, even minor changes in currency rates matter significantly. Here, basis points come in handy to quantify relatively small alterations. Changes in exchange
rates are enumerable in basis points, making it easier for traders and investors to understand market movements and make informed decisions.
Precision and Utility of Basis Points
As seen through these examples, basis points function as a standard unit of measure for dealing with percentages, particularly small ones. By using basis points, financial professionals sidestep
potential confusion arising from terms like 'percentage point' and 'percent'.
A change from 2% to 3% can justifiably be described as an increase of one percentage point or 50 percent, which can lead to misinterpretation. In the world of finance, ambiguity can be expensive. In
such a situation, one could instead plainly state a 100 basis point increase, providing better precision and clarity.
Furthermore, the use of basis points helps convey the magnitude of a change in a way that is easier to grasp. For instance, a change of 0.10% might seem insignificant; however, expressing it as a
move of 10 basis points emphasises its potential effect on financial markets and investments.
In conclusion, basis points serve as a valuable tool in the financial world. They offer the dual benefit of precision and simplicity, enabling fluid communication and comprehension of financial
changes and conditions.
Basis Points vs. Percentage Points
A major distinction between basis points and percentage points lies in their precision, particularly when documenting alterations in financial instruments. Basis points offer more accuracy when it
comes to tracking small changes that percentage points might overlook.
Basis Points and Precision
A single basis point denotes a one-hundredth of a percentage point (0.01%). Consequently, it's immensely useful for detailing small fluctuations. For instance, a move from 3.00% to 3.03% yields a 3
basis point increase, which a plain percentage point change might round off or even exclude in a broad analysis. This precision makes basis points invaluable in cases where even minor shifts can have
a significant impact, such as interest rate changes in a large loan, or small improvements in the return on a multi-million dollar investment portfolio.
Understanding the Difference with Percentage Points
By contrast, percentage points give a simplified, generalized view, glossing over smaller details. In the percentage point system, an increase from 3% to 4% is an increase of 1 percentage point, or
100 basis points. The percentage point system can be more approachable, particularly for those less familiar with financial jargon. However, it's less precise and might not convey the full story in
all scenarios, especially when the details matter.
In summary, while percentage points are commonly used to describe larger, more easily noticeable changes, basis points offer a detailed, nuanced perspective, capturing even the minor shifts. By being
precise, basis points help in capturing the smallest of financial swings and provide a more accurate measurement tool in the financial world.
Basis Points in CSR Reporting
Potential of Reporting CSR Metrics in Basis Points
There is an emerging trend of businesses that are expressing key performance indicators (KPIs) related to Corporate Social Responsibility (CSR) in basis points. By doing so, businesses aim to detail
their sustainability initiatives more accurately, with granularity and precision often proving instrumental in accurately assessing their impact.
Benefits of Using Basis Points in CSR Reporting
The primary advantage of using basis points for CSR metrics lies in the enhanced scope that it provides for nuances. Since a basis point is a unit that equals 1/100th of a percent, it is
intrinsically scalable and allows for a more refined presentation of data.
One key metric, for instance, could be the reduction of the company's carbon footprint. If the company reduces its carbon emissions by 0.25% in a year, it represents a 25-basis point reduction which
sounds more significant.
Another area where basis points prove useful is in measuring social impact. If a company increases its charitable contributions by, say, 0.1%, it can report a 10 basis point increase instead. Higher
positive values reflected in CSR reporting could potentially increase investor interest and boost the company's reputation.
Drawbacks of This Approach
However, it's not all upside. The use of basis points may prove to be somewhat problematic, primarily because it's not widely understood outside of financial circles. This could create a barrier in
communication for stakeholders who aren’t as financially savvy.
Furthermore, it could be misleading. While 10 basis points sound more significant than a 0.1% increase, they are the same thing. This could be viewed as an attempt to inflate numbers and misrepresent
actual progress.
Finally, the shift to basis points could also increase the complexity of CSR reporting. Companies may have to explain their choice and its implications, taking up valuable time and resources.
Ultimately, using basis points in CSR reporting brings both complexity and clarity. Given their potential to enhance or distort the true picture, businesses must be mindful of these factors and make
informed decisions.
Basis Points in Risk Management
In risk management, basis points serve as valuable tools for evaluating and comparing potential risks. Many entities—be it banks, investment firms, or enterprises—use them to determine the level of
risk associated with various financial decisions or instruments.
Assessing Potential Risks
Risk managers often use basis points to measure changes in interest rates, exchange rates, or the yield of a bond. A single basis point change in interest rates, for instance, may seem minute, but
when dealing with large sums of money, this tiny change can significantly alter the risk profile of an investment.
For example, if a bank lends millions of dollars at a certain interest rate, a rise or drop of just a few basis points in that rate can impact the amount of interest the bank receives. This, in turn,
affects the bank’s risk of not achieving its desired return.
Comparing Risks
When it comes to comparing risks, basis points provide a common scale, making the comparison of financial products or investments more straightforward. They give an exact measure of change, free of
the ambiguities that can accompany terms like 'a small increase' or 'a significant decrease'. Therefore, basis points help risk managers make more accurate comparisons, such as the comparative risk
of two different bonds.
For instance, if Bond A yields 5.00% and Bond B yields 5.05%, the yield difference can be easily expressed as 5 basis points. This comparison allows financial managers to better understand potential
risk and returns.
The Impact of Slight Changes
While the change of a single basis point represents a very small percentage change, it's crucial to understand that these small changes can heavily impact the overall risk profiles of financial
investments. Given the large sums involved in financial transactions, a single basis point change can translate into significant absolute amounts.
For example, in a $200 million loan agreement, a 1 basis point increment in interest rates represents an additional cost or income of $20,000 annually. Consequently, the compounding effect of these
tiny changes over longer periods can be substantial, demonstrating how basis points, though small, can heavily influence overall risk calculations.
Therefore, understanding and managing the effect of change in basis points is crucial in financial risk management.
The Role of Basis Points in Monetary Policy
Central banks, primarily responsible for managing a country's money supply, frequently use basis points when implementing monetary policy. The idea is to affect the cost and availability of money, in
turn influencing inflation rates, economic growth, and stability.
Use in Interest Rate Policy
One of the most pivotal ways central banks use basis points is in setting interest rates, particularly the benchmark interest rate. A movement in the rate is usually in terms of basis points. For
instance, when you hear news stating the Federal Reserve raised rates by 25 basis points, it means they've hiked the rate by 0.25%.
Alterations in this rate influence other interest rates throughout the economy, including those for mortgages, car loans, and corporate bonds. Higher rates make borrowing more expensive, often
leading to decreased spending and investment, which can slow inflation. Conversely, lower interest rates make borrowing cheaper, which can encourage spending, potentially stimulating the economy.
Impact on Inflation
Basis points play an intriguing role in combating inflation or deflation. If inflation – a general rise in prices over time – is high, the central bank can raise interest rates. Hiking the rate by a
certain number of basis points discourages borrowing due to higher costs, helping cool down the economy and curb inflation.
In the case of deflation – a general fall in prices – the central bank can lower the interest rates, expressed as a reduction in basis points. The lower borrowing costs can bolster spending and
investment, helping to stop or prevent deflation.
Economic Stability
Lastly, basis points can be utilized as a tool for achieving economic stability. Monitoring and adjusting interest rates in terms of basis points can help maintain a balance between inflation and
economic growth. Too much economic growth can cause high inflation, while too little can lead to a recession. By finely tuning the basis points in their policies, central banks endeavor to foster an
environment of sustainable economic growth.
Basis Points in Sustainability Reporting
In sustainability reporting, the role of basis points becomes particularly valuable when tracking changes, over time, in a company's Environmental, Social, and Governance (ESG) performance. Given
that a basis point equals 0.01%, they serve as a straightforward and easily communicable measure for incremental changes, particularly in the context of ESG metrics, which often involve large
numerical values.
Using Basis Points for Environmental Performance
For instance, a company tracking its carbon footprint might report a decrease of 50 basis points in its Co2 emissions from one year to the next. This means a reduction of 0.50% from the previous
year's emission levels. They might also report an increase of 10 basis points in their renewable energy use, signifying an increase of 0.10% in their use of green power sources.
Basis Points Reflecting Social Impact
Similarly, when quantifying social impact, a company may choose to communicate the decrease of workplace incidents through the measure of basis points. If the incident rate falls by 20 basis points,
the company is indicating a 0.20% drop in workplace incidents.
Role of Basis Points in Governance Reporting
In the framework of governance reporting, basis points might quantify changes such as the representation of women and minorities on an organization's board. For example, an increase of 100 basis
points in the representation of women could signal a 1% increase in female board members over the last reporting period.
Basis points provide a versatile tool for businesses to communicate incremental improvements in sustainability reporting. They translate abstract, large-scale measures into understandable
percentages, empowering stakeholders to gauge and appreciate the company's ongoing efforts towards being an accountable and responsible business enterprise.
A Practical Guide to Calculating Basis Points
To begin the process of calculating basis points, follow these straightforward steps:
Step 1: Understand the Numbers
First, understand the numbers you are working with. If you're dealing with a shift from a 5% interest rate to a 5.5% rate, the change in terms of basis points is what you need to find out.
Step 2: Calculate the Difference
Next, compute the difference between the two interest rates. In our example, subtract 5% from 5.5% to get 0.5%. This represents the change in interest rate.
Step 3: Convert to Basis Points
Convert the percentage change you got from Step 2 into basis points. Remember that one basis point is equal to 0.01%, so multiply your result by 100. In our example, multiplying 0.5% by 100 provides
the result of 50. Therefore, the change in interest rate of 0.5% is equivalent to a change of 50 basis points.
Common Pitfalls to Avoid
While calculating basis points might seem easy, there are some common miscalculations that people often make. Here are few of them you ought to avoid:
Mistaking Percentage Points and Basis Points
One of the common mistakes is to confuse percentage points and basis points. Remember that 1% equals 100 basis points — not 1 basis point. In our above example, a 0.5% change in interest rate equates
to a 50 basis point change, not 0.5 basis points.
Misplacing Decimal Point
Misplacing the decimal could lead to a calculation error. Always double-check where you've placed your decimal point. In all our calculations, we used percentages; however, in a decimal format, a 5%
rate would be represented as 0.05, not 0.5.
Neglecting to Multiply by 100
When converting the difference in rates from a percentage to basis points, the key is to remember to multiply by 100. Missing this step might cause you to understate the difference drastically.
Keep these potential pitfalls in mind while calculating basis points to ensure the accuracy of your calculations.
Basis Points in Exchange Rate Movements
Basis points are a commonly used unit of measure in the exchange rate market, providing an accurate means to convey changes in currency value. When exchange rates move, even by a small fraction, it
can lead to significant outcomes in the international trade and investment sectors.
Generally speaking, a move of one basis point in an exchange rate is equivalent to a 1/100th of 1% change in the value of one currency against another. For example, if the exchange rate between the
US dollar and Euro moved from $1.2000 to $1.2001, it would represent a one basis point move.
Exchange Rate and International Trade
In the realm of international trade, the importance of basis points cannot be understated. A mere shift of a few basis points can dramatically impact the cost of importing or exporting goods. For
instance, a depreciation in a country's currency (say, a negative movement of 100 basis points or 1%) makes its exports cheaper and imports more expensive. This could boost the competitive advantage
of a country's export industry but simultaneously increase the cost of imported goods and services.
Exchange Rate and Investment Exposure
In terms of investment exposure, understanding basis points is crucial in managing foreign exchange risk. When an investor has money invested in another country, changes in the exchange rate
(expressed as basis points) can significantly impact their return on investment.
For instance, assume an investor who has purchased foreign bonds worth $1,000,000. If the foreign currency appreciates against the investor’s local currency by 100 basis points, the value of the bond
investment rises to $1,010,000 when converted back to the investor's local currency. Conversely, if the foreign currency depreciates by 100 basis points, the value falls to $990,000.
Hence, keeping a vigilant eye on the movement of basis points in exchange rates can offer global investors insights into potential profits or losses, serving as a crucial guide in managing investment
Marking changes in exchange rates using basis points allows traders and investors to quantify risks and opportunities, making it a notable tool in the world of finance and international trade.
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Basic Finite Element Mesh Explained
5 min read
Author: JC Sun
Publish Date: 28 Dec, 2021
To ensure stable analysis performance and reliable result approximation, meshing is important. Meshing is the process to create finite elements and to connect those elements to formulate a set of
functions. Finite elements are created by separating the known geometry with imaginary lines, and the elements are then connected with each other by specifying nodal connectivity at the element
boundaries. Every element can be represented by a set of matrices (shown later), and connecting the elements essentially compiles the individual matrices into one structural matrix.
• What is Finite Element Mesh?
Any non-time-dependent finite element analysis contains the following steps,
1. Meshing
2. Assigning boundary conditions
3. Applying loads
4. Numerical analysis
5. Postprocessing
To ensure stable analysis performance and reliable result approximation, meshing is important. Meshing is the process to create finite elements and to connect those elements to formulate a set of
functions. Finite elements are created by separating the known geometry with imaginary lines, and the elements are then connected with each other by specifying nodal connectivity at the element
boundaries. Every element can be represented by a set of matrices (shown later), and connecting the elements essentially compiles the individual matrices into 1 structural matrix.
Any non-time-dependent finite element analysis contains the following steps, Based on the shapes, there are the following types of finite elements:
1. One-dimensional elements
2. Two-dimensional elements
3. Three-dimensional elements
In this article, we are going over each type of the finite element listed above, explaining its pros and cons, and when to use each type.
One-dimensional elements (line elements) include bar elements and beam elements, and the difference between the two depends on their load-bearing capabilities. Bar elements are suitable for modeling
trusses because their load-bearing capabilities are limited in the axial directions. On the other hand, beam elements are suitable for modeling frames because they can resist bending, twisting, as
well as axial forces. A beam that is continuous over two or more supports can usually be modeled using one beam element per span between supports without using several beam elements to model one
individual span. Thus, when a one-dimensional element is used for static analysis, the discretization phase of modeling becomes trivial, and for stress analysis, the name “matrix methods of
structural mechanics” may be used in preference to “FEA” [1].
To represent a bar in axial tension, shown in figure 1, the following considerations can be made to compile the stiffness matrix.
Figure 1. A prism bar with one end fixed and the other end subjected to a force P.
A prismatic bar of length L, cross-section area A, and elastic modulus E is subjected to a force P at one end and with the other end fixed. The tensile elongation of the bar can be represented as
Figure 2. A two-node bar element.
The equilibrium at each individual node gives the following:
Which can be written in the matrix format as:
further reduction gives:
Where [k] is the characteristic matrix (stiffness matrix), {d} is the displacement vector, and {r} is the loads associated with each individual node.
Figure 3. A beam with two nodes (a) with two degrees of freedom at each node, translational in y and rotation in z, (b) with vertical loads and rotational moments at each node.
Figure 3 shows a beam element with two nodes. Each node is subjected to two degrees of freedom (figure 3a) and two nodal forces (figure 3b). Using the Euler-Bernoulli beam theory, the following
matrix equation can be formed:
The same element stiffness matrix can be obtained by calculating using interpolation and shape functions,
Where [B] is the strain-displacement matrix obtained from the shape functions [N]. We can also use shape function interpolation to obtain the stiffness matrix of 2D and 3D elements as well, as shown
later in the article.
Line elements are widely used in general structural analysis to obtain an overview of structural behavior, as well as in the detailed analysis in conjunction with other types of elements as
connections, stiffeners, etc. In bridge engineering, the general structural analysis using 1D elements gives engineers a comprehensive understanding of the bridge structural behavior, as well as
provides member forces and moments for design code checking for various standards.
However, to obtain more insights about localized structural zones, or when analyzing complex bridge geometries, local refinement would be required. To satisfy those higher analysis requirements,
two-dimensional elements and three-dimensional elements offer better result approximation. Figure 4 (left) shows the general analysis result using the 1D element, and figure 4 (right) shows the
refined analysis for the high moment elements using 2D elements. The nodal moment loads in the localized detailed analysis are extracted from the general analysis results.
Figure 4. General analysis using 1D element and using its element force-moment results as initial loads for the 2D detailed analysis.
To solve two-dimensional (2D) problems, 2D elements are needed. Similar to how stiffness matrix is constructed for 1D elements as shown in equation 6, 2D elements’ stiffness matrix can also be
constructed from shape function and interpolation,
where again, [k] is the stiffness matrix, [B] is the strain-displacement matrix obtained from the shape functions, [E] is the constitutive matrix, t is the thickness of the element, and A is the area
of the element.
Common 2D problems include plane stress, plane strain, shell, axisymmetric solid, geogrid 2D, and gauging shell elements. Plain strain and axisymmetric solid elements are 2D shape elements, but they
are used to express 3D stress states [2]. The common 2D element shape used is the triangular element with 3 nodes, shown in figure 5a, this type of element is called a constant strain triangle (CST)
because, in stress analysis, a linear displacement field produces a constant strain field [1]. The CST elements do not work very well, because the “locking” effect can make the mesh overly stiff [3].
Even though refining the mesh can help with the accuracy, it also does increase the analysis solving time. The inaccuracy due to “locking” can be improved with the 6 node triangular elements. As
shown in figure 5b, they have middle nodes between the vertices. 6 node triangular elements are also known as linear strain triangles (LST) or quadratic triangles.
Figure 5. (a) Constant-strain triangle element, (b) Linear strain triangle element.
A simple but less used 2D element is the 4-node rectangular element (Q4) whose sides are parallel to the global coordinate systems. This system is easy to construct automatically but it is not well
suited to approximate inclined boundaries [4]. The Q4 elements experience the same “locking” effects as the CST elements, however, the issue can be improved with quadratic rectangular elements (Q8,
Q9) by adding mid edge nodes.
• Three-Dimensional Elements
As shown in figure 6(a), tetrahedron has 4 nodes and is the most basic 3D finite element. Figure 6(b) shows a pentahedral (pyramid) element, figure 6(c) shows an 8 node rectangular solid element
(Q8), and figure 6(d) shows an 8 node hexahedral isoparametric element.
Figure 6. (a) 4 node tetrahedron element, (b) 5 node pentahedral (pyramid) element, (c) 8 node rectangular solid element, d) 8 node hexahedral isoparametric element.
Similar to rectangular Q4 elements and CST elements, Q8 also has the disadvantage of shear locking. Also similar to Q4 elements, Q8 rectangular solid elements are difficult to mesh irregular
geometries, especially for geometries with varying mesh densities. To reduce shear locking, adding mid-edge nodes to the Q8 element would help, and making the elements isoparametric will help with
meshing flexibilities. Isoparametric formulation permits quadrilateral and hexahedral elements to have non-rectangular shapes [1]. When combining the two efforts, adding mid-edge nodes to the linear
isoparametric hexahedral elements would make them more versatile and produce better analysis results.
3D geometries are needed to perform 3D mesh and they come from CAD models, which take longer and more effort to produce. Furthermore, analysis containing 3D elements usually contains more nodes and
elements thus would take longer to solve. 2D analysis, however, requires less time to mesh and less time to solve. The 2D analysis also provides a more sufficient amount of structural information
than 1D analysis and produces results close to 3D analysis. However, 2D analysis can only replace 3D analysis when structures can be represented by plate elements. When a structure is more complex,
as shown in figure 7, 3D elements need to be used.
Figure 7. A lug and pin model.
Engineers sometimes model structural elements using combined 1D/2D/3D mesh to take advantage of the benefit of each type of element while saving analysis time, as shown in the bridge model in figure
Figure 8. A bridge model utilizing hybrid element types with merged nodes at the element boundaries
[1] R. Cook, D. Malkus, M. Plesha, R. Witt, Concepts and Applications of Finite Element Analysis, Fourth Edition, 2001, John Wiley & Sons Inc., New York, NY.
[2] Midas Information Technology, Analysis Reference Midas FEA NX, Chapter 3, 2021.
[3] Th. Zimmermann, S. Commend, Stabilized Finite Element Applications in Geomechanics, Laboratory of Structural and Continuum Mechanics, Department of Civil Engineering Swiss Federal Institute of
Technology, 2001, Lausanne-EPFL, Switzerland.
[4] S.S. Bhavikatti, Finite Element Analysis, 2005, New Age International Limited, New Delhi.
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MATH 107 University of Maryland Global Campus College Algebra Final Exam - Course Help Online
I have attached the assignments and two answer sheets you can use. Please show all of your work neatly and write all of your answers on the Answer Sheet and if use an extra sheet of paper for solving
problems please attach as well.
You may have to write steps of the solution or justify your answer to get credit. You may type your work using plain-text formatting or an equation editor, or you may hand-write your work and scan
your work. In either case, show work neatly and correctly, following standard mathematical conventions. Each step should follow clearly and completely from the previous step. If necessary, you may
attach extra pages. If you do the work by hand, please make sure that the scans include all of your work, are completely readable, and are submitted right-side up. Most scanners have a setting that
will allow you to create one PDF document from all the pages of your Answer Sheet β please make use of this option if it is available on your scanner. You must submit a single file in commonly used
formats, for example word processing or PDF.
Math 107 Final Exam Spring 2020
Professor: Dr. Kamal Hennayake
This is an open-book exam. You may only refer to your text and other course materials as you work on the
exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor
consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1 β 12 are Multiple Choice. (Work not required to be shown)
Problems $13 β 25 are Short Answer. (Work not required to be shown)
Problems #26 β 30 are Short Answer with work required to be shown.
1) Find the coordinates of the vertex of the graph of the function.
π (π ₯) = β 2π ₯ 2 β 4π ₯ + 8
A. (β 1, 10)
B. (β 1, 14)
C. (1, 2)
D. (10, β 1)
E. (2, 1)
2) What is the end behavior of the graph?
left -negative, right -positive
left -positive, right β positive
left -negative, right β negative
left -positive, right -negative
3) Determine if the function is Odd, Even, or neither.
π (π ₯) = β
A. Odd
B. Even
+ 4π ₯
3π ₯
C. Neither
Math 107
Final Exam
4) What is the least possible degree of the polynomial graphed below?
4) Select the choice that is a graph of the function.
π (π ₯) = 6
π ₯β 1
(π ₯ + 3)(π ₯ β 2)
6) Solve the equation.
3π ₯ = 81
A. 2
B. 3
C. 4
D. 5
Math 107
Final Exam
7) Starting with the graph of π (π ₯) = 7π ₯ , write the equation of the graph that results from shifting f(x) 8 units to
the left.
A. π (π ₯) = 7π ₯+8
B. π (π ₯) = 7π ₯β 8
C. π (π ₯) = 7π ₯ + 8
D. π (π ₯) = 7π ₯ β 8
E. π (π ₯) = 8 β 7π ₯
8) What is the number of real zeros of the polynomial graphed below?
A. 3
B. 4
C. 5
D. 6
E. 7
9) Match the function with its graph. π (π ₯) = β 2(π ₯ + 1)2 + 5
Math 107
Final Exam
10) Which of the following criteria must be true about a function in order for that function to have an inverse?
π β 1 (π ₯) = π (π ₯)
π (π β 1 (π ₯)) = π ₯ and π β 1 (π (π ₯)) = π ₯
The function passes the horizontal line test
The function is one-to-one
A. ii only
B. ii, iii, iv
C. ii, iii
D. i, ii, iii, and iv
11) Which of the following statements indicate when c is a zero of a polynomial function f(x)?
c is the x-value of an x-intercept of π (π ₯).
c is a solution to f(x) = 0; that is f(c) = 0.
c is the function value of f(0); that is, f(0) = c.
c produces a remainder of 0 when f(x) is divided by x β c; that if π (π ₯) = (π ₯ β π )π (π ₯)
A. i, iii, iv
B. i, ii, iv
C. i, ii, iii, and iv
D. i, ii
12) Use the graph of the rational function to find the requested information.
As π ₯ β 3β , π (π ₯) β ?
β β
β 3
SHORT ANSWER (Work not required to be shown)
13) Give the domain of π (π ₯) = 3 + 7β 14 β 4π ₯ in interval notation.
14) The point (β 12, β 18) is on the graph of π ¦ = π (π ₯)
a. A point on the graph of π ¦ = π (π ₯), where π (π ₯) = 3 π (π ₯) + 15
b. A point on the graph of π ¦ = π (π ₯), where π (π ₯) = π (12 β π ₯)
c. A point on the graph of π ¦ = π (π ₯), where π (π ₯) = β 2π (β 3π ₯)
15) Tell whether the vertex is a maximum or a minimum. π (π ₯) = β 10π ₯ 2 β 81π ₯ β 811
Math 107
Final Exam
16) Consider the function graphed below. Answer the followings. Join multiple intervals with a union if
needed. For example: the domain of the function is (β β , β ).
a. Give the interval(s) where the function is increasing.
b. Give the interval(s) where the function is decreasing.
c. Give the interval(s) where the function is constant.
d. Give the range of the function using interval notation.
17) Complete the description of the piecewise function graphed below.
π π
π (π ₯) =
π π
π π
18) Consider the function π (π ₯) = 102|π ₯ + 2| β 20400 to find the intercepts.
a. Find the x-intercept(s) (if any as point(s)).
b. Find the y-intercept (if any as a point).
19) Determine an equation for the pictured graph. Write your answer in factored form. Do not expand the
20) Write an equation for a rational function with:
Vertical asymptotes at x = 6 and x = 5. The x intercepts at x = -5 and x = 3. Horizontal asymptote at y = 5
Math 107
Final Exam
21) Solve the following quadratic inequality. β π ₯ 2 + π ₯ β € 132. Write your answer in interval notation.
22) Write log π 2 = β 2 in exponential form.
23) Solve the equation log π π β 21 = π ₯ for x.
24) The number of bacteria in a culture is given by the function π (π ‘) = 9050π 0.45π ‘ . Where time is measured in
a. What is the relative rate of growth of this bacterium population?
b. What is the initial population of the culture?
c. How many bacteria will the culture contain in five days?
25) Given that π (π ₯) = β π ₯ 2 + 5π ₯ and π (π ₯) = π ₯ β 7 calculate the followings.
a. (π β π )(β 5)
b. (π β π )(β 5)
26) Solve the equation for x? Show work clearly.
log 3 (π ₯ + 3) = log 3 π ₯ + log 3 3
27) The population of the world in 1987 was 5 billion and the relative growth rate was estimated at 2 percent
per year. Assuming that the world population follows an exponential growth model, find the projected world
population in 2025. Round your answer to 2 decimal places in billions. Show work clearly.
28) Let π (π ₯) = π ₯β 3 and π (π ₯) = π ₯β 5.
a. Find (π β π )(π ₯) Show work clearly.
b. Find and the domain of (π β π )(π ₯). Justify your answer.
29) Solve the rational inequality. Write your answer in interval notation. Show work clearly.
π ₯β 8
β €1
π ₯+9
2π ₯
30) Find the inverse function of π (π ₯) = π ₯+7. Show work clearly.
Math 107 Final Examination
Spring, 2020
Math 107 College Algebra
Final Examination: Spring, 2020
Instructor __________________________
Answer Sheet
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator.
Record your answers and work in this document.
There are 30 problems.
Problems #1-12 are multiple choice. Record your choice for each problem.
Problems #13-25 are short answer. Record your answer for each problem.
Problems #26-30 are short answer with work required. When requested, show all work and write
all answers in the spaces allotted on the following pages. You may type your work using plaintext formatting or an equation editor, or you may hand-write your work and scan it. In either
case, show work neatly and correctly, following standard mathematical conventions. Each step
should follow clearly and completely from the previous step. If necessary, you may attach extra
You must complete the exam individually. Neither collaboration nor consultation with
others is allowed. Your exam will receive a zero grade unless you complete the following
honor statement.
Please sign (or type) your name below the following honor statement:
I have completed this final examination myself, working independently and not consulting
anyone except the instructor. I have neither given nor received help on this final examination.
Name _____________________
Math 107 Final Examination
Spring, 2020
MULTIPLE CHOICE. Record your answer choices.
SHORT ANSWER. Record your answers below.
Math 107 Final Examination
Spring, 2020
Math 107 Final Examination
Spring, 2020
SHORT ANSWER with Work Shown. Record your answers and work.
Math 107 Final Examination
Work for part (a):
Justification for part (b):
Spring, 2020
Math 107 Final Examination
Spring, 2020
Math 107 College Algebra
Final Examination: Spring, 2020
Instructor __________________________
Answer Sheet
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator.
Record your answers and work in this document.
There are 30 problems.
Problems #1-12 are multiple choice. Record your choice for each problem.
Problems #13-25 are short answer. Record your answer for each problem.
Problems #26-30 are short answer with work required. When requested, show all work and write
all answers in the spaces allotted on the following pages. You may type your work using plaintext formatting or an equation editor, or you may hand-write your work and scan it. In either
case, show work neatly and correctly, following standard mathematical conventions. Each step
should follow clearly and completely from the previous step. If necessary, you may attach extra
You must complete the exam individually. Neither collaboration nor consultation with
others is allowed. Your exam will receive a zero grade unless you complete the following
honor statement.
Please sign (or type) your name below the following honor statement:
I have completed this final examination myself, working independently and not consulting
anyone except the instructor. I have neither given nor received help on this final examination.
Name _____________________
Math 107 Final Examination
Spring, 2020
MULTIPLE CHOICE. Record your answer choices.
SHORT ANSWER. Record your answers below.
Math 107 Final Examination
Spring, 2020
Math 107 Final Examination
Spring, 2020
SHORT ANSWER with Work Shown. Record your answers and work.
Math 107 Final Examination
Work for part (a):
Justification for part (b):
Spring, 2020 | {"url":"https://coursehelponline.com/math-107-university-of-maryland-global-campus-college-algebra-final-exam-2/","timestamp":"2024-11-13T21:17:32Z","content_type":"text/html","content_length":"53764","record_id":"<urn:uuid:0f83e713-bcb3-4415-9529-e13b2311f916>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00554.warc.gz"} |
Data aggregation (numbers) | JavaScript
Free JavaScript course.
for tracking progress →
JavaScript: Data aggregation (numbers)
One particular class of tasks that cannot be done without loops is data aggregation. These problems include searching for the maximum and minimum values, as well as finding sums and averages. Their
main feature is that their result depends on the whole data set. Calculating the sum requires you to add all the numbers together, and calculating the maximum requires you to compare all the numbers.
Anyone who deals with numbers, e.g., accountants or marketers, will be familiar with these tasks. It's usually done using spreadsheet software like Microsoft Excel or Google Tables.
Let's consider a simple example: the sum of a set of numbers. We can implement a function to add numbers in a given range, including bounds. Here we have a range of numbers from a minimum value
(lower bound) to a maximum (upper bound). For example, the range [1, 10] includes all integers from 1 to 10.
sumNumbersFromRange(5, 7); // 5 + 6 + 7 = 18
sumNumbersFromRange(1, 2); // 1 + 2 = 3
// The range [1, 1] is also a range
// it includes just one number, the bound of the range itself
sumNumbersFromRange(1, 1); // 1
sumNumbersFromRange(100, 100); // 100
To implement this function, we need a loop because adding numbers is an iterative process (an iteration for each number), and the number of iterations depends on the size of the range. Before looking
at the code, try answering the questions below:
• Which value should you initialize the counter with?
• How will it change?
• When should the loop stop?
Try and think through these questions first and then have a look at the code below:
const sumNumbersFromRange = (start, finish) => {
// You can, of course, change the 'start' value
// But the input arguments must be left unchanged
// It makes the code easier to analyze
let i = start;
let sum = 0; // Sum initialization
while (i <= finish) { // Move to the end of the range
sum = sum + i; // Calculate sum for each number
i = i + 1; // Go to the next number in the range
// Return loop result
return sum;
The general structure of the loop here is standard. A counter initialized with a start value for the range, a loop with a condition to stop at the end of the range and, finally, a counter change at
the end of the loop. The number of iterations in this type of loop is finish - start + 1. Thus, for the range from 5 to 7, it is 7 - 5 + 1, or three iterations.
The main difference from regular processing is related to the logic of computing the result. In aggregation tasks, there is always a variable that stores the result of the loop. In the code above, it
is sum. With each loop iteration, this variable changes, adding another number from the range: sum = sum + i. The whole process looks like this:
// Calling sumNumbersFromRange(2, 5);
let sum = 0;
sum = sum + 2; // 2
sum = sum + 3; // 5
sum = sum + 4; // 9
sum = sum + 5; // 14
// 14 – the result of adding the numbers in the range [2, 5]
The sum variable has an initial value of 0. Why do you need to set the value at all? Any iterative operation starts with a value. You can't just declare an empty variable and start working with it
within a loop. It will lead to an incorrect result:
// when there is no initial value
// js sets it to undefined
let sum;
// first iteration
sum = sum + 2; // ?
It will result in NaN, i.e. not a number, in sum. It occurs due to an attempt to add 2 and undefined. So you need to have an initial value. Why is 0 chosen in the code above? Well, it is easy to
check that all the other options would lead to the wrong result. If the initial value is 1, the sum will be 1 more than it should be.
In mathematics, there is a concept of an identity element/neutral element, an element for each type of operation. Its meaning is easy to grasp. An operation with this element doesn't change the
operand. For addition, any number plus zero results in the number itself. The same goes for subtraction. Even concatenation has a neutral element, which is an empty string: '' + 'one' will be 'one'.
Self-check. What is the neutral element of a multiplication operation?
Write the multiplyNumbersFromRange() function that multiplies numbers in a given range, including its bounds. An example:
multiplyNumbersFromRange(1, 5); // 1 * 2 * 3 * 4 * 5 = 120
multiplyNumbersFromRange(2, 3); // 2 * 3 = 6
multiplyNumbersFromRange(6, 6); // 6
The exercise doesn't pass checking. What to do? 😶
If you've reached a deadlock it's time to ask your question in the «Discussions». How ask a question correctly:
• Be sure to attach the test output, without it it's almost impossible to figure out what went wrong, even if you show your code. It's complicated for developers to execute code in their heads, but
having a mistake before their eyes most probably will be helpful.
In my environment the code works, but not here 🤨
Tests are designed so that they test the solution in different ways and against different data. Often the solution works with one kind of input data but doesn't work with others. Check the «Tests»
tab to figure this out, you can find hints at the error output.
My code is different from the teacher's one 🤔
It's fine. 🙆 One task in programming can be solved in many different ways. If your code passed all tests, it complies with the task conditions.
In some rare cases, the solution may be adjusted to the tests, but this can be seen immediately.
I've read the lessons but nothing is clear 🙄
It's hard to make educational materials that will suit everyone. We do our best but there is always something to improve. If you see a material that is not clear to you, describe the problem in
“Discussions”. It will be great if you'll write unclear points in the question form. Usually, we need a few days for corrections.
By the way, you can participate in courses improvement. There is a link below to the lessons course code which you can edit right in your browser. | {"url":"https://code-basics.com/languages/javascript/lessons/aggregation-numbers","timestamp":"2024-11-03T09:09:29Z","content_type":"text/html","content_length":"32696","record_id":"<urn:uuid:61bce53e-f37e-4814-a414-0ac8df5eb0e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00544.warc.gz"} |
s Continuous Probability Distributions
Discrete Versus Continuous Probability Distributions
All probability distributions can be classified as discrete probability distributions or as continuous probability distributions, depending on whether they define probabilities associated with
discrete variables or continuous variables.
Discrete vs. Continuous Variables
If a variable can take on any value between two specified values, it is called a continuous variable; otherwise, it is called a discrete variable.
Some examples will clarify the difference between discrete and continuous variables.
• Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire
fighter's weight could take on any value between 150 and 250 pounds.
• Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity.
We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.
Just like variables, probability distributions can be classified as discrete or continuous.
Discrete Probability Distributions
If a random variable is a discrete variable, its probability distribution is called a discrete probability distribution.
An example will make this clear. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the random variable X represent
the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable.
The probability distribution for this statistical experiment appears below.
Number of heads Probability
0 0.25
1 0.50
2 0.25
The above table represents a discrete probability distribution because it relates each value of a discrete random variable with its probability of occurrence. On this website, we will cover the
following discrete probability distributions.
Note: With a discrete probability distribution, each possible value of the discrete random variable can be associated with a non-zero probability. Thus, a discrete probability distribution can always
be presented in tabular form.
Continuous Probability Distributions
If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution.
A continuous probability distribution differs from a discrete probability distribution in several ways.
• The probability that a continuous random variable will assume a particular value is zero.
• As a result, a continuous probability distribution cannot be expressed in tabular form.
• Instead, an equation or formula is used to describe a continuous probability distribution.
Most often, the equation used to describe a continuous probability distribution is called a probability density function. Sometimes, it is referred to as a density function, a PDF, or a pdf. For a
continuous probability distribution, the density function has the following properties:
• Since the continuous random variable is defined over a continuous range of values (called the domain of the variable), the graph of the density function will also be continuous over that range.
• The area bounded by the curve of the density function and the x-axis is equal to 1, when computed over the domain of the variable.
• The probability that a random variable assumes a value between a and b is equal to the area under the density function bounded by a and b.
For example, consider the probability density function shown in the graph below. Suppose we wanted to know the probability that the random variable X was less than or equal to a. The probability that
X is less than or equal to a is equal to the area under the curve bounded by a and minus infinity - as indicated by the shaded area.
Note: The shaded area in the graph represents the probability that the random variable X is less than or equal to a. This is a cumulative probability. However, the probability that X is exactly equal
to a would be zero. A continuous random variable can take on an infinite number of values. The probability that it will equal a specific value (such as a) is always zero.
On this website, we cover the following continuous probability distributions. | {"url":"https://stattrek.com/probability-distributions/discrete-continuous","timestamp":"2024-11-06T10:27:52Z","content_type":"text/html","content_length":"50853","record_id":"<urn:uuid:98d9b701-e2e2-4215-ad17-42f2907da6aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00707.warc.gz"} |
Closed Kinematic Chains
4.4 Closed Kinematic Chains
This section continues the discussion from Section 3.4. Suppose that a collection of links is arranged in a way that forms loops. In this case, the C-space becomes much more complicated because the
joint angles must be chosen to ensure that the loops remain closed. This leads to constraints such as that shown in (3.80) and Figure 3.26, in which some links must maintain specified positions
relative to each other. Consider the set of all configurations that satisfy such constraints. Is this a manifold? It turns out, unfortunately, that the answer is generally no. However, the C-space
belongs to a nice family of spaces from algebraic geometry called varieties. Algebraic geometry deals with characterizing the solution sets of polynomials. As seen so far in this chapter, all of the
kinematics can be expressed as polynomials. Therefore, it may not be surprising that the resulting constraints are a system of polynomials whose solution set represents the C-space for closed
kinematic linkages. Although the algebraic varieties considered here need not be manifolds, they can be decomposed into a finite collection of manifolds that fit together nicely.^4.11
Unfortunately, a parameterization of the variety that arises from closed chains is available in only a few simple cases. Even the topology of the variety is extremely difficult to characterize. To
make matters worse, it was proved in [489] that for every closed, bounded real algebraic variety that can be embedded in , there exists a linkage whose C-space is homeomorphic to it. These troubles
imply that most of the time, motion planning algorithms need to work directly with implicit polynomials. For the algebraic methods of Section 6.4.2, this does not pose any conceptual difficulty
because the methods already work directly with polynomials. Sampling-based methods usually rely on the ability to efficiently sample configurations, which cannot be easily adapted to a variety
without a parameterization. Section 7.4 covers recent methods that extend sampling-based planning algorithms to work for varieties that arise from closed chains.
Subsections Steven M LaValle 2020-08-14 | {"url":"https://lavalle.pl/planning/node168.html","timestamp":"2024-11-03T06:53:36Z","content_type":"text/html","content_length":"7877","record_id":"<urn:uuid:e8cc17e0-4752-4a20-98d5-f08c4c714587>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00356.warc.gz"} |
CS228 Programming Assignment #9—Learning with Incomplete Data
1 Introduction
In PA8, you learned tree-structured Bayesian networks to represent body poses. In this assignment, you will be using the network you learned for human poses from PA8 to perform action
recognition on real-world KinectTMdata. The KinectTMdata was collected by performing actions
and extracting the human pose data associated with each action. We have collected data for the
following three actions: “clap”, “high kick”, and “low kick”. Your goal in this assignment is to
build a model that is able to identify the action performed.
The approach that we will take in this assignment is to model an action as a sequence of poses.
However, unlike in PA8, we do not have the class labels for these poses. More specifically, in PA8
we knew whether a pose belonged to a human or an alien, but in the KinectTMdata, we do not
know in advance what poses comprise an action. In fact, we do not even know what the possible
poses that comprise the various actions are. As such, we will have to build and learn models
that incorporate a latent variable for pose classes, which makes this a more challenging task than
the classification problem in PA8. You will make extensive use of the Expectation-Maximization
(EM) algorithm to learn these models with latent variables.
2 Data Representation
We have processed the action data into a sequence of constituent poses. Each action has a
di↵erent number of poses in the sequence depending on the length of the action. The constituent
poses are in the same format used in PA8: that is, each pose has the 10 body parts each with
the three (y, x, ↵) components representing the pose. Refer to Section 2 and Figure 1 from PA8
for the representation and notation of each of the parts.
More formally, you are given a set of actions a1,…,an. Each action ai is a sequence of
poses pi
m, where the number of poses m is the length of the action. A pose pi
j can be
described by its body parts {Ok}10
k=1, with Ok = (yk, xk, ↵k). A pose is stored as a 10 x 3 matrix.
We will also use two of the data structures you saw in PA8, so take a moment to re-familiarize
yourself with them:
• P — This structure holds the model parameters, and is the same as the one used in PA8.
Slight changes are made to this structure for the section on HMM’s, which we describe
when they come up.
• G — This is the same graph structure used in PA8 for defining the tree structure dependencies between parts of a pose. Remember to handle all possible parameterizations of G
in your code.
CS228 Programming Assignment #9 2
3 Bayesian Clustering of the Poses
You will be experimenting with two latent variable models. The first is a Bayesian clustering
model, which you will use to cluster poses from di↵erent actions: we will combine the constituent
poses from all the action data and cluster them into groups of similar poses. Ideally, we will
automatically discover meaningful pose clusters where the majority of poses in the each cluster
are part of a single action.
The structure of the model is exactly as shown in Figure 1. This is similar to the model from
PA8, with a key di↵erence that the class variables C are now hidden variables; these variables
were observed in PA8 and given as part of the training data. As such, we no longer have a
completely labeled dataset to learn the model from, and we will use the EM algorithm to learn
the model from the partially labeled data.
In the clustering model, each cluster, or class, is associated with its own set of conditional
linear Gaussian (CLG) parameters for the tree-structured human pose network. If we have K
classes, then we will have K sets of CLG parameters. Our goal in the EM procedure is to learn
the CLG parameters for each class without knowledge of the true class labels for each pose. This
is similar to what you did in PA8, except that now we are not given the class label for each of
the poses. We describe the EM algorithm for learning the model in the following section.
O2 O3
Figure 1: Graphical model for Bayesian clustering of poses. The class variable is hidden here,
and we’ve only shown the first 6 body parts for clarity.
3.1 The EM algorithm for Bayesian Clustering
The general concept of EM is to generate a set of soft assignments for your hidden variables
(E-step) and use these assignments to update your estimates of the model parameters (M-step).
Each of these steps is done to maximize the model’s likelihood taking either parameters or unknown data as given, and we iterate back and forth between these steps until we have reached a
local maxima and can no longer improve the likelihood. We’ll take a look at the details of these
steps below:
E-step: In the E-step our goal is to do soft assignment of poses to classes. Thus, for each
pose, we infer the conditional probabilities for each of the K class labels using the current model
CS228 Programming Assignment #9 3
parameters. This can be done by first computing the joint probability of the class assignment
and pose (body parts), which decomposes as follows:
P(C = k, O1, ··· , O10) = P(C = k)
P(Oi|C = k, Opa(i)) (1)
where pa(i) is the index of part i’s parent. Each of the probabilites in the product can be
computed as described in PA8, and vary depending on the tree structure used. Once we have
computed the joint probability of the class assignment and pose, we can then condition on the
pose to obtain the conditional class probability (which are our expected sucient statistics):
P(C = k|O1, ··· , O10) (2)
Note that this is the conditional class probability for a single pose, and you will need to compute
this probability for all poses in the dataset.
M-step: In M-step, we seek to find the maximum likelihood CLG parameters given our soft
assignments of poses to classes (conditional class probabilities) from the E-step. Thus, for each
class k, we fit the CLG parameters with each pose weighted by its conditional probability as
given in Equation 2. The weight of a particular pose for class k is equivalent to the current
estimated probability that the pose belongs to the class. This step is identical to the parameter
fitting step in PA8, with the only di↵erence that each pose now makes a weighted contribution
to the estimation of the CLG parameters for all classes (as opposed to only for the given class
in PA8).
3.2 Data Structures and Code
Now let’s get to the implementation. We have provided you with the general structure for the
EM algorithm and the expected input and output parameters. It’s up to you to implement the
Note: if you need to normalize probabilities, do this in the log probability space before converting
to probabilities. This is required to avoid numerical issues.
• EM cluster.m (35 points) — This is the function where we run the overall EM procedure
for clustering. This function takes as input the dataset of poses, a known graph structure
G, a set of initial probabilities, and the maximum number of iterations to be run. Descriptions of the input and output parameters are given as comments in the code. Follow the
comments and structure given in the code to complete this function. Note that we start
with the M-step first.
Note on decreasing likelihood: You may notice that the log likelihood can decrease sometimes. This is because we are only printing out the log likelihood of the data. Since we smooth
the variances of our Gaussians to avoid numerical issues, the log likelihood of this smoothing
prior can a↵ect things. If we calculated the log likelihood of the data together with the prior,
that number should always increase. The same holds in the following sections as well.
You may find the following functions that we’ve provided useful in your implementation:
• FitGaussianParameters.m — This is similar to the function you wrote in PA8, while
allowing for weighted training examples. The weights control the influence of each example
on the parameters learned.
CS228 Programming Assignment #9 4
• FitLinearGaussianParameters.m — Similar to the function you wrote in PA8, with the
same addition of weights as in FitGaussianParameters.m.
• lognormpdf.m — Computes the natural logarithm of the normal probability density function.
• logsumexp.m — Computes the log(sum(exp(X))) of the input X. This function’s implementation avoids numerical issues when we must add probabilities that are currently in
log space.
Relevant Data Structures:
• poseData — N x 10 x 3 matrix that contains the pose data, where N is the number of
poses, and each pose is associated with a 10 x 3 matrix that describes its parts, similar to
PA8. The poseData is created by randomly sampling poses from all of the actions.
• ClassProb — N x K matrix that contains the conditional class probabilities of the N
examples to the K classes. ClassProb(i, j) is the conditional probability that example i
belongs to class j.
Sample Input/Output: To test your code, a seperate file PA9SampleCases.mat contains the input and correct output for each of the sample cases that we test for in the submit script (Similar to PA8).
For argument j of the function call in part i, you should use
exampleINPUT.t#ia#j (replacing the #j with j). For output, look at exampleOUTPUT.t#io#j
for argument j of the output to part i.
3.3 Evaluating the Clusters
With the implementation complete for the clustering algorithm, we can proceed to see how well
the clustering algorithm works in practice. We have given you a pre-extracted pose dataset,
a graph structure, and a set of initial probabilities in the file PA9Data.mat. You can also
experiment on your own with di↵erent graph structures and initial probabilities.
Try running the following command, which clusters the poses into 3 clusters:
[P loglikelihood ClassProb] = EM_cluster(poseData1, G, InitialClassProb1, 20)
With the posterior class probabilities ClassProb, we can obtain the cluster that each pose
associates most strongly with, and visualize the poses in each of these clusters. You can visualize
a set of poses using the function VisualizeDataset.m. These two steps can be done with the
following commands to visualize poses in cluster i:
[maxprob, assignments] = max(ClassProb, [], 2)
VisualizeDataset(poseData1(assignments == i, :, :))
We can see from the visualized poses that the clusters are not perfect, as the KinectTMdata is
not as well-behaved as the synthetic data used in PA8. In addition, we can look at the ground
truth action labels in the labels1 vector, which gives the action class that each pose was taken
from. This can be done for poses in cluster i as follows:
labels1(assignments == i)’
CS228 Programming Assignment #9 5
We can see that the clusters typically do not consist of poses from a single action. This makes intuitive sense, as actions may share similar poses. For example, in the “high kick” and “low kick”
actions, it’s possible that there are poses that both have the leg raised to the waist. We’ve also
provided a set of initial probabilities for clustering into 6 clusters using the following command:
[P loglikelihood ClassProb] = EM_cluster(poseData1, G, InitialClassProb2, 20)
What happens when we have twice the number of clusters? Are the clusters more concentrated
with a single action class? Do the visualizations look better?
4 A Bayesian Classifier for Action Recognition
In the action recognition problem, we are given a set of N training examples D = {(ai, ci)}
consisting of actions ai and their associated classes ci; there are 3 di↵erent types of actions in
the training set, so ci 2 {1, 2, 3}, where 1 is “clap”, 2 is “high kick” and 3 is “low kick”. We
wish to build a model to classify unseen actions. To do so, we will use the general framework of
a Bayesian classifier: we model the joint distribution over an action A and its class C as
P(A, C) = P(A|C)P(C) (3)
The intuition behind this model is that each action has a set of distinct characteristics that we
can capture using the class conditional action model P(A|C).
We can classify an action a by picking the class assignment c⇤ with the highest posterior
probability P(c⇤|a) given the action:
c⇤ = arg max c
P(c|a) = arg max c
P(a) = arg max c
P(a|c) (4)
The last equality follows because we have a uniform class distribution in the training set, and
the denominator P(a) is the same for each of the classes. Thus, given an unseen action, we can
classify it by simply computing P(a|c) over each class, and picking the class whose model yields
the highest likelihood.
We have yet to specify a key component of our classifier: the class conditional action model
P(A|C). What model should we use? One possibility would be to fit a Bayesian clustering model
(a mixture of poses model) for each action to model the types of poses that appear in each action
class. However, we can guess that this will probably not work, as suggested by the clustering
results in the previous section where many of these actions shared poses that look similar. Thus,
it is likely that the class conditional action models will be similar and the classifier will not
perform well.
A key observation about actions that will help us build a better action model is that though
the poses comprising an action may look similar, or even be the same, the sequence in which
these poses occur defines the action. For example, in the “low kick” action, we would expect a
sequence of poses in which the foot is lifted, kicked in a direction, and then returned back to
the original position. Thus, we should try to leverage the temporal nature of action poses in our
model. The mixture of poses model does not account for this temporal nature of actions, so we
will turn to a di↵erent model that allows us to leverage this information.
4.1 HMM action models
Using a Hidden Markov Model (HMM) for the action model will allow us to capture the sequential
nature of the data. Given an action ai consisting of a sequence of m poses pi
m, we
CS228 Programming Assignment #9 6
can construct a HMM of length m with hidden state variables S1,…,Sm for each pose that
correspond to the unknown pose classes. The HMM action model defines a joint distribution
over the hidden state variables and the poses comprising an action of length m:
P(S1,…,Sm, P1,…,Pm) = P(S1)P(P1|S1)
Since the HMM is a template model, it is parameterized by 3 CPDs – the prior distribution over
the initial states P(S1), the transition model P(S0
|S), and the emission model P(P|S). The first
two CPDs P(S1) and P(S0
|S) are table CPDs, while the emission model for pose Pj with parts
k=1 is
P(Pj |S) = Y
P(Oi|S, Opa(i))
where pa(i) denotes the parent of node i, is actually the pose model that you worked with in
Section 2.4 of PA8, with the skeletal structure for humans that you learned in the assignment.
This equation is very similar to Equation 1, with the only di↵erence here being that we aren’t
accounting for prior class probabilities anymore. Figure 2 shows an example HMM for an action
consisting of a sequence of 3 poses, where we have explicitly shown the first 6 body parts that
comprise a pose.
O2 O3
O2 O3
O2 O3
Figure 2: Graphical model for the HMM. In our HMM, each state variable represents the class
of the underlying pose. The emission probabilities for each pose are computed using the learned
4.2 Learning the HMM action model using EM
Since the state variables are hidden, we will use the EM algorithm to learn the parameters of the
HMM. The general EM framework (iterating between estimating assignments and parameters)
does not change, but the specifics of these steps can be tricky so we will now describe each of
the steps in detail.
E-step: In the E-step, we would like to compute the expected sucient statistics needed to
estimate our parameters. The two sets of expected sucient statistics needed are the marginals
over each individual state variable Si, as well as the pairwise marginals over consecutive state
variables Si and Si1.
Though computing these marginals may seem daunting, the E-step of the algorithm can be implemented using clique tree inference, which you wrote in PA4 and used in PA7. We include
CS228 Programming Assignment #9 7
optimized solution code for the functions from PA4 to help with inference in your HMM model,
which you can call using ComputeExactMarginalsHMM.m. In particular, we have modified
it to work in log space to avoid numerical issues.
As described in Section 4.1, the HMM consists of 3 types of factors (CPDs) that you will
need to include in your clique-tree for performing inference in the E-step. Note that for a single
action ai with a sequence of m observed poses pi
m, you will be able to reduce the factors (Pj , Sj ) = P(Pj |Sj ) by the observed poses pi
j to obtain singleton factors over Sj alone,
(Sj ) = (pi
j , Sj ). With these factors, we can run inference on the HMM for our action ai.
After running inference, we can extract the expected sucient statistics Mj [sj ] = P(Sj =
sj |ai) and M[s0
, s] = Pm
j=2 P(Sj = s0
, Sj1 = s|ai) from the calibrated clique tree for estimating
the initial state prior and the transition CPD, as explained below. Note that these statistics are
described for a single action, and you will need to aggregate these statistics across the entire
dataset of actions. We also extract the conditional class probabilities P(Sj = sj |ai) for each
state to estimate the emission CPD, as explained below. Also, as in PA7, we can compute the
likelihood of an action by marginalizing out the probabilities of any of the cliques.
Using the aggregated expected sucient statistics, we can re-estimate the parameters in the
• Initial state prior: The prior P(S1) can be re-estimated as:
P(S1 = s1) = M1[s1]
k=1 M1[k]
• Transition CPD: We can re-estimate these using the expected sucient statistics as:
P(S0 = s0
|S = s) = M[s0
, s]
k=1 M[k, s]
• Emission CPD: The parameters for the emission probabilities are the CLG parameters
used in pose clustering, and can be fit in the same way as the previous section, where we
treat each pose as having a weight equal to its conditional class probability, which reduces
to the M-step from pose clustering. Essentially, in this step you are doing pose clustering
over all the poses in all the actions.
4.3 Data Structures and Code
In practice, there are many numerical issues that arise when implementing the HMM. Thus, for
many of these functions, we would like to keep our probabilities in log space whenever possible,
and to always normalize in log space to avoid numerical issues. Before you start writing this
function, be sure to go over the details of the useful data structures and functions in Sections 5
and 6.
• EM HMM.m (35 points) — This is the function where you will implement the EM procedure
to learn the parameters of the action HMM. Many parts of this function should be the
same as portions of EM cluster.m. This function takes as input the dataset of actions, the
dataset of poses, a known graph structure G, a set of initial probabilities, and the number
of iterations to be run. Descriptions for the input and output parameters are given as
comments in the code. Follow the comments and structure given in the code to complete
this function. Note that we start with the M-step first.
CS228 Programming Assignment #9 8
4.4 Recognizing Actions
Now that you’ve implemented the EM algorithm to learn an HMM, we can now move on to our
end goal of action recognition. As discussed in Section 4, we will be using a Bayesian classifier,
which means that we will train a separate HMM for each action type, then classify actions based
on which HMM gives the highest likelihood for the action.
• RecognizeActions.m (10 points) — In this function, you should train an HMM for each
action class using the training set datasetTrain. Then, classify each of the instances in
the test set datasetTest and compute the classification accuracy. Details on the datasetTrain and datasetTest data structures can be found in Section 6.
At this point you have successfully implemented an action recognition system. Let’s see how
well this performs in practice. Run the command:
load PA9Data;
[accuracy, predicted_labels] = RecognizeActions(datasetTrain1, datasetTest1, G, 10)
If everything is implemented correctly, you should obtain an accuracy of 82%, which is
excellent accuracy on the recognition task; in comparison, random guessing will yield an accuracy
of 33%. You can use the VisualizeDataset.m function to visualize the actions. Try visualizing
some of the actions used to train the HMMs as well as the unknown actions that were classified
using the HMMs. Also, we have set the cardinality of the hidden state variables to be 3 for faster
4.4.1 E↵ect of EM Initialization on Recognition Accuracy
In the lectures, we learnt that the initialization of the EM algorithm can significantly a↵ect
the quality of the model that it returns. You will now see this e↵ect for yourself. Execute the
following command, which uses the same datasets for training and testing, but uses a di↵erent
set of initialializations:
[accuracy, predicted_labels] = RecognizeActions(datasetTrain2, datasetTest2, G, 10)
You should see that the accuracy drops to 73%, which is 9% less than that obtained with
the previous initialization! Both of the initial model parameters were randomly sampled from
a uniform distribution, but the di↵erence in accuracy is significant. This occurs because of
the multiple local maxima present in the likelihood function: the latter initialization caused the
algorithm to converge to a di↵erent local maximum. At this particular local maxima, the learned
models find it harder to distinguish between the di↵erent actions. This example illustrates the
importance of initialization in learning HMMs, and more generally speaking, latent variable
models. Typically, we will want to construct an initial set of model parameters using knowledge
about the latent variables in our model, rather than picking these parameters randomly.
4.5 Extending the Model
Your final task is to refine and extend the code you’ve written for recognizing actions. We
have provided you with the variables datasetTrain3 and datasetTest3 in the PA9Data.mat
file. Your task is to train models on the datasetTrain3 data to recognize the actions in
datasetTest3. We’ve given you a random set of initial probabilities in datasetTrain3 that
don’t perform very well. Note that we do not give you the labels in datasetTest3, which is
similar to the extra credit scenario in PA7.
CS228 Programming Assignment #9 9
• RecognizeUnknownActions.m (20 points) — In this function, you should train a model
for each action class using the training set datasetTrain3. Then, classify each of the instances in the test set datasetTest3 and save the classifier predictions by calling SavePredictions.m.
This function is left empty for you to be creative in your approach (we give some suggestions
below). When you are done, write a short description of your method in YourMethod.txt,
which is submitted along with your predictions in the submit script. Your score for this
part will be determined by the percentage of unknown action instances you successfully
recognize. If you obtain an accuracy of x%, you will receive 0.2x points for this part.
To help you get started, here are some suggestions. Try to be creative in your approach, leveraging the ideas you have learned over the entire class. Good luck!
• Better initializations — Can you devise a better method for initializing the model’s
parameters that will help EM to find a good local maxima? For example, you can try
initializing the probabilities by clustering the poses together using a simple method like
K-means or Hierarchical Agglomerative Clustering (HAC). To evaluate your initalizations,
you may want to split your dataset into a training set and a validation set, so you can
check your performance on the validation set.
• Hidden state variables — In the sample data we had you run, we fixed the hidden
state variables Si to have cardinality of 3. However, it’s possible that there are more than
3 underlying pose classes. Is there an optimal cardinality for the hidden state variables?
Again, a validation set might be useful to evaluate your performance.
• Extending the HMM — Is the HMM the best model for our action data? You might try
implementing a more complicated model, such as a duration HMM. Another possibility is
to place restrictions on the states and their transitions. For example, we might be certain
that the first state is always the same pose, and that the transitions typically occur in a
fixed order. How can we encode this in the initial state probabilities and transition matrix?
• Generating more data — The amount of data you are given to learn the HMMs is not
large. Given that we are learning a large number of parameters, more data could be very
helpful. Can you think of ways of warping the existing instances you have to obtain more
training data?
5 Useful Functions for HMM
List of functions we have written for you that you may find useful:
• ComputeExactMarginalsHMM.m — Similar to the ComputeExactMarginalsBP.m function you wrote in PA4. This function takes as input a set of factors in log space for an
HMM, and runs clique tree inference in log space to avoid numerical issues. The function
returns marginals (normalized) for each variable, as well as the calibrated clique tree (unnormalized). Note that the messages are unnormalized, as this may be helpful in computing
the log likelihood.
• SavePrediction.m — Call this function to save your predictions for the unknown actions.
This will prepare the predictions in a mat file so that they can be submitted in the submit
• VisualizeDataset.m — Helps to visualize a dataset of N poses of size N x 10 x 3.
CS228 Programming Assignment #9 10
6 Useful Data Structures for HMM
• P — This structure holds the model parameters. There are 2 changes made to this structure
for HMM’s that make this di↵erent from PA8 and the pose clustering section. First, there
is now a field P.transMatrix, which stores the K x K transition matrix, where K is the
number of possible classes. Second, the field P.c now contains the initial state prior
probabilities, instead of the prior class probability for all states.
• actionData — This is an array of structs that contains the actions. Each element of the
array is an instance of an action, with the fields action, marg ind, and pair ind defined
as follows:
– action — a string with the name of this action.
– marg ind — a set of indices that index the first dimensions in the associated poseData and ClassProb matrices that correspond to the sequence of poses that make up
this action. Thus, if marg ind= [10, 11,…, 19] then poseData(10, :, :), poseData(11,
:, :), …, poseData(19, :, 🙂 give the coordinates of this action’s 19-10+1=10 component poses. Additionally, rows 10 through 19 of ClassProb give the corresponding
class assignment probabilities of each of these individual poses.
– pair ind — a set of indices giving the rows in the PairProb data structure that
correspond to the pairwise transition probabilities for each pair of consecutive states
Si and Si1 in the action. Thus, if pair ind=[1, 2,… 9] then the pairwise transition
probability between the first and second poses in this action will be in row 1 of
PairProb, the transition between the 2nd and 3rd will be in row 2, and so forth.
• ClassProb — N x K matrix that contains the conditional class probabilities of the N
poses to the K classes. Note that N here is the total number of poses over all the actions.
ClassProb(i, j) is the conditional probability that pose i belongs to state j. Rows of this
matrix are indexed by the actionData structure. Using this matrix, you should be able
to estimate parameters in the M-step for the initial state prior and the emission CPDs.
• PairProb — V x K2 matrix that contains the pairwise transition probabilities (expected
sucient statistics for the transition CPD, M[s0
, s]) for every pair of consecutive states Si
and Si1 in all actions. Thus, every pair of consecutive states connected by an edge has
an entry in this matrix. V is the number of these edges. Rows of this matrix are indexed
by the actionData structure.
• datasetTrain — This is an array of structs that contains actions used for training. Each
element of the array is a struct for a di↵erent action, and contains the fields actionData,
poseData, InitialClassProb, and InitialPairProb, which are described above. The first
element of the array is “clap”, the second is “high kick”, and the third is “low kick”.
• datasetTest — This is the struct that contains actions used for testing. There are 3 fields:
actionData, poseData, and labels. The actionData and poseData are described above, and
the labels field is an N x 1 vector with the labels for each of the N action instances in
the datasetTest (with “clap” = 1, “high kick” = 2, “low kick” = 3). Note that for the
unknown test data, the labels field is not provided.
CS228 Programming Assignment #9 11
7 Conclusion
Congratulations! You’ve completed the final programming assignment for CS228! In this programming assignment, you’ve put together many of the ideas you’ve learned in class, ranging
from the basic factor data structures to advanced concepts like the Expectation-Maximization
algorithm, to create a system that is able to recognize human actions in real-world data. We
hope that you’ve enjoyed the assignment, and good luck for the final! | {"url":"https://codingprolab.com/answer/cs228-programming-assignment-9-learning-with-incomplete-data/","timestamp":"2024-11-14T00:56:40Z","content_type":"text/html","content_length":"166366","record_id":"<urn:uuid:c3d492ee-cc7a-474d-ab5d-d4c1a76a0eb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00186.warc.gz"} |
Valid Parentheses - LeetCode Solution
Valid Parentheses - LeetCode Problem
Problem Description
Welcome to another fun coding challenge! Today, we'll be tackling the Valid Parentheses problem.
Given a string containing just the characters '(', ')', '{', '}', '[' and ']', determine if the input string is valid. An input string is valid if:
1. Open brackets must be closed by the same type of brackets.
2. Open brackets must be closed in the correct order.
That's it! Now, let's dive into the solution.
First, we'll start by initializing an empty stack. Then, we'll iterate through each character in the input string. If the current character is an opening bracket, we'll push it onto the stack. If the
current character is a closing bracket, we'll check if the stack is empty. If it is, then the input string is invalid. Otherwise, we'll pop the top character from the stack and check if it matches
the closing bracket. If it doesn't match, then the input string is invalid. Finally, we'll check if the stack is empty. If it is, then the input string is valid. Otherwise, it's invalid.
Here's the Python code for our solution:
def isValid(s: str) -> bool:
stack = []
brackets = {')': '(', '}': '{', ']': '['}
for char in s:
if char in brackets.values():
elif char in brackets.keys():
if not stack or brackets[char] != stack.pop():
return False
return not stack
Let's go through this code step-by-step.
1. We start by initializing an empty stack and a dictionary of bracket pairs.
stack = []
brackets = {')': '(', '}': '{', ']': '['}
The dictionary brackets maps each closing bracket to its corresponding opening bracket. For example, ')' maps to '(', '}' maps to '{', and ']' maps to '['.
1. We iterate through each character in the input string s.
for char in s:
1. If the current character is an opening bracket, we'll push it onto the stack.
if char in brackets.values():
1. If the current character is a closing bracket, we'll check if the stack is empty. If it is, then the input string is invalid.
elif char in brackets.keys():
if not stack or brackets[char] != stack.pop():
return False
The if not stack condition checks if the stack is empty. If it is, then we know that there's no matching opening bracket for the current closing bracket, so the input string is invalid. The brackets
[char] != stack.pop() condition checks if the top of the stack contains the corresponding opening bracket for the current closing bracket. If it doesn't match, then the input string is invalid.
1. Finally, we'll check if the stack is empty. If it is, then the input string is valid. Otherwise, it's invalid.
return not stack
The not stack expression returns True if the stack is empty, and False if it's not empty. Since we're checking for validity, we want to return True if the stack is empty (i.e., all brackets have been
matched), and False if it's not empty (i.e., there are unmatched opening brackets).
Complexity Analysis
Now that we have our solution, let's analyze its time and space complexity.
Time Complexity
Our solution iterates through each character in the input string. Since we're doing constant time operations for each character (pushing and popping from a stack, and checking if a key is in a
dictionary), the time complexity of our solution is O(n), where n is the length of the input string.
Space Complexity
Our solution uses a stack to keep track of the opening brackets. The maximum size of the stack is n/2, where n is the length of the input string. This happens when the input string consists entirely
of opening brackets, followed by their corresponding closing brackets. Therefore, the space complexity of our solution is also O(n).
And there you have it! We've successfully tackled the Valid Parentheses problem. I hope you found this blog post helpful and informative. Remember, practice makes progress. Happy Coding! 😎. | {"url":"https://blog.eyucoder.com/valid-parentheses-leetcode-problem","timestamp":"2024-11-06T04:52:04Z","content_type":"text/html","content_length":"136983","record_id":"<urn:uuid:c618bed1-9576-457c-be23-256a340bbc1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00483.warc.gz"} |
The Mathematics of Datasearch
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A co-publication of the AMS, IAS/Park City Mathematics Institute, and Society for Industrial and Applied Mathematics
Hardcover ISBN: 978-1-4704-3575-2
Product Code: PCMS/25
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Click above image for expanded view
The Mathematics of Data
A co-publication of the AMS, IAS/Park City Mathematics Institute, and Society for Industrial and Applied Mathematics
Hardcover ISBN: 978-1-4704-3575-2
Product Code: PCMS/25
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
eBook ISBN: 978-1-4704-4990-2
Product Code: PCMS/25.E
List Price: $112.00
MAA Member Price: $100.80
AMS Member Price: $89.60
Hardcover ISBN: 978-1-4704-3575-2
eBook ISBN: 978-1-4704-4990-2
Product Code: PCMS/25.B
List Price: $237.00 $181.00
MAA Member Price: $213.30 $162.90
AMS Member Price: $189.60 $144.80
• IAS/Park City Mathematics Series
Volume: 25; 2018; 325 pp
MSC: Primary 15; 52; 60; 62; 65; 68; 90
Data science is a highly interdisciplinary field, incorporating ideas from applied mathematics, statistics, probability, and computer science, as well as many other areas. This book gives an
introduction to the mathematical methods that form the foundations of machine learning and data science, presented by leading experts in computer science, statistics, and applied mathematics.
Although the chapters can be read independently, they are designed to be read together as they lay out algorithmic, statistical, and numerical approaches in diverse but complementary ways.
This book can be used both as a text for advanced undergraduate and beginning graduate courses, and as a survey for researchers interested in understanding how applied mathematics broadly defined
is being used in data science. It will appeal to anyone interested in the interdisciplinary foundations of machine learning and data science.
This volume is a co-publication of the AMS, IAS/Park City Mathematics Institute, and Society for Industrial and Applied Mathematics
Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.
Graduate students and researchers interested in applied mathematics of data.
□ Articles
□ Petros Drineas and Michael Mahoney — Lectures on randomized numerical linear algebra
□ Stephen Wright — Optimization algorithms for data analysis
□ John Duchi — Introductory lectures on stochastic optimization
□ Per-Gunnar Martinsson — Randomized methods for matrix computations
□ Roman Vershynin — Four lectures on probabilistic methods for data science
□ Robert Ghrist — Homological algebra and data
□ What should you expect from a book titled 'The Mathematics of Data'? Nearly anything. There are numerous elementary books with similar titles that don't go far beyond showing the reader how
to compute the standard deviation. But what if you saw that the book was published by AMS and SIAM? That changes everything. You know it won't be elementary, and it will probably be high
quality, which is indeed the case here.
John D. Cook, MAA Reviews
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 25; 2018; 325 pp
MSC: Primary 15; 52; 60; 62; 65; 68; 90
Data science is a highly interdisciplinary field, incorporating ideas from applied mathematics, statistics, probability, and computer science, as well as many other areas. This book gives an
introduction to the mathematical methods that form the foundations of machine learning and data science, presented by leading experts in computer science, statistics, and applied mathematics.
Although the chapters can be read independently, they are designed to be read together as they lay out algorithmic, statistical, and numerical approaches in diverse but complementary ways.
This book can be used both as a text for advanced undergraduate and beginning graduate courses, and as a survey for researchers interested in understanding how applied mathematics broadly defined is
being used in data science. It will appeal to anyone interested in the interdisciplinary foundations of machine learning and data science.
This volume is a co-publication of the AMS, IAS/Park City Mathematics Institute, and Society for Industrial and Applied Mathematics
Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute.
Graduate students and researchers interested in applied mathematics of data.
• Articles
• Petros Drineas and Michael Mahoney — Lectures on randomized numerical linear algebra
• Stephen Wright — Optimization algorithms for data analysis
• John Duchi — Introductory lectures on stochastic optimization
• Per-Gunnar Martinsson — Randomized methods for matrix computations
• Roman Vershynin — Four lectures on probabilistic methods for data science
• Robert Ghrist — Homological algebra and data
• What should you expect from a book titled 'The Mathematics of Data'? Nearly anything. There are numerous elementary books with similar titles that don't go far beyond showing the reader how to
compute the standard deviation. But what if you saw that the book was published by AMS and SIAM? That changes everything. You know it won't be elementary, and it will probably be high quality,
which is indeed the case here.
John D. Cook, MAA Reviews
You may be interested in...
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Λ<sup>⁎</sup>(1405)-matter: Stable or unstable?
A recent suggestion by Akaishi and Yamazaki (2017) [3] that purely-Λ^⁎(1405) nuclei provide the absolute minimum energy in charge-neutral baryon matter for baryon-number A≳8, is tested within RMF
calculations. A broad range of Λ^⁎ interaction strengths, commensurate with (K¯K¯NN)[I=0] binding energy assumed to be of order 100 MeV, is scanned. It is found that the binding energy per Λ^⁎, B/A,
saturates for A≳120 with values of B/A considerably below 100 MeV, implying that Λ^⁎(1405) matter is highly unstable against strong decay to Λ and Σ hyperon aggregates. The central density of Λ^⁎
matter is found to saturate as well, at roughly twice nuclear matter density. Moreover, it is shown that the underlying very strong K¯N potentials, fitted for isospin I=0 to the mass and width values
of Λ^⁎(1405), fail to reproduce values of single-nucleon absorption fractions deduced across the periodic table from K^− capture-at-rest bubble chamber experiments.
Bibliographical note
Publisher Copyright:
© 2018 The Authors
• Kaonic atoms
• RMF
• Strange matter
• Λ(1405) resonance
Dive into the research topics of 'Λ^⁎(1405)-matter: Stable or unstable?'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/%CE%BBsupsup1405-matter-stable-or-unstable","timestamp":"2024-11-06T04:18:13Z","content_type":"text/html","content_length":"50040","record_id":"<urn:uuid:4d7a9f50-a7a6-4d14-a508-b7c0a5504fa0>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00152.warc.gz"} |
Acreage Calculator - calculator
Acreage Calculator
Acreage Calculator: When dealing with land transactions, property development, or agricultural planning, accurately calculating acreage is crucial. This is where an Acreage Calculator comes into
play. In this article, we will explore what an Acreage Calculator is, how to use it, its advantages and disadvantages, and how you can make the most of it for your needs.
Acreage Calculator
Unit Value
Acres (ac)
Square Millimeters (mm²)
Square Centimeters (cm²)
Square Decimeters (dm²)
Square Meters (m²)
Square Kilometers (km²)
Square Inches (in²)
Square Feet (ft²)
Square Yards (yd²)
Square Miles (mi²)
Ares (a)
Decares (da)
Hectares (ha)
Soccer Fields (sf)
What is an Acreage Calculator?
An Acreage Calculator is a tool designed to help users determine the size of a piece of land in acres. Acreage, a unit of measurement used to quantify large areas of land, is especially relevant in
real estate, agriculture, and land management. This calculator can handle various input formats and dimensions, making it versatile for different types of land measurements.
What is the Acreage Calculator Website?
The term “Acreage Calculator Calculator website” refers to online platforms that provide an Acreage Calculator tool. These websites are designed to offer users a straightforward interface for
entering land dimensions and receiving instant calculations of acreage. Examples include tools available on websites like “calculator.net” or “convertunits.com.”
Example Websites:
• Calculator.net: A popular site for various calculation needs, including acreage. It provides easy-to-use tools for converting land measurements.
• ConvertUnits.com: Offers a range of conversion tools, including an acreage calculator that caters to different input types.
How to Use an Acreage Calculator Website
Using an Acreage Calculator website is typically straightforward. Follow these steps to calculate acreage effectively:
Visit the Calculator Website: Open a reliable Acreage Calculator website. Search for “Acreage Calculator” in your browser, and select a reputable site.
Input Dimensions: Enter the dimensions of the land you want to measure. Depending on the tool, you might need to input measurements in feet, meters, or other units.
Select the Measurement Unit: Choose the appropriate unit of measurement if the calculator supports multiple units. Ensure that your dimensions are consistent with the chosen unit.
Calculate: Click the “Calculate” or equivalent button. The calculator will process your inputs and display the acreage result.
Review the Result: The calculated acreage will be shown, often with options to convert it into other units if needed.
Formula for Acreage Calculation
The basic formula to calculate acreage depends on the shape of the land. For a rectangular or square plot, the formula is:
Acreage=Length (in feet)×Width (in feet)/43,560
• Length and Width are the dimensions of the land in feet.
• 43,560 is the number of square feet in one acre.
For other shapes, such as irregular plots, the calculation might involve more complex formulas or methods, including breaking the plot into smaller sections and summing their areas.
Advantages and Disadvantages of an Acreage Calculator
1. Accuracy: These calculators provide precise results based on the input dimensions, helping users avoid manual calculation errors.
2. Efficiency: Instant results save time compared to manual calculations or using traditional methods.
3. User-Friendly: Most online calculators are designed to be intuitive, requiring minimal input from users.
4. Versatility: Many calculators support various input units and land shapes, making them adaptable to different needs.
1. Input Accuracy: The accuracy of the results depends on the accuracy of the input measurements. Incorrect dimensions will lead to incorrect acreage calculations.
2. Complex Shapes: For highly irregular plots, the calculator may require advanced inputs or methods that might not be straightforward.
3. Dependence on Technology: Online calculators require a stable internet connection and may not be accessible in all areas.
4. Potential Errors: Some calculators may not account for all measurement units or might have bugs, leading to potential inaccuracies.
Additional Information Related to Acreage Calculators
Applications of Acreage Calculators
1. Real Estate: Real estate agents and property buyers use acreage calculators to determine the size of plots and compare them with other properties.
2. Agriculture: Farmers and landowners use these tools to plan crop areas, manage land resources, and optimize agricultural practices.
3. Land Development: Developers use acreage calculators for planning construction projects, understanding land usage, and complying with zoning regulations.
4. Environmental Planning: Environmentalists use these calculators to assess land for conservation, habitat restoration, and other ecological projects.
Common Mistakes to Avoid
1. Incorrect Units: Ensure that you are using consistent units for length and width. Mixing units can lead to incorrect calculations.
2. Overlooking Shape Complexity: For irregularly shaped land, consider breaking it into simpler shapes and summing their areas for accurate results.
3. Neglecting Conversion Factors: If converting between units, double-check conversion factors to ensure accuracy.
4. Ignoring Calculator Limits: Be aware of the calculator’s limitations and verify results if the plot size is exceptionally large or small.
Related Calculator-
What is the difference between an Acreage Calculator and a Land Area Calculator?
While both calculators determine land size, an Acreage Calculator specifically converts measurements into acres, whereas a Land Area Calculator might provide results in various units (e.g., square
feet, square meters).
Can I use an Acreage Calculator for irregular-shaped land?
Yes, but you may need to use advanced methods or break the land into simpler shapes for accurate results. Some calculators handle irregular shapes better than others.
Are Acreage Calculators free to use?
Many online Acreage Calculators are free, though some websites might offer premium features or advanced tools for a fee.
How accurate are online Acreage Calculators?
Most online Acreage Calculators are quite accurate if correct dimensions are provided. However, always double-check results for critical applications and use reliable tools.
Can Acreage Calculators be used for commercial purposes?
Yes, Acreage Calculators are used in various commercial applications, including real estate, agriculture, and development projects. Ensure that the tool meets your needs and provides reliable | {"url":"https://calculatordna.com/acreage-calculator/","timestamp":"2024-11-12T10:37:53Z","content_type":"text/html","content_length":"103152","record_id":"<urn:uuid:0fb9260e-c172-4b7e-ab5c-6a641f953de4>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00203.warc.gz"} |
Student Perspectives: An Introduction to Graph Neural Networks (GNNs)
A post by Emerald Dilworth, PhD student on the Compass programme.
This blog post serves as an accessible introduction to Graph Neural Networks (GNNs). An overview of what graph structured data looks like, distributed vector representations, and a quick description
of Neural Networks (NNs) are given before GNNs are introduced.
An Introductory Overview of GNNs:
You can think of a GNN as a Neural Network that runs over graph structured data, where we know features about the nodes – e.g. in a social network, where people are nodes, and edges are them sharing
a friendship, we know things about the nodes (people), for instance their age, gender, location. Where a NN would just take in the features about the nodes as input, a GNN takes in this in addition
to some known graph structure the data has. Some examples of GNN uses include:
• Predictions of a binary task – e.g. will this molecule (which the structure of can be represented by with a graph) inhibit this given bacteria? The GNN can then be used to predict for a molecule
not trained on. Finding a new antibiotic is one of the most famous papers using GNNs [1].
• Social networks and recommendation systems, where GNNs are used to predict new links [2].
What is a Graph?
A graph, $G = (V,E)$, is a data structure that consists of a set of nodes, $V$, and a set of edges, $E$. Graphs are used to represent connections (edges) between objects (nodes), where the edges can
be directed or undirected depending on whether the relationships between the nodes have direction. An $n$ node graph can be represented by an $n \times n$ matrix, referred to as an adjacency matrix.
Idea of Distributed Vector Representations
In machine learning architectures, the data input often needs to be converted to a tensor for the model, e.g. via 1-hot encoding. This provides an input (or local) representation of the data, which
if we think about 1-hot encoding creates a large, sparse representation of 0s and 1s. The input representation is a discrete representation of objects, but lacks information on how things are
correlated, how related they are, what they have in common. Often, machine learning models learn a distributed representation, where it learns how related objects are; nodes that are similar will
have similar distributed representations. If each node is represented by an input vector, when the model learns a distributed vector, this is typically a much smaller vector. The diagram below gives
a visual representation of how the model goes from a local to distributed representation [3]. As an example, the skip-gram model is a method which learns distributed vector representations to capture
syntactic and semantic word relationships [4].
Brief Overview of what a Neural Network does
A neural network is a machine learning model comprised of interconnected nodes (called artificial neurons), which are organised into layers. These neurons process and transmit information through
weighted connections, where the weights are tunable parameters. The layer-wise structure allows the mappings to create more and more abstract representations of the inputs. A classic example of a NN
is for image classification. The NN is trained on a large dataset of images, where each image is labelled with a specific category, e.g. “dog” or “cat”. The NN learns the features and patterns in
the images that help it to classify the image into a category. Then when presented with an unseen image, it will output probabilities of which category the image belongs to. They are trained in an
iterative manner, and there are lots of adjustable choices for the model which can effect the prediction accuracy. By training NNs on large amounts of data to learn, they are useful to make
predictions, classify data, and recognise patterns.
Graph Neural Networks (GNNs)
The inputs to a GNN are:
• An $n \times d$ feature matrix of the nodes, $X$ ($n$ nodes, $d$ input features). $X$ could for example be some 1-hot encoding matrix of features about the nodes.
• An $n \times n$ adjacency matrix, $A$.
You can think of a GNN as taking in an initial (input) representation of each node, $X$, and a graph structure the nodes belong to, $A$, and outputting some output representation of each node, $H$.
The initial representation has information about each node individually, and the output represents how each node belongs within the context of the graph given its features. The GNN exploits local
interactions made by nodes to update the features of each node and maps to a latent feature space, $H$. $H$ is an $n \times f$ matrix, where $f$ is the number of output features per node, which can
be thought of as the distributed representation in the context of distributed vector representations. Usually $f << d$.
$\underset{inputs}{(X, A)} \xrightarrow{\text{GNN}} \underset{outputs}{(H, A)}$
If you are familiar with embedding methods, you may think of $H$ as the embedding vectors of the graph after passing it through the GNN. Similar nodes will be closer to one another in the embedding
The data used for training the GNN is split like it would be for a NN, into a train-validation-test split. At each layer in the GNN, the values of $H$ are updated in a train-validation setup, and
once all layers are passed through, the test data is used to evaluate the performance of the model. If there are $L$ layers of the GNN, then there are $L$ values of $H$ calculated, $H^{(0)}$,…,$H^
{(L)}$, where $H^{(0)} = X$. But how is $H$ calculated?
Let’s start by considering the case where the graph $A$ is a binary undirected and unweighted graph – i.e. $A_{ij} = A_{ji} = \begin{cases} 0, & \text{if}\ i \leftrightarrow j \\ 1, & \text
{otherwise} \end{cases}$. Each neural network layer can be thought of as a non-linear function update of the previous layer:
$H^{(\ell +1)} = f(H^{(\ell)}, A)$
The particular GNN model choice is decided by the choice of $f(\cdot, \cdot)$.
If we consider the simple update rule: $H^{(\ell+1)} = \sigma(A H^{(\ell)} W^{(\ell)})$, where $W^{(\ell)}$ is a learnable node-wise shared linear transformation (the weight matrix for the $\ell^{th}
$ layer) and $\sigma$ is a non-linear activation function (e.g. ReLU), this model encounters two limitations listed below. Most models which are used in practice make two adjustments to the input to
circumvent possible problems/limitations:
1. If a node does not have a self loop in the graph ($A_{ii} = 0$), when any matrix is multiplied with A, for every node the feature vectors of all neighbouring nodes are summed up, but the node
itself is not included. This means that the node is not considering the information it has about itself. To fix this, $A$ is updated to have every node connect to itself, by $\tilde{A} = A + I$,
where $I$ is the $n \times n$ identity matrix.
2. If $A$ or $\tilde{A}$ is multiplied by another matrix, it can change the scale of the output features. Therefore it is usual to normalise avoid this problem – e.g. multiplying $\tilde{A}$ by the
degree matrix $\tilde{D}$, where $\tilde{D}_{ii} = \sum\limits_j \tilde{A}_{ij}$. This adjustment is sometimes referred as a mean pooling update rule.
A popular choice of GNN is the Graph Convolutional Network (GCN) [5]. We shall look at the update rules for this model in a bit more detail.
Graph Convolutional Network (GCN)
For this model, the input graph used is $\tilde{A}$ as described above, however the normalisation rule used is the symmetric normalisation: $\tilde{D}^{-\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}
{2}}$. The update rule for a GCN thus looks like:
$H^{(\ell+1)} = \sigma ( \tilde{D}^{-\frac{1}{2}} \tilde{A} \tilde{D}^{-\frac{1}{2}} H^{(\ell)} W^{(\ell)})$
where for each node this looks like:
$\textbf{h}_i^{(\ell+1)} = \sigma \left( \sum\limits_{j \in N_i} \frac{1}{\sqrt{|N_i||N_j|}} W^{(\ell)} \textbf{h}_j^{(\ell)} \right)$
$N_i$ denotes the set of neighbours of node $i$. At each layer, node $i$ is being updated by a non-linear transformation ($\sigma$) of the average of all its neighbours features in that layer. At the
beginning each node knows about itself. In the first layer, it can learn about its distance 1 neighbours, in the second layer it begins to learn something about its distance 2 neighbours . At each
layer the nodes learn more about how they belong in the graph, there is an increasing perceptive field of what you know about each node.
More generally, a GCN can be expressed as an explicit version of the below:
$\textbf{h}_i^{(\ell+1)} = \sigma \left( \sum\limits_{j \in N_i} \alpha_{ij} W^{(\ell)} \textbf{h}_j^{(\ell)} \right)$
where $\alpha_{ij}$ is explicitly defined as $\frac{1}{\sqrt{|N_i||N_j|}}$ in a GCN. $\alpha_{ij}$ says something about the importance of node $j$‘s features for node $i$.
A popular benchmark/example of implementing GNNs is on the Cora dataset [6]. [6] describes the dataset as: “The Cora dataset consists of 2708 scientific publications classified into one of seven
classes. The citation network consists of 5429 links. Each publication in the dataset is described by a 0/1-valued word vector indicating the absence/presence of the corresponding word from the
dictionary. The dictionary consists of 1433 unique words.” There is a graph structure in the citation network and feature vectors for each node (of words in the dictionary) which can both be used in
a GNN model. One example of use is to train the model to be able to classify papers that were not used to train the model into one of the seven classes based on the words present in their feature
vector (words in the dictionary). [6] provides more examples of uses and comparisons of different GNN models based on their accuracy.
[1] Stokes, Jonathan M., et al. “A deep learning approach to antibiotic discovery.” Cell 180.4 (2020): 688-702.
[2] Fan, Wenqi, et al. “Graph neural networks for social recommendation.” The world wide web conference. 2019.
[3] MSR Cambridge, AI Residency Advanced Lecture Series. An Introduction to Graph Neural Networks: Models and Applications, 2020.
[4] Mikolov, Tomas, et al. “Distributed representations of words and phrases and their compositionality.” Advances in neural information processing systems 26 (2013).
[5] Kipf, Thomas N., and Max Welling. “Semi-supervised classification with graph convolutional networks.” arXiv preprint arXiv:1609.02907 (2016).
[6] https://paperswithcode.com/dataset/cora | {"url":"https://compass.blogs.bristol.ac.uk/2023/02/17/student-perspectives-an-introduction-to-graph-neural-networks-gnns/","timestamp":"2024-11-05T09:35:28Z","content_type":"text/html","content_length":"71311","record_id":"<urn:uuid:a6f5e4fd-9b0f-4f29-a963-cfdc5abd20de>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00196.warc.gz"} |
558 Sign/Square Week to Arcsec/Square Minute
Sign/Square Week [sign/week2] Output
558 sign/square week in degree/square second is equal to 4.5764833711262e-8
558 sign/square week in degree/square millisecond is equal to 4.5764833711262e-14
558 sign/square week in degree/square microsecond is equal to 4.5764833711262e-20
558 sign/square week in degree/square nanosecond is equal to 4.5764833711262e-26
558 sign/square week in degree/square minute is equal to 0.00016475340136054
558 sign/square week in degree/square hour is equal to 0.59311224489796
558 sign/square week in degree/square day is equal to 341.63
558 sign/square week in degree/square week is equal to 16740
558 sign/square week in degree/square month is equal to 316502.64
558 sign/square week in degree/square year is equal to 45576379.52
558 sign/square week in radian/square second is equal to 7.9874702988922e-10
558 sign/square week in radian/square millisecond is equal to 7.9874702988922e-16
558 sign/square week in radian/square microsecond is equal to 7.9874702988922e-22
558 sign/square week in radian/square nanosecond is equal to 7.9874702988922e-28
558 sign/square week in radian/square minute is equal to 0.0000028754893076012
558 sign/square week in radian/square hour is equal to 0.010351761507364
558 sign/square week in radian/square day is equal to 5.96
558 sign/square week in radian/square week is equal to 292.17
558 sign/square week in radian/square month is equal to 5524.01
558 sign/square week in radian/square year is equal to 795457.88
558 sign/square week in gradian/square second is equal to 5.0849815234736e-8
558 sign/square week in gradian/square millisecond is equal to 5.0849815234736e-14
558 sign/square week in gradian/square microsecond is equal to 5.0849815234736e-20
558 sign/square week in gradian/square nanosecond is equal to 5.0849815234736e-26
558 sign/square week in gradian/square minute is equal to 0.00018305933484505
558 sign/square week in gradian/square hour is equal to 0.65901360544218
558 sign/square week in gradian/square day is equal to 379.59
558 sign/square week in gradian/square week is equal to 18600
558 sign/square week in gradian/square month is equal to 351669.6
558 sign/square week in gradian/square year is equal to 50640421.68
558 sign/square week in arcmin/square second is equal to 0.0000027458900226757
558 sign/square week in arcmin/square millisecond is equal to 2.7458900226757e-12
558 sign/square week in arcmin/square microsecond is equal to 2.7458900226757e-18
558 sign/square week in arcmin/square nanosecond is equal to 2.7458900226757e-24
558 sign/square week in arcmin/square minute is equal to 0.0098852040816327
558 sign/square week in arcmin/square hour is equal to 35.59
558 sign/square week in arcmin/square day is equal to 20497.96
558 sign/square week in arcmin/square week is equal to 1004400
558 sign/square week in arcmin/square month is equal to 18990158.13
558 sign/square week in arcmin/square year is equal to 2734582770.92
558 sign/square week in arcsec/square second is equal to 0.00016475340136054
558 sign/square week in arcsec/square millisecond is equal to 1.6475340136054e-10
558 sign/square week in arcsec/square microsecond is equal to 1.6475340136054e-16
558 sign/square week in arcsec/square nanosecond is equal to 1.6475340136054e-22
558 sign/square week in arcsec/square minute is equal to 0.59311224489796
558 sign/square week in arcsec/square hour is equal to 2135.2
558 sign/square week in arcsec/square day is equal to 1229877.55
558 sign/square week in arcsec/square week is equal to 60264000
558 sign/square week in arcsec/square month is equal to 1139409487.88
558 sign/square week in arcsec/square year is equal to 164074966255.1
558 sign/square week in sign/square second is equal to 1.5254944570421e-9
558 sign/square week in sign/square millisecond is equal to 1.5254944570421e-15
558 sign/square week in sign/square microsecond is equal to 1.5254944570421e-21
558 sign/square week in sign/square nanosecond is equal to 1.5254944570421e-27
558 sign/square week in sign/square minute is equal to 0.0000054917800453515
558 sign/square week in sign/square hour is equal to 0.019770408163265
558 sign/square week in sign/square day is equal to 11.39
558 sign/square week in sign/square month is equal to 10550.09
558 sign/square week in sign/square year is equal to 1519212.65
558 sign/square week in turn/square second is equal to 1.2712453808684e-10
558 sign/square week in turn/square millisecond is equal to 1.2712453808684e-16
558 sign/square week in turn/square microsecond is equal to 1.2712453808684e-22
558 sign/square week in turn/square nanosecond is equal to 1.2712453808684e-28
558 sign/square week in turn/square minute is equal to 4.5764833711262e-7
558 sign/square week in turn/square hour is equal to 0.0016475340136054
558 sign/square week in turn/square day is equal to 0.94897959183673
558 sign/square week in turn/square week is equal to 46.5
558 sign/square week in turn/square month is equal to 879.17
558 sign/square week in turn/square year is equal to 126601.05
558 sign/square week in circle/square second is equal to 1.2712453808684e-10
558 sign/square week in circle/square millisecond is equal to 1.2712453808684e-16
558 sign/square week in circle/square microsecond is equal to 1.2712453808684e-22
558 sign/square week in circle/square nanosecond is equal to 1.2712453808684e-28
558 sign/square week in circle/square minute is equal to 4.5764833711262e-7
558 sign/square week in circle/square hour is equal to 0.0016475340136054
558 sign/square week in circle/square day is equal to 0.94897959183673
558 sign/square week in circle/square week is equal to 46.5
558 sign/square week in circle/square month is equal to 879.17
558 sign/square week in circle/square year is equal to 126601.05
558 sign/square week in mil/square second is equal to 8.1359704375577e-7
558 sign/square week in mil/square millisecond is equal to 8.1359704375577e-13
558 sign/square week in mil/square microsecond is equal to 8.1359704375577e-19
558 sign/square week in mil/square nanosecond is equal to 8.1359704375577e-25
558 sign/square week in mil/square minute is equal to 0.0029289493575208
558 sign/square week in mil/square hour is equal to 10.54
558 sign/square week in mil/square day is equal to 6073.47
558 sign/square week in mil/square week is equal to 297600
558 sign/square week in mil/square month is equal to 5626713.52
558 sign/square week in mil/square year is equal to 810246746.94
558 sign/square week in revolution/square second is equal to 1.2712453808684e-10
558 sign/square week in revolution/square millisecond is equal to 1.2712453808684e-16
558 sign/square week in revolution/square microsecond is equal to 1.2712453808684e-22
558 sign/square week in revolution/square nanosecond is equal to 1.2712453808684e-28
558 sign/square week in revolution/square minute is equal to 4.5764833711262e-7
558 sign/square week in revolution/square hour is equal to 0.0016475340136054
558 sign/square week in revolution/square day is equal to 0.94897959183673
558 sign/square week in revolution/square week is equal to 46.5
558 sign/square week in revolution/square month is equal to 879.17
558 sign/square week in revolution/square year is equal to 126601.05 | {"url":"https://hextobinary.com/unit/angularacc/from/signpw2/to/arcsecpmin2/558","timestamp":"2024-11-11T07:52:39Z","content_type":"text/html","content_length":"113168","record_id":"<urn:uuid:e6b75f30-9ca4-4095-9321-ba2807cc8039>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00546.warc.gz"} |
CuPy is an open-source array library for GPU-accelerated computing with Python. CuPy utilizes CUDA Toolkit libraries including cuBLAS, cuRAND, cuSOLVER, cuSPARSE, cuFFT, cuDNN and NCCL to make full
use of the GPU architecture. The figure shows CuPy speedup over NumPy. Most operations perform well on a GPU using CuPy out of the box. CuPy speeds up some operations more than 100X. Read the
original benchmark article Single-GPU CuPy Speedups on the RAPIDS AI Medium blog.
CuPy's interface is highly compatible with NumPy and SciPy; in most cases it can be used as a drop-in replacement. All you need to do is just replace numpy and scipy with cupy and cupyx.scipy in your
Python code. The Basics of CuPy tutorial is useful to learn first steps with CuPy. CuPy supports various methods, indexing, data types, broadcasting and more. This comparison table shows a list of
NumPy / SciPy APIs and their corresponding CuPy implementations.
>>> import cupy as cp >>> x = cp.arange(6).reshape(2, 3).astype('f') >>> x array([[ 0., 1., 2.], [ 3., 4., 5.]], dtype=float32) >>> x.sum(axis=1) array([ 3., 12.], dtype=float32)
The easiest way to install CuPy is to use pip. CuPy provides wheels (precompiled binary packages) for Linux and Windows. Read the Installation Guide for more details. CuPy can also be installed from
Conda-Forge or from source code.
# For CUDA 11.2 ~ 11.x pip install cupy-cuda11x # For CUDA 12.x pip install cupy-cuda12x # For AMD ROCm 4.3 pip install cupy-rocm-4-3 # For AMD ROCm 5.0 pip install cupy-rocm-5-0
You can easily make a custom CUDA kernel if you want to make your code run faster, requiring only a small code snippet of C++. CuPy automatically wraps and compiles it to make a CUDA binary. Compiled
binaries are cached and reused in subsequent runs. Please read the User-Defined Kernels tutorial. And, you can also use raw CUDA kernels via Raw modules.
>>> x = cp.arange(6, dtype='f').reshape(2, 3) >>> y = cp.arange(3, dtype='f') >>> kernel = cp.ElementwiseKernel( ... 'float32 x, float32 y', 'float32 z', ... ''' ... if (x - 2 > y) { ... z = x * y;
... } else { ... z = x + y; ... } ... ''', 'my_kernel') >>> kernel(x, y) array([[ 0., 2., 4.], [ 0., 4., 10.]], dtype=float32) | {"url":"https://cupy.dev/?featured_on=pythonbytes","timestamp":"2024-11-07T17:25:12Z","content_type":"text/html","content_length":"17728","record_id":"<urn:uuid:6fc8a9ae-3854-42fb-9a30-6188a06dfd13>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00732.warc.gz"} |
Sales and Revenue
Let's consolidate acquired knowledge with a practical task!
If you want to multiply each element of the row to a specific value (i.e., not one number for all elements), you need to have the condition: the number of rows of the matrix equals the
multiplying vector length satisfied.
You have the selling data for a local furniture store stored in the sellings matrix.
Month Sofa Armchair Dining table Dining chair Bookshelf
March 16 21 30 23 10
April 40 39 13 21 16
May 11 21 36 32 16
And vector of prices named prices.
Sofa Armchair Dining table Dining chair Bookshelf
Your tasks are:
1. Output the total number of items sold each month.
2. Transpose of the sellings matrix and reassign the result to the sellings variable.
3. Find out revenue by each good by multiplying sellings and prices. Save the result to the income variable.
4. Output the monthly revenues.
5. Output the total three months' revenue.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Let's consolidate acquired knowledge with a practical task!
If you want to multiply each element of the row to a specific value (i.e., not one number for all elements), you need to have the condition: the number of rows of the matrix equals the
multiplying vector length satisfied.
You have the selling data for a local furniture store stored in the sellings matrix.
Month Sofa Armchair Dining table Dining chair Bookshelf
March 16 21 30 23 10
April 40 39 13 21 16
May 11 21 36 32 16
And vector of prices named prices.
Sofa Armchair Dining table Dining chair Bookshelf
Your tasks are:
1. Output the total number of items sold each month.
2. Transpose of the sellings matrix and reassign the result to the sellings variable.
3. Find out revenue by each good by multiplying sellings and prices. Save the result to the income variable.
4. Output the monthly revenues.
5. Output the total three months' revenue.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Let's consolidate acquired knowledge with a practical task!
If you want to multiply each element of the row to a specific value (i.e., not one number for all elements), you need to have the condition: the number of rows of the matrix equals the
multiplying vector length satisfied.
You have the selling data for a local furniture store stored in the sellings matrix.
Month Sofa Armchair Dining table Dining chair Bookshelf
March 16 21 30 23 10
April 40 39 13 21 16
May 11 21 36 32 16
And vector of prices named prices.
Sofa Armchair Dining table Dining chair Bookshelf
Your tasks are:
1. Output the total number of items sold each month.
2. Transpose of the sellings matrix and reassign the result to the sellings variable.
3. Find out revenue by each good by multiplying sellings and prices. Save the result to the income variable.
4. Output the monthly revenues.
5. Output the total three months' revenue.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Let's consolidate acquired knowledge with a practical task!
If you want to multiply each element of the row to a specific value (i.e., not one number for all elements), you need to have the condition: the number of rows of the matrix equals the
multiplying vector length satisfied.
You have the selling data for a local furniture store stored in the sellings matrix.
Month Sofa Armchair Dining table Dining chair Bookshelf
March 16 21 30 23 10
April 40 39 13 21 16
May 11 21 36 32 16
And vector of prices named prices.
Sofa Armchair Dining table Dining chair Bookshelf
Your tasks are:
1. Output the total number of items sold each month.
2. Transpose of the sellings matrix and reassign the result to the sellings variable.
3. Find out revenue by each good by multiplying sellings and prices. Save the result to the income variable.
4. Output the monthly revenues.
5. Output the total three months' revenue.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Switch to desktop for real-world practiceContinue from where you are using one of the options below | {"url":"https://codefinity.com/courses/v2/19992f09-74f9-4908-abde-61ecd3a95a27/878e6c6f-1772-40f2-a941-c0aa1dae6a3a/bd4aa8b9-3bc9-4b91-acb8-48ce6ca526d1","timestamp":"2024-11-06T13:42:18Z","content_type":"text/html","content_length":"353222","record_id":"<urn:uuid:e3005a18-2755-49a7-9827-aedd3730fb8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00099.warc.gz"} |
This is why your quotes of T getting CMB initially ranged away from journalist so you can creator
7. Whether or not we know that the notion of thermal balance try valid even for infinite apartment and you can discover FRW designs, why don’t we remember your amazing Big-bang brand of Lemaitre are
a cool model, without attendant concept of one “thermal harmony” or “temperature” (T) of your radiation.
The very thought of Gorgeous Big bang model are because of Gammow, therefore turns out you to definitely having a thought rays dominated market, temperature
That however try not to dictate T(t) out of any basic principle since ongoing off proportionality regarding foregoing equation is not understood.
You can still designate value of the new T(t) with presumptions in regards to the desired design, and you will that a sense is some types of tautology.
On top of that, accident try missing simply for a mathematical fluid termed as “DUST” with no tension, zero temperatures
8. We understand one, the key to the origin regarding kinematical stress and you will temperature away from a fluid is the common collisions certainly one of their constituent dust.
In the ideal BBC, all test particles (galaxies in present era) are receding away from one another without any mutual collision. Thus in the ideal BBC, which involves assumptions of perfect
homogeneity and isotropy, the fluid is a dust. This has been shown specifically by expressing g[00] in terms of pressure and density (Mitra 2011a,b, 2012). The fact that g[00]=1 for the ideal BBM,
leads to p=0. Then in the absence of any collision, temperature of the BBM fluid T=0 too. Thus ideal BBM should be COLD and not HOT.
nine. Energy sources are protected to have a system which includes a good timelike Destroying vector. By listing one to FRW metric does not have any eg Destroying vector, it’s possible to declare
that overall energy of the FRW world you desire not protected. But not, people system can be gain otherwise cure time only by the getting together with the rest of the World.
Following Einstein’s definition of Lagrangian density and gravitational field energy density (pseudo tensor), Tolman (1930, 1962) derived a general formula for the total matter plus gravitational
field energy (P[0]) of an arbitrary self-gravitating system (Landau Lifshitz 1962). And by using the Tolman ansatz, in a detail study, I worked http://www.datingranking.net/es/citas-birraciales out
an expression for P[0] for the FRW universe (Mitra 2010). It was found that
1. So matter as well as gravitation times impetus thickness is actually status independent, just the k=0 flat model will be admissible.
2. To ensure a no cost losing observer notices zero gravitational industry in the regional Lorentz Body type, Cosmological Lingering can be no: Lambda =0.
3. When the time momentum preservation is broken, there can be emergence away from endless times anyplace each time away from nothingness. Just in case energy conservation has to be honored, FRW
design should be Static no contraction otherwise expansion.
1. FRW world is basically static
2. In the event the mathematically, one would design an energetic FRW model, then, you should tacitly features a vacuum design with rho =0, Lambda =0.
3. Quite simply, real bodily market which have inhomogeneous shipments out-of lumpy matter can not be demonstrated by the BBM.
1. Tension of your own water p=0 (as the received earlier), or
2. World try fixed: (overdot Roentgen) =0
Once the we are these are a SCALAR, it contraction can not be informed me aside because the one “accentuate perception”. On top of that, such as a contradiction is going to be fixed from the
comprehending that this new adhoc factor
But also for the newest Universe, there is absolutely no Other countries in the Universe, which means the complete opportunity (count as well as gravitation) must be stored
Due to the fact most likely applicant having Ebony Energy of the LCDM design are the one and only new Cosmological Lingering, cuatro independent evidences one L=0 strongly means that the so-titled
“Ebony Energy” are a fantasy arising from the latest try to determine cutting-edge lumpy market by the an enthusiastic oversimplified model and therefore needs finest homogeneity and you will
isotropy (Mitra 2013b, 2014a). | {"url":"http://www.flytymetransport.com/this-is-why-your-quotes-of-t-getting-cmb-initially/","timestamp":"2024-11-13T01:56:10Z","content_type":"text/html","content_length":"55437","record_id":"<urn:uuid:86a83578-21a5-4e21-b78c-0d6f881fdc9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00839.warc.gz"} |
Etale bundles and sheaves
1640 views
I am now going through Isham's book Modern differential geometry for physicists and got stuck with the notions of etale bundle, presheaf and sheaf. Could someone please suggest some other, more
intuitive and more accessible references on etale bundles and sheaves, preferably the ones giving more motivation and sufficiently many explicit (and worked-out) examples and, preferably, accessible
to theoretical physicists (i.e., not just mathematicians)?
P.S. To make things clear, a few math texts I have managed to find so far like Godement's and Bredon's Sheaf Theory (two books with the same title) seem way too tough for me. The part on sheaves in
Arapura's Algebraic Geometry over Complex Numbers is somewhat better but still a bit too fast going and with too few examples and not too much motivation. Pretty much the same applies to the part on
sheaves (which is too brief anyway) in the Clay Institute volume Mirror Symmetry. If there are no suitable books, are there perhaps some good lecture notes on the subject accessible to physicists
rather than just mathematicians, from which one get a reasonable intuition on sheaves and stuff?
This post imported from StackExchange Physics at 2014-03-30 15:28 (UCT), posted by SE-user just-learning
Why have you removed the "intuitive" part? You state in the question that you want an intuitive resource.
This post imported from StackExchange Physics at 2014-03-30 15:28 (UCT), posted by SE-user Dimensio1n0
Here is a motivation for the general notion of sheaf and sheaf cohomology:
A general introduction to differential geometry as needed in physics in terms of sheaves is at
More along these lines is in section 1.2 of arXiv:1310.7930, which describes physics in terms of sheaves (and higher sheaves) on smooth manifolds (and variants thereof).
This post imported from StackExchange Physics at 2014-03-30 15:28 (UCT), posted by SE-user Urs Schreiber
+1 I'm sure this joke has been made before but what a truly awesome schreiber you are to have put together your contibutions to the nLab! What a wonderful project!
This post imported from StackExchange Physics at 2014-03-30 15:28 (UCT), posted by SE-user WetSavannaAnimal aka Rod Vance | {"url":"https://www.physicsoverflow.org/10405/etale-bundles-and-sheaves","timestamp":"2024-11-04T10:24:34Z","content_type":"text/html","content_length":"123202","record_id":"<urn:uuid:cfb8295f-c46c-4b2b-a640-0c6bf4b12000>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00763.warc.gz"} |
Physics OCR Unit 1 (PIA G491)
• Created by: emilyanne
• Created on: 13-05-12 18:09
converging lenses:
light radiates as wavefronts which are curved. curvature decreases as they travel away from source. LENSES are to INCREASE or DECREASE curvature.
CONVERGING LENSES add curvature and focus them to a point (distance between point and lends is FOCAL LENGTH) and form a real image.
digital storage:
a PIXEL is what makes up a digital image and has an assigned value (the number of BITS which is 0 or 1.)
the number of values=2^(number of bits)
rates of information of transfer: bits per second.
quality of digital image increased by taking average of pixels around each pixel (including the one itself) and averaging them.
1 of 6
digitising a signal from analogue: samples are taken at regular intervals and stored as a number of bits. (if not enough samples then it will not be reproduced very clearly)
sampling rate: samples per second (same as frequency) in Hz.
spectrum: the presence of a range of frequencies.
fundamental frequency: lowest frequency which makes up the signal.
polarisation of waves:
TRANSVERSE WAVES (perpendicular to direction of energy) can be polarised (which means waves vibrate in the same direction)
2 of 6
terminology and equations:
byte is 8 bits.
magnification is (image height)/(object height).
resolution is smallest physical change a sensor can detect.
sensitivity is the amount of electrical change per physical change outputted.
refractive index is (speed of light in material)/(speed of light in vacuum).
amount of info in image: no of pixels x bits per second.
v(elocity) = f(requency) x λ (wavelength).
min rate of sampling should be bigger than or equal to 2 x max freq of signal.
rate of transmission of digital information = samples per second x bits per sample.
3 of 6
terminology, concepts and equations:
current, I, is flow of charged particles.
potential difference, V, is energy per unit charge.
resistance is caused by electrons colliding with other electrons and particles.
conductance is how much an electrical component carries an electircal current.
IN SERIES, total R = R1 + R2 + R3 + ...
IN PARALLEL, total G = G1 + G2 + G3 + ...
emf is pd across a cell.
Ohm's law is that V = IR, and for potential dividers V2 = V(total in) x R2/(R1+R2).
load resistance is resistance across a power source.
4 of 6
stiffness is resistance of a material to deformation.
ductile is that it can be shaped without breaking (undergoes deformation before breaking).
hardness is that it cannot easily be scratched.
toughness is when a material is resistant to fracturing. (area under a stress/strain graph and the opposite of brittle).
elastic: returns to original dimensions when force is removed.
plastic: dimensions have changed when force is removed.
stress: measurement of strength. =F(orce)/A(rea cross sectional).
strain: how much a material stretchs. =x (extension)/ L (original length)
5 of 6
young's modulus: measure of stiffness. stress/strain.
the graph has a steep gradient to start (elastic region) and then is flatter (plastic region).
METALS: ductile, elastic to a point, stiff (because of positive ions and negative electron attractions)
POLYMERS: long chain molecules. insulator (no free electrons), elastic (long chains), stiff (long chains tangle up).
6 of 6
refractive index is wrong on card 3:
(refractive index) = (speed of light in vacuum) / (speed of light in material) | {"url":"https://ws.getrevising.co.uk/revision-cards/physics_ocr_unit_1_pia_g491","timestamp":"2024-11-13T19:43:10Z","content_type":"text/html","content_length":"44145","record_id":"<urn:uuid:0bedba41-d790-4e9c-824e-74012b453fdd>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00310.warc.gz"} |
Course Listing
Find UMTYMP courses descriptions and topics for both the High School Component (Algebra I, Algebra II, Geometry, and Precalculus), and the Calculus Component (Calculus I, Calculus II, and Calculus
High School Component
The UMTYMP high school component is two years long. In the first-year students take Algebra 1 and Algebra 2. During the second, students take Geometry in the fall semester and Math Analysis
(PreCalculus) in the spring.
Algebra 1
Algebra 1
• Number systems and properties
• Polynomials and factoring
• Equations and inequalities involving linear functions, polynomials, multiple variables and/or absolute values
• Rational expressions and functions
• Exponents and radicals
• Distance, slope and equations of lines
Algebra 2
Algebra 2
• Functions and graph transformations
• Combining functions
• Polynomial division, remainder and factor theorem
• Graphing rational functions
• Exponential functions, inverse functions and logarithms
• Linear and nonlinear systems of equations
• Matrices, matrix algebra, matrix multiplication and inverse matrices
• Conic sections
• Sequences and Series
• Probability and statistics
• Permutations, combinations
• Binomial Theorem
• Mathematical Induction
• Proof based geometry course with emphasis on problem solving
• Points, lines, and planes
• Angles, triangles and congruency theorems
• Perimeter and area
• Similar triangles
• Right triangles and Pythagorean theorem
• Triangle centers, including centroid, orthocenter, incenter and circumcenter
• Quadrilaters and polygonals, including classification, properties and areas
• Circles and power of a point
• Polyhedra and other 3D solids
• Isometries
• Analytic geometry
• Right triangle trigonometry
• Compass and straightedge constructions are used throughout
• Equations and inequalities
• Polynomial, rational, exponential and logarithmic functions
• Unit circle trigonometry, including graphs, inverse trigonometric functions, trigonometric equations and trigonometric identities
• Polar coordinates and graphs
• Parametric functions and graphs
• Vectors with both algebraic and geometric approaches
• Conic sections
• Factoring polynomials
• Linear and nonlinear systems of equations
• Matrices and matrix algebra
• Sequences, series and mathematical induction
Many of these topics are covered in algebra; in Precalculus we review and extend that material.
Calculus Component
The Calculus component lasts for three years. Courses are named by year, not by content.
At most colleges and universities, for example, Calculus 1 and Calculus 2 refer to the first and second semesters of single-variable calculus. In UMTYMP, single-variable calculus is covered during
the fall and spring semesters of Calculus 1, sometimes referred to as UMTYMP Calculus 1A and UMTYMP Calculus 1B. In UMTYMP, Calculus 2 refers to the second-year course, which covers linear algebra
and other topics.
UMTYMP Calculus 1
UMTYMP Calculus 1
Course Content
Fall (Math Functions of one variable; limits; continuity; derivatives, including applications and the geometric interpretation of first and second derivatives; mean value theorem and extended mean
1471) value theorem; extreme values; linear approximations; optimization. Proofs of major results, such as the product rule, chain rule, and L’Hospital’s rule.
Spring Integration, including definitions, applications, and techniques, with more exposure to proofs and formal reasoning. Rigorous treatment of sequences and series.
(Math 1472)
UMTYMP Calculus 2
UMTYMP Calculus 2
Course Content
Fall (Math Introduction to differential equations, including first and second-order linear differential equations; systems of linear equations; logic, set theory, and methods of proof; precise
1473) definition of limits of sequences and functions; 3D coordinates; dot and cross products; equations of lines and planes in 3D; linear transformations.
Spring Theoretical course in linear algebra, including Euclidean space and general vector spaces, including function spaces; eigenvalues and discrete dynamical systems.
(Math 2471)
UMTYMP Calculus 3
UMTYMP Calculus 3
Course Content
Fall (Math Multivariable functions; differential geometry of curves in Euclidean space; parametric surfaces; partial and directional derivatives; total derivative matrix and linear approximations;
2472) chain rule; quadratic forms, Sylvester’s Theorem, Taylor’s Theorem, and multivariable optimization; Lagrange multipliers.
Spring (Math Multiple integration; integrals on parametric curves and surfaces; classical theorems in vector analysis, stressing a conceptual and geometric approach. | {"url":"https://cse.umn.edu/mathcep/course-listing","timestamp":"2024-11-11T05:17:46Z","content_type":"text/html","content_length":"84331","record_id":"<urn:uuid:1e1904ab-cc18-43a5-a9e8-3f0dc6c5917f>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00297.warc.gz"} |
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How to Calculate and Solve for Critical Mill of Speed | Ball Mill ... Jul 18, 2021Find the diameter of balls when the critical speed of mill is 15 and the mill diameter is 10. This implies that;
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ME EN 7960 Precision Machine Design Ball Screw Calculations 413 Permissible Speed • When the speed of a ball screw increases, the ball screw will approach its natural frequency, causing a
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Figure The effect of mill speed on the power drawn by a rotating mill. The liner profile and the stickiness of the pulp in the mill can have a significant effect on the actual critical velocity.
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The ball mill rotates clockwise at various constant fractions N of the critical speed of rpm, at. Charge behaviour. Fig. 1 shows typical charge shapes predicted for our 'standard' 5 m ball mill
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The approximate horsepower HP of a mill can be calculated from the following equation: HP = (W) (C) (Sin a) (2π) (N)/ 33000. where: W = weight of charge. C = distance of centre of gravity or
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A crash course on applied statistics with a focus on statistical modelling
A nutshell of little (statistics) stars
This book is written in English. However, your browser will easily translate the text to your favorite language. Please check your browser’s documentation for details. It should be done with one or
two clicks. \(\square\)
This is an introductory course on statistical modeling. Welcome!
The focus of this course is on how to specify a theoretical idea (possibly vague) in a testable statistical model.
Please read me
In order to benefit as much as possible from this course, it is essential to read this preface information. Yoda agrees (s. Figure 1).
Figure 1: Yoda finds you should read the manual
Use the print button of your browser to print the html page into a PDF page.
Course description
Analyzing research data can broadly be classified in three parts: explorative data analysis, modeling (including inference), and visualization. Either part is pivotal in its own right, but it can be
argued that modeling is at the core of the scientific endeavor. However, in practice, modeling, visualization, and data exploration is heavily intertwined, so that three parts may be recognized (as
individual entities) but not usefully separated from each other. This idea provides the rationale of this course: Data exploration, data visualization and data modeling is discussed as an integrated
The focus is on practical data analysis; theoretical concepts are, where mentioned, second class citizens due to time constraints and the didactic aims of the course.
For example, statistical inference – such as p-values and confidence intervals – are not more than touched briefly, as the instructor believes that modeling, not inference, is of prime importance for
the auditorium.
We will use the R environment for all computations (freely available). Please bring your own Laptop with R and RStudio installed (installation guides are provided). Data and R code will be provided.
We’re on a crash course
The course is set-up as a “crash course” which indicates that we’ll rather try to cover a breadth of steps rather than digging deep at certain particular points. The rationale of this approach is
that before digging deep, it is necessary to gain an overview of the territory. In addition, if one particular topic is not of interest to a given student (perhaps to difficult/simple), not much time
is lost.
Be warned! Compare this crash course to a dancing crash course right before your wedding: A lot can be achieved by such a course in some instances, or rather, the worst consequences (of not knowing
how to dance) may be fenced off, but one should not expect to be a dancing queen (king) thereafter.
More on modelling
Models and modeling are of pivotal importance in many sciences, not only for providing an explanation of nature en miniature (theoretical models), but also for gauging how closely the empirical data
at hand match the theoretical model. Translating a theoretical model into statistical language is called statistical modeling and provides the guiding principle in this introductory course.
Regression models will be presented as a lingua franca of statistical modeling, and we will learn that many empirical questions can (comfortably) be analyzed using a regression framework. Depending
on the background and aims of the participants (and time permitting), we will shed light on some standard topics such as model comparison, classification models, and typical pitfalls. Given a more
advanced auditorium, we may want to explore how causal and non-causal associations can be translated and tested using simple linear statistical models. Foundational ideas of statistical modeling will
be accompanied by short examples and case studies to facilitate transfer and practical application after the course.
Course prerequisites
Basic computer usage knowledge is needed (downloading materials from the internet, operating a PC, etc). Basic R knowledge is needed. Basic knowledge of statistical concepts (such as descriptive
statistics) is needed. Willingness to learn is essential.
Learning objectives
Upon successful completion of this course, students should be able to:
• select the right statistical visualization for a variety of data contexts
• “crunch” or “wrangle” data
• explain what statistical modeling means
• formulate basic statistical models
• differentiate between predictive and explanatory modeling
• apply the methods to own datasets
Course Literature
This course builds on the freely available e-book ModernDive. Each topic is paralleled by an accompanying chapter from ModernDive. A hard copy can be purchased here. The book is for sale in print
Course logistics
This course can be presented as a one-day seminar or split-up in two or more blocks.
The course can be held in English or German.
For students
Please bring your own computer and read the notes regarding course logistics in advance. Note that some upfront preparation is needed from the learners.
R and RStudio^1 will be needed throughout the course. Please make sure that the IT is running. In case of technical difficulties with R feel free to use RStudio Cloud; free plans are available.
All learning materials (such as literature, code, data) will be provided in electronic format.
For organizers
The following technical equipements is needed for courses in classroom:
• WiFi access for students and instructor(s)
• electricity/power plugs for students and instructors
• Projector
• seat and desk for each student and instructor
UPFRONT student preparation
• Install R and RStudio, see ModernDive Chap. 1.1. In case you have your R running on your system, please make sure that you’re uptodate. If outdated, download and install the most recent versions
of the software. Similarly, hit the “Update” button in RStudio’s “Packages” tab to update your packages if you have not done so for a couple of months.
• Sign-in at RStudio Cloud. It’s super helpful because I as the techer can provide you with an environment where all R stuff is ready to use (packages installed etc).
• Install the necessary R packages as used in the book chapters covered in this course (see the sections on “Needed packages” in each chapter). If in doubt, see here the instructions on how to
install R packages. Here’s the actual list on the R packages we’ll need.
• Students new to R are advised to learn the basics, see ModernDive, Chap 1.2 - 1.5
• Bring your own laptop
• Make sure your internet connection is stable and your loudspeaker/headset is working; a webcam is helpful.
• Students are advised to review the course materials after each session.
• I recommend that you carefully check the course description to make sure the course fits your needs (not too advanced/basic).
Didactic outline
This course can rather be considered a workshop in the sense that the instructor uses a dialogue-based approach to teaching and that there are numerous exercises during the course. Instead of
providing long talks to the students, the instructor feels obligated to engage students in back-and-forth conversations. Similarly, the presentation of a large number of Powerpoint slide is avoided.
Instead, a thorough course literature is available (free online), so that students will have no barrier in diving deeply into the materials and ideas presented. However, during class it is more
important to transmit the pivotal ideas; details need to be read and worked by the students individually after (and before) the course. As an alternative to presenting a lot of text on slides, in
this course there will be a (electronic) whiteboard where concepts are developed dynamically and in pace of the teaching conversation thereby adjusting the “dose” of new thoughts to the actual pace
of the instruction.
Please not that the focus and the amount covered in a course strongly depends on the background, aims and prior knowledge of audience.
Overview on topics covered
• Data visualization building the grammar of graphics and ggplot2
• Data wrangling based on the tidyverse in R
• Basic concepts of statistical modelling
• Primer on causal inference
• Introduction to regression analysis
• Quick refresher of statistical inference
Sebastian Sauer works as a professor at Ansbach university, teaching statistics and related stuff. Analyzing data to answer questions related to social phenomena is one of his major interests. He is
trying to help raising the methological (and particularly statistical) skills in the sciences (ie., scientists). The programming language “R” is one of his favorite tools. He sees himself as a
learner, and is particularly interested learning more on quantitative approaches to understand nature. Open Science is a hot topic to him. He hopes to contribute to pressing social problems such as
populism by bringing in his statistical and psychological know-how. He writes a blog which serves as a sketchpad for stuff in his mind (not immune to thought updates) at https://data-se.netlify.app/.
Sebastian is the author of “Moderne Datenanalyse mit R” (Sauer, 2019). His publication list is available on Google Scholar.
Check-out my personal homepage here.
Feel free to contact me via email at any time at profsebastiansauerATgmail.com.
Assessment and grades
There is no assessment, there are no grades!
Talk to me
It’s my goal to make this an excellent course and a stimulating and enjoyable experience for all of us. So that I can find out if this is happening, I encourage feedback—be it positive or negative—on
all aspects of the course at any time. For example, if something I’m doing is making it difficult for you to learn, then let me know before it’s too late; if you particularly enjoyed something we did
in class, say so so that we can do it again.
Course materials
Most of the materials as presented below is made available through the course book ModernDive. Please check the relevant chapters of the book before the course to make sure you have all materials
R Packages
All R packages are accessible through the course book; please consult the relevant chapters. Please install all R packages used before the course. Here’s a tutorial on how to install R packages.
The most important R packages for this course are:
The following packages are useful for data access (but not strictly mandatory):
• palmerpenguins
• ggpubr
• rio
• vtree
• visdat
• dataexplorer
• tableOne
• flextable
• gapminder
• nycflights13
• fivethirtyeight
• skimr
• ISLR
For the Bayes models you’ll need some extra software (free, save and stable), but somewhat more hassle to install. Using Bayes in this course is optional. You don’t miss a lot if you don’t use it.
For the R package {rstanarm} to run, you’ll need to install RStan. On Windows, this amounts to installing RTools. On Mac, you’ll need to install the XCode CLI^2.
In sum, follow the instructions on the RStan website. It’s unfortunately a bit complicated.
All data are accessible through the course book; please consult the relevant chapters.
Labs (case studies)
Practical data analysis skills can be practiced using these labs; in addition Chapter 11 provides two cases studies. Note that such content may be used as homework.
There are a lot of case studies scattered on the internet.
Sketching causal models
Dagitty is great tool for sketching causal graphs (DAGs), it can be usd in your browser or as R package. Here’s an example of a collider bias. Check out this post for an intuitive explanation.
German introductary course
Readers who speak German may check out this Blitzkurs into data analysis using R.
Where are the slides?
There are none. I feel that slides are not optimal for learning. In class, slides can be detrimental if they are too wordy because that distracts from that the dialogue with the instructor, and I
hold this very dialogue as essential. Outside of class, slides are neither helpful. Instead, a good book is much more beneficial, because in a book, there’s enough room to patiently explain in
sufficient details, an endeavor which is impossible for a slide deck.
To underline my messages to you, dear learners, I will use some sketches on a virtual whiteboard, some interactive apps, live coding, and some (pre-prepared) diagrams. That’s a bit similar to what
happens at Khan Academy. You might have noticed that many courses at Coursera follow a similar approach.
I readily confess that this approach is novel to many learners in these days, learners who are accustomed to hundreds of Powerpoint slides. Please be open and I think you will appreciate this
didactic style.
Technical Details
Last update: of this page: r Sys.time()
1. Desktop version, not the server↩︎
2. possibly you need also a Fortran compiler, but maybe that’s optional↩︎ | {"url":"https://stats-nutshell.netlify.app/","timestamp":"2024-11-05T08:49:01Z","content_type":"application/xhtml+xml","content_length":"57318","record_id":"<urn:uuid:4259da83-37a8-47dc-bfa3-468c9100c356>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00360.warc.gz"} |
Concept information
p-xylene number concentration
• Number concentration means number of molecule per unit volume, and is used in the construction molecular_concentration_of_X_in_Y, where X is a material constituent of Y. A chemical species
denoted by X may be described by a single term such as 'nitrogen' or a phrase such as 'nox_expressed_as_nitrogen'. The chemical formula for p-xylene is C8H10. P-xylene is a member of the group of
hydrocarbons known as aromatics. The IUPAC name for p-xylene is 1,4-xylene.
{{#each properties}}
{{toUpperCase label}}
{{#each values }} {{! loop through ConceptPropertyValue objects }} {{#if prefLabel }}
{{#if vocabName }}
{{ vocabName }}
{{/if}} {{/each}} | {"url":"https://vocabulary.actris.nilu.no/skosmos/actris_vocab/en/page/p-xylenenumberconcentration","timestamp":"2024-11-09T19:54:04Z","content_type":"text/html","content_length":"20602","record_id":"<urn:uuid:2671d9b7-16fa-4d74-b950-7a7877867102>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00673.warc.gz"} |
Cantilever Beam With Couple Moment
Cantilever Beam With Couple Moment Calculator
Calculate the Cantilever beam with couple moment with our free online tool using the input parameters
Utilize our user-friendly online calculator to compute the slope and deflection of a cantilever beam subjected to a couple moment.
Send the result to an email
• Slope at free end = ML / EI
• Deflection at any section = Mx^2 / 2EI
The variables used in the formula are:
M is the couple moment at the free end,
E is the Elastic Modulus,
I is the Area moment of Inertia,
L is the Length of the beam and
x is the position of the load. | {"url":"https://calchub.xyz/cantilever-beam-with-couple-moment/","timestamp":"2024-11-06T10:48:12Z","content_type":"text/html","content_length":"42939","record_id":"<urn:uuid:76114617-e848-4642-bf11-7ebf0fba1221>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00667.warc.gz"} |
Divide 3-digit numbers by 5 - Division Maths Worksheets for Year 5 (age 9-10) by URBrainy.com
Divide 3-digit numbers by 5
A step by step approach for short division, using 3-digit numbers and dividing by 5.
4 pages
Divide 3-digit numbers by 5
A step by step approach for short division, using 3-digit numbers and dividing by 5.
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© Copyright 2011 - 2024 Route One Network Ltd. - URBrainy.com 11.5.3 | {"url":"https://urbrainy.com/get/2921/divide-digits-by-7470","timestamp":"2024-11-12T22:52:26Z","content_type":"text/html","content_length":"114145","record_id":"<urn:uuid:6559a7e2-2104-49b2-ae28-510f0cb8dfbb>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00105.warc.gz"} |
Entropy Change During Expansion Into Vacuum
Task number: 3940
Determine the entropy change of ideal gas with temperature of 20 °C, pressure of 100 kPa and volume of 2 l, provided that the gas expands into vacuum to twice its original volume. Consider the
process to be isothermal.
• Hint 1 – What process is it?
The process of gas expanding into vacuum is called a spontaneous expansion.
A spontaneous expansion is an irreversible adiabatic process during which work done on and by the gas is zero. Therefore it applies: Q = W = 0. From the First Law of Thermodynamics, the following
applies for the internal energy of gas: ΔU = 0. Since the internal energy of gas depends only on its temperature and not on its volume, its temperature remains constant during a spontaneous
During a spontaneous expansion, the pressure and volume of the gas changes unpredictably. Only the initial and final states are in equilibrium.
• Hint 2 – Entropy
Entropy S is a state function. What does it mean?
• Hint 3 – Entropy Change
Entropy change ΔS of a system during a process starting in equilibrium state A and ending in equilibrium state B is described as
\[\mathrm{\Delta} S=S_B-S_A=\int\limits_{A}^{B}\frac{dQ}{T},\]
where Q is heat tranferred in or out of the system during the process, and T is thermodynamic temperature of the system.
During spontaneous expansion, however, pressure, temperature and volume of the gas changes unpredictably. Therefore, we cannot find a relation between heat Q and temperature T which would enable
us to perform given integration. How do we find the unknown entropy change then?
• Hint 4 – Entropy Change of Reversible Isothermal Expansion
Entropy change ΔS of reversible isothermal expansion is described as follows:
\[\Delta S=\frac{Q}{T},\]
where Q is total heat and T is thermodynamic temperature during this process.
• Given Values
t = 20 °C => T = 293.15 K gas temperature
p[1] = 100 kPa = 10^5 Pa gas pressure
V[1] = 2 l = 2·10^−3 m^3 gas initial volume
V[2] = 2V[1] gas final volume
ΔS = ? entropy change
• Analysis
When gas expands into a vacuum, this process is called a spontaneous expansion. It is an irreversible adiabatic process during which the gas does not perform work and no work is supplied to the
system. Furthermore, pressure and volume of the gas change unpredictably. Therefore, we cannot use the same formulas as with irreversible processes during calculation.
We use the fact that entropy is a state function. Its change between the initial and final state therefore depends only on these two states and not on the way the system got there. We substitute
the irreversible spontaneous expansion with reversible isothermal expansion with the same initial and final state. This process is convenient because temperature does not change during
spontaneous expansion of ideal gas.
Then we determine the entropy change for this chosen process as a ratio of total exchanged heat and thermodynamic temperature during this process. The heat supplied during reversible isothermal
expansion is equal to work done by the gas. To determine this work we need to use integration due to the fact that pressure is a volume function. Pressure as a volume function is expressed from
Boyle's Law.
Since entropy is a state variable, the determined entropy change is the same as the entropy change during spontaneous expansion.
• Solution
Change in entropy ΔS of a system in a process starting in an equilibrium state A and ending in an equilibrium state is described as:
\[\mathrm{\Delta} S=S_B-S_A=\int\limits_{A}^{B}\frac{dQ}{T},\]
where Q is heat transferred to or from the system during the process and T is thermodynamic temperature.
We need to determine entropy change during a spontaneous epxansion. During this process, however, pressure, temperature and volume changes unpredictably. Only the initial state A and final state
B are at equilibrium. Therefore, we cannot find a relation between heat Q and temperature T that would enable us to perform the integration in the above mentioned formula.
However, we can use the fact that entropy is a state function. This means that its change between initial and final state depends only on these two states and not on the way the system got from
one state to the other.
Let us replace the irreversible spontaneous expansion with a convenient reversible process. Since temperature does not change during a spontaneous expansion of ideal gas, we choose a reversible
isothermal expansion with the same initial state A and final state B.
For entropy change during this process, the following simplified relation applies
\[\mathrm{\Delta} S=\frac{Q}{T}.\]
Now we determine it.
During a reversible isothermal process, the internal energy of gas does not change which according to the First Law of Thermodynamics means that accepted heat Q is equal to work W done by the
system during expansion.
This work can be determined by a relation
where V[1] and V[2] is the initial and final volume of gas and p its pressure that changes during the expansion (it is a volume function).
We determine the pressure as a volume function from Boyle's Law:
From here we can determine pressure p:
\[p = \frac{p_1V_1}{V}.\]
Now we can perform the integration:
\[W = \int\limits_{V_1}^{V_2}p\, \text{d}V = \int\limits_{V_1}^{2V_1}\frac{p_1V_1}{V}\, \text{d}V =\]
we factor the constant values out of the integral
\[=p_1V_1 \int\limits_{V_1}^{2V_1}\frac{1}{V}\, \text{d}V = \]
then we perform the integration an substitute the limits
\[=p_1V_1[\ln V]_{V_1}^{2V_1} = p_1V_1 \ln \frac{2V_1}{V_1}= p_1V_1 \ln 2.\]
The entropy change we are looking for is:
\[\mathrm{\Delta} S=\frac{Q}{T}=\frac{W}{T} = \frac{p_1V_1\ln{2}}{T}.\]
As we have already stated above, entropy is a state function. Therefore this determined entropy change is the same as entropy change during a spontaneous expansion.
• Numerical Solution
\[\Delta S= \frac{p_1V_1\ln{2}}{T}= \frac{10^5\cdot{ 2}\cdot{ 10^{-3}}\cdot \ln{2}}{293.15} \,\mathrm{JK^{-1}}\dot{=} 0.47\,\mathrm{JK^{-1}}\]
• Answer
Entropy increased by approximately 0.47 JK^−1 during the expansion. | {"url":"https://physicstasks.eu/3940/entropy-change-during-expansion-into-vacuum","timestamp":"2024-11-11T17:22:52Z","content_type":"text/html","content_length":"35059","record_id":"<urn:uuid:c6788b20-9b64-4bd7-956b-118de501e9c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00672.warc.gz"} |