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Did Humans Invent Mathematics, or Is It a Fundamental Part of Existence?
Physics24 November 2021
Many people think that mathematics is a human invention. To this way of thinking, mathematics is like a language: it may describe real things in the world, but it doesn't 'exist' outside the minds of
the people who use it.
But the Pythagorean school of thought in ancient Greece held a different view. Its proponents believed reality is fundamentally mathematical.
More than 2,000 years later, philosophers and physicists are starting to take this idea seriously.
As I argue in a new paper, mathematics is an essential component of nature that gives structure to the physical world.
Honeybees and hexagons
Bees in hives produce hexagonal honeycomb. Why?
According to the 'honeycomb conjecture' in mathematics, hexagons are the most efficient shape for tiling the plane. If you want to fully cover a surface using tiles of a uniform shape and size, while
keeping the total length of the perimeter to a minimum, hexagons are the shape to use.
Charles Darwin reasoned that bees have evolved to use this shape because it produces the largest cells to store honey for the smallest input of energy to produce wax.
The honeycomb conjecture was first proposed in ancient times, but was only proved in 1999 by mathematician Thomas Hales.
Cicadas and prime numbers
Here's another example. There are two subspecies of North American periodical cicadas that live most of their lives in the ground. Then, every 13 or 17 years (depending on the subspecies), the
cicadas emerge in great swarms for a period of around two weeks.
Why is it 13 and 17 years? Why not 12 and 14? Or 16 and 18?
One explanation appeals to the fact that 13 and 17 are prime numbers.
Imagine the cicadas have a range of predators that also spend most of their lives in the ground. The cicadas need to come out of the ground when their predators are lying dormant.
Suppose there are predators with life cycles of 2, 3, 4, 5, 6, 7, 8 and 9 years. What is the best way to avoid them all?
Above: P1–P9 represent cycling predators. The number-line represents years. The highlighted gaps show how 13- and 17-year cicadas manage to avoid their predators.
Well, compare a 13-year life cycle and a 12-year life cycle. When a cicada with a 12-year life cycle comes out of the ground, the 2-year, 3-year and 4-year predators will also be out of the ground,
because 2, 3, and 4 all divide evenly into 12.
When a cicada with a 13-year life cycle comes out of the ground, none of its predators will be out of the ground, because none of 2, 3, 4, 5, 6, 7, 8, or 9 divides evenly into 13. The same is true
for 17.
It seems these cicadas have evolved to exploit basic facts about numbers.
Creation or discovery?
Once we start looking, it is easy to find other examples. From the shape of soap films, to gear design in engines, to the location and size of the gaps in the rings of Saturn, mathematics is
If mathematics explains so many things we see around us, then it is unlikely that mathematics is something we've created. The alternative is that mathematical facts are discovered: not just by
humans, but by insects, soap bubbles, combustion engines, and planets.
What did Plato think?
But if we are discovering something, what is it?
The ancient Greek philosopher Plato had an answer. He thought mathematics describes objects that really exist.
For Plato, these objects included numbers and geometric shapes. Today, we might add more complicated mathematical objects such as groups, categories, functions, fields, and rings to the list.
Plato also maintained that mathematical objects exist outside of space and time. But such a view only deepens the mystery of how mathematics explains anything.
Explanation involves showing how one thing in the world depends on another. If mathematical objects exist in a realm apart from the world we live in, they don't seem capable of relating to anything
Enter Pythagoreanism
The ancient Pythagoreans agreed with Plato that mathematics describes a world of objects. But, unlike Plato, they didn't think mathematical objects exist beyond space and time.
Instead, they believed physical reality is made of mathematical objects in the same way matter is made of atoms.
If reality is made of mathematical objects, it's easy to see how mathematics might play a role in explaining the world around us.
In the past decade, two physicists have mounted significant defenses of the Pythagorean position: Swedish-US cosmologist Max Tegmark and Australian physicist-philosopher Jane McDonnell.
Tegmark argues reality just is one big mathematical object. If that seems weird, think about the idea that reality is a simulation. A simulation is a computer program, which is a kind of mathematical
McDonnell's view is more radical. She thinks reality is made of mathematical objects and minds. Mathematics is how the Universe, which is conscious, comes to know itself.
I defend a different view: the world has two parts, mathematics and matter. Mathematics gives matter its form, and matter gives mathematics its substance.
Mathematical objects provide a structural framework for the physical world.
The future of mathematics
It makes sense that Pythagoreanism is being rediscovered in physics.
In the past century physics has become more and more mathematical, turning to seemingly abstract fields of inquiry such as group theory and differential geometry in an effort to explain the physical
As the boundary between physics and mathematics blurs, it becomes harder to say which parts of the world are physical and which are mathematical.
But it is strange that Pythagoreanism has been neglected by philosophers for so long.
I believe that is about to change. The time has arrived for a Pythagorean revolution, one that promises to radically alter our understanding of reality.
Sam Baron, Associate professor, Australian Catholic University.
This article is republished from The Conversation under a Creative Commons license. Read the original article. | {"url":"https://www.sciencealert.com/was-math-always-here-wild-new-paper-puts-spin-on-an-ancient-pythagorean-idea","timestamp":"2024-11-01T19:23:08Z","content_type":"text/html","content_length":"151843","record_id":"<urn:uuid:b19c507e-1eee-4d0b-b83b-5e7b30c58dca>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00267.warc.gz"} |
That shape/space/measures bit of the menu is looking awfully bare. Not as bare as the data/probability section, but still.
Simple PowerPoint here.
Little starter
Followed by some example problem pairs and some questions
I am still loving fill in the gap resources. Interestingly, my group managed to do this without seeing the formula triagle/rearranging the formula. They just used thinking skills to work backwards.
Not sure that the reflect/expect/reflect bit of these questions works. I’ve tried to make sure they compare it to the first question each time, but I think it is better when it’s more of a ‘journey’
and you compare it to the previous question in each case.
Still A* for formatting. It looks good.
Then I’ve made a blooket for some checking, some exam questions and a plenary.
This worked well with my group but it maybe needs some mini whiteboard work.
Have a lovely weekend
Mean from a list
Some data handling for once!
I have deliberately just stuck to mean here. I like to focus on it, rather than doing mean, median and mode all in one lesson.
I quite like these questions. Just because there’s loads to talk about when going through them. Things like
• On g to h : I added one to each value. What happened to the mean?
• b and c. I’ve increased one value by 1 and decreased another by 1. What happens to the mean?
• e) I’ve multiplied all values by 10. What has happened to the mean?
Lots of things to generalise and investigate. I’ve even chucked in a bit of algebra.
I’ve extended things by including some ‘missing number’ questions. Only 6.
Then there’s a learning check.
That’s it.
And that’s a resource every single week of this half term. The menu is looking a lot more full these days. Roll on 2023.
Function Notation (substitution with numbers and algebra)
A very simple resource, this. But there’s not much function notation stuff out there.
I find pupils’ responses to function notation interesting. Stuff they can do like 2x+3 = 5x – 6 becomes suddenly impossible when presented in the form f(x) = 2x + 3 , g(x) = 5x – 6, find f(x) = g(x).
I’m not sure why.
Anyone found anything awesome on function notation?
ChristMATHS 2022
My annual Christmas maths quiz is here. I’ve made it early this year to give people plenty of time to find it. (Looking back, I’ve made some sort of this quiz for 6 years now!) This isn’t just a
quiz, it’s a quiz with lots of links to maths.
I know some people like to teach up until the last second, but I’ve always liked to finish the term off on a nice little maths quiz. I think it’s OK for us sometimes to let our hair down a little. I
say this. I’m bald.
The rounds this year include some repeats from last year. Like the Shakin’ Stevens video round.
I think it’s nice to kick off with a song. And I’m sorry but I find this video hilarious. It’s utterly cursed. I’m usually cackling all the way through.
Round two this year is inspired by Linkee. I love a quiz. I love Linkee.
The questions were really hard to write!
Round three is a cipher type thing.
Let them use a calculator. Go on. Be nice. It’s Christmas.
Round four is the Chris Moyles Quiz night round. These are always a good laugh.
Round five is sudoko. I found this worked well last year. So I kept it. Also I couldn’t think of a round to replace it.
That’s Christmaths!
Other thoughts
• Just two more weeks to go of term. Then I’d have uploaded a resource every week of term one. That’s pretty good going if you ask me.
• If you’ve used any of my resources this term and liked them, and have some spare cash about, please do me a favour. Please consider a gift to KidsOut. Their online shop is filled with gifts that
you can purchase to send to children in refuges. Read all about them. I genuinely get a little teary each time I do. Here is a real challenge. Take a second to read their website and stay dry
• I hope you are having a wonderful Christmastime. Teaching is stressful and difficult and I hope you get the rest and relaxation you deserve.
Words and numbers
I think this resource is quite good.
One thing I’ve noticed with pupils, is that literacy gets in the way of the mathematics more often then you would realise. There’s literacy issues with things like ‘range’ etc having very strict
mathematical definitions, but we also use loads of different words for the same things.
Let’s take multiplication for instance. If we wanted to talk about 4 multiplied by 3, we might say : 4 times 3, 4 lots of 3, 4 groups of 3, four threes, etc. There’s loads and I think it can be quite
a lot for a pupil to get their head around.
So I created 6 little starters that all use the same numbers but switch out the wording.
I think using them has been really helpful. There’s a few ambiguous ones in there, too. I love ambiguous cases, they provide great discussions.
Download these and have a look. I think they’re one of my best ideas.
Solving by factorising
This is a follow up to my factorising quadratics lesson. I taught this two weeks ago, but I was cleaning up the slides to be put online. I really think the exercise that goes along with it is nicely
thinky. Some variation and slow build up of activity.
3 leads on to these problem solve-y questions
Not a huge lot here, but I thought this lesson and my other quadratics lesson here sequenced well.
Other thoughts
• I’ve been doing barvember. The pupils in my year 7 class have kinda hated it, which proves just how valuable and needed it was. These questions are so high quality, and it’s lovely to show how
you can do some of these problems just with bars and basic four operations knowledge. I laminated some responses from the pupils, and they loved seeing their work up on the board. Great stuff.
• This blog is going to break it’s usual format for the next few weeks. Next week I’ve got some starters for you, the week after … ChristMaths is going to be published early.
• As always, thoughts and insights to @ticktockmaths on Twitter.
Substitution with positive integers
Sometimes a resource benefits from you deleting stuff, not adding stuff.
I’ve got a much better understanding of what I want these resources to be, and how they’re helpful these days.
Case in point. This used to be a massive PowerPoint stuff with activities, just for the sake of doing activities. But there’s way better stuff on resourceaholic.com if you want nice worksheets.
I’ve cut right down on that. Now I know what I want. An example problem pair, some good questions and a base to build from. Other people have made interesting and fun substitutions tasks. I don’t
need to put bad approximations of them in my slides. These are the bones of a lesson.
One thing I DO need is lots of feedback opportunities. Thus I’ve added in loads of mini whiteboard work.
These don’t move as quick as the gif 🙂 Clicking controls the speed.
And some questions whose progression makes sense
And that’s pretty much it. There’s some exam questions and a plenary, but I genuinely think that by deleting loads of stuff I’ve got much closer to the actual inquisition of the skill.
Sometimes, less is more.
Factorising Quadratics
No solving here. Just concentrating on the core skill of factorising. I started with an example and then this
Giving them one bracket so we build up the skill. But wait.
It doesn’t matter which way around the brackets go but on this exercise it kinda makes it look like it does
Did this this cause an issue with my students? No. Because I pointed out this to them. Sometimes sharing your thought processes when planning is a really useful thing.
Then there’s a regular exercise, and then this
Yes. I’ve had someone make emojis of my own face to use in my lesson resources. This might be one of the most egotistical things any maths teacher has done. Which leads into…
I’d rather do this as an extension rather than solving as I believe solving is this whole other thing. I’m going to do a PowerPoint wholly devoted to solving quadratics. I really do believe the
skills are better split up this way.
Other thoughts
• This only deals with quadratics where a=1. I’ve done a whole over PowerPoint on a>1 here.
• Thinking about factorising quadratics in depth, you could easily do 2 lessons on factorising, 1/2 on non-monic quadratics and 1/2 on solving after you’ve factorised. And that’s without even
talking much about graphs. My current year 11 scheme of work has ‘solving by factorising’ in one lesson. There is too much content in GCSE/IGCSE.
I apologise for the over branding 🙂
Final resource of this half term.
I’ve packaged together some of the starters that I’ve been putting on Twitter challenging misconceptions I’ve seen in lessons. Stuff that I see a lot but doesn’t necessarily have a place in a SoW.
Stuff like the difference between a half and two
Or when we need to write trailing and leading zeroes.
I haven’t included the answers deliberately. These are meant for discussion. For instance, 12 here could be a currency.
I did the same with 1s.
I would do these with any class that have had algebra introduced. Even doing it with my 11 Higher class caught some misconceptions.
I would maybe only do this with lower years, though. I am trying to get at the idea that the digits stay in the same arrangement. I have found that I have two groups this year who find multiplying
and dividing by 10, 100 and 1000 really difficult. On a recent assessment more than a few tried to do 2401/100 using the bus stop method.
I’ve reinforced it and reinforced it, but there are about 2 or 3 that still lack the skill and I’m running out of ways to address it. Any ideas to @ticktockmaths on twitter.
There’s also some that struggle with halving and doubling quickly. Again, really struggling to embed this.
This one tries to get at the unitary thing. It was interesting hearing student responses, especially as the answer depends so much on what x is. There’s a version with 3 if you want something more
The last one is identifying if the answer is a quarter (like half or two. I was kinda losing steam by this point.)
Also this page isn’t editable because of some stupid saving between computers bug 🙁
Right, that’s it for this half term.
I’ve managed to fill quite a lot of gaps.
Behind the scenes I’ve also got a lot of PowerPoints half started, ready to be finished up and polished up for uploading here. The goal of covering most things seems achievable, if still long way
No resource next week. It’s half term and I want a rest.
Addition and Subtraction of Decimals
Quite a simple lesson here, but I’ve tried to fill it with some more thinky questions.
My teaching has become far too scripted into example problem pair – some questions – a plenary. I’m not building in enough opportunities for problem solving and struggle time.
Part of that is a scheme of work that forces pace. Part of that is my own routines, comfort zone and, quite frankly, a bit of tiredness. I massively admire those people on Twitter who keep putting
out amazing task after amazing task. *cough* https://mathshko.com/ *cough*. Sometimes I question the usefulness of putting these PowerPoints online. Most people can write an example on the board, and
the questions aren’t groundbreaking, just well thought out.
Anyway, I tried to add some stuff like filling in the gaps.
Some thoughts for the week
• It’s important to think through your examples. I recently gave this example problem pair to year 11.
I was trying to teach them what happens when we have a non repeating digit in a recurring decimal to fractional problem. Anyone else see why this is a trash example problem pair? | {"url":"https://ticktockmaths.co.uk/page/2/","timestamp":"2024-11-06T08:03:29Z","content_type":"text/html","content_length":"179672","record_id":"<urn:uuid:0fccffa8-0cfa-4f20-ae36-e1d8ded1cfcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00175.warc.gz"} |
Time series analysis of Stock data
What is Time Series?
Time series is a set of observations or data points taken at specified time usually at equal intervals and it’s used to predict the future values based on the previous observed values. In time series
there must be one variable, that is Time. Time series are very frequently plotted via line charts.Time series are used in statistics, stock market forecasting, signal processing, pattern recognition,
weather forecasting, earthquake prediction, astronomy, and largely in any domain of applied science and engineering which involves the measurements should be in equal intervals like a day, a week, a
month and a year.
Components of Time Series Analysis:
1. Trend
2. Seasonality
3. Irregularity
4. Cyclic
ARIMA Model:
The ARIMA model is a form of Regression analysis. An ARIMA model can be better understood by looking into its individual components: Auto regression (AR), Integrated (I) and Moving Averages (MA). In
AR model, Partial Auto Correlation Function(PACF) graph is used to find P value and in MA model, Auto Correlation Function(ACF) graph to find q value. Integration Function is used to find the d value
. ie, the differentiation
What is Stationarity?
Time series is said to be stationary if its statistical properties such as mean, variance remain constant over time. A model that shows stationarity is one that shows there is constancy to the data.
Most economic and market data show trends, so the purpose of differencing is to remove any trends or seasonal structures. If not Stationary, It has to perform transformations on the data to make it
Stationary. For check stationarity it have:
1. Rolling Statistics
2. Dickey-Fuller test
Auto correlation and Partial Auto-Correlation Functions
The ACF graph is drawn to determine the q value and the PACF graph is drawn to determine the p value.
From the above graphs it can be see that the value of p is 2 and the value of q is 2 respectively.
Now these values have to be substituted in the ARIMA model to get the predictions.
The ARIMA model captures the movement of data correctly.
Residual Sum Of Squares(RSS) should be less as possible.
The RSS value has been less as compared to AR and MA models.
Last updated 2020-10-09 23:03:51 by Kennedy Waweru | {"url":"https://dasclab.uonbi.ac.ke/analytics/blog/time-series-analysis-of-stock-data","timestamp":"2024-11-02T07:44:34Z","content_type":"text/html","content_length":"11891","record_id":"<urn:uuid:3b8e26a1-95b8-4814-ac8c-2075e076e84c>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00262.warc.gz"} |
Essential parameters are underlined. These parameters must be correct in order to get a valid result. Although a computation may be useful, even with an incorrect setting, it is necessary to be
intimately familiar with the underlying algorithms in order to interpret the result correctly.
It is also possible to use an overly conservative setting, which may not harm the result. This usually comes at the expense of using much more resources (time and memory) than necessary.
Guidance Parameters
These parameters do not directly affect the computation. Instead they control how much information to display during the computation, guide bbsvdopt in choosing the actual parameters, and control the
This set of parameters is by far the most useful for a normal user.
Parameter: trip [true | false]
Determine whether singular triplets should be computed. bbsvds returns empty singular matrices if trip=false.
Default: determined from the output in bbsvds, true for bbsvdf and bbsvdopt.
Parameter: atol [scalar]
Absolute tolerance of convergence. In most cases one would only set atol manually if the default value is known to be very conservative. See below for a discussion of this parameter.
Default: the value of tau specified for the black-box operator.
Parameter: disp [-1 | 0 | 1 | 2]
Verbosity level. The idea is that disp=-1 should quench any output, while disp=0 only shows error conditions. The normal level show progress information and disp=2 is good for debugging.
Default: disp=1.
Parameter: mem [scalar]
Available memory, in megabytes. You should not rely too much on this; it is the last priority that will be honored. Use bbsvdmem to estim ate the actual usage.
The default behaviour is to use as much memory as necessary to get convergence in most cases.
Default: unlimited.
Parameters specific to bbsvds
The following paramaters are specific to bbsvds, and does not affect anything else.
Parameter: sort [true | false]
Determine if triplets will sorted in order of decreasing singular values, before being returned.
Default: true.
Parameter: tol [scalar]
This parameter is present for compatibility with and can only be used for Matlab matrices. It will set atol in a way that is similar to svds.
We strongly recommend not to use this parameter, except for situations where compatibility with svds is vital.
Restarting Strategy
The best choice of restarting strategy is problem dependent. A restart occurs when the toolbox exhausts the memory allocated for the computation or some triplets converged. The restarting strategy
decides what triplets to throw away before starting a new iteration.
There are two reasons that keeping all triplets is a bad idea. First, this will often exhaust the memory rather quickly. Secondly, a restart uses a dense SVD-computation, which uses time that is
cubic in the number of triplets at the end of the iteration. Therefore we may be able to keep 100 triplets, but if we keep 200 a restart will take 8 times longer.
The downside is that throwing a triplet away means that we probably need to find it again some time later. Therefore, if the matrix-vector products can be computed fast (say, a very sparse matrix)
then we may want to use a low value. On the other hand, if the matrix-vector product is slow (say, built from a series of Fourier transforms and complex processing) we may want to set this somewhat
Parameter: K [integer]
This is the number of triplets kept in memory for the purpose of converging new triplets. When this many triplets has been the reached, a restart is forced. Generally, one would almost
always want to set this to the highest possible value allowed by the available memory.
Parameter: rkeep [integer]
When a restart occurs, the toolbox will keep at least this many triplets (if there are that many). However, it may keep somewhat more than this number, because it also keeps triplets that
are important for the convergence of the chosen triplets. The current version of the toolbox will try to keep K at least twice as big as rkeep.
Parameter: rmaxi [integer]
Maximum number of restarts without convergence. It is possible that a high value (e.g. rmaxi=50) may extract more triplets than the default setting. Generally speaking, however, the effort
spent when several restarts are required for each triplet is rarely justified.
Deflation Strategy
The deflation strategy controls how converged triplets are handled. After convergence, a triplet is explicitly removed from space used to search for new triplets. This is necessary to guarantee that
triplets correspond to unique triplets of the operator, allow non-simple triplets to converge, and to keep triplets orthogonal to each other.
Parameter: dgreedy [true | false]
In a greedy strategy, a triplet will be deflated as soon as it converges to the specified tolerance. A problem with the greedy strategy is that triplets does not converge in any particular
order. It is usually the case that some triplets, corresponding to small singular values, will converge at about the same time as a few triplets in a cluster of singular values. To prevent
this, a non-greedy strategy is normally preferable for SVD computations. This will only deflate triplets corresponding to a "run" of triplets.
The downside with this approach is that some convrged triplets may be kicked out of memory. This may add a significant penalty, since it is likely that these triplets will be found a bit
Applications, such as equation solving, that relies on a particular starting-vector should always set dgreedy=true.
Default: false
Parameter: dtail [integer]
Triplets with nearly identical singular values must explicitly be kept orthogonal to each other. The "tail" is used to ensure that triplets corresponds to unique triplets of the operator
and to keep triplets with nearly identical singular values nearly orthogonal.
It is also used to create a gap so converged triplets, that was found previously, can be ignored. If dtail is too small you risk skipping clustered singular triplets.
Parameter: dlead [integer]
This is the number of the leading triplets that is kept in memory. This is needed to keep rounding errors in check. If the computer worked in exact arithmetic, you could safely set dlead=
Generally, this should be set large enough to absorb a set of dominating singular triplets of an operator. The number of triplets the software can extract from an operaton may, in the
worst case, be limited to about dlead+dtail.
Parameter: dsides [1 | 2]
Some applications does not require that individual triplets are accurate. For instance one may be interested in just the singular values or a low-rank approximation of the operator.
In this case it is possible to save an enormous amount of effort and memory by using a one-sided deflation strategy. This means that only the vectors corresponding to the smallest
dimension of the operator are saved in the leading triplets. This implies that the amount of memory needed scales with the smallest dimension of the operator.
Generally one should set dsides=1 if the application can tolerate it.
Parameter: dorth [scalar]
This option only affects two-sided deflation, i.e. dsides=2. The software attempts to keep triplets orthogonal simply by keeping them accurate. However, this is only possible for triplets
with large singular values. Therefore triplets must be kept orthogonal when the singular values drops low enough. The option dorth determines how orthogonal these triplets needs to be.
Note that the number of triplets that can converge may be limited if this value is too small. In most cases the level of orthogonality is sufficient for practical purposes.
Default: dorth=1e-4
atol specifies the absolute accuracy of the singular values before a triplet is deemed to converge. A computed singular value, σ, satisfies |σ-σ[0]|≤atol, where σ[0] is an exact singular value.
The calculation does not take rounding errors into account. It is often possible to drive the tolerance below the accuracy of the calculation, although this tolerance is fictive. However, if an
operator is more accurate than a single value of tau (usually the case), then it is often possible to get surprisingly accurate singular values.
If singular vectors of the operator bb are computed, and dsides=2, then the following holds (with similar comments on rounding errors):
norm(bb*v-sig*u) ≤ atol
norm(bb'*u-sig*v) ≤ atol
abs(v1'*v2) ≤ dorth
Two-sided deflation
BBTools keeps triplets orthogonal implicitly, i.e. by keeping them accurate. This scheme accounts for the fact that more triplets can be computed than will fit in memory. It starts to fall apart for
small singular values, because of rounding errors. Therefore, pairs of singular triplets are explicitly kept orthogonal to each other when the singular values start approaching the level of the
rounding errors. This place where this starts to take effect is given by dorth.
One-sided deflation
For one-sided deflation, i.e. dsides=1, the norm of the residual in the "long space" is approximately atol*(norm(bb)/σ). The orthogonality of the triplets are impaired similarly, but the singular
values are accurate and so are low-rank approximations of the operator.
To be more precise, assume that length(v)<length(u):
norm(bb*v-sig*u) ≤ atol
abs(v1'*v2) ≤ atol
norm(bb'*u-sig*v) ≤ atol*(norm(bb)/sig)
abs(u1'*u2) ≤ atol*(norm(bb)/sig) | {"url":"https://xtra.nru.dk/bbtools/help/toolbox/bbtools/parameters_svd.html","timestamp":"2024-11-09T03:01:01Z","content_type":"text/html","content_length":"14651","record_id":"<urn:uuid:3a8cc1e2-d30b-4024-b375-19b96d29eecb>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00867.warc.gz"} |
Unveiling the Correct Spelling: How to Spell Triangle with Precision
Unveiling The Correct Spelling: How To Spell Triangle With Precision
Triangle, a polygon with three sides, derives its spelling from the Greek word “trigonon,” meaning “three-cornered.” To spell “triangle,” arrange the letters in the correct sequence: t-r-i-a-n-g-l-e.
Pronounce each letter clearly, and remember the phonetics: /ˈtraɪ.æŋ.ɡəl/. Understanding triangle’s etymology, classifications, properties, and significance in fields like mathematics and engineering
enhances our understanding of its geometry and use in problem-solving.
The Triangle: A Geometric Cornerstone
In the realm of geometry, the triangle stands as a paramount figure, its significance extending far beyond its humble appearance. It embodies the essence of polygons, with its three sides and three
angles forming a shape that has captivated scholars, artists, and scientists for centuries.
What is a Triangle?
A triangle is the simplest polygon, defined by three line segments that intersect at their endpoints, creating three distinct angles. These angles may vary in measure, giving rise to different types
of triangles, such as right, acute, and obtuse triangles.
Importance of Triangles
Triangles are omnipresent in the world around us, from the towering peaks of mountains to the intricate patterns of snowflakes. In architecture, they provide structural stability to buildings and
bridges, while in engineering, they form the basis of trusses and other load-bearing structures. They even find application in art, as artists use them to create dynamic compositions and convey
Components of a Triangle: Sides, Angles, and Polygon
A triangle, as the name suggests, is a closed figure defined by three straight lines. These lines are referred to as sides. Each side connects two of the triangle’s three corners, known as vertices.
The length of a side is the distance between the two vertices it connects.
At each vertex where two sides meet, an angle is formed. A triangle has three angles, each measured in degrees. The sum of the interior angles of any triangle is always 180 degrees.
A polygon is a closed figure with straight sides. Triangles are polygons, specifically classified as 3-sided polygons. They possess the properties of polygons, such as having a definite perimeter
(the sum of the lengths of all sides) and area (the space enclosed within the figure).
Properties of Triangles as Polygons
As a polygon, triangles exhibit certain properties:
• Convexity: All angles are less than 180 degrees, giving the triangle a non-overlapping interior.
• Rigidity: The shape of a triangle cannot be deformed without changing the lengths of its sides or the measures of its angles.
• Symmetry: Depending on the lengths of their sides and angles, triangles can exhibit various types of symmetry, including reflectional, rotational, or translational symmetry.
The Process of Spelling “Triangle”: Breaking It Down Letter by Letter
Have you ever wondered how to spell the seemingly complex word, triangle? It may sound intimidating, but spelling this fundamental geometric shape is as simple as putting together a puzzle. Let’s
break it down into its individual letters and understand the correct sequence in which they should be arranged.
Starting with the first letter, we have T. It’s a tall and sturdy letter that stands firmly at the beginning of the word. Next comes R, the curved letter that adds a bit of flair to the mix.
Following closely behind is I, a slim and elegant letter that adds a touch of height.
Moving on to the middle part of the word, we encounter A. This broad letter brings width to the word and helps connect the first and last parts. Then comes N, a diagonal letter that gives the word a
sense of balance.
Finally, we reach the end of the word with G, a curved letter that brings a gentle closure. And there you have it – the word triangle spelled out in its entirety.
Remember, the key to spelling triangle accurately is to focus on the sequence of the letters. T-R-I-A-N-G-L-E. Once you have the order right, you’ll be able to spell it confidently every time. So, go
ahead, give it a try and impress your friends with your newfound spelling skills!
Pronunciation and Phonetics: Unveiling the Sounds of “Triangle”
The word “triangle,” with its seemingly straightforward spelling, may surprise you with its pronunciation. Let’s delve deeper into the intriguing sounds that make up this geometric term.
Properly pronouncing “triangle” is essential for effective communication. It consists of three syllables: “tri” + “an” + “gle”. The emphasis falls on the first syllable, “tri”. Each syllable is
pronounced as follows:
• “tri”: This syllable is pronounced like the word “tree.” Your tongue should touch the roof of your mouth just behind your upper front teeth.
• “an”: Here, we have the short “a” sound, as in the word “hat.”
• “gle”: This syllable rhymes with the word “bell.” Your tongue should rise towards the roof of your mouth as you pronounce it.
Understanding the phonetic components of “triangle” is crucial for accurate pronunciation. Phonetics, a branch of linguistics, focuses on the study of speech sounds. By understanding the individual
sounds that make up a word, we can pronounce it correctly.
In the case of “triangle,” the sounds represented by the letters are as follows:
• “t”: A voiceless alveolar stop. Pronounced by placing the tip of your tongue on the alveolar ridge (the bony ridge behind your upper front teeth) and releasing it quickly.
• “r”: A voiced alveolar trill. Formed by vibrating the tip of your tongue against the alveolar ridge.
• “i”: A high front vowel. Pronounced by raising the front of your tongue towards the roof of your mouth.
• “a”: A low back vowel. Pronounced by lowering the back of your tongue and spreading it wide.
• “n”: A voiced alveolar nasal. Created by allowing air to flow through your nose while touching the tip of your tongue to the alveolar ridge.
• “g”: A voiced velar plosive. Pronounced by raising the back of your tongue towards the velum (the soft palate) and releasing it.
• “l”: A voiced alveolar lateral. Pronounced by placing the tip of your tongue on the alveolar ridge and allowing air to flow around the sides.
• “e”: A mid front vowel. Pronounced by raising the front of your tongue towards the roof of your mouth, slightly lower than for “i.”
By breaking down “triangle” into its phonetic components, we gain a deeper understanding of its pronunciation. This knowledge empowers us to communicate clearly and confidently, ensuring accurate
comprehension in any conversational setting.
The Etymology and Historical Evolution of the Term “Triangle”
Prepare to embark on a linguistic journey as we delve into the etymology and historical development of the term “triangle.” This seemingly simple word holds a rich history that has shaped its meaning
and significance in our world.
The word “triangle” traces its roots back to ancient Greek, where it originated from the term “trigonon.” This word is composed of two Greek elements: “tri,” meaning three, and “gonos,” meaning
angle. Thus, “trigonon” directly translates to “three angles.”
The concept of a triangle as a geometric shape emerged in ancient Greece alongside the development of geometry as a mathematical discipline. Early Greek mathematicians such as Pythagoras and Euclid
recognized the distinctive properties and relationships of triangles, leading to the establishment of fundamental principles in triangle geometry.
Over time, the term “trigonon” was borrowed into Latin as “triangulum,” preserving its original Greek meaning. As Latin became the lingua franca of Western Europe, “triangulum” spread throughout the
Roman Empire and beyond.
During the Renaissance, the Latin term “triangulum” was adopted into English as “triangle.” This word has remained in use ever since, becoming an integral part of our vocabulary and mathematical
The historical evolution of the term “triangle” mirrors the development of geometry as a field of knowledge. From its humble beginnings in ancient Greece to its widespread adoption in modern
languages, this word has witnessed the evolution of mathematics and its impact on our understanding of the world around us. Today, the term “triangle” stands as a testament to the enduring legacy of
ancient Greek thinkers and the pervasive influence of geometry in our lives.
Classification of Triangles: Exploring the Diverse World of Three-Sided Shapes
In the realm of geometry, triangles reign supreme as one of the most fundamental shapes, captivating the minds of mathematicians, engineers, and artists alike. Understanding the various types of
triangles is essential for mastering this geometric marvel and unlocking its potential.
Types of Triangles Based on Side Lengths
Triangles can be classified based on the lengths of their sides. Three distinct categories emerge:
• Equilateral triangles: All three sides are equal in length, forming a perfect triangle.
• Isosceles triangles: Two sides are equal in length, creating a symmetrical shape.
• Scalene triangles: All three sides are different in length, resulting in an asymmetrical triangle.
Types of Triangles Based on Angle Measures
Another classification system for triangles revolves around the measures of their interior angles. This division yields three primary types:
• Right triangles: One angle measures exactly 90 degrees, forming a “right” angle.
• Acute triangles: All three angles are less than 90 degrees, creating a sharp-cornered figure.
• Obtuse triangles: One angle is greater than 90 degrees, resulting in a “blunt” angle.
Visualizing the Variations
To fully grasp the diversity of triangles, visual examples are indispensable. Consider the following:
• Equilateral triangle: Imagine a perfectly equilateral triangle with all three sides measuring the same length. It resembles a flawless equilateral hexagon.
• Isosceles triangle: Picture a triangle with two identical sides, forming a symmetrical shape. It resembles a house with a pitched roof.
• Scalene triangle: Visualize a triangle with all three sides and angles differing in length and measure. It appears more irregular and asymmetrical.
• Right triangle: Imagine a triangle with one angle forming a perfect right angle, like a carpenter’s square.
• Acute triangle: Think of a triangle with all three angles less than 90 degrees, creating a sharp-pointed shape.
• Obtuse triangle: Picture a triangle with one angle exceeding 90 degrees, resulting in a triangle with a wider angle.
Understanding the various classifications of triangles is fundamental for geometry enthusiasts, students, and anyone seeking to master the intricacies of this geometric icon.
Properties and Characteristics: Unraveling the Essence of Triangles
In the realm of geometry, triangles stand apart as captivating shapes, adorned with an array of unique properties. These features, like shimmering gems, illuminate the essence of triangles and
empower us to solve problems with precision.
The Sum of Interior Angles: An Unwavering Principle
Every triangle whispers a secret that the sum of its interior angles is always 180 degrees. This fundamental truth holds constant, regardless of the triangle’s size, shape, or orientation. It’s a
guiding light, illuminating our understanding of angles and their relationships within a triangle.
The Pythagorean Theorem: A Triumph of Logic
For right triangles, another gem shines—the Pythagorean theorem. This equation, named after the legendary mathematician Pythagoras, reveals a profound connection between the sides: the square of the
hypotenuse (longest side) is equal to the sum of the squares of the other two sides. This theorem transforms triangles into logical puzzles, where solving for unknown sides becomes an exercise in
mathematical artistry.
Area Formula: Unveiling Triangular Spaces
Triangles, like ethereal sails, enclose areas that can be measured with a simple formula: 0.5 * base * height. This formula empowers us to calculate the extent of triangular spaces, whether they
adorn a canvas or capture our imagination in real-world scenarios.
Understanding the properties and characteristics of triangles is a cornerstone of geometrical literacy. These properties weave together to form a tapestry of knowledge, empowering us to unravel the
mysteries of angles, decipher the relationships between sides, and conquer problems with ease. Embracing this knowledge, we unlock the secrets that triangles hold, illuminating our path towards
geometrical mastery.
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Mortgage Amortization Calulator
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Master Books Homeschool Curriculum - Principles of Mathematics Book 1 Set
Inspire Confidence in Math
Mathematical terms and concepts can seem so complex and intimidating as students transition from the upper elementary years to Jr High. Katherine Hannon's down-to-earth style and conversational tone
sets students at ease and inspires them to approach all those terms and concepts with confidence. She imparts an understanding that each simply describe a useful tool we can put to use in everyday
life, history, science, art, music, and more!
Engaging, Conversational Lessons
Principles of Mathematics focuses on mathematical concepts and emphasizes the practical application of it. Students learn both the ‘why’ and ‘how’ of math through engaging, conversational lessons.
This approach firms up foundational math concepts and prepares students for upper-level math in a logical, step-by-step way. This helps the student understand concepts, build problem-solving skills,
and see how different aspects of math connect.
Incredibly Faith Building
“Every time you solve a math problem, you’re relying on the underlying consistency present in math. Any time you see that math still operates consistently, it’s testifying that God is still on His
throne, faithfully holding all things together.” – Katherine Hannon
In Principles of Mathematics Book 1, each lesson and exercise leads the student towards understanding the character and attributes of God through the study of math history, concepts, and application.
Principles of Mathematics Book 1 goes beyond adding a Bible verse or story to math instruction, it actively teaches and describes how the consistencies and creativity we see in mathematical concepts
proclaim the faithful consistency of God Himself and points students towards understanding math through a Biblical worldview.
Excellent High School Preparation
Principles of Mathematics Book 1 lays a solid foundation—both academically and spiritually—as your student prepares for High School math! Students will study concepts of arithmetic and geometry,
further develop their problem-solving skills, see how mathematical concepts are applied in a practical way to everyday life, and strengthen their faith!
In Principles of Mathematics Book 1 your student will:
• Explore arithmetic & geometry
• Strengthen critical thinking skills
• Transform their view of math through a Biblical Worldview
• Find the height of a tree without leaving the ground
• Understand mathematical concepts, history, and their practical application
• and so much more!
Principles of Mathematics Book 1 Teacher Guide Includes:
• Convenient Daily Schedule—saving you time!
• Optional Accelerated Daily Schedule
• Student Worksheets
• Quizzes
• Tests
• Answer Key
• 3-hole-punched, perforated pages
Course Features:
• 30-45 minutes per lesson, 4-5x per week
• Recommended Grade Level: 7th - 8th
□ High school students may also complete both Principles of Mathematics 1 & 2 in a year using the accelerated schedule included in the Teacher Guide
□ 1 Math credit earned
□ Pre-Algebra
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What is N-
The N-ary tree data structure is a complex binary tree data structure that allows us to have multiple children at each node, similar to the binary tree data structure, but with n number of children.
This structure is slightly more complex than the prevalent binary trees. The only difference between an N-ary and a binary tree is in their shapes. In an N-ary, we can add and remove leaves (and
therefore branches) from the root of the system during its construction.
A famous example of an N-ary tree would be London’s famous London Eye Ferris wheel.
N-ary Trees hold certain advantages over Binary Trees, namely that it takes up significantly less space when there is no more room to grow vertically in a Binary Tree. This also allows for linear
storage of data rather than the tree-like structure used in Binary Trees, making it perfect for database files where you would like to save as much space as possible without sacrificing too much
speed or efficiency.
General Idea
N-ary trees are a variety of binary trees. They differ by having a single node at the top which holds multiple children. These children may have their children, themselves being n-ary trees with one
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Multiplying Negative Rational Numbers Worksheet
Multiplying Negative Rational Numbers Worksheet work as foundational tools in the realm of mathematics, giving a structured yet functional platform for students to discover and grasp numerical
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Equivalent expressions with negative numbers multiplication and division Negative numbers multiplication and division Quiz 3 Negative numbers multiplication and division Unit test
Understanding Multiplication with Negative Integers Practice multiplying negative integers 1 Find each product Then describe any patterns you notice 3 2 7 5 2 2 7 5 1 2 7 5 0 2 7 5 2 1 2 7 5 2 2 2 7
5 2 3 2 7 5 2 Solve each problem Explain how you determined the sign of the products
At their core, Multiplying Negative Rational Numbers Worksheet are cars for theoretical understanding. They encapsulate a myriad of mathematical concepts, directing learners via the labyrinth of
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Multiply And Divide Positive And Negative Rational Numbers Worksheet Jack Cook s
Multiply And Divide Positive And Negative Rational Numbers Worksheet Jack Cook s
The Corbettmaths Textbook Exercise on Multiplying Negatives and Dividing Negatives
Worksheet by Kuta Software LLC 2 11 3 2 4 5 12 3 4 1 3 Find each product 13 5 8 5 14 3 9 0 8 15 9 5 2 3 16 5 5 7 48 Find each quotient 17 5 6 3 5 18 7 1 0 5 19 1 9 1 20 6 12 3
The heart of Multiplying Negative Rational Numbers Worksheet lies in growing number sense-- a deep comprehension of numbers' significances and affiliations. They motivate exploration, welcoming
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From Theory to Real-World Application
Multiplying Rational Numbers Worksheet
Multiplying Rational Numbers Worksheet
Unit 13 Add and subtract fractions different denominators Unit 14 Multiply and divide multi digit numbers Unit 15 Divide fractions Unit 16 Multiply and divide decimals Unit 17 Exponents and powers of
ten Unit 18 Add and subtract negative numbers Unit 19 Multiply and divide negative numbers Course challenge
Negative numbers are numbers with a value of less than zero They can be fractions decimals rational and irrational numbers 13 2 6 4 and 123 are all negative numbers We have a page dedicated to
learning about negative numbers below What are Negative Numbers
Multiplying Negative Rational Numbers Worksheet serve as conduits connecting theoretical abstractions with the palpable facts of everyday life. By infusing functional circumstances into mathematical
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Versatility is inherent in Multiplying Negative Rational Numbers Worksheet, using a collection of pedagogical devices to deal with varied knowing styles. Aesthetic help such as number lines,
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Multiplying Negative Rational Numbers Worksheet chart a training course towards mathematical fluency. They instill determination, critical reasoning, and analytical skills, vital features not only in
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Verdict: Embracing the Magic of Numbers
Multiplying Negative Rational Numbers Worksheet represent the magic inherent in maths-- a captivating trip of expedition, exploration, and proficiency. They go beyond standard rearing, functioning as
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Multiplying Negative Numbers Worksheet
Multiplying Negative Numbers Worksheet
Check more of Multiplying Negative Rational Numbers Worksheet below
Multiplying Rational Expressions Worksheet
Multiplying And Dividing Rational Numbers Worksheet
Multiplying Negative Numbers Worksheet
Multiplying Rational Numbers Worksheet Pdf Worksheet
Multiplying Rational Numbers Worksheet
8 Best Images Of Rational Numbers 7th Grade Math Worksheets Algebra 1 Worksheets Rational
Grade 7 Mathematics Lowndes County School District
Understanding Multiplication with Negative Integers Practice multiplying negative integers 1 Find each product Then describe any patterns you notice 3 2 7 5 2 2 7 5 1 2 7 5 0 2 7 5 2 1 2 7 5 2 2 2 7
5 2 3 2 7 5 2 Solve each problem Explain how you determined the sign of the products
Multiplying And Dividing Positives And Negatives Date Period
Multiplying and Dividing Positives and Negatives Date Period Find each quotient 1 10 5 2 24 12 3 20 2 4 300 20 5 65 5 6 66 6 7 75 15 8 56 14 9 102 17 10 72 4 11 153 17 12 12 3 13 48
Understanding Multiplication with Negative Integers Practice multiplying negative integers 1 Find each product Then describe any patterns you notice 3 2 7 5 2 2 7 5 1 2 7 5 0 2 7 5 2 1 2 7 5 2 2 2 7
5 2 3 2 7 5 2 Solve each problem Explain how you determined the sign of the products
Multiplying and Dividing Positives and Negatives Date Period Find each quotient 1 10 5 2 24 12 3 20 2 4 300 20 5 65 5 6 66 6 7 75 15 8 56 14 9 102 17 10 72 4 11 153 17 12 12 3 13 48
Multiplying Rational Numbers Worksheet Pdf Worksheet
Multiplying And Dividing Rational Numbers Worksheet
Multiplying Rational Numbers Worksheet
8 Best Images Of Rational Numbers 7th Grade Math Worksheets Algebra 1 Worksheets Rational
Multiplying Rational Numbers Worksheet
Multiplying Negative Numbers Worksheet
Multiplying Negative Numbers Worksheet
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Interaction between neutrino flavor oscillation and Dark Energy as a super-luminal propagation
by Marco Lelli
Direct Download
As it is well known a recent series of experiments, conducted in collaboration between CERN laboratories in Geneva and the Gran Sasso National Laboratory for Particle Physics, could have decreed the
discovery of the transmission of a beam of super-luminal particles.
Experimental data indicate that the distance between two laboratories (approximately 730 km) was covered by a beam of neutrinos with an advance of approx 60 nanoseconds with respect to a signal
travelling at the relativistic limit speed c (which takes a time interval of the order of 2,4.10-3 s to perform the way).
Neutrino beam starts from CERN and after travelling 730 km through the Earth’s crust, affects lead atoms of the OPERA detector at Gran Sasso laboratories. Production of neutrino beam is due by the
acceleration and collision of protons and heavy nuclei. This event produces pions and kaons, which then decay into muons and νμ.
The initial energy of neutrino beam is 17 GeV and its composition is almost entirely due to νμ.
Publication of the OPERA experimental data immediately got a deep world mass-media echoes: the possible confirmation of the results of the experiment seems to imply an explanation leading to change
our current thoughts about theory of relativity and, therefore, the intimate space-time nature. In this assumption c may not be considered a speed limit on the quantum scale investigation.
In this paper we try to show how the uncertainty principle and the oscillation in flavor eingenstates of neutrino beam may provide a possible explanation for OPERA’s data.
Our research assumes two basic hypotheses.
First approximation: approximation in number of flavor eigenstates (and then in mass eigenstates) within is supposed to play neutrino oscillation.
We consider this oscillation between two flavor eigenstates. Then we assume that each component of the neutrino beam can be described by a linear combination of two eigenstates of flavor. These two
eigenstates are: μ flavor (the flavor of neutrino beam generation) and τ flavor.
Oscillations in this two flavor was already observed in first half of 2010 within the same OPERA experimental series.
Although, as it is known, the neutrino oscillation cover three mass eigenstates for its complete description, we assume here an approximation for dominant mass of neutrino τ, which reduces the
description of neutrino propagation in a linear combination of only two mass eigenstates.
In this approximation we can now describe the propagation of each neutrino produced at CERN as a combination of two mass eigenstates as follows:
Flavor and mass eigenstates are related by a unitary transformation which implies a mixing angle in vacuum similar to Cabibbo mixing angle for flavor of quarks:
Second approximation: we suppose that propagation of neutrino beam is in vacuum. The propagation in vacuum is determined by the temporal evolution of the mass eigenstates
We can consider valid this assumption, at least in first approximation, because matter interacts in particular with νe and less with νμ and ντ.νe weakly interacts with matter by W± and Z° bosons
while νμ and ντ only by Z° bosons. So the principal possible effect consists in a massive transformation of νe in the |νμ› eigenstate.
Considering the small number of νe in starting beam we can neglect this effect.
Assuming that in the initial state only νμ are present in the beam, through a series of elementary steps, we can get
then we can obtain the probability
In the approximation mμ « Eμ we can write
and finally the transition probabilities between eigenstates of flavor
νμ beam produced at CERN propagates as a linear superposition of mass eingestates given by the following relation
This superposition generates an uncertainty in propagating mass neutrino that grows over time and is equal to
This uncertainty in the mass eigenstates of the neutrino implies an uncertainty in the energy of propagation.
Given the relativistic equation
taking the momentum of propagation p=cost, the uncertainty linked to neutrino mass eigenstate is linearly reflected in an uncertainty in the propagation energy:
Therefore we have
Following the uncertainty principle we have
so the uncertainty (12), about the value of νμ energy of propagation, causes a corresponding uncertainty in its time of flight between the point of production and the point of arrival.
This uncertainty is expressed as follows:
In OPERA case available experimental data are:
12=1, in analogy with the value attributed to Cabibbo quark mixing angles, and a value for Δm12 ≈ 10-²eV ≈ 1,6.10-²¹ J we have
(14) shows that the advance on the propagation of neutrino beam, detected in the execution OPERA experiment, is between the range determined by the uncertainty principle.
The advance Δt is then interpreted by the uncertainty principle and the neutrino flavor oscillation during propagation. This oscillation implies an uncertainty in the neutrino propagation energy, due
to the linear superposition of its mass eigenstates, which affects the uncertainty of its flight time.
According to this interpretation, therefore, the results of OPERA experiment, if confirmed, would represent not a refusal of the condition of c as a relativistic speed limit, but rather a stunning
example of neutrino flavor oscillation according to physics’s laws known today (uncertainty principle and speed limit c).
The range indicated in (14) depends on the competition of two factors. On one hand, the intrinsic nature of inequality of the uncertainty principle, on the other our fuzzy knowledge of Δm12 between
mass eigenstates of neutrinos with different flavors.
One of the most convincing experimental proofs of flavor neutrino oscillation is the lack of solar electron neutrinos measured experimentally respect to the theoretically expected flow.
OPERA, as well as other tests, was designed to observe possible flavor oscillation in a neutrino beam running along the earth’s subsurface. Any oscillation can be found by observing a change of
flavor in a fraction of neutrinos in the arrive.
However, if this happens, neutrino mass eigenstate is described by a linear superposition of mass eigenstates of pure muon neutrino and tau neutrino.
This condition generates an uncertainty on the propagation energy, which translates into an uncertainty on the flight time.
This is directly proportional to the total flight time and the square of the difference between the mass values of the different flavors of neutrinos, while it is inversely proportional to the total
energy of the beam.
In this interpretation, therefore, the advance of the flight time of the neutrino beam with respect to the velocity c, far from being a refutation of the relativistic speed limit, is a good
demonstration of neutrino flavor oscillation.
So we could use the advantage Δt in an attempt to determine, more accurately, the value of Δm12.
On the other hand, examples of physical effects equivalent to a super-luminal propagation of particles are considered in other fields of contemporary theoretical physics. Hawking effect about the
emission temperature of a Black Hole is, under this respect, a very significant example.
Cosmic neutrinos flavor oscillations. We can now consider what could be the value of the advantage Δt respect to the time of flight of c in the case of neutrinos coming, for example, from a SuperNova
In this case the average energy of neutrinos νe is of the order of 10^7 eV and the time of flight, for example in the case of SuperNova 1987a, of the order of 10¹² s.
Under these conditions we have
and it is conceivable that it may start a continuous sequence of oscillations in mass eigenstates.
The logical consequence of this situation is a superposition of two equally probable mass eigenstates.
So the uncertainty in mass eigenstates exists with respect to the state of arrival of the neutrino and a mixing of mass eigenstates with the same probability equal to ½.
In this hypothesis we have
therefore an advantage Δt of approx six orders of magnitude lower than in the OPERA case.
Interpretation of the principle of uncertainty used above. The uncertainty principle is commonly intended as an aid to explanation for the impossibility of determining, by observation,
contemporarily the position and momentum of a physical system, with absolute precision, because the one excludes the other.
Assuming this interpretation the uncertainty principle could explain , in the case of OPERA, a set of measures centered on an advance Δt=0 with a spread on the obtained measurement results in the
order of (14).
In contrast, the experimental measurements provided by OPERA appears to be centered on a value of Δt ≈ 60 ns in advance respect to the time of flight of c!
Which explanation is therefore possible to give to the application of the uncertainty principle to justify the consistency of the data provided by OPERA with the fundamental laws of physics known
The most coherent interpretation seems to be as follows: the temporal evolution of the neutrino mass eigenstate introduces a temporal evolution in the state of total energy that interacts with
space-time producing a reduction of the time of flight. This interaction has to be coherent with the uncertainty principle.
Energy gained or released by neutrino, during oscillation, must be released or gained by space-time, according to the principle of conservation of energy.
A more accurate explanation will require the introduction of some new hypotheses.
We suppose below that space-time possesses a quantized structure. We define a fundamental 1D string element that has the dimension of a length or a time. This fundamental element is a 1D vector in
the 2D string wolrdsheet: we call this element the quantum of space-time.
To each 1D of space-time is associated a 1D energy-momentum vector (the total energy associated to a quantum of space-time) that is related to the module of the 1D quantum of space-time with a
relation of constraint that we define below.
To introduce the basic unit of space-time we introduce the Polyakov 2D string action and we proceed to its quantization finding the 1D elementary quantum of space-time
Now we want to consider (17) in the limit n -> 1. The infinitesimal parameters dσ and dτ take the meaning of physically limit movement along, respectively, the spatial direction and temporal
direction of the 2D string worldsheet.
We can call these limit movement as follows
Ω^x e Ω^0 take the meaning of quantum of space-time in space direction and time direction in the 2D string worldsheet.
Therefore, in this case, to each spatial direction of the elementary string element corresponds a temporal direction that, in a Minkowski’s manifold, is orthogonal to the space direction. The
relation (18) binds the module of the element of string along the spatial direction with respect to temporal direction, in the case of a Minkowski’s manifold, and have the values lp and lp/c.
Double differentiation
appearing in (17) must now be rewritten taking into account that in a Minkowski’s manifold, for relations (18), we can write
Since it is possible to show that 2D string worldsheet action of Polyakov coincides with Nanbu-Goto action
given the relation
and because we have
we can rewrite (18) as follows
In (20) with Tμν we have indicated relation Tμν = Tημν. So we indicate string tension in 2 dimensions as a tensor of rank 2.
In a Minkowski’s manifold we have:
So the string tension in a Minkowski’s manifold can be written as a tensor of rank 2 whose product with the module of the fundamental string elements (the quantum of space-time) in spatial and
temporal direction is constant and equal to Planck’s quantum of action. Contracting one of the two indices of tension with one of the two vectors Ω^μ or Ω^ν we get the 2D energy-momentum vector for
the string element along the direction μ and ν respectively,
it is now possible to define the following relation
Relation (23) was obtained in a Minkowski’s manifold: it is therefore valid in a region of space-time in which the action of gravitational energy is negligible. Under these conditions (23) defines a
relation of constraint: the product of the 1D length of the fundamental string element (the length of the module of the quantum of space-time) and the 2D energy-momentum vector of 2D string
worldsheet associated with this element is constant and equal to Planck’s constant.
2D energy- momentum vector Eν thus defines the expectation value of energy of empty space that corresponds to the amount of energy needed to increase string length of an element of length lp along ν
Similarly we can define Eν as the 2D energy-momentum vector associated with the increase of a quantum of space-time along ν direction. For these reasons, in a Minkowski’s manifold, (23) takes the
valids in each quantum of space-time.
Calculation of the anticipation Δt in the time of flight. (24) can be written taking into account variations in the 2D string worldsheet fundamental element:
multiplying the two members is obtained the variational relation of least action for the elementary 2D string worldsheet:
so we have
and then
From (28) we obtain (13) and the result (14). In (28) the term is an appropriate constant of integration that take in to account vacuum fluctuations of energy of magnitude for the system under
Conclusions. Conducing our analysis in 2D we quantize the 2D Polyakov string worldsheet action, obtaining a constraint relation that relates 2D energy -momentum vector and the module of 2D elementary
string element (the quantum of space-time).
We have therefore assumed that the neutrino flavor oscillation interacts with the energy associated with each element of the 2D worldsheet string (or the space-time) exchanging energy. This exchange
is obeying the law of conservation of energy.
This kind of interaction does not require any hypothesis of fifth force, and may, on the contrary, be assumed of gravitational type, in the sense that the energy due to the neutrino mass eigenstates
interacts with the energy of the elementary string element with an easy phase overlapping, just as it is with a gravitational mass.
We can therefore assume that neutrino, through the temporal evolution of its mass eigenstates, exchanges energy with space-time. This exchange causes a change, a contraction in the length of the 2D
fundamental string element. Integration of this contractions along the path of neutrino flight produces as a result the observed advantage in the time of the flight.
The energy associated with each elementary quantum of 2D string worldsheet in a Minkowski’s manifold corresponds to the energy of empty space-time, ie the vacuum energy of the gravitational field in
absence of gravitational source. The target of a forthcoming work will be to show how this vacuum energy is able to produce effects phenomenological equivalent to hypothesis of dark energy and dark
matter under certain conditions.
Basing on the assumptions here introduced the same uncertainty principle, from first and irreducible principle of physics, assumes the rank of derived condition through (25) – (28) by a more
fundamental principle that is (23).
[1] B. M. Pontecorvo, Sov. Phys. Usp., 26 (1983) 1087.
[2] L. Wolfenstein, Phys. Rev. D, 17 (1978) 2369.
[3] S. P. Mikheev e A. Yu. Smirnov, Il Nuovo Cimento C, 9 (1986) 17.
[4] S. Braibant, G.Giacomelli, M. Spurio, Particelle ed interazioni fondamentali, Springer, 2010.
[5] J. N. Bahcall, “Neutrino astrophysics” (Cambridge, 1989); http://www.sns.ias.edu/~jnb
[6] http://www.arcetri.astro.it/science/SNe/sn1987a.jpg
[7] H. A. Bethe e J. R. Wilson, Astrophys. J., 295 (1985) 14.
[8] G. Pagliaroli, F. Vissani, M. L. Costantini e A. Ianni, Astropart. Phys., 31 (2009) 163.
[9] V. S. Imshennik e O. G. Ryazhskaya, Astron. Lett., 30 (2004) 14.
[10] W. Baade e F. Zwicky, Proc. Natl. Acad. Sci. U.S.A., 20 (1934) 259.
[11] A.M.Polyakov, Gauge Fields and Strings, Harwood academic publishers, 1987.
[12] Measurement of the neutrino velocity with the OPERA detector in the CNGS beam, arXiv:1109.4897.
[13] F. L. Villante e F. Vissani, Phys. Rev. D, 76 (2007) 125019.
[14] F. L. Villante e F. Vissani, Phys. Rev. D, 78 (2008) 103007.
[15] M. A. Markov, “The Neutrino” (Dubna) 1963.
by Marco Lelli
Direct download
Dear H. Hansson:
First, have a camel, then cure his cough.
Warm Regards,
Dear Mr. Rossi,
Regarding you answer to Mr. Martin.
I think you are doing the right thing. But still, I would not put my bet on it. The theory of how things becomes fuc**ed are universal (if it can it will). It’s not good to become too successful.
Both the U.S.A and EU have anti-trust laws. If your goal becomes a reality, your business will be targeted by authorities on both sides of the Atlantic.
Dear dr. Rossi, very good news from you about domestic E-Cat!
Time is coming “fast and furious” like a movie (but this isn’t fiction, it is a wonderful reality) and soon the whole world will see the New Era of energy.
You are explaining your selling strategy:
“…Our sales will go exclusively to our Customers and our Licensees and we will be very much aware of where our E-Cats will go…”
But what about us, the single persons that have already made a preorder of one unit?
Thank you
Italo R.
Dear Andrea Rossi,
I’m sorry I have to bring this up, but you say that not even pictures of the E-Cat will be available before they are on sale.
While the E-Cat may be a revolutionary device, how can you expect a consumer to buy a product that is from a company that nobody knows about or with performance that is truly revolutionary but has
not been independently verified. Where I come from you don’t even buy a melon before you pick it up and smell it.
What kinds of consumer assurance will be provided before the E-Cat is on sale?
Thanks and continued good luck with your invention and hard work. We are really pulling for you but…
P.S. Now may not be the time to answer this question and if you drop or ignore it, I will understand. However it does need to be addressed at some point before Autumn, IMO.
Dear Mr. Rossi,
In some of your answers, you wrote that companies that want to distribute and sell e-cats, have to mail you and give a detailed description of their company. You also wrote that licenses will be
given for limited geografical area’s, so here my questions:
i) Is Europe covered by one or only a few licenses, or do you give seperate licenses for every country?
ii) As I live in Belgium, how about Belgium? Do you already co-operate with a company in Belgium that will distribute the home-E-cats?
iii) If yess, could you disclose the name of that company (in Belgium)? Whenn will the list of your licensees (companies worldwide that represent you) be given?
iv) Concerning the industrial 1MW plants (and more), are these treated seperately from the home-units? (I think yess) And will industrial clients always have to contact your headquarters (Leonardo
Corporation)? (I think yess)
v) Do you already co-operate with Belgian companies/clients (for industrial plants)? Of course I respect if you want to keep this secret…
vi) In autumn last year (2011), I thought ‘maybe I could ask him to represent him in Belgium’, but I am a single person, and I don’t own a distribution company, so I guess this is not possible, also
because you probably work via existing and very well established distribution companies? (And don’t worry, I appreciate that, because it proves that you work very professionaly…)
Kind Regards,
Dear Martin:
We started the tests of the domestic reactor. The technology is completely changed, new patents have been applied for, an intense testing program is going on. With very good results. The data are
totally covered, we are talking of the unit that will me made in 1 million pcs/year, we will not give any data before the next Autumn. No pictures will be available until the product will be for
sale. The strategy with which we will annichilate the competition must remain secret to the last moment. By the way: if there is out there some clown that thinks to buy a bulk of E-Cats, change the
body and say”here is our product! We copied it from Rossi because we are the Mandrake of the copycat!” better forget it from now. Our sales will go exclusively to our Customers and our Licensees and
we will be very much aware of where our E-Cats will go: no bulk sales outside our sales network, and a price too low to allow a profitable reverse-engineering. By the way: we are receiving very
strange and pretty much clownesque requests of pre-order for tens of thousands of E-Cats from the Aegean area…. By the way: all the investors who think to make money investing in companies who are
copying our patents, both pending and granted, better prepare an army of attorneys: we will not only make extremely competitive prices, but also will defend in all the competent Courts our
Intellectual Property. Better to know this from the very beginning.
Warm Regards,
Dear Andrea,
Mats Lewan told about à test on 20 februari. Can you tell us some more
About this test? Goal, pictures etc
Best regards,
Dear Stefano:
1- your pre order has been accepted, of course.
2- Thank you for your suggestion,
Warm Regards,
Dear Ing. Daniel De Caluwe’:
Thank you.
Warm regards,
Dear Roberto 1963:
Our E-Cat module of 10 kW is good for 100 mq.
Warm Regards,
To M a r i o
February 27th, 2012 at 8:57 AM
answer to Rob :
1m3 of natural gas delivers 10kw/hour of thermal energy and costs 0,60euro x 24 hours = 14,4euro per day.
for 6 months of continuous operation : 14,4 x 180 = 2592euro.
Rob says that in his country 1kWh cost 0,60 euro
We often see statements that from 1kWh of heat we can get o.3kWh of electric energy.
I wonder how much heat we are getting from 1kWh of electricity from conventional (resistance) heaters.
COP = 6:1 is not completely clear to understand the efficiency, since on one side we have 1kW of electricity, on other side 1kW of heat.
The clear good side of E-Cat is, that it can replace conventional dirty sources of energy, especially coal.
And with possibility of producing electricity it can make household independent from delivery energy from outside sources on daily basis.
If ECAT will be able to heat only 3 liters of water per minute, I wonder if it will be able to heat the house, because it seems to me that it is too low thermal efficiency. At my house the pump has a
capacity of at least 0.5 cubic meter per hour.
Dear Mr. Rossi,
On february 25th, 2012 at 12:33 PM, user ‘Rends’ mentioned a critical article in Forbes magazine:
Well, this is what I quickly wrote in response to it, one day ago (You can read my answer also on page 5 of the comment-section, below the article in higher link):
“””Daniel De Caluwé 1 day ago
1. (I think starting from january 2011) Untill 28 oktober of last year (2011), several tests has been done on the E-cats, and as a civil electromechanical engineer (from Belgium), I verified some of
the previous tests, and they were very convincing to me.
2. On his blog http://www.journal-of-nuclear-physics.com/ , Andrea Rossi explained several times why he doesn’t organise any more tests:
– Several tests already have been done, with good result(s), and the latest test (of 28 oktober 2011) was done by an important customer, a company or organisation that doesn’t want to reveal its name
(Mr. Rossi has a NDA-agreement with this entity), but who approved the 1MW container plant, and who buyed it. And together with this company or organisation, they improved the control system and the
instrumentation, and they are also working on the issue of generating electricity with the 1 MW plant. Also, based on these tests and his coöperation with this major organisation, Mr. Rossi got a lot
of investors, that support his company Leonardo Corporation.
– His competitors are very interested in his secrets, especially in the right chemical formula of the catalyser, that is used to enhance the reaction. And because pattent-approvals are still pending,
Mr. Rossi doesn’t want to take the risk of doing more tests, that could reveal the secrets to his competitors.
– With his team and the companies that are choosen by his investors, he ‘s working 16 hours a day (7 day’s a week) to also make electricity (starting with the industrial 1 MW plant), and also on an
automated production line for the home-ecats, that could be delivered within 18 months (worldwide), hopefully before next winter (in the northern hemisphere), but planned within 18 months. So, Mr.
Rossi and his team and co-operators, have no time to do further tests. International approval of his patent(s) still is pending, so he still is not protected by it, and therefore het puts all his
efforts in being the first to deliver good working home-ecats, not revealing the secrets of the E-cat to possible competitors. But as soon as pattents are approved, Mr. Rossi wrote he could give more
information about the exact working of the E-cats…
Ir. Daniel De Caluwé
I quickly wrote that message (last sunday) in response (and to your defence) to the critical article in Forbes magazine.
As I believe 100% in what you do, I hope you roughly agree with above answer, and that I didn’t make too much errors?
Kind Regards,
Ir. Daniel De Caluwé
Good morning Mr Rossi,
we mailed each other previously about tesla turbines.
Lately, I have been in contact with Mr Pierluigi Paoletti; he is a financial analyst expert of the present economical crysis (his speeches can be found on Youtube and on http://www.scecservice.org:
look for the pdf flyer on homepage, italian only).
Since many of his (social and economical aspects) and Your (scientific and technological with social-economical feed-backs) efforts are aiming toward a similar goal, I would be glad of making each
other aware of both Your achivements.
At the same time, I would extend this invite to all like-minded people!
Thank You for Your attention and Your work,
Best Regards,
PS My preorder for a home ecat was accepted?
Dear Joe:
1- there is no reason to postpone the launch of the domestic E-Cats
2- nobody can give this assurance
Warm Regards,
Dr Rossi,
1. Would you consider postponing the production of the domestic E-Cats in favor of the MW plants if the MW plants show themselves capable of generating electrical power very soon? The reason for this
is that modern society is structured around electricity and not around hot water or warm air. Such a decision might prove to be much more beneficial for Leonardo Corporation.
2. Have you received strong assurances from your customer that your MW plant will be protected against industrial espionage?
All the best,
answer to Rob :
1m3 of natural gas delivers 10kw/hour of thermal energy and costs 0,60euro x 24 hours = 14,4euro per day.
for 6 months of continuous operation : 14,4 x 180 = 2592euro.
have a nice day
Dear Gio:
The 1 MW plants are made with another technology, respect the new 10 kW E-Cats, but they too need the drive.
Warm Regards,
Dear Don Witcher:
I go to study this.
Warm Regards,
Dear H.Hansson:
Yes, the remote control can be made by internet or by phone.
I suggest anyway to let alone the food stuff.
Warm Regards,
Dear ing. rossi
I read your reply( here is your answer:..”we need a concentration of thermal energy that the system you proposed does not allow…”) to S. Broenink (here is a part of his question:”…Would it be
possible to make a self sustained closed loop of two E-Cats? ……
The 1 MW e-cat has got the same problem and it is not possible to make a self sustained closed loop?
Dear Mr. Rossi,
I think Larry is touching something important.. As many users of eCat will choose to stay “off grid” there are a increasing need to do monitoring on distance, in real time (iPhone??). This will be of
importance when you starts to offer electric generators. If the eCat for some reasons makes an emergency stop and there are no back-up units. A power black-out can ruin not only the food in your
refrigerator,.. but cause damage to other system as well.
I think this is “the” accessory that will be sold along with your eCat.. even if a basic remote system is included….
Dr. Rossi
Have you considered using a commercial flywheel power system in conjunction with ecats for self sustaining operation? Its a well developed technology that should be adaptable. http://
www.vyconenergy.com/pages/flywheeltech.htm Don Witcher
Dear Clauba:
Demineralized water is the solution.
Warm Regards,
Dear Dr. Rossi,
I was pretty sure that according your declaration, electric power is necessary to start the reaction inside the e-cat.
Answers to S. Broenink and Stephen T. confirm the above.
Nevertheless, to gio’s question “So can i use my gas boiler, with no need to change it, to start up and re-start e-cat ?”, you answered yes.
Therefore, one possible interpretation is that it would be feasible to run the e-cat without electric power (the 1/6 of the thermal power) by using the thermal power of a non-modified water
circulating gas-heater (with a normal output of 70 °C water, very common in Italy), to start and control the e-cat.
Maybe it is only a misunderstanding, and I am sorry to dig into contradictions, but could you kindly clarify the meaning of your answer?
Many thanks in advance
Dear Larry:
Good question, the answer is yes. It will be possible also to turn it on and off from remote.
Warm Regards,
Dear Claud,
I am not able to answer, I do not know the issue: for example, I think that our certification will not allow any use of the E-Cat for tasks not explicitly granted .
It is not just a matter of sterilization ( on this you are right, the steam sterilizes) but I suppose that to deal with food there are requirements that we do not know.
Warm Regards,
Dear H. Hansson:
Yes: the air conditioned generator; eventually, the electricity generator.
Warm Regards,
Dear Enio Burgos:
Your pre-order has been accepted, Thank you!
Warm Regards,
Mr. A. Rossi,
Wladimir Guglinski is a friend, I am from Brazil too, and I´d like to order one ecat (10 kW), please. Is it possible? Please, let know how to do…
Thank you, best wishes.
Enio Burgos
Dear Mr Rossi,
Every product is associated with some kind of accessory. Like,.. if you sells tennis rackets you can also expect to sell tennis balls to the same customer.
Do you foresee any must-have-accessories for eCat customers??
Dear Mr. Rossi, the steam sterilizes itself the piping as far as no trace of poisoning material outcomes from the exaust. E-cat could therefore be perfectly suitable for steam-cooking. Do you agree?
Dear Dr Rossi
Thank you for all your efforts make cold fusion a reality.
Will your eCat controller have a wireless capability that would let the customer monitor it with a computer on the same home network. It would be reassuring for your customers to be able to read the
temperatures and current draw in real time and could be useful in diagnosing problems and applying software updates.
Thank You
Dear Mr. Rossi,
have you had any problems with hard permanent water in your devices ?
if any, what is your solution?
Dear Rampado Dr Roberto:
Much more. But of course this has nothing top do with products for sale: as I said, it’s like the difference between a racing car and a regular one.
Warm Regards,
Dear Stefano:
Please send yourself an info to the guys of the website.
Thank you for your kind attention,
Warm Regards,
Dear Claud:
I think that the steam from the E-Cat can go in contact with food, but you should maintain the piping perfectly clean for hygienic reasons. I didn’t think to this application before, anyway probably
to use something for food there are requirements that , honestly, I do not know.
Warm Regards,
p.s. I ban nobody, sometimes comments are spammed by the robot mistakingly.
Dear Mr. Rossi, few days ago I asked whether the steam produced by e-cats can come in direct contact with foodstuff or not. I didn’t find my question in the blog. Did you banned it?
Dear Mr Rossi,
just an information for your informatics: on the ecat.com site, i believe that the “ECAT Energy Cost Calculator” doesn’t work properly. It doesn’t show the costs clicking on the “calculate” key.
Thank you
best regards and good work
Caro Dr Rossi,
L’E-Cat verrà da te venduto con COP minimo di 6 ed è già un grandissimo risultato oltre che pratico anche scientifico, perchè se non altro dimostra che le reazioni LENR termiche esistono e sono
Ma se non sono indiscreto, qual’è il COP massimo raggiunto nei tuoi esperimenti ?
Anche senza dare i numeri precisi, basterebbe una risposta “qualitativa”, tipo poco/ abbastanza/ molto/ moltissimo di più…
Ti si è mai squagliato, fuso o esploso un E-Cat?
Dear Mr. Rossi,
you will sold the E-cat with COP=6 minimum. COP=6 is a great scientific result. At minimum it shows that the thermal LENR exist and they are useful.
But if I am not too intrusive, which is the maximum COP reached in your experiments?
Even without giving specific numbers, a “qualitative” answer such as little / somewhat / very / much more would be enough.
Just for curiosity. Have you ever been able to melt or to explode an E-cat?
Thank you.
Dear Gio:
Warm Regards,
Dear ing. rossi
So can i use my gas boiler, with no need to change it, to start up and re-start e-cat ?
Dear psi:
I will visit personally many of the first-year Customers, to check personally how’s going on. When you will have the E-Cat installed send me a memo of this comment.
Warm Regards,
Dear S. Broenink:
Yes, you can easily.
Warm Regards,
Dear S.Broenink:
You never bugged me.
No, for thermodynamical reasons: we need a conventration of thermal energy that the system you proposed does not allow.
I wish you were right.
Warm Regards,
Dear Stephen T.:
Thank you, we are working with focus on the electric power production: it’s it the way which will bring us to the total self sustaining.
Warm Regards,
Dear Gio:
dear ing. rossi
i read the question of bob from Netherlands and your answer too.
As you say the costs are different in each country, so the set up of e-cat allows to use different energy sources,selecting the most economic .
Dear Andrea Rossi,
Your newest cooperation with the large turbine/electric company is very good news. I hope they will help you develop something small like this turbine that can fit in your hand. (a bit larger
I spoke to a Siemens representative back in July 2011 about a 300 kW turbine for the 1MW container but there was nothing appropriate to help you then. You have made much progress.
Best Wishes,
Sorry to bug you again Mr. Rossi, but I was thinking: Would it be possible to make a self sustains closed loop of two E-Cats? 1/6 of the output of E-cat A will be used to power E- Cat B and 1/6 the
power output of E-Cat B will be used to power E-Cat A..
Thanks again!
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Solution: Closest Points
Solution for the Closest Points Problem.
We'll cover the following
This computational geometry problem has many applications in computer graphics and vision. A naive algorithm with quadratic running time iterates through all pairs of points to find the closest pair.
Our goal is to design an $O(n\cdot$$log(n))$ time divide-and-conquer algorithm.
To solve this problem in time $O(n$$\cdot$$log(n))$, let’s first split the given $n$ points by an appropriately chosen vertical line into two halves, $S_{1}$ and $S_{2}$, of size $\frac{n}{2}$
(assume for simplicity that all $x$-coordinates of the input points are different). By making two recursive calls for the sets $S_{1}$ and $S_{2}$, we find the minimum distances $d_{1}$ and $d_{2}$
in these subsets. Let $d$ = min{$d_{1}$, $d_{2}$}.
Level up your interview prep. Join Educative to access 80+ hands-on prep courses. | {"url":"https://www.educative.io/courses/algorithmic-problem-solving-preparing-for-a-coding-interview/solution-closest-points","timestamp":"2024-11-10T21:40:25Z","content_type":"text/html","content_length":"853184","record_id":"<urn:uuid:f5733893-c10b-425a-aa29-8a4c0680da3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00377.warc.gz"} |
The Family of Finite-Tailed Distributions
cdfft {cdfquantreg} R Documentation
The Family of Finite-Tailed Distributions
Density function, distribution function, quantile function, and random generation of variates for a specified cdf-quantile distribution.
cdfft(q, sigma, theta, fd, sd, mu = NULL, inner = TRUE, version)
pdfft(y, sigma, theta, fd, sd, mu = NULL, inner = TRUE, version)
qqft(p, sigma, theta, fd, sd, mu = NULL, inner = TRUE, version)
rqft(n, sigma, theta, fd, sd, mu = NULL, inner = TRUE, version)
q vector of quantiles.
sigma vector of standard deviations.
theta vector of skewness.
fd A string that specifies the parent distribution. At the moment, only "arcsinh", "cauchit" and "t2" can be used. See details.
sd A string that specifies the child distribution. At the moment, only "arcsinh", "cauchy" and "t2" can be used. See details.
mu vector of means if 3-parameter case is used.
inner A logic value that indicates if the inner (inner = TRUE) case or outer (inner = FALSE) will be used.
version A string indicates that which version will be used. "V" is the tilt parameter function while "W" indicates the Jones Pewsey transformation.
y vector of quantiles.
p vector of probabilities.
n Number of random samples.
pdfft gives the density, rqft generates random variate, qqft gives the quantile function, and cdfft gives the cumulative density of specified distribution.
version 1.3.1-2 | {"url":"https://search.r-project.org/CRAN/refmans/cdfquantreg/html/cdfft.html","timestamp":"2024-11-05T23:34:54Z","content_type":"text/html","content_length":"3780","record_id":"<urn:uuid:61552441-4820-4a7c-b043-ca297ce74494>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00416.warc.gz"} |
manufacturing process by kestoor praveen milling machine
Manufacturing Process 2 By Kestoor Praveen Manufacturing Process 2 By Kestoor Praveen Thats something that will lead you to comprehend even more in the zone of the earth, knowledge, certain
locations, past era, enjoyment, and a lot more?. ... kestoor praveen milling machine manufacturing process 2 by kestoor praveen pdf | {"url":"https://hebergement-perouges.fr/Jun/04-8890.html","timestamp":"2024-11-04T14:13:55Z","content_type":"text/html","content_length":"45816","record_id":"<urn:uuid:bbb8e1d9-bb1a-4aa2-947c-6c808af6f1f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00635.warc.gz"} |
Finding the Value that Completes the Square | sofatutor.com
Finding the Value that Completes the Square
Basics on the topic Finding the Value that Completes the Square
Quadratic equations can be solved by using a certain method known as “completing the square”.
When solving quadratic equations by completing the square, first rewrite the function in standard form, ax² + bx + c = 0. If the value of a is greater than zero (a>0), then divide the whole equation
by a such that the coefficient of x² is equal to one. The quadratic equation is now of the form x² + b/ax + c/a = 0.
Next, put the constant c/a on the right-hand side of the equation to get x² + b/ax = -c/a. To complete the square, write the left-hand side of the equation (x² + b/ax) as a perfect square trinomial
by adding (b/2a)² to both sides of the equation, so that you have the expression x² + b/ax + (b/2a)² = -c/a+(b/2a)². To continue this process, factor and evaluate the equation.
Extract the square of the squared binomial by taking the square root of both sides of the equation; the result is a linear equation. Finally, solve the equation by simply isolating the variable x.
Notice that some quadratic equations have two solutions, some have one solution, and some even have two imaginary solutions.
Solve quadratic equations in one variable.
Transcript Finding the Value that Completes the Square
General Good is staking out a warehouse where a high-priority target is located. General Good's mission, should she choose to accept, is to find the values that complete the square.
Let's take a closer look at the General's mission objectives. She must rescue the hostage, Prince Fluffington IV and return him safely to his family. To rescue the prince, General Good must stay in
the shadows as she jumps from the roof of one building to another with the aid of her trusty jetpack. The flight path of the jetpack is in the shape of a parabola and can be written as a quadratic
function, so before she can rescue the hostage, General Good needs to make some calculations.
f(x) and h(x)
The jetpack has two different settings, f(x) and h(x). So to rescue the prince, she has to select the correct one so she can land her jump right inside the airshaft. Let’s draw a coordinate system.
We can plot her path on the x- and y-axis. Lucky for us, we know the equation of the two possible flight paths.
Let's take a look at the first one. Since she needs to solve for the roots she sets the quadratic function equal to zero. Okay! Now to complete the square. General Good modifies the equation so the
coefficient of the highest power is equal to one. Next, move the constant to the right side of the equal sign and then find the c-value that changes the left side of the equation to have the format
of a perfect square trinomial: a² + 2ab+b².
To do this, divide the coefficient of the x-term by 2 and then square it, so -80 divided by 2 and squared is equal to 1600. This is the value that *completes the square. Remember, whatever we add or
subtract on one side of the equation, we also must do to the other. Watch and learn!
Factoring and solving for x
Now we can factor the trinomial. Then take the square root of each side and solve for x. As you can see, if General Good chooses this flight path, it'll land her short of the airshaft. So, let’s take
a look at the second option. If it doesn't work, I don't know what General Good'll do. Again, we have to complete the square. Set the equation equal to zero. Modify the equation so that the
coefficient of the highest power is equal to 1.
Use opposite operations to write the constant on the right side of the equal sign. Find a c-value that creates a perfect square trinomial by dividing the coefficient of the x-value in half and
squaring it. Add the c-value to both sides of the equation. Now, factor the trinomial. Take the square root of each side and solve. x is equal to 10 and 90. Thank goodness! The second flight path
will work! Whew, I thought jumping between the buildings would be the difficult part of this story.
As expected, General Good lands her jump and rescues the prince, but he’s not exactly what she expected.
Finding the Value that Completes the Square exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Finding the Value that Completes the Square.
• Explain how to solve the given function.
The standard form of a quadratic equation is given by $ax^2+bx+c=0$.
Keep the following in mind:
$(a+b)^2=a^2+2ab-b^2$ or $(a-b)^2=a^2-2ab-b^2$.
$x^2+bx+c=0$ is equivalent to $x^2+bx+\left(\frac b2\right)^2=-c+\left(\frac b2\right)^2$.
Factoring, we get $x^2+bx+\left(\frac b2\right)^2=\left(x+\frac b2\right)^2$.
How should one solve such an equation?
1. First rewrite the equation so that the coefficient of the highest power is equal to $1$: we have to multiply by $-45$. This leads to $x^2-80x+700=0$.
2. Next move the constant to the right using opposite operations: $x^2-80x=-700$.
3. Now find the $c$ value to form a perfect square trinomial, namely $c=\left(\frac{-80}2\right)^2=40^2=1600$.
4. Add this value on both sides of the equation and combine the terms on the right-hand side: $x^2-80x+1600=-700+1600=900$.
5. Factor the trinomial on the left-hand side: $(x-40)^2=900$.
6. Almost done: take the square root on both sides to get $x-40=\pm 30$.
7. Last but not least you get the solutions by adding $40$ on both sides: $x=30+40=70$ or $x=-30+40=10$.
• Solve the given quadratic equation.
Let's have a look at $x^2+4x-5=0$:
1. Add $5$ to $x^2+4x=5$.
2. $c=\left(\frac42\right)^2=4$
3. Add $c=4$ to $x^2+4x+4=9$.
Looking at $(a+b)^2=a^2+2ab+b^2$ and the trinomial $x^2+4x+4$, we can conclude that $a=x$ and $b=2$. We then have that
Now we can proceed as follows: $x^2+4x+4=9$ is equivalent to
Now we take the square root to get $x+2=\pm 3$.
Last we subtract $2$ to get the solutions
Let's practice finding the solution of a quadratic equation by finding the value that completes the square:
1. Divide the equation on both sides by $-0.01$ to get the coefficient of the highest power equal to $1$: $x^2-100x+900=0$.
2. Move the constant to the right: $x^2-100x=-900$.
3. Now we have to do the main part of the work: find the $c$ value to form a perfect square trinomial, namely $c=\left(\frac{-100}2\right)^2=50^2=2500$.
4. Add this value to both sides of the equation: $x^2-100x+2500=-900+2500=1600$.
5. Factor the trinomial: $(x-50)^2=1600$.
6. Take the square root: $x-50=\pm 40$.
7. Add $50$ to get the solutions $x=40+50=90$ or $x=-40+50=10$.
• Find the missing value $c$ that completes the square.
Take a look at a quadratic binomial
The missing value completes this term to
In general the missing term is given as the square of the coefficient of the linear term divided by $2$.
Let's have a look at another example, $x^2+6x$.
The missing term is $\left(\frac62\right)^2=3^2=9$.
For $x^2+bx$, the missing value $c$ is given by $c=\left(\frac b2\right)^2$.
Adding this value to the binomial above we get a perfect square trinomial which we can factor out.
1. $x^2+12x$ - here we have $b=12$. Squaring half of it gives us $c=\left(\frac{12}2\right)^2=6^2=36$.
2. $x^2-8x$ - we can conclude with $b=12$ that $c=\left(\frac{-8}2\right)^2=4^2=16$.
3. $x^2+16x$ - with $b=12$ we get $c=\left(\frac{16}2\right)^2=8^2=64$.
4. $x^2+18x$ - here we have $b=18$. Thus we get $c=\left(\frac{18}2\right)^2=9^2=81$.
• Decide which equation(s) will give General Good's jetpack the right distance to jump.
All the trinomials on the left-hand sides are given in perfect square form.
For example, we have $x^2+140x+4900=(x+70)^2$.
After factoring, take the square of both sides.
Let's have a look at an example:
This leads to $x+4=\pm 4$.
Subtracting $4$ on both sides leads to $x=-8$ or $x=0$.
Here we have the distance $|-8-0|=8$.
We solve each equation above and subtract the resulting solutions to check if the distance is $80$.
Each trinomial is in a perfect square form:
1. $x^2-120x+3600=1600$ is equivalent to $(x-60)^2=1600$. Taking the square root gives us $x-60=\pm 40$ and thus $x=40+60=100$ or $x=-40+60=20$. The difference of both solutions is $100-20=80$
2. $x^2-120x+3600=900$ is equivalent to $(x-60)^2=1600$. Again taking the square root leads to $x-60=\pm 30$ and thus $x=30+60=90$ or $x=-30+60=30$. The difference of both solutions is $90-30=
60$. This equation would not help her get from tree to tree.
3. $x^2+80x+1600=2500$ - we rewrite this as $(x+40)^2=2500$ and take the square root to $x+40=\pm 50$. The resulting solutions are $x=50-40=10$ or $x=-50-40=-90$. The difference of both
solutions is $10-(-90)=100$. This equation would not help her get from tree to tree.
4. $x^2+80x+1600=1600$ - the difference to the equation above is the right side: $(x+40)^2=1600$. Taking the square root results in $x+40=\pm 40$ and the solutions are $x=40-40=0$ or $x=-40-40=
-80$. The difference of both solutions is $0-(-80)=80$ $~~~~$✓
5. $x^2+140x+4900=1600$ - factoring leads to $(x+70)^2=1600$. Now we take the square root to $x+70=\pm 40$ to get the solutions $x=40-70=-30$ or $x=-40-70=-110$. The difference of those
solutions is $-30-(-110)=80$ $~~~~$✓
6. $x^2-140x+4900=2500$ leads to $(x-70)^2=4900$ and, by taking the square root, to $x-70=\pm70$. This gives us the solutions $x=70-70=0$ or $x=-70-70=-140$ with the difference $0-(-140)=140$.
This equation would not help her get from tree to tree.
• Identify the polynomial which is equal to $(a+b)^2$.
Factor out the squared binomial $(a+b)^2=(a+b)(a+b)$ using the FOIL method.
For example, $x^2+6x+9=(x+3)^2$.
Finding the value that completes the square leads to the following:
So if you already have a binomial like $x^2+6x$ you have to find a value that completes this binomial to a perfect trinomial which can be factored.
This value for $x^2+6x$ is $\left(\frac62\right)^2=3^2=9$, giving us that
The trinomial on the left $x^2+6x+9$ is the desired perfect trinomial.
• Find the two solutions of the following equation.
You have to divide both sides of the equation by the coefficient of the highest power, unless it is already $1$.
The value $c$ completing the square of $x^2+bx$ is given by $c=\left(\frac b2\right)^2$.
You can check your solutions by putting them in the equation above.
Step by step we solve the equation:
1. Divide by $4$ to get a coefficient of the highest power equal to $1$: $x^2+8x-20=0$.
2. Move the constant to the right: $x^2+8x=20$.
3. The value that completes the square on the left to a perfect square trinomial is given by $c=\left(\frac82\right)^2=4^2=16$.
4. Add $c=16$: $x^2+8x+16=20+16=36$.
5. Factor the trinomial: $(x+4)^2=36$.
6. Take the square root: $x+4=\pm 6$.
7. Last but not least, you get the solutions by subtracting $4$ on both sides: $x=6-4=2$ or $x=-6-4=-10$. | {"url":"https://us.sofatutor.com/math/videos/finding-the-value-that-completes-the-square","timestamp":"2024-11-12T22:27:15Z","content_type":"text/html","content_length":"161293","record_id":"<urn:uuid:25da098d-602a-4e26-a2c9-7de77820fd92>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00177.warc.gz"} |
What Is a Strike Price? Options for Beginners
In options trading, the strike price is the price at which the owner of a call or put option can exercise their contract and convert it into the underlying asset.
Options contracts give the owner the right, but not the obligation, to exercise their option at the strike price. If an option is exercised, the seller (or writer) of the option must deliver the
underlying asset at that strike price.
• Strike Price: The agreed-upon price to buy (calls) or sell (puts) the underlying asset.
• Fixed Strike Price: The strike price stays the same, but the stock price moves, determining whether an option is ITM, ATM, or OTM.
• Profit Calculation: Strike prices are crucial to determining profit, loss, and breakeven points.
• Option Moneyness: The relationship between the strike price and underlying asset price defines an option's moneyness and assignment odds.
What Is a Strike Price?
The term 'strike' comes from Old English, when two parties would 'strike' a deal or come to an agreement. Before options were standardized, contracts involved two trusting parties agreeing on terms,
with the most important part being the price—hence the term "strike price" today!
In options trading, the strike price is the price at which the owner of a call option can buy, or the owner of a put option can sell, the underlying asset. The key here is that options owners don’t
have to exercise their contract—they have the ‘option’ to. Get it?
Strike Price: The Underlying Asset
Options are derivatives. You can think of them as side bets. Their value is ‘derived’ from a separate, underlying asset. Underlying assets include, but are not limited to:
• Stocks
• ETFs
• Commodities
• Currencies
Understanding the underlying asset helps traders assess how likely (or unlikely) an option’s strike price will be reached, which impacts an option’s profit or loss scenarios.
Strike Prices: Calls and Puts
Options are divided into two categories on options chains: calls and puts.
When you buy a call option, you’re making a bullish bet on the underlying market. When you buy a put option, you’re betting the underlying market will go down.
• Holders of long call options have the right to buy the underlying asset at the strike price anytime before expiration.
• Holders of long put options have the right to sell the underlying asset at the strike price anytime before expiration.
Strike Price vs Stock Price
A strike price is a fixed number that doesn’t change throughout the life of the option contract.
This is in contrast to the price of the underlying asset, like stocks, which constantly fluctuate. The position of your option's strike price relative to the stock price tells you how your option is
The above chart would be promising to a call owner, but disappointing to the owner of a long put option.
📖 Understanding Option Liquidity
Strike Price: Calculating Profit & Loss
When you buy a call option, you want the stock price to rise above the strike price. When you buy a put option, you want the stock price to fall below the strike price.
In order to understand how your position is faring, understanding strike prices is a must. The below graphs show us how strike prices help us determine the profit, loss and break-even on call and put
To explore this deeper, here are the equations to determine profit, loss and break-even on various option positions:
Strike Prices & Option Moneyness
At any point in time, an options contract will be in one of three "money" states. Moneyness is determined by the relationship between the stock price and the option's strike price.
Understanding the "moneyness" of an option is essential because it indicates where the option stands in relation to the stock price and influences the likelihood of it being exercised.
For example, if you’re short an option that is "in-the-money," your chances of being assigned goes up dramatically. This means you may be required to deliver the stock, so make sure you have the
funds ready! I’ve personally seen traders get wiped out by not paying attention to the moneyness state of their options.
Here’s a guide to help you determine moneyness. This may be confusing at first, but it becomes intuitive after a few trades - promise!
How to Choose a Strike Price
So, what strike price should you choose? If you're buying single call or put options, it depends on how bullish or bearish you are.
The further out-of-the-money you go, the lower the probability an option has of becoming profitable. Because of this, the further out-of-the-money you go, the cheaper the option becomes.
Here’s how this looks for call options:
And the opposite is true for put options:
Because of something called "time decay," buying out-of-the-money options is often a losing proposition. These options are popular with retail traders because they’re cheaper and have a high payout
potential, but the truth is, most out-of-the-money options expire worthless.
Strike Prices: Credit and Debit Spreads
Both buying and selling single options carry a lot of risk. That’s why most seasoned traders prefer trading spreads, which involves both buying and selling options of varying strike prices at the
same time.
The most basic type of spread is a vertical spread. In a debit, or "bullish," vertical call spread, you buy one call option and sell another, further out-of-the-money call option.
The same applies to vertical put spreads: you buy one put option and sell another, further out-of-the-money put option.
Call Debit Vertical Spread (bullish):
• +1 Call Option (buy at a lower strike price)
• -1 Call Option (sell at a higher strike price)
Put Debit Vertical Spread (bearish):
• +1 Put Option (buy at a higher strike price)
• -1 Put Option (sell at a lower strike price)
You can both buy and sell vertical spreads—when you buy them, they’re called debit spreads, but when you sell them, they’re called credit spreads.
Strike prices are crucial when trading spreads because they determine both your risk and profit potential. Let’s take a look at an example.
Example: 2-Point Vertical Call Spread
You want to buy the 101 strike price call option on XYZ. The current price for this option is $1.50, so you will need $150 to place this trade (options are priced in multiples of 100).
You decide that $150 is too much risk for you. To reduce your risk, you sell another, further out-of-the-money option at the same time.
Let’s say you decide to buy the 101 call and sell the 103 call, creating a call vertical spread, as seen below.
This is called a ‘2-point’ spread because the difference between the strike prices is 2 (103 - 101=2).
You bought the 101 call for $1.50 and sold the 103 call for $0.50. The net cost, and the most you can lose, is therefore $1.00 ($1.50 - $0.50).
Let’s break down the P/L scenarios:
• Max Profit = Width of Spread - Net Debit
Example: 2 - 1 = $1 ($100)
• Max Loss = Net Debit Paid
Example: $1 ($100)
What’s important to know here is the narrower the spread, the lower your cost, but also the lower your potential profit. In this 2-point spread, the most you can make is $2, or $200, minus your $100
cost, leaving you a maximum profit of $100. Since your max profit equals your max loss, the market is saying this trade has a 50% chance of success.
The closer the strike prices, the smaller the risk and reward, so it’s all about finding that sweet spot based on where you think the underlying is headed.
What if we widen out our strike prices?
⚠️ Besides the initial debit paid, it is essential to consider the commissions and fees associated with most options transactions when calculating the net profit or loss. These fees can significantly
impact the overall return on investment.
Example: 4-Point Vertical Call Spread
Let’s say you are feeling bullish and want to risk a little bit more. You therefore widen the spread, buying the 99 call and selling the 103 call. The difference between these strike prices is 4, so
this is a 4-point spread.
Here’s the premium we paid and received:
• 99 Call: $2.50 debit
• 103 Call: $0.50 credit
We’re buying the 99 call for $2.50 and selling the 103 call for $0.50, which makes our net cost (or debit) $2.00.
Because this four-point spread costs $2, the most we can make is $4, or $400, minus our $200 debit paid, giving us a max profit of $200. The wider the spread, the bigger the potential reward.
However, wider spreads also cost more, so picking the right strike prices is crucial.
Here are the P/L calculations for the above 4-pointer:
Max Profit = Width of Spread - Net Debit
Example: 4 - 2 = $2 ($200)
Max Loss = Net Debit Paid
Example: $2 ($200)
Can you sell an option before it hits the strike price?
You can both buy and sell options whenever the market is open - you do not have to wait for the strike price to be reached.
Do options automatically exercise at strike price?
Long in-the-money options are typically automatically exercised by brokers at expiration, unless you instruct them otherwise.
Do you pay the strike price of an option?
You don’t pay the strike price when you enter a long option, you pay a premium. The strike price is only paid if you choose to exercise the option to buy (calls) or sell (puts) the underlying asset.
What happens if an option hits the strike price?
When an option hits its strike price, it’s considered at-the-money. For long options, as long as it hasn’t expired, nothing happens automatically. However, short positions risk being assigned if they
move past at-the-money and become in-the-money.
What is the best strike price of an option?
Strike prices don’t have any value on their own; their value comes from the price of the underlying asset. Out-of-the-money options are cheaper to buy but have a lower chance of profit, while
in-the-money options cost more but have higher intrinsic value.
What is the difference between strike price and exercise price?
Strike price and exercise price mean the same thing—they are both the price you’ll pay to buy (call) or sell (put) the underlying asset if you choose to exercise the option.
What is the strike price of an option at the money?
At-the-money options have strike prices that match the current price of the underlying asset, like the stock price.
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An Etymological Dictionary of Astronomy and Astrophysics
Fr.: idée
A thought, conception, or notion existing in the mind as a result of mental understanding, awareness, or activity. See also → thought, → concept.
Idea, from L. idea "idea," pre-Platonic Gk. idea "form, semblance, nature, fashion," in Plato "a timeless, universal archetype of existents; ideal prototype," literally "look, form," from idein "to
see," from PIE *wid-es-ya-, suffixed form of base *weid- "to know, to see;" cf. Pers. bin- "to see" (present stem of didan); Mid.Pers. wyn-; O.Pers. vain- "to see;" Av. vaēn- "to see;" Skt. veda "I
Miné "idea," related to Pers. maneš "disposition, temperament, greatness of soul," minu "heaven, paradise," also equivalent to Ger. Geist in recent philosophical translations, došman "enemy," pašimân
"penitent, regretful," pežmân "sad, mournful," šâdmân "joyful, cheerful, pleased," ârmân "desire; → ideal;" dialectal (Šuštar) mana "(he) thinks, imagines," (Tarq-e Natanz) môna "to imagine, suppose;
" Mid.Pers. mênidan "to think, consider," mên "thought, idea," mênišn "thought, thinking, mind, disposition," mênitâr "thinker," mênôg "spiritual, immaterial, heavenly," from Av. man- "to think,"
mainyeite "he thinks," manah- "mind, thinking, thought; purpose, intention," mainyu- "mind, mentality, mental force, inspiration," traditionally translated as "spirit," Angra Mainyu "hostile
mentality" (Mod.Pers. Ahriman); O.Pers. maniyaiy "I think," Ardumaniš- (proper noun) "upright-minded," Haxāmaniš- (proper noun, Hellenized Achaemenes, founder of the Achaemenian dynasty) "having the
mind of a friend;" cf. Sogdian mân "mind;" Skt. man- "to think," mánye "I think," manyate "he thinks," mánas- "intelligence, understanding, conscience;" Gk. mainomai "to be angry," mania "madness,"
mantis "one who divines, prophet;" L. mens "mind, understanding, reason," memini "I remember," mentio "remembrance;" Lith. mintis "thought, idea;" Goth. muns "thought," munan "to think;" Ger. Minne
"love," originally "loving memory;" O.E. gemynd "memory, thinking, intention;" PIE base *men- "to think, mind; spiritual activity."
۱) آرمان، مینهوار؛ ۲) آرمانی، مینهای، مینهوار
1) (n.) ârmân (#), minevâr; 2) (adj.) ârmâni (#), mineyi, minevâr
Fr.: idéal
1) (n.) A standard of perfection, beauty, or excellence.
Math.: A subset of a ring that is closed under addition and multiplication by any element of the ring.
2) (adj.) Existing only in the imagination; not real or actual.
Conforming exactly to an ideal, law, or standard; perfect. → ideal gas.
M.E. ydeall, from L.L. idealis "existing in idea," from L. → idea.
Ârmân "ideal" in Mod.Pers., traditionally "desire; hope; grief," variants armân, urmân, prefixed from mân, "thought, mind," → idea. The first element may be related to Av. armaē- "in peace, still;
quietly;" PIE base *er[ə]- "to be still" (cf. Skt. īrmā (adv.) "quiet, still, being in the same place;" Gk. erôé "calm, peace;" O.H.G. rouwa "rest"), as in Av. armaē.šad- "sitting quietly,"
armaē.štā- "standing still, stagnant." Therefore, Pers. ârmân may be related to Av. *armaē.manah- (PIE *ermen-) "thought in peace, quiet mind."
Mineyi, minevâr, adj. from miné, → idea. | {"url":"https://dictionary.obspm.fr/?showAll=1&formSearchTextfield=idea","timestamp":"2024-11-09T02:45:01Z","content_type":"text/html","content_length":"31473","record_id":"<urn:uuid:0b6e0f1c-9d1b-483d-8cfb-973149a1f564>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00794.warc.gz"} |
Depth first traversal of Binary Trees in Javascript
Depth First Traversal in Binary Trees
In an effort to teach myself fundamentals that I might have missed in my rudimentary yet effective bootcamp experience, I'm going to cover some basics in a series about data structures and
algorithms. As you might have surmised, in this post we're going to be discussing depth first traversal
A depth first traversal is a way of accessing every node in graph or binary tree.
A depth first traversal is characterized by the direction of the traversal.
In other words, we traverse through one branch of a tree until we get to a leaf, and then we work our way back to the trunk of the tree.
In this post, I'll show and implement three types of depth first traversal.
Inorder traversal
As 'depth first' implies, we will reach the 'leaf' (a node with no children) by traveling downward in a recursive manner.
Inorder traversal adheres to the following patterns when traveling recursively:
1. Go to left-subtree
2. Visit Node
3. Go to right-subtree
We can illustrate this concept with the follow gif
Let's code!
For the following examples we'll be using the Tree class I've defined below.
class Tree {
constructor(value, left, right) {
this.value = value;
this.left = left;
this.right = right;
Let's go ahead and create the tree we see in the example gif
tree = new Tree(
new Tree(2, new Tree(4, new Tree(8)), new Tree(5)),
new Tree(3, new Tree(6, new Tree(9), new Tree(10)), new Tree(7))
Finally, let's implement inorder traversal:
const inOrderTraversal = (node, cb) => {
if (node !== undefined) {
inOrderTraversal(node.left, cb);
inOrderTraversal(node.right, cb);
inOrderTraversal(tree, console.log);
// 8, 4, 2, 5, 1, 9, 6, 10, 3, 7
As you can see, the code mimics the steps outlined above.
The trickiest bit of this visualization is imagining the recursion until you hit the left-most leaf. I personally groan at the sight of recursion, but it's just something that has to be confronted
with depth first traversal.
Fun fact:
Inorder Traversal of Binary Search Tree will always give you Nodes in sorted manner
Preorder traversal
Preorder traversal adheres to the following patterns when traveling recursively:
1. Visit Node
2. Go to left-subtree
3. Go to right-subtree
In other words, preorder is extremely similar to inorder except for the fact it will visit the root of the node first.
Let's implement preorder traversal:
const preOrderTraversal = (node, cb) => {
if (node !== undefined) {
preOrderTraversal(node.left, cb);
preOrderTraversal(node.right, cb);
preOrderTraversal(tree, console.log);
// 1, 2, 4, 8, 5, 3, 6, 9, 10, 7
Postorder traversal
Post traversal adheres to the following patterns when traveling recursively:
1. Go to left-subtree
2. Go to right-subtree
3. Visit Node
Once again, postorder is extremely similar to the others except for the fact it will visit the left subtree, then the right subtree and finally the node itself.
Let's implement postorder traversal:
const postOrderTraversal = (node, cb) => {
if (node !== undefined) {
postOrderTraversal(node.left, cb);
postOrderTraversal(node.right, cb);
postOrderTraversal(tree, console.log);
// 8, 4, 5, 2, 9, 10, 6, 7, 3, 1
That wraps it up for depth first traversal...as far as I know. Please let me know if you've learned anything or I've made any egregious mistakes!
Till next time, cheers!
Top comments (0)
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/ggenya132/depth-first-traversal-in-javascript-3ehp","timestamp":"2024-11-13T15:23:35Z","content_type":"text/html","content_length":"75247","record_id":"<urn:uuid:2f896c00-961c-4be5-b598-0c4cd275b6a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00767.warc.gz"} |
A model for the constant-density boundary layer surrounding fire whirls
This paper investigates the steady axisymmetric structure of the cold boundary-layer flow surrounding fire whirls developing over localized fuel sources lying on a horizontal surface. The inviscid
swirling motion found outside the boundary layer, driven by the entrainment of the buoyant turbulent plume of hot combustion products that develops above the fire, is described by an irrotational
solution, obtained by combining Taylor's self-similar solution for the motion in the axial plane with the azimuthal motion induced by a line vortex of circulation 2πΓ. The development of the boundary
layer from a prescribed radial location is determined by numerical integration for different swirl levels, measured by the value of the radial-to-azimuthal velocity ratio σ at the initial radial
location. As in the case σ=0, treated in the seminal boundary-layer analysis of Burggraf et al. (Phys. Fluids, vol. 14, 1971, pp. 1821–1833), the pressure gradient associated with the centripetal
acceleration of the inviscid flow is seen to generate a pronounced radial inflow. Specific attention is given to the terminal shape of the boundary-layer velocity near the axis, which displays a
three-layered structure that is described by matched asymptotic expansions. The resulting composite expansion, dependent on the level of ambient swirl through the parameter σ, is employed as boundary
condition to describe the deflection of the boundary-layer flow near the axis to form a vertical swirl jet. Numerical solutions of the resulting non-slender collision region for different values of σ
are presented both for inviscid flow and for viscous flow with moderately large values of the controlling Reynolds number Γ/ν. The velocity description provided is useful in mathematical formulations
of localized fire-whirl flows, providing consistent boundary conditions accounting for the ambient swirl level.
Dive into the research topics of 'A model for the constant-density boundary layer surrounding fire whirls'. Together they form a unique fingerprint. | {"url":"https://research.manchester.ac.uk/en/publications/a-model-for-the-constant-density-boundary-layer-surrounding-fire-","timestamp":"2024-11-12T13:52:05Z","content_type":"text/html","content_length":"57810","record_id":"<urn:uuid:ebf2fdd2-a782-4cb3-a2b9-7abf17b13b96>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00635.warc.gz"} |
BrierScore: Brier Score for Assessing Prediction Accuracy in AndriSignorell/DescTools: Tools for Descriptive Statistics
Calculate Brier score for assessing the quality of the probabilistic predictions of binary events.
x either a model object if pred is not supplied or the response variable if it is.
pred the predicted values
scaled logical, defining if scaled or not. Default is FALSE.
... further arguments to be passed to other functions.
either a model object if pred is not supplied or the response variable if it is.
The Brier score is a proper score function that measures the accuracy of probabilistic predictions. It is applicable to tasks in which predictions must assign probabilities to a set of mutually
exclusive discrete outcomes. The set of possible outcomes can be either binary or categorical in nature, and the probabilities assigned to this set of outcomes must sum to one (where each individual
probability is in the range of 0 to 1).
\frac{1}{n} \cdot \sum_{i=1}^{n}\left ( p_{i}-o_{i} \right )^2 \; \; \; \textup{where} \; p_{i} predicted probability \; \textup{and} \; o_{i} observed value out of (0,1)
The lower the Brier score is for a set of predictions, the better the predictions are calibrated. Note that the Brier score, in its most common formulation, takes on a value between zero and one,
since this is the largest possible difference between a predicted probability (which must be between zero and one) and the actual outcome (which can take on values of only 0 and 1). (In the original
(1950) formulation of the Brier score, the range is double, from zero to two.)
Brier, G. W. (1950) Verification of forecasts expressed in terms of probability. Monthly Weather Review, 78, 1-3.
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/github/AndriSignorell/DescTools/man/BrierScore.html","timestamp":"2024-11-13T21:47:39Z","content_type":"text/html","content_length":"33469","record_id":"<urn:uuid:d0e1a8e3-4a7d-4a86-8daa-caae5676a53a>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00294.warc.gz"} |
Nadaptive and non adaptive routing algorithms-pdf
When a router uses a nonadaptive routing algorithm it consults a static table in order to determine to which computer it should send a packet of data. In dynamically changing networks source routing
is not deployed and destination. Continuouslyadaptive discretization for messagepassing cadmp is a new messagepassing algorithm for approximate inference. The nonadaptive routing algorithm is an
algorithm that constructs the static table to determine which node to send the packet. Difference between adaptive and non adaptive routing. We can estimate the flow between all pairs of routers.
This is also known as static routing as route to be taken is computed in advance and downloaded to routers when router is booted. On the stability of adaptive routing in the presence of congestion.
Routing algorithms distance vector, link state study notes. When a node is unavailable in a static routing environment, the packet must either wait for the node to become available again or the
packet will fail to be delivered. Adaptive routing algorithms for alloptical networks 1.
Most messagepassing algorithms approximate continuous probability distributions using either. Routing algorithms distance vector, link state study. Classification of routing algorithms geeksforgeeks.
In non adaptive routing algorithms, the basis of routing decisions are static tables. An adaptive probabilistic routing algorithm iit kanpur. Continuously adaptive discretization for messagepassing
cadmp is a new messagepassing algorithm for approximate inference. Nonadaptive algorithms nonadaptive algorithms do not modify their routing decisions when they have been preferred.
Continuouslyadaptive discretization for messagepassing.
Adaptive routing is an alternative to nonadaptive, static routing, which requires network engineers to manually configure fixed routes for packets. Instead the route to be taken in going from one
node to the other is computed in advance, offline, and downloaded to the routers when the network is booted. Recursive adaptive algorithms for fast and rapidly time. Adaptive routing algorithms for
alloptical networks. Adaptive routing algorithm is an algorithm that constructs the routing table based on the network conditions. It takes into account both the topology and the load in this routing
algorithm. The thesis presents a routing algorithm, loadsensitive adaptive routing lsar, which. This is in contrast to an adaptive routing algorithm, which bases its decisions on data which reflects
current traffic conditions. Routing algorithm at a router decides which output line an incoming packet should. Flooding and random walks are the types of non adaptive routing algorithms.
Centralized, isolated and distributed are the types of adaptive routing algorithms. Implementations of adaptive routing can cause adverse effects if care is not taken in analyzing the behavior of the
algorithm under different scenarios concentrated traf. This class of algorithms includes not only the simple minmax algorithm using a linear constraint solver, but also other solutions that optimize
routes through the. Improved adaptive routing algorithm in distributed data centers.
Therefore, it is often not possible to simplify the problem of routing information when using an adaptive algorithm to a problem of shipping flow through the network. So, new path to node dcj
contains this link, has length esti mate dj,ix. Many ft routing algorithms build on the principles above to ensure seamless. Adaptive algorithms for neural network supervised learning 1931 changed so
that it is more likely to produce the correct response the next time that the input stimulus ispresented. The result shows that, compared with nonadaptive ospfisis routing. For example,
thisisachieved bychanging the nth connection weight. These algorithms do not base their routing decisions on measurements and estimates of the current traffic and topology. From the known average
amount of traffic and the average length of a packet, you can compute the mean packet delays using queuing theory. | {"url":"https://tersmarrovi.web.app/1626.html","timestamp":"2024-11-01T22:05:57Z","content_type":"text/html","content_length":"9360","record_id":"<urn:uuid:78403e43-3740-49cd-99db-2d842df96936>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00726.warc.gz"} |
Histograms As A Tool To Describe Data
Student Teacher
Students make a histogram of their typical-student data and then write a summary of what the histogram shows.Students are introduced to histograms, using the line plot to build them. They
investigate how the bin width affects the shape of a histogram. Students understand that a histogram shows the shape of the data, but that measures of center or spread cannot be found from the
graph.Key ConceptsA histogram groups data values into intervals and shows the frequency (the number of data values) for each interval as the height of a bar.Histograms are similar to line plots
in that they show the shape and distribution of a data set. However, unlike a line plot, which shows frequencies of individual data values, histograms show frequencies of intervals of values.We
cannot read individual data values from a histogram, and we can't identify any measures of center or spread.Histograms sometimes have an interval with the most data values, referred to as the
mode interval.Histograms are most useful for large data sets, where plotting each individual data point is impractical.The shape of a histogram depends on the chosen width of the interval, called
the bin width. Bin widths that are too large or too small can hide important features of the data.Goals and Learning ObjectivesLearn about histograms as another tool to describe data.Show that
histograms are used to show the shape of the data for a wider range of data.Compare a line plot and histogram for the same set of data.
Statistics and Probability
Middle School
Grade 6
Material Type:
Lesson Plan
Date Added:
Media Format: | {"url":"https://oercommons.org/courseware/lesson/2287","timestamp":"2024-11-06T01:44:41Z","content_type":"text/html","content_length":"72222","record_id":"<urn:uuid:88b385f5-2429-4747-9c32-5cbab0ef6326>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00692.warc.gz"} |
Matter Time, Aethertime
The two bubbles of spacetime quantum and general relativity with their conjugates of momentum and position exist within the aethertime universe as shown in the figure below. Aethertime unifies charge
and gravity as time-scaled versions of the same aether decay and so a shrinking aethertime universe is what unifies quantum charge and general relativity's gravity forces. Unification has eluded many
very smart people for nearly a century, but aethertime unifies charge and gravity forces with the same time scaling that also unifies an antimatter antiverse with our matter universe into a the
growth and decay of a single aether pulse in time. As a result, aether decay shows that it is the disparate notions of spacetime itself that precludes unification with a pernicious elusiveness that
even very smart people simply have not yet been able to figure out. That does not mean that I am smarter than everyone else, just luckier...and grateful for the many very precise measurements of
objects and time that show aether decay.
Einstein's general relativity (GR) describes motion with a continuous gravity force that moves objects along determinate paths called geodesics in the void of empty and continuous space and time. The
green ellipse in the figure shows the predictions of GR. In contrast, quantum theory as the red ellipse moves objects with discrete expectation values of momentum and position with charge force
mediated by photon exchange. Quantum excitations result in motion along uncertain paths in that same space, which is now full of vacuum oscillators, that space where gravity motion involves
determinate paths.
While the continuous spin of the earth tells a continuous gravity time with discrete aether decay, the discrete ticks of atomic clocks and the discrete spin of atomic matter tell a much more precise
quantum time. Since there is not a quantum expectation value for time, quantum time is discrete and fully reversible even while gravity leads to a determinate and continuous expanding universe with a
continuous gravity time.
In GR, space, motion, and time all become undefined at the event horizons of large matter accretions known as black holes and just beyond the CMB as well as at very small scale, the Planck scale.
Spacetime has recognized these limitations for many years, but spacetime still does not show any other way to predict the futures of objects without continuous space, motion, and time. The very
definition of space is odd since space is a void of nothing that we cannot sense in a continuum of time and space. Spacetime only assumes that space exists because all of the objects moving around
need space as a way to be different from each other and so it is our sensations of objects that inform us about space.
The evidence that spacetime has for the nothing of empty space depends, then, on objects not changing and being constant. Objects may move from one place to another in spacetime and emit or absorb
energy and mass, but total object mass and energy must always conserved in any action in empty space. Of course, if an object loses mass by emission, the motion of that mass involves kinetic energy,
which increases the mass relative to the rest frame and so the accounting of constant relativistic mass must be done very carefully.
Both charge and gravity forces act through space, but while photons mediate charge force, gravity force simply cannot have a carrier particle since gravity is by its very nature continuous in a
continuous space and time. Since there is only one time dimension in GR, atomic time, the continuum of gravity force means that space and time become undefined in the universe at both very small and
very large mass accretions, including at beyond the CMB, which is now at 0.9991c. Once the CMB transitions to 1.0c, it will move beyond the event horizon of spacetime, which seems like a really
unlikely universe.
Aethertime is an alternative way to keep track of objects with discrete aether and time delay and as a bonus, aether decay becomes the genesis of all force. Discrete aether, matter exchange, and time
delay represent a more general aethertime reality that augments and wraps around spacetime's more limited continuous space, motion, and time and so aethertime does not suffer from the same
limitations of spacetime. Aethertime predicts motion with discrete aether exchanges in a fully quantum action and discrete time delays for even very large and very small objects, and now aethertime
augments the limitations of the continuous motion of continuous space and time with a complete universe.
Instead of predicting motion with fields of force in the vacuum of space, aethertime predicts motion with matter exchange and aether decay. The decay of discrete aether is now an inherent property of
aethertime from which emerge the fields of force that are inherent properties of continuous space, motion, and time. Unlike the nothing of empty space which we never by definition measure or sense,
there is plenty of evidence for discrete aether decay in the many measurements of object mass decays that show the same very slow decay over time, 0.26 ppb/yr. These measurements include the IPK
primary mass standard, the earth spin, the earth-moon orbital period, milky-way to Andromeda time delay, and the average decay of several thousand neutron stars as millisecond pulsars.
While objects in spacetime appear to have constant mass over atomic time, all objects in aethertime decay along with the rest of the universe over the very long time scales of aethertime. And of
course, the natures of both charge and gravity forces derive from the same decay of aether, which appears to act through space. However, aether does not exist in space and so space has zero density,
but aether decay does result in a pressure for space. A pressure for space with zero density is equivalent to an incompressible aether.
The period of atomic time also decays over aether time as do c, h, and α, which really changes how we interpret the distant objects of deep space including the decays of millisecond pulsar neutron
stars and the red shifts of distant galaxies. The decay of discrete aether mediates both charge and gravity forces in aether time, which for charge is by exchange of the dipolar photon while for
gravity is by the exchange of mono/quadrupole photon pairs scaled by the time size of the universe. Therefore, charge and gravity forces are both mediated by the same aether exchange and decay with
the scaling of the universe time size. In addition to atomic time, there is a second time dimension, aether time, and discrete aether and time now predict action for all of universe, including both
very small and very large mass accretions.
Unlike the lonely void of empty space, objects are never alone in a universe of discrete aether. We exist as part of the discrete aether whose exchange connects us with all other objects. Discrete
ether has both mass and spin and is the basic building block and aether decay is the binding force of the entire universe, not just for observable particles. Our notion of two particles in relative
motion in continuous space emerges from the discrete aether exchange between those particles and their time delays. While neither atomic time nor continuous space have meaning at the event horizon of
a black hole, aether time represents an event horizon as simply a transition from fermion matter and atomic time to discrete boson matter and time delay. | {"url":"https://www.discreteaether.com/2015_08_22_archive.html","timestamp":"2024-11-15T02:21:54Z","content_type":"text/html","content_length":"136077","record_id":"<urn:uuid:9283c116-bd2e-4e8e-8b43-311fdca69b28>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00169.warc.gz"} |
Diophantine exponents for standard linear actions of ${\rm SL}_2$ over discrete rings in $\mathbb {C}$
Acta Arithmetica 177 (2017), 53-73 MSC: Primary 11J20, 11J13, 11A55; Secondary 22Fxx. DOI: 10.4064/aa8370-6-2016 Published online: 23 December 2016
We give upper and lower bounds for various Diophantine exponents associated with the standard linear actions of ${\mathrm{SL}_{2}( \mathcal 0_K )}$ on the punctured complex plane $\mathbb C^2 \
setminus \{ \mathbf{0} \}$, where $K$ is a number field whose ring of integers $\mathcal O_K$ is discrete and any complex number is within a unit distance of some element of $\mathcal O_K$. The
results are similar to those of Laurent and Nogueira (2012) for the ${\mathrm{SL}_2(\mathbb{C})}$ action on $\mathbb R^2 \setminus \{ \mathbf{0} \}$, albeit our uniformly nice bounds are obtained
only outside of a set of null Lebesgue measure. | {"url":"https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/177/1/91957/diophantine-exponents-for-standard-linear-actions-of-rm-sl-2-over-discrete-rings-in-mathbb-c","timestamp":"2024-11-05T04:14:57Z","content_type":"text/html","content_length":"44781","record_id":"<urn:uuid:88dc8f7f-2ca7-4d62-9249-9c4c6c47384b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00270.warc.gz"} |
2 vars limit with WolframAlpha
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2 vars limit with WolframAlpha
Hello everybody!
Does anybody know why WolframAlpha gives the answer "0" to
limit[(x*y^2)/(x^2 + y^4),{x,y}->{0,0}]
against the fact that, although on many paths the limit is 0, in each neighborhood of the origin there are:
• an infinite number of points (of the kind x = y^2) on which the function simplifies to the constant 1/2
• an infinite number of points (of the kind x = - y^2) on which the function simplifies to the constant - 1/2.
In my opinion the right answer should then be "limit does not exist, is path dependent or cannot be determined", (the same WolframAlpha gives, for example, to limit[(x*y)/(x^2 + y^2),{x,y}->{0,0}],
which can't be "0" because the function simplifies to the constant 1/2 if you set y = x)
Infact both functions have a discontinuity in the origin, and this can also be shown by Plot3D (using more PlotPoints makes it clearer)
Plot3D[(x y^2)/(x^2 + y^4), {x, -.5, .5}, {y, -.5, .5},
PlotRange -> {-.5, .5}, PlotPoints -> 150]
Plot3D[(x y)/(x^2 + y^2), {x, -.5, .5}, {y, -.5, .5},
PlotRange -> {-.5, .5}]
Am I somehow wrong or is it a bug? Does Mathematica 8, 9 or 10 calculate this kind of limits? (I'm only up to version 7...)
To understand better, I also asked WolframAlpha for
limit[(x^a*y^(a*b))/(x^(2*a) + y^(2*a*b)), {x, y} -> {0, 0}]
with a = 1,2,3,4,5 and b = 1,2,3,4 and still got the answer "0" (except for b = 1) against the fact that the function simplifies to 1/2 on the points where x = y^b (and to the constant -1/2 for x =
-y^b if a is odd).
It looks as if the algorithm "checked" only the paths like x = y and not those of the kind x = y^b.
The same fact occurs if you change x and y and ask for
limit[(x^(a*b) y^a)/(x^(2*a*b) + y^(2*a)), {x, y} -> {0, 0}]
Thank you very much, bye!
Mario Gianini
1 Reply
It's a weakness in that Alpha limit code.
Mathematica does not at this time have code dedicated to multivariate limits. | {"url":"https://community.wolfram.com/groups/-/m/t/365144","timestamp":"2024-11-02T14:20:47Z","content_type":"text/html","content_length":"95621","record_id":"<urn:uuid:9588408b-d3ed-4fc3-a477-e0ae5f98b5df>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00264.warc.gz"} |
Quantum information and gravity cutoff in theories with species
We show that lowering of the gravitational cutoff relative to the Planck mass, imposed by black hole physics in theories with N species, has an independent justification from quantum information
theory. First, this scale marks the limiting capacity of any information processor. Secondly, by taking into the account the limitations of the quantum information storage in any system with species,
the bound on the gravity cutoff becomes equivalent to the holographic bound, and this equivalence automatically implies the equality of entanglement and Bekenstein-Hawking entropies. Next, the same
bound follows from quantum cloning theorem. Finally, we point out that by identifying the UV and IR threshold scales of the black hole quasi-classicality in four-dimensional field and high
dimensional gravity theories, the bound translates as the correspondence between the two theories. In case when the high dimensional background is AdS, this reproduces the well-known AdS/CFT
relation, but also suggests a generalization of the correspondence beyond AdS spaces. In particular, it reproduces a recently suggested duality between a four-dimensional CFT and a flat
five-dimensional theory, in which gravity crosses over from four to five dimensional regime in far infrared.
ASJC Scopus subject areas
• Nuclear and High Energy Physics
Dive into the research topics of 'Quantum information and gravity cutoff in theories with species'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/quantum-information-and-gravity-cutoff-in-theories-with-species","timestamp":"2024-11-06T20:35:43Z","content_type":"text/html","content_length":"54055","record_id":"<urn:uuid:98cec1a5-2182-45a4-8873-9e45a1002272>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00865.warc.gz"} |
Data Structures in C++—STL Containers
Data Structures in C++ — STL Containers
Source: ProgrammerCave
Understanding data structures and how they are manipulated as we add, remove and modify data is important to help make better decisions when writing code.
Choosing the right data structure depending on its purpose may help achieve optimal runtime. Although performance should always be measured, time and space complexity acts as a guide for what to
expect from our code.
Due to time constraints, we can’t always profile every possible solution to a problem, and choose the best based on the results (depends on the scope and significance of the problem of course).
Therefore, understanding the options you have and their costs can be a great asset to write better code.
Big O — short prerequisite
The big O notation is often used to denote the worst-case time and space complexity i.e. complexity may be faster or equal to the value.
The most common misconception of this notation is that it measures how fast or how much space an algorithm consumes. It is important to understand that big O describes how an algorithm scales.
For instance, algo A with time O(log n) scales better than an algo B with time O(n), but this doesn’t necessarily mean that operation A is faster than operation B under all circumstances (depends on
factors like number of objects in container, memory locality, etc.). Therefore, it is important to always measure when it comes to performance.
Common Data Structures in C++
Source: ProgrammerCave
STL provides us with a number of data structures (known formally as containers). These general-purpose containers provide a simple way to construct and interface with common data structures in RAII
style — which alleviates worry about managing resources.
Side note before we begin:
There are obviously more complex data structures not available in the STL such as tries, ropes and graphs, which we will not be looking at in this overview. But most of these contain building blocks
or are extensions of the data structures we will be looking at. Thus, understanding these will definitely help with the apprehension of other more complex data structures.
Static arrays
Array elements stored in contiguous memory.
Arrays are containers that store elements in contiguous memory. This is available through std::array in the STL, which is effectively a light wrapper around a raw C array.
The main advantage of using std::array over raw C arrays is that it provides a modern interface just like any other STL containers e.g. size() and at(). It also has friendly value semantics, allowing
itself to be passed or returned by value (it doesn’t decay into a pointer implicitly).
Dynamic Arrays
Copying over elements during “resizing” is O(n)
Available through std::vector, these are similar to static arrays other than being able to resize. However, this resize does come at the cost of having to allocate a larger array and copying all the
elements over — this process of copying over is linear in time i.e. O(n).
More on Arrays
The main advantage of arrays is its constant O(1) access time, as each element can be accessed directly through indexing. However, random insertion and deletion are O(n), due to the need for shifting
Back insertion and deletion can both be constant time, but this is amortised for insertion due to the need to resize the array which is linear in time — by doubling in size every time, it can be
amortised to infinity as O(1).
The logical size and physical size or the array can be accessed through size() and capacity() respectively.
Searching arrays is also linear in time, though sorted arrays can be knocked down to O(log n) through binary search.
It is worth noting that front insertion and deletion can be improved to O(1) by using circular arrays (or ring buffers). An example of this is Boost’s Circular buffer.
Linked List
Doubly-linked list, each node holds a pointer to previous and next node.
Available through std::forward_list (singly-linked) and std::list (doubly-linked) — linked lists offer constant time insertion and deletion at the cost of knowing where you would like to operate.
Accessing an element requires traversal i.e. O(n), which is the same for searching, regardless of it being sorted.
std::deque — best of Linked List and Array
A std::deque, or a double-ended queue is most commonly implemented as a sequence of fixed-size arrays.
Source: StackOverflow
This allows it to have constant time insertions at both ends of the structure, as it avoids having to copy over elements to a larger array like std::vector has to.
This also allows O(1) access through indexing, though this is through another layer of indirection of dereferencing a pointer provided by the bookkeeping data structure. Again, this is why it is
important to remember that time complexity notation refers to the scalability. If what you’re after is performance, always measure!
Essentially being linked arrays, the deque also suffers from linear time O(n) random insertion and deletion due the need to shift elements. Deques also typically have larger minimal memory cost, as a
deque holding just one element has to allocate the full internal array.
Array of null-terminated (\0) characters.
Available as std::string, strings are arrays of null-terminated characters. This means they share similar time complexities on operations as arrays.
C++17’s note on strings — std::string_view :
It’s worth mentioning that from C++17 onwards, consider the usage of std::string_view to improve efficiency in your code. A std::string_view is an object that acts as a non-owning/read-only view of
an existing string.
This avoids unnecessary copying or allocation of data when passing around strings, while still providing access to common string functions such as substr() and comparison operators.
An example is preferring it over const std::string& as a function parameter when passing string literals. This way, we get to avoid allocations and exceptions introduced by having to construct a
std::string object first.
Example of where std::string_view would be preferred
Stacks and Queues
Stack — Last In First Out (LIFO)
Queue — First In First Out (FIFO)
The container adaptors std::stack and std::queue are LIFO and FIFO data structures respectively. These are called adaptors as they act as wrappers around the aforementioned data structures.
By default, both are implemented with std::deque, but std::stack can be created with a std::vector or std::list through its template parameter.
For completeness, std::queue couldn’t be implemented with std::vector, because front deletion isn’t constant.
Pushing and popping of data are done in constant time, with the caveat that pushing being an amortised O(1) for std::stack implement with a std::vector due to resizing.
Heaps are implemented as an array, but could be represented in binary tree form — Source: Wikipedia
A heap is a binary tree where each parent node is smaller (min heap) / larger (max heap) or equal to its children nodes.
Available through std::priority_queue, these can be implemented with either std::vector or std::deque due to the need for O(1) random access.
Appropriately named a priority queue, it allows the queue to be sorted in order of priority with the choice of the lowest or highest being in the front (or top of the heap), no matter the order of
the data being pushed.
Pushing and popping is done in O(log n) time, due to the need to heapify every time we remove or add data.
Heapifying process — Source: Codecademy
STL also provide heapifying algorithms through std::make_heap, std::push_heap and std::pop_heap.
Hash Tables
Hash table — using chaining as means to handle hashing collision.
Hash tables are available in the STL’s unordered associative containers, std::unordered_map, std::unordered_set, and their multi counterparts that allow duplicate keys — std::unordered_multiset and
std::unordered_multimap .
Though hash tables are known for their O(1) lookups, it is important to be cognizant that the worst case is O(n).
Worst case for hash tables — when all elements are stored in a linked-list-like fashion, due to collision.
An example scenario (depends on implementation) might be where our hash table ends up becoming more of a linked list due to chaining as means to handle hashing collision.
The same applies to insertion and deletion, where by average it would be O(1), but in the worst case scenario, it could be linear in time O(n).
Avoiding Collisions:
Fortunately, we do not have to be hashing algorithm experts to help avoid collisions as member functions like max_load_factor() and reserve() are made available.
max_load_factor() sets the maximum load facto, which is the ratio between number of elements to number of buckets that decides when the hash table would automatically increase its number of buckets.
This defaults to 1.
reserve() resizes the number of buckets to contain the specified number of elements, while still conforming to the max load factor, and also rehashes the container.
Binary tree structure.
Just like their unordered counterpart, associative containers std::map, std::set also have multi versions that support duplicate keys — std::multimap and std::multiset.
However, just like the name suggests, these containers keep stored data in a sorted order in a binary tree structure i.e. traversals will see elements in sorted order.
But just like hash tables in the worst case, the lookup time complexity of a binary tree can also be O(n), when the tree degenerates into a linear linked-list like structure.
Worst-case for binary tree, nodes form a linked-list-like structure.
This is why most implementations of these containers use some type of self-balancing binary tree like the red-black tree or AVL tree.
Red-black tree keeps tree balanced to maintain logarithmic time complexity — Source: Javatpoint
These trees adhere to specific rules to keep the tree balance, and thus maintain logarithmic time complexity for searches.
This also means that insertion and deletion happen in O(log n) time, which comes from placing the new element in the right position + rebalancing the tree.
It is important to take time to decide which data structure best fit its purpose in your code — consider where insertion and deletion most often happen? are quick lookups important? do we need these
data to be sorted or does it not matter? is quick direct access is a priority?
In most cases, the more complex the data structure, the more expensive it gets (memory + runtime). So, when deciding which data structure suits best, it is always good to start from basic arrays,
working your way up to more complex data structures.
Feel free to leave a comment if there are any questions or something you would like to add! | {"url":"https://ryonaldteofilo.medium.com/data-structures-in-c-stl-containers-b81574855950?responsesOpen=true&sortBy=REVERSE_CHRON&source=user_profile---------0----------------------------","timestamp":"2024-11-02T11:25:20Z","content_type":"text/html","content_length":"207492","record_id":"<urn:uuid:c09a2985-a955-46c6-91ce-c7516a7261e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00341.warc.gz"} |
What is a 23 sided 3d shape called?
What is a 23 sided 3d shape called?
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
Is there a 100-sided shape?
A 100-sided polygon is called a hectogon.
What is a 200000 sided shape called?
The regular megagon has Dih1,000,000 dihedral symmetry, order 2,000,000, represented by 1,000,000 lines of reflection.
What is a shape with 21 sides called?
geometry, faces, Geometric, polygon, closed figure, 21 sides, twenty-one sides, icosikaihenagon, icosihenagon, 21-sided.
What is a shape with 1 billion sides called?
Gigagon. A gigagon is a two-dimensional polygon with one billion sides. It has the Schläfli symbol.
What is a 999 sided shape called?
This is still my favourite thing I discovered in maths. It’s a polygon name builder. A 35-sided polygon is called a “triacontakaipentagon.“ A 672-sided polygon is a “hexahectaheptacontakaidigon.“ and
999 = enneahectaenneacontakaienneagon Kind of useless but fun.
What is a 100 sided 3d shape called?
Zocchihedron is the trademark of a 100-sided die invented by Lou Zocchi, which debuted in 1985. Rather than being a polyhedron, it is more like a ball with 100 flattened planes. It is sometimes
called “Zocchi’s Golfball”.
What is a billion sided shape?
A gigagon is a two-dimensional polygon with one billion sides.
What is a million sided shape?
Regular megagon
Regular megagon
Coxeter diagram
Symmetry group Dihedral (D 1000000 ), order 2×1000000
Internal angle (degrees) 179.99964°
Dual polygon Self
What is a 10000 sided shape?
In geometry, a myriagon or 10000-gon is a polygon with 10,000 sides.
What is a 99 sided shape?
What is a 99 sided shape called? In geometry, an enneacontagon or enenecontagon or 90-gon is a ninety-sided polygon. Thus, the 99 sided shape is called nonacontakainonagon or enneacontanonagon. Here,
enneaconta is the prefix for the number of sides numbered from 90 to 99.
Is a myriagon a circle?
A myriagon, is a polygon with ten thousand sides, and cannot be visually distinguished from a circle.
Is a megagon a circle?
A Megagon is a polygon with 1,000,000 sides and angles. Even if drawn at the size of the earth, it would still be very hard to visually distinguish from a circle.
What is a Googolgon?
googolgon (plural googolgons) (geometry) A polygon with a googol number of sides (virtually indistinguishable from a circle)
Is there an infinite sided shape?
In geometry, an apeirogon (from the Greek words “ἄπειρος” apeiros: “infinite, boundless”, and “γωνία” gonia: “angle”) or infinite polygon is a generalized polygon with a countably infinite number of
sides. Apeirogons are the two-dimensional case of infinite polytopes.
What is a shape with 10000 sides?
In geometry, a myriagon or 10000-gon is a polygon with 10,000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.
How many sides does a Nonanonacontanonactanonaliagon have?
What do you call a 9999-sided polygon? A nonanonacontanonactanonaliagon. While polygons are important in computer graphics, what’s special about a 9999-gon?
What do you call a 99999 sided polygon?
What do you call a 9999-sided polygon? A nonanonacontanonactanonaliagon.
How many sides does a Icosikaioctagon have?
In geometry, an icositetragon (or icosikaitetragon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon’s interior angles is 3960 degrees.
What does a hendecagon look like?
In geometry, a hendecagon (also undecagon or endecagon) or 11-gon is an eleven-sided polygon. (The name hendecagon, from Greek hendeka “eleven” and –gon “corner”, is often preferred to the hybrid
undecagon, whose first part is formed from Latin undecim “eleven”.)
What shape is a undecagon?
An 11-sided polygon (a flat shape with straight sides). | {"url":"https://www.camomienoteca.com/research-paper-help/what-is-a-23-sided-3d-shape-called/","timestamp":"2024-11-09T03:57:18Z","content_type":"text/html","content_length":"60700","record_id":"<urn:uuid:135a3a09-f547-4132-b87c-987933a495d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00391.warc.gz"} |
Continuous 1-D wavelet transform
wt = cwt(x) returns the continuous wavelet transform (CWT) of x. The CWT is obtained using the analytic Morse wavelet with the symmetry parameter, gamma ($\gamma$), equal to 3 and the time-bandwidth
product equal to 60. cwt uses 10 voices per octave. The minimum and maximum scales are determined automatically based on the energy spread of the wavelet in frequency and time.
The cwt function uses L1 normalization. With L1 normalization, if you have equal amplitude oscillatory components in your data at different scales, they will have equal magnitude in the CWT. Using L1
normalization shows a more accurate representation of the signal. See L1 Norm for CWT and Continuous Wavelet Transform of Two Complex Exponentials.
wt = cwt(x,wname) uses the analytic wavelet specified by wname to compute the CWT.
[wt,f] = cwt(___,fs) specifies the sampling frequency, fs, in hertz, and returns the scale-to-frequency conversions f in hertz. If you do not specify a sampling frequency, cwt returns f in cycles per
[wt,period] = cwt(___,ts) specifies the sampling period, ts, as a positive duration scalar. cwt uses ts to compute the scale-to-period conversions, period. period is an array of durations with the
same Format property as ts.
[wt,f,coi] = cwt(___) returns the cone of influence, coi, in cycles per sample. Specify a sampling frequency, fs, in hertz, to return the cone of influence in hertz.
[wt,period,coi] = cwt(___,ts) returns the cone of influence, coi, as an array of durations with the same Format property as ts.
[___,fb,scalingcfs] = cwt(___) returns the scaling coefficients for the wavelet transform.
[___] = cwt(___,Name=Value) specifies one or more additional name-value arguments. For example, wt = cwt(x,TimeBandwidth=40,VoicesPerOctave=20) specifies a time-bandwidth product of 40 and 20 voices
per octave.
cwt(___) with no output arguments plots the CWT scalogram. The scalogram is the absolute value of the CWT plotted as a function of time and frequency. Frequency is plotted on a logarithmic scale. The
cone of influence showing where edge effects become significant is also plotted. Gray regions outside the dashed white line delineate regions where edge effects are significant. If the input signal
is complex-valued, the positive (counterclockwise) and negative (clockwise) components are plotted in separate scalograms.
If you do not specify a sampling frequency or sampling period, the frequencies are plotted in cycles per sample. If you specify a sampling frequency, the frequencies are in hertz. If you specify a
sampling period, the scalogram is plotted as a function of time and periods. If the input signal is a timetable, the scalogram is plotted as a function of time and frequency in hertz and uses the
RowTimes as the basis for the time axis.
To see the time, frequency, and magnitude of a scalogram point, enable data tips in the figure axes toolbar and click the desired point in the scalogram.
Continuous Wavelet Transform Using Default Values
Obtain the continuous wavelet transform of a speech sample using default values.
load mtlb;
w = cwt(mtlb);
Continuous Wavelet Transform Using Specified Wavelet
Load a speech sample.
Loading the file mtlb.mat brings the speech signal, mtlb, and the sample rate, Fs, into the workspace. Display the scalogram of the speech sample obtained using the bump wavelet.
load mtlb
Compare with the scalogram obtained using the default Morse wavelet.
Continuous Wavelet Transform of Earthquake Data
Obtain the CWT of the Kobe earthquake data. The data are seismograph (vertical acceleration, nm/sq.sec) measurements recorded at Tasmania University, Hobart, Australia on 16 January 1995 beginning at
20:56:51 (GMT) and continuing for 51 minutes. The sampling frequency is 1 Hz.
Plot the earthquake data.
xlabel("Time (mins)")
ylabel("Vertical Acceleration (nm/s^2)")
title("Kobe Earthquake Data")
grid on
axis tight
Obtain the CWT, frequencies, and cone of influence.
[wt,f,coi] = cwt(kobe,1);
View the scalogram, including the cone of influence.
Obtain the CWT, time periods, and cone of influence by specifying a sampling period instead of a sampling frequency.
[wt,periods,coi] = cwt(kobe,minutes(1/60));
View the scalogram generated when specifying a sampling period.
Continuous Wavelet Transform of Two Complex Exponentials
Create two complex exponentials, of different amplitudes, with frequencies of 32 and 64 Hz. The data is sampled at 1000 Hz. The two complex exponentials have disjoint support in time.
Fs = 1e3;
t = 0:1/Fs:1;
z = exp(1i*2*pi*32*t).*(t>=0.1 & t<0.3)+2*exp(-1i*2*pi*64*t).*(t>0.7);
Add complex white Gaussian noise with a standard deviation of 0.05.
wgnNoise = 0.05/sqrt(2)*randn(size(t))+1i*0.05/sqrt(2)*randn(size(t));
z = z+wgnNoise;
Obtain and plot the cwt using a Morse wavelet.
Note the magnitudes of the complex exponential components in the colorbar are essentially their amplitudes even though they are at different scales. This is a direct result of the L1 normalization.
You can verify this by executing this script and exploring each subplot with a data cursor.
Sinusoid and Wavelet Coefficient Amplitudes
This example shows that the amplitudes of oscillatory components in a signal agree with the amplitudes of the corresponding wavelet coefficients.
Create a signal composed of two sinusoids with disjoint support in time. One sinusoid has a frequency of 32 Hz and amplitude equal to 1. The other sinusoid has a frequency of 64 Hz and amplitude
equal to 2. The signal is sampled for one second at 1000 Hz. Plot the signal.
frq1 = 32;
amp1 = 1;
frq2 = 64;
amp2 = 2;
Fs = 1e3;
t = 0:1/Fs:1;
x = amp1*sin(2*pi*frq1*t).*(t>=0.1 & t<0.3)+...
amp2*sin(2*pi*frq2*t).*(t>0.6 & t<0.9);
grid on
xlabel("Time (sec)")
Create a CWT filter bank that can be applied to the signal. Since the signal component frequencies are known, set the frequency limits of the filter bank to a narrow range that includes the known
frequencies. To confirm the range, plot the magnitude frequency responses for the filter bank.
fb = cwtfilterbank(SignalLength=numel(x),SamplingFrequency=Fs,...
FrequencyLimits=[20 100]);
Use cwt and the filter bank to plot the scalogram of the signal.
Use a data cursor to confirm that the amplitudes of the wavelet coefficients are essentially equal to the amplitudes of the sinusoidal components. Your results should be similar to the ones in the
following figure.
Using CWT Filter Bank on Multiple Time Series
This example shows how using a CWT filter bank can improve computational efficiency when taking the CWT of multiple time series.
Create a 100-by-1024 matrix x. Create a CWT filter bank appropriate for signals with 1024 samples.
x = randn(100,1024);
fb = cwtfilterbank;
Use cwt with default settings to obtain the CWT of a signal with 1024 samples. Create a 3-D array that can contain the CWT coefficients of 100 signals, each of which has 1024 samples.
cfs = cwt(x(1,:));
res = zeros(100,size(cfs,1),size(cfs,2));
Use the cwt function and take the CWT of each row of the matrix x. Display the elapsed time.
for k=1:100
res(k,:,:) = cwt(x(k,:));
Elapsed time is 0.928160 seconds.
Now use the wt object function of the filter bank to take the CWT of each row of x. Display the elapsed time.
for k=1:100
res(k,:,:) = wt(fb,x(k,:));
Elapsed time is 0.393524 seconds.
CUDA Code from CWT
This example shows how to generate a MEX file to perform the continuous wavelet transform (CWT) using generated CUDA® code.
First, ensure that you have a CUDA-enabled GPU and the NVCC compiler. See The GPU Environment Check and Setup App (GPU Coder) to ensure you have the proper configuration.
Create a GPU coder configuration object.
cfg = coder.gpuConfig("mex");
Generate a signal of 100,000 samples at 1,000 Hz. The signal consists of two cosine waves with disjoint time supports.
t = 0:.001:(1e5*0.001)-0.001;
x = cos(2*pi*32*t).*(t > 10 & t<=50)+ ...
cos(2*pi*64*t).*(t >= 60 & t < 90)+ ...
Cast the signal to use single precision. GPU calculations are often more efficiently done in single precision. You can however also generate code for double precision if your NVIDIA® GPU supports it.
Generate the GPU MEX file and a code generation report. To allow generation of the MEX file, you must specify the properties (class, size, and complexity) of the three input parameters:
• coder.typeof(single(0),[1 1e5]) specifies a row vector of length 100,000 containing real single values.
• coder.typeof('c',[1 inf]) specifies a character array of arbitrary length.
• coder.typeof(0) specifies a real double value.
sig = coder.typeof(single(0),[1 1e5]);
wav = coder.typeof('c',[1 inf]);
sfrq = coder.typeof(0);
codegen cwt -config cfg -args {sig,wav,sfrq} -report
Code generation successful: View report
The -report flag is optional. Using -report generates a code generation report. In the Summary tab of the report, you can find a GPU code metrics link, which provides detailed information such as the
number of CUDA kernels generated and how much memory was allocated.
Run the MEX file on the data and plot the scalogram. Confirm the plot is consistent with the two disjoint cosine waves.
[cfs,f] = cwt_mex(x,'morse',1e3);
axis tight
xlabel("Time (Seconds)")
ylabel("Frequency (Hz)")
title("Scalogram of Two-Tone Signal")
Run the CWT command above without appending the _mex. Confirm the MATLAB® and the GPU MEX scalograms are identical.
[cfs2,f2] = cwt(x,'morse',1e3);
Change Default Frequency Axis Labels
This example shows how to change the default frequency axis labels for the CWT when you obtain a plot with no output arguments.
Create two sine waves with frequencies of 32 and 64 Hz. The data is sampled at 1000 Hz. The two sine waves have disjoint support in time. Add white Gaussian noise with a standard deviation of 0.05.
Obtain and plot the CWT using the default Morse wavelet.
Fs = 1e3;
t = 0:1/Fs:1;
x = cos(2*pi*32*t).*(t>=0.1 & t<0.3)+sin(2*pi*64*t).*(t>0.7);
wgnNoise = 0.05*randn(size(t));
x = x+wgnNoise;
The plot uses a logarithmic frequency axis because frequencies in the CWT are logarithmic. In MATLAB, logarithmic axes are in powers of 10 (decades). You can use cwtfreqbounds to determine what the
minimum and maximum wavelet bandpass frequencies are for a given signal length, sampling frequency, and wavelet.
[minf,maxf] = cwtfreqbounds(numel(x),1000);
You see that by default MATLAB has placed frequency ticks at 10 and 100 because those are the powers of 10 between the minimum and maximum frequencies. If you wish to add more frequency axis ticks,
you can obtain a logarithmically spaced set of frequencies between the minimum and maximum frequencies using the following.
numfreq = 10;
freq = logspace(log10(minf),log10(maxf),numfreq);
Next, get the handle to the current axes and replace the frequency axis ticks and labels with the following.
AX = gca;
AX.YTickLabelMode = "auto";
AX.YTick = freq;
In the CWT, frequencies are computed in powers of two. To create the frequency ticks and tick labels in powers of two, you can do the following.
AX = gca;
freq = 2.^(round(log2(minf)):round(log2(maxf)));
AX.YTickLabelMode = "auto";
AX.YTick = freq;
Change Scalogram Coloration
This example shows how to scale scalogram values by maximum absolute value at each level for plotting.
Load in a signal and display the default scalogram. Change the colormap to pink(240).
load noisdopp
Take the CWT of the signal and obtain the wavelet coefficients and frequencies.
[cfs,frq] = cwt(noisdopp);
To efficiently find the maximum value of the coefficients at each frequency (level), first transpose the absolute value of the coefficients. Find the minimum value at every level. At each level,
subtract the level's minimum value.
tmp1 = abs(cfs);
t1 = size(tmp1,2);
tmp1 = tmp1';
minv = min(tmp1);
tmp1 = (tmp1-minv(ones(1,t1),:));
Find the maximum value at every level of tmp1. For each level, divide every value by the maximum value at that level. Multiply the result by the number of colors in the colormap. Set equal to 1 all
zero entries. Transpose the result.
maxv = max(tmp1);
maxvArray = maxv(ones(1,t1),:);
indx = maxvArray<eps;
tmp1 = 240*(tmp1./maxvArray);
tmp2 = 1+fix(tmp1);
tmp2(indx) = 1;
tmp2 = tmp2';
Display the result. The scalogram values are now scaled by the maximum absolute value at each level. Frequencies are displayed on a linear scale.
t = 0:length(noisdopp)-1;
shading interp
xlabel("Time (Samples)")
ylabel("Normalized Frequency (cycles/sample)")
title("Scalogram Scaled By Level")
Changing the Time-bandwidth Product
This example shows that increasing the time-bandwidth product ${P}^{2}$ of the Morse wavelet creates a wavelet with more oscillations under its envelope. Increasing ${P}^{2}$ narrows the wavelet in
Create two filter banks. One filter bank has the default TimeBandwidth value of 60. The second filter bank has a TimeBandwidth value of 10. The SignalLength for both filter banks is 4096 samples.
sigLen = 4096;
fb60 = cwtfilterbank(SignalLength=sigLen);
fb10 = cwtfilterbank(SignalLength=sigLen,TimeBandwidth=10);
Obtain the time-domain wavelets for the filter banks.
[psi60,t] = wavelets(fb60);
[psi10,~] = wavelets(fb10);
Use the scales function to find the mother wavelet for each filter bank.
sca60 = scales(fb60);
sca10 = scales(fb10);
[~,idx60] = min(abs(sca60-1));
[~,idx10] = min(abs(sca10-1));
m60 = psi60(idx60,:);
m10 = psi10(idx10,:);
Since the time-bandwidth product is larger for the fb60 filter bank, verify the m60 wavelet has more oscillations under its envelope than the m10 wavelet.
grid on
hold on
hold off
xlim([-30 30])
title("TimeBandwidth = 60")
grid on
hold on
hold off
xlim([-30 30])
title("TimeBandwidth = 10")
Align the peaks of the m60 and m10 magnitude frequency responses. Verify the frequency response of the m60 wavelet is narrower than the frequency response for the m10 wavelet.
cf60 = centerFrequencies(fb60);
cf10 = centerFrequencies(fb10);
m60cFreq = cf60(idx60);
m10cFreq = cf10(idx10);
freqShift = 2*pi*(m60cFreq-m10cFreq);
x10 = m10.*exp(1j*freqShift*(-sigLen/2:sigLen/2-1));
plot([abs(fft(m60)).' abs(fft(x10)).'])
grid on
legend("Time-bandwidth = 60","Time-bandwidth = 10")
title("Magnitude Frequency Responses")
Plot CWT Scalogram in Subplot
This example shows how to plot the CWT scalogram in a figure subplot.
Load the speech sample. The data is sampled at 7418 Hz. Plot the default CWT scalogram.
Obtain the continuous wavelet transform of the signal, and the frequencies of the CWT.
[cfs,frq] = cwt(mtlb,Fs);
The cwt function sets the time and frequency axes in the scalogram. Create a vector representing the sample times.
tms = (0:numel(mtlb)-1)/Fs;
In a new figure, plot the original signal in the upper subplot and the scalogram in the lower subplot. Plot the frequencies on a logarithmic scale.
axis tight
title("Signal and Scalogram")
xlabel("Time (s)")
axis tight
shading flat
xlabel("Time (s)")
ylabel("Frequency (Hz)")
Input Arguments
x — Input signal
vector | timetable
Input signal, specified as a vector or a single-variable regularly sampled timetable. The input x must have at least four samples.
Data Types: single | double
Complex Number Support: Yes
wname — Analytic wavelet
"morse" (default) | "amor" | "bump"
Analytic wavelet used to compute the CWT. Valid options for wname are "morse", "amor", and "bump", which specify the Morse, Morlet (Gabor), and bump wavelet, respectively.
The default Morse wavelet has symmetry parameter gamma ($\gamma$) equal to 3 and time-bandwidth product equal to 60.
Data Types: char | string
fs — Sampling frequency
positive scalar
Sampling frequency in hertz, specified as a positive scalar. If you specify fs, then you cannot specify ts. If x is a timetable, you cannot specify fs. fs is determined from the RowTimes of the
Data Types: single | double
ts — Sampling period
scalar duration
Sampling period, also known as the time duration, specified as a scalar duration. Valid durations are years, days, hours, minutes, and seconds. You cannot use calendar durations. If you specify ts,
then you cannot specify fs. If x is a timetable, you cannot specify ts. ts is determined from the RowTimes of the timetable when you set the PeriodLimits name-value argument.
Example: wt = cwt(x,hours(12))
Data Types: duration
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Example: wt = cwt(x,"bump",VoicesPerOctave=10) returns the CWT of x using the bump wavelet and 10 voices per octave.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: wt = cwt(x,"ExtendedSignal",true,"FrequencyLimits",[0.1 0.2]) extends the input signal symmetrically and specifies frequency limits of 0.1 to 0.2 samples per cycle.
ExtendSignal — Extend input signal symmetrically
true or 1 (default) | false or 0
Option to extend the input signal symmetrically by reflection, specified as one of these:
• 1 (true) — Extend symmetrically
• 0 (false) — Do not extend symmetrically
If ExtendSignal is false, the signal is extended periodically. Extending the signal symmetrically can mitigate boundary effects.
If you want to invert the CWT using icwt with scaling coefficients and approximate synthesis filters, set ExtendSignal to false.
Data Types: logical
FrequencyLimits — Frequency limits
two-element scalar vector
Frequency limits to use in the CWT, specified as a two-element vector with positive strictly increasing entries.
• The first element specifies the lowest peak passband frequency and must be greater than or equal to the product of the wavelet peak frequency in hertz and two time standard deviations divided by
the signal length.
• The second element specifies the highest peak passband frequency and must be less than or equal to the Nyquist frequency.
• The base-2 logarithm of the ratio of the upper frequency limit, freqMax, to the lower frequency limit, freqMin, must be greater than or equal to 1/NV, where NV is the number of voices per octave:
log[2](freqMax/freqMin) ≥ 1/NV.
For complex-valued signals, (-1) × flimits is used for the anti-analytic part, where flimits is the vector specified by FrequencyLimits.
If you specify frequency limits outside the permissible range, cwt truncates the limits to the minimum and maximum valid values. Use cwtfreqbounds to determine frequency limits for different
parameterizations of the CWT.
Example: wt = cwt(x,1000,VoicesPerOctave=10,FrequencyLimits=[80 90])
Data Types: double
PeriodLimits — Period limits
two-element duration array
Period limits to use in the CWT, specified as a two-element duration array with strictly increasing positive entries.
• The first element must be greater than or equal to 2 × ts, where ts is the sampling period.
• The maximum period cannot exceed the signal length divided by the product of two time standard deviations of the wavelet and the wavelet peak frequency.
• The base-2 logarithm of the ratio of the minimum period, minP, to the maximum period, maxP, must be less than or equal to -1/NV, where NV is the number of voices per octave:
For complex-valued signals, (-1) × plimits is used for the anti-analytic part, where plimits is the vector specified by PeriodLimits.
If you specify period limits outside the permissible range, cwt truncates the limits to the minimum and maximum valid values. Use cwtfreqbounds to determine period limits for different
parameterizations of the wavelet transform.
Example: wt = cwt(x,seconds(0.1),VoicesPerOctave=10,PeriodLimits=[seconds(0.2) seconds(3)])
Data Types: duration
VoicesPerOctave — Number of voices per octave
10 (default) | integer from 1 to 48
Number of voices per octave to use for the CWT, specified as an integer from 1 to 48. The CWT scales are discretized using the specified number of voices per octave. The energy spread of the wavelet
in frequency and time automatically determines the minimum and maximum scales.
TimeBandwidth — Time-bandwidth product of the Morse wavelet
60 (default) | scalar greater than or equal to 3 and less than or equal to 120
Time-bandwidth product of the Morse wavelet, specified as a scalar greater than or equal to 3 and less than or equal to 120. The symmetry parameter, gamma ($\gamma$), is fixed at 3. Wavelets with
larger time-bandwidth products have larger spreads in time and narrower spreads in frequency. The standard deviation of the Morse wavelet in time is approximately sqrt(TimeBandwidth/2). The standard
deviation of the Morse wavelet in frequency is approximately 1/2 × sqrt(2/TimeBandwidth).
If you specify TimeBandwidth, you cannot specify WaveletParameters. To specify both the symmetry and time-bandwidth product, use WaveletParameters instead.
In the notation of Morse Wavelets, TimeBandwidth is P^2.
WaveletParameters — Symmetry and time-bandwidth product of the Morse wavelet
[3,60] (default) | two-element vector of scalars
Symmetry and time-bandwidth product of the Morse wavelet, specified as a two-element vector of scalars. The first element is the symmetry, $\gamma$, which must be greater than or equal to 1. The
second element is the time-bandwidth product, which must be greater than or equal to $\gamma$. The ratio of the time-bandwidth product to $\gamma$ cannot exceed 40.
When $\gamma$ is equal to 3, the Morse wavelet is perfectly symmetric in the frequency domain and the skewness is 0. When $\gamma$ is greater than 3, the skewness is positive. When $\gamma$ is less
than 3, the skewness is negative.
For more information, see Morse Wavelets.
If you specify WaveletParameters, you cannot specify TimeBandwidth.
FilterBank — CWT filter bank
cwtfilterbank object
CWT filter bank to use to compute the CWT, specified as a cwtfilterbank object. If you set FilterBank, you cannot specify any other options. All options for the computation of the CWT are defined as
properties of the filter bank. For more information, see cwtfilterbank.
If x is a timetable, the sampling frequency or sampling period in fb must agree with the sampling frequency or sampling period determined by the RowTimes of the timetable.
Example: wt = cwt(x,FilterBank=cfb)
Output Arguments
wt — Continuous wavelet transform
Continuous wavelet transform, returned as a matrix of complex values. By default, cwt uses the analytic Morse (3,60) wavelet, where 3 is the symmetry and 60 is the time-bandwidth product. cwt uses 10
voices per octave.
• If x is real-valued, wt is an Na-by-N matrix, where Na is the number of scales, and N is the number of samples in x.
• If x is complex-valued, wt is a 3-D matrix, where the first page is the CWT for the positive scales (analytic part or counterclockwise component) and the second page is the CWT for the negative
scales (anti-analytic part or clockwise component).
The minimum and maximum scales are determined automatically based on the energy spread of the wavelet in frequency and time. See Algorithms for information on how the scales are determined.
Data Types: single | double
f — Scale-to-frequency conversions
Scale-to-frequency conversions of the CWT, returned as a vector. If you specify a sampling frequency, fs, then f is in hertz. If you do not specify fs, cwt returns f in cycles per sample. If the
input x is complex, the scale-to-frequency conversions apply to both pages of wt.
period — Scale-to-period conversions
Scale-to-period conversions, returned as an array of durations with the same Format property as ts. Each row corresponds to a period. If the input x is complex, the scale-to-period conversions apply
to both pages of wt.
coi — Cone of influence
array of real numbers | array of durations
Cone of influence for the CWT. If you specify a sampling frequency, fs, the cone of influence is in hertz. If you specify a scalar duration, ts, the cone of influence is an array of durations with
the same Format property as ts. If the input x is complex, the cone of influence applies to both pages of wt.
The cone of influence indicates where edge effects occur in the CWT. Due to the edge effects, give less credence to areas that are outside or overlap the cone of influence. For additional
information, see Boundary Effects and the Cone of Influence.
fb — CWT filter bank
cwtfilterbank object
CWT filter bank used in the CWT, returned as a cwtfilterbank object. See cwtfilterbank.
scalingcfs — Scaling coefficients
real- or complex-valued vector
Scaling coefficients for the CWT, returned as a real- or complex-valued vector. The length of scalingcfs is equal to the length of the input x.
More About
Analytic Wavelets
Analytic wavelets are complex-valued wavelets whose Fourier transform vanish for negative frequencies. Analytic wavelets are a good choice when doing time-frequency analysis with the CWT. Because the
wavelet coefficients are complex-valued, the coefficients provide phase and amplitude information of the signal being analyzed. Analytic wavelets are well suited for studying how the frequency
content in real world nonstationary signals evolves as a function of time.
Analytic wavelets are almost exclusively based on rapidly decreasing functions. If $\psi \left(t\right)$ is an analytic rapidly decreasing function in time, then its Fourier transform $\stackrel{^}{\
psi }\left(\omega \right)$ is a rapidly decreasing function in frequency and is small outside of some interval $\alpha <\omega <\beta$ where $0<\alpha <\beta$. Orthogonal and biorthogonal wavelets
are typically designed to have compact support in time. Wavelets with compact support in time have relatively poorer energy concentration in frequency than wavelets which rapidly decrease in time.
Most orthogonal and biorthogonal wavelets are not symmetric in the Fourier domain.
If your goal is to obtain a joint time-frequency representation of your signal, we recommend you use cwt or cwtfilterbank. Both functions support the following analytic wavelets:
• Morse Wavelet Family (default)
• Analytic Morlet (Gabor) Wavelet
• Bump Wavelet
For more information regarding Morse wavelets, see Morse Wavelets. In the Fourier domain, in terms of angular frequency:
• The analytic Morlet is defined as
where is the indicator function of the interval [0,∞).
• The bump wavelet is defined as
where ϵ = 2.2204×10^-16.
If you want to do time-frequency analysis using orthogonal or biorthogonal wavelets, we recommend modwpt.
When using wavelets for time-frequency analysis, you usually convert scales to frequencies or periods to interpret results. cwt and cwtfilterbank do the conversion. You can obtain the corresponding
scales associated by using scales on the optional cwt output argument fb.
For guidance on how to choose the wavelet that is right for your application, see Choose a Wavelet.
• The syntax for the old cwt function continues to work but is no longer recommended. Use the current version of cwt. Both the old and current versions use the same function name. The inputs to the
function determine automatically which version is used. See cwt function syntax has changed.
• When performing multiple CWTs, for example inside a for-loop, the recommended workflow is to first create a cwtfilterbank object and then use the wt object function. This workflow minimizes
overhead and maximizes performance. See Using CWT Filter Bank on Multiple Time Series.
Minimum Scale
For the analytic wavelets supported by the cwt function, the Fourier transforms are real valued and equivalent to the magnitude frequency response (see Analytic Wavelets). The wavelet filters the cwt
function uses are normalized so that the peak magnitudes for all passbands are approximately equal to 2. This is so that a unit-magnitude oscillation in the data shows up with a scalogram magnitude
equal to 1.
Let $\stackrel{^}{\psi }\left(\omega \right)$ denote the mother wavelet in the Fourier domain. For a fixed real-valued scalar c between 0 and 1, there exists a frequency ξ such that
$\stackrel{^}{\psi }\left(\xi \right)=2c.$
The minimum scale ${s}_{0}$ is $\xi /\pi$.
By default, the cwt function sets the fraction of the peak magnitude at the Nyquist frequency at 50% (c equals 0.5) for the Morse wavelet and 10% (c equals 0.1) for the analytic Morlet and bump
wavelets. You can change the fraction by changing the frequency or period limits to use in the CWT. For more information, see cwtfreqbounds and, in particular, the Cutoff name-value argument. See
also the cwtfilterbank object function scales.
Maximum Scale
The cwt function uses the energy spread of the wavelet in time to determine the maximum scale.
To determine the maximum scale, the cwt function first obtains ${\sigma }_{t}$, the time standard deviation of the wavelet. If a wavelet has a time standard deviation of ${\sigma }_{t}$, then
dilating that wavelet by $s$ gives a standard deviation of $s\text{\hspace{0.17em}}{\sigma }_{t}$.
The maximum scale is the value $s$ such that $2\text{\hspace{0.17em}}s\text{\hspace{0.17em}}{\sigma }_{t}=N$, where N is the signal length. The cwt function constrains the largest scale to be that
value so that two time standard deviations of the wavelet span the data length. This ensures that at the largest scale, the wavelet oscillates and that cwt uses a valid wavelet.
Analytic expressions for time standard deviation of the Morse wavelet exist. For the bump and analytic Morlet wavelets, the cwt function estimates the standard deviation of the mother wavelet in time
by obtaining its inverse Fourier transform, normalizing it to be a valid probability density function, and then obtaining the square root of the second central moment.
Scales and Voices Per Octave
Wavelet transform scales are powers of 2 and are denoted by ${s}_{0}{\left({2}^{\frac{1}{NV}}\right)}^{j}$. NV is the number of voices per octave, and j ranges from 0 to the largest scale. For a
specific small scale, ${s}_{0}$:
${s}_{0}{\left({2}^{\frac{1}{NV}}\right)}^{j}\le \frac{N}{2{\sigma }_{t}}$
Converting to base-2 logarithm:
$j{\mathrm{log}}_{2}\left({2}^{\frac{1}{NV}}\right)\le {\mathrm{log}}_{2}\left(\frac{N}{2{\sigma }_{t}{s}_{0}}\right)$
$j\le NV{\mathrm{log}}_{2}\left(\frac{N}{2{\sigma }_{t}{s}_{0}}\right)$
Therefore, the maximum scale is
${s}_{0}{\left({2}^{\frac{1}{NV}}\right)}^{floor\left(NV{\mathrm{log}}_{2}\left(\frac{N}{2{\sigma }_{t}{s}_{0}}\right)\right)}$
L1 Norm for CWT
In integral form, the CWT preserves energy. However, when you implement the CWT numerically, energy is not preserved. In this case, regardless of the normalization you use, the CWT is not an
orthonormal transform. The cwt function uses L1 normalization.
Wavelet transforms commonly use L2 normalization of the wavelet. For the L2 norm, dilating a signal by 1/s, where s is greater than 0, is defined as follows:
The energy is now s times the original energy. When included in the Fourier transform, multiplying by $1}{\sqrt{s}}$ produces different weights being applied to different scales, so that the peaks at
higher frequencies are reduced more than the peaks at lower frequencies.
In many applications, L1 normalization is better. The L1 norm definition does not include squaring the value, so the preserving factor is 1/s instead of $1}{\sqrt{s}}$. Instead of high-frequency
amplitudes being reduced as in the L2 norm, for L1 normalization, all frequency amplitudes are normalized to the same value. Therefore, using the L1 norm shows a more accurate representation of the
signal. See example Continuous Wavelet Transform of Two Complex Exponentials.
[1] Lilly, J. M., and S. C. Olhede. “Generalized Morse Wavelets as a Superfamily of Analytic Wavelets.” IEEE Transactions on Signal Processing 60, no. 11 (November 2012): 6036–6041. https://doi.org/
[2] Lilly, J.M., and S.C. Olhede. “Higher-Order Properties of Analytic Wavelets.” IEEE Transactions on Signal Processing 57, no. 1 (January 2009): 146–160. https://doi.org/10.1109/TSP.2008.2007607.
[3] Lilly, J. M. jLab: A data analysis package for MATLAB^®, version 1.6.2. 2016. http://www.jmlilly.net/jmlsoft.html.
[4] Lilly, Jonathan M. “Element Analysis: A Wavelet-Based Method for Analysing Time-Localized Events in Noisy Time Series.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Sciences 473, no. 2200 (April 30, 2017): 20160776. https://doi.org/10.1098/rspa.2016.0776.
Extended Capabilities
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Usage notes and limitations:
• Single- and double-precision input signal are supported. The precision must be set at compile time.
• Timetable input signal is not supported.
• Only analytic Morse ('morse') and Morlet ('amor') wavelets are supported.
• The following input arguments are not supported: Sampling period (ts), PeriodLimits name-value pair, NumOctave name-value pair, and FilterBank name-value pair.
• Scaling coefficient output and filter bank output are not supported.
• Plotting is not supported.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The cwt function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU
(Parallel Computing Toolbox).
Version History
Introduced in R2016b
R2020a: cwt supports GPU acceleration
The cwt function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU
(Parallel Computing Toolbox).
R2020a: GPU Code Generation: Single-precision support
You can generate CUDA^® code from the cwt function that supports single-precision input data. You must have MATLAB Coder™ and GPU Coder™ to generate CUDA code.
R2018a: 'NumOctaves' name-value argument will be removed
The NumOctaves name-value argument will be removed in a future release. Use either:
• Name-value argument FrequencyLimits to modify the frequency range of the CWT.
• Name-value argument PeriodLimits to modify the period range of the CWT.
See cwtfreqbounds for additional information.
R2016b: cwt function syntax has changed
This release provides an updated version of the continuous wavelet transform, cwt. With the new and simplified syntax, you can easily choose wavelets best suited for continuous wavelet analysis,
frequency or period ranges, and voices per octave. Default values for wavelet and scaling are provided so they need not be specified.
The syntax for the old cwt function continues to work but is no longer recommended. Use the updated version of cwt. Both the old and updated versions use the same function name. The inputs to the
function determine automatically which version is used.
Functionality Use This Instead Compatibility Considerations
Old cwt Updated cwt Update all instances of cwt to use the updated cwt syntax.
See Also | {"url":"https://es.mathworks.com/help/wavelet/ref/cwt.html","timestamp":"2024-11-02T07:56:44Z","content_type":"text/html","content_length":"223180","record_id":"<urn:uuid:f50677c4-36b1-41e6-8561-14b006789e7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00870.warc.gz"} |
Force, Mass And Acceleration Homework Packet - Zara Homework
Homework Help
Force, mass and acceleration homework packet
Questions 1-2 refer to a toy car that can move in either direction along a horizontal line (the + position axis).
Assume that friction is so small that it can be ignored. A force toward the right of constant magnitude is applied to the car.
1. Sketch on the axes below using a solid line the shape of the car’s acceleration—time graph.
2. Suppose that the mass of the car were twice as large. The same constant force is applied to the car. Sketch on the axes above using a dashed line the car’s acceleration—time graph. Explain any
differences in this graph compared to the car’s acceleration—time graph with the original mass.
3. When a force is applied to an object with mass equal to the standard kilogram, the acceleration of the mass is 3.25 m/s2. (Assume that friction is so small that it can be ignored.) When the same
magnitude force is applied to another object, the acceleration is 2.75 m/s2. What is the mass of this object? What would the second object’s acceleration be if a force twice as large were applied to
it? Show your calculations.
4. Given an object with mass equal to the standard kilogram, how would you determine if a force applied to it has magnitude equal to one newton? (Assume that frictional forces are so small that they
can be ignored.)
In Question 5, assume that friction is so small that it can be ignored.
5. The spring scale in the diagram below reads 10.5 N. If the cart moves toward the right with an acceleration also toward the right of 3.25 m/s2, what is the mass of the cart? Show your calculations
and explain.
In Questions 6 – 8, friction may not be ignored.
6. The force applied to the cart in Question 5 by spring scale F1 is still 10.5 N. The cart now moves toward the right with a constant velocity. What are the magnitude and direction of the frictional
force? Show your calculations and explain your reasoning.
7. The force applied to the cart in Question 5 by spring scale F1 is still 10.5 N. The cart now moves toward the right with an acceleration also toward the right of 1.75 m/s2. What are the magnitude
and direction of the frictional force? Show your calculations and explain.
8. The force applied to the cart by spring scale F1 is 10.5 N. The cart now moves toward the right with a constant velocity. The frictional force has the same magnitude as in Question 7. What does
spring scale F2 read? Show your calculations and explain. | {"url":"https://zarahomework.com/force-mass-and-acceleration-homework-packet/","timestamp":"2024-11-03T12:54:32Z","content_type":"text/html","content_length":"68723","record_id":"<urn:uuid:9b430434-e732-4cf4-988f-8264b4aa4692>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00592.warc.gz"} |
Table driven interpolation module with set function. Supports linear and smooth modes
This module performs table lookup using unevenly spaced X and Y values.
The module supports linear and DSPC interpolation. Internally the interpolation is performed using a piece wise polynomial approximation. With linear interpolation a first order polynomial is used;
for 4th order, a polynomial is used. The 4th order interpolation preserves the shape of the function as opposed as overshoots on a spline. This new interpolation method developed by DSP Concepts Inc.
developed uses a reduced set of 3 coefficients.
The input to the module serves as the X value in the lookup and the output equals the Y value derived from doing the interpolation. The module is configured by setting the internal matrix .XY. The
first row of .XY represents X values and the second row Y values.
At instantiation time you specify the total number of points (MAXPOINTS) in the table. MAXPOINTS determines the amount of memory allocated to the table. Then at run time, you can change the number of
points actively used by the table (.numPoints). .numPoints must be less than or equal to MAXPOINTS.
The variable .order specifes either linear (=2) or cubic (=4) interpolation.
The module also provides a custom inspector for easy configuration from the Audio Weaver Designer GUI. The GUI translates the XY table into a piece wise polynomial segments. The matrix .polyCoeffs
contains 4 x (MAXPOINT-1) values. Each column represents the coefficients for each polynomial. Row 4 is the X^3 coefficent, row 3 the X^2 coefficient, row 2 the X coefficient and row 1 the constant
If the input x falls outside of the range of values in the XY table then the input is clipped to the allowable range.
Type Definition
typedef struct _ModuleTableInterpRuntime
ModuleInstanceDescriptor instance; // Common Audio Weaver module instance structure
INT32 maxPoints; // Maximum number of values in the lookup table. The total table size is [maxPoints 2].
INT32 numPoints; // Current number of interpolation values in use.
INT32 order; // Order of the interpolation. This can be either 2 (for linear) or 4 (for pchip).
INT32 updateActive; // Specifies whether the poly coefficients are updating (=1) or fixed (=0).
FLOAT32* XY; // Samples of the lookup table. The first row is the x values and the second row is the f(x) values
FLOAT32* polyCoeffs; // Interpolation coefficients returned by the grid control
} ModuleTableInterpRuntimeClass;
Name Type Usage isHidden Default value Range Units
maxPoints int const 0 8 4:1:1000
numPoints int parameter 0 4 4:1:8
order int parameter 0 2 2:2:4
updateActive int parameter 1 1 0:1
XY float* parameter 0 [2 x 8] Unrestricted
polyCoeffs float* state 0 [4 x 7] Unrestricted
Input Pins
Name: in
Description: audio input
Data type: float
Channel range: Unrestricted
Block size range: Unrestricted
Sample rate range: Unrestricted
Complex support: Real
Output Pins
Name: out
Description: audio output
Data type: float
MATLAB Usage
File Name: table_interp_runtime_module.m
M=table_interp_runtime_module(NAME, MAXPOINTS)
This Audio Weaver module performs interpolation using a lookup table together
with a configurable interpolation order. The table contains (X,Y) value pairs
and can be unevenly spaced.
For 4th order, this module uses a new interpolation method developed by
Holguin at DSP Concepts Inc.. It corresponds to a variation of the
Polynomial Hermit Interpolation or to the Akima Interpolation methods.
The interpolation method is different than in table_interp_module. The inspector
function is borrowed from such module and the response is slightly
different. This module however saves 25% of the memory for coefficients
and includes a runtime set function on the target.
NAME - name of the module.
MAXPOINTS - maximum number of points allocated to the lookup table. This
is set at design time and has a minimum value of 4. At run
time, you can change the number of values in the lookup table
from 4 to MAXPOINTS.
The module can be configured to perform linear or pchip interpolation. | {"url":"https://documentation.dspconcepts.com/awe-designer/latest-version/tableinterpruntime","timestamp":"2024-11-08T01:35:45Z","content_type":"text/html","content_length":"40013","record_id":"<urn:uuid:e87fc8f3-1931-4eef-9f92-114ec5fd6a2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00285.warc.gz"} |
Quantum Computing - Ceylon First UKQuantum Computing
Quantum computing is a field of computer science that uses the principles of quantum theory to build computing devices and algorithms. Unlike classical computers, which store and process information
using bits represented as binary digits (0s and 1s), quantum computers use quantum bits, or qubits. Qubits have some unique properties, such as being able to exist in multiple states simultaneously,
that allow quantum computers to perform certain types of computations much faster than classical computers.
Quantum computers have the potential to solve complex problems that are currently beyond the reach of classical computers, such as simulating quantum systems, optimizing large-scale systems, and
cracking certain encryption algorithms. However, building and using a quantum computer is a challenging task, due to the highly sensitive and fragile nature of qubits and the difficulty in
controlling and manipulating them. Nevertheless, significant progress has been made in the field in recent years, and there are now several companies and research organizations working to develop
practical quantum computing systems.
Why do we need quantum computers?
Quantum computers have the potential to solve a wide range of problems that are difficult or impossible for classical computers. Here are a few examples:
1. Simulating quantum systems: One of the key benefits of quantum computers is their ability to simulate quantum systems, such as molecules and materials, with greater accuracy and efficiency than
classical computers. This could have important applications in fields such as chemistry, material science, and drug discovery.
2. Optimizing large-scale systems: Quantum computers can use quantum algorithms to solve optimization problems, such as the traveling salesman problem, more efficiently than classical computers.
This could have important applications in fields such as logistics, finance, and energy management.
3. Cryptography: Certain encryption algorithms that are currently used to secure communications and protect sensitive information, such as RSA and Elliptic Curve Cryptography, can be broken by a
large enough quantum computer. On the other hand, quantum computers can also be used to create new, more secure forms of encryption.
4. Machine learning: Quantum computers have the potential to perform machine learning tasks, such as pattern recognition and data analysis, faster and more efficiently than classical computers.
Overall, quantum computers are a new and powerful tool that have the potential to impact many fields and enable breakthroughs in areas where classical computers have reached their limits.
Where are quantum computers used?
Quantum computers are still in the early stages of development, and their use is currently limited to research and experimentation. However, there are already several applications and domains where
quantum computers are being used or could be used in the future, including:
1. Scientific simulations: Quantum computers can be used to simulate quantum systems, such as molecules and materials, with greater accuracy and efficiency than classical computers. This could have
important applications in fields such as chemistry, material science, and drug discovery.
2. Optimization problems: Quantum computers can use quantum algorithms to solve optimization problems, such as the traveling salesman problem, more efficiently than classical computers. This could
have important applications in fields such as logistics, finance, and energy management.
3. Cryptography: Certain encryption algorithms that are currently used to secure communications and protect sensitive information, such as RSA and Elliptic Curve Cryptography, can be broken by a
large enough quantum computer. On the other hand, quantum computers can also be used to create new, more secure forms of encryption.
4. Machine learning: Quantum computers have the potential to perform machine learning tasks, such as pattern recognition and data analysis, faster and more efficiently than classical computers.
5. Financial modeling: Quantum computers can be used to model complex financial systems, such as stock markets, more efficiently than classical computers.
6. Supply chain optimization: Quantum computers can be used to optimize supply chain management, such as finding the most efficient routes for shipping goods.
7. Artificial intelligence: Quantum computers have the potential to enhance artificial intelligence algorithms, such as deep learning, by providing faster and more efficient data processing.
These are just a few examples of the potential applications of quantum computers. As the field continues to advance, it is likely that new and exciting uses for quantum computers will be
Why quantum computers are faster?
Quantum computers are faster than classical computers for certain types of computations because they use quantum bits, or qubits, which have unique properties that allow them to perform certain
operations more efficiently. Here are a few reasons why quantum computers are faster:
1. Parallelism: One of the key properties of qubits is that they can exist in multiple states simultaneously, known as superposition. This allows a quantum computer to perform many calculations in
parallel, without the need for sequential processing.
2. Interference: Another important property of qubits is that they can interfere with each other, which allows a quantum computer to amplify or cancel out certain results based on their relative
phase. This allows a quantum computer to find the solution to a problem much faster than a classical computer.
3. Quantum entanglement: Quantum entanglement is a phenomenon in which two or more qubits become correlated in such a way that the state of one qubit cannot be described independently of the other
qubits. This allows a quantum computer to perform certain operations faster than a classical computer, by encoding information in the correlations between qubits rather than in the state of
individual qubits.
Overall, these properties of qubits allow quantum computers to perform certain types of computations much faster than classical computers, by allowing them to take advantage of the properties of
quantum mechanics in ways that classical computers cannot. However, it is important to note that quantum computers are not faster than classical computers for all types of computations, and there are
still many challenges to be overcome in order to build a practical and scalable quantum computer.
How do quantum computers work?
Quantum computers work by harnessing the principles of quantum mechanics to perform calculations. The basic building block of a quantum computer is the quantum bit, or qubit, which can exist in
multiple states simultaneously, known as superposition. Unlike classical bits, which can only be either 0 or 1, qubits can represent a range of values between 0 and 1, allowing them to store and
process more information than classical bits.
In a quantum computer, the qubits are used to represent the states of a quantum system, such as an atom or a photon. By applying quantum operations, or gates, to the qubits, the state of the quantum
system can be manipulated to perform a calculation. For example, a simple quantum operation might involve rotating the state of a qubit by a certain angle, or entangling the state of two qubits so
that their properties become correlated.
Once the calculation is complete, the state of the qubits is measured to obtain the result. Because qubits can exist in multiple states simultaneously, the result of a quantum calculation is a
probability distribution over possible outcomes, rather than a single definite result. In order to obtain a definite result, the calculation must be repeated many times and the results averaged to
obtain an estimate of the true result.
Quantum algorithms are designed to take advantage of the unique properties of qubits, such as superposition and entanglement, to perform calculations that are difficult or impossible for classical
computers. Some well-known quantum algorithms include Shor’s algorithm for factorizing large integers, Grover’s algorithm for searching an unsorted database, and the HHL algorithm for solving linear
systems of equations.
Quantum computers are still in the early stages of development, and there are many challenges that must be overcome in order to build a practical and scalable quantum computer. However, the field is
advancing rapidly, and researchers are making progress towards the development of a useful and usable quantum computer.
Making quantum computers useful?
Making quantum computers useful and practical for real-world applications is a major goal of the field of quantum computing. There are several challenges that must be overcome in order to achieve
this goal, including:
1. Scalability: Currently, quantum computers are limited in the number of qubits they can use and the size of problems they can solve. In order to make quantum computers useful, they must be scaled
up to have more qubits and be able to solve larger and more complex problems.
2. Reliability: Quantum computers are still relatively prone to errors, due to the delicate nature of quantum systems and the difficulties in controlling and manipulating qubits. In order to make
quantum computers useful, the reliability and stability of qubits must be improved to the point where quantum computers can perform complex calculations without producing significant errors.
3. Noise and decoherence: Quantum systems are susceptible to noise and decoherence, which can cause errors in quantum calculations and reduce the accuracy of the results. In order to make quantum
computers useful, the impact of noise and decoherence on qubits must be reduced, either by improving the hardware or by developing algorithms that are robust to these effects.
4. Integration with classical systems: Quantum computers must be integrated with classical computing systems, in order to perform the types of calculations that are required in real-world
applications. This includes developing software and programming languages that are suitable for quantum computing, as well as interfacing quantum computers with classical data storage and
processing systems.
5. Ease of use: Finally, in order to make quantum computers useful, they must be made accessible to a wide range of users, including researchers, engineers, and scientists who may not have a
background in quantum computing. This requires developing user-friendly software and interfaces, as well as providing training and support to help users get started with quantum computing.
These are just a few of the challenges that must be overcome in order to make quantum computers useful. However, the field of quantum computing is rapidly advancing, and researchers are making
progress towards these goals. By working to address these challenges, it is hoped that quantum computers will one day become an important tool for solving complex problems in a wide range of fields
and applications.
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You know you're a programmer when...
Last night I was trying to decide where to plant some flowers. There were some restrictions - for one thing, I didn't want any plant to be hidden behind a taller one. The morning glory had to go next
to the railing, so it would have something to climb. Identical plants should be adjacent, but similar colors shouldn't be. And something conspicuous should probably go in front, so no one would
trample the flowerbed by accident. Obviously this was a constraint problem, and one too hard to solve in my head. So, trowel in hand, I set about figuring out how to solve it by machine.
It is a valuable habit for a programmer to hand any problem over to a program at the first sign of difficulty. Or at least it's valuable when you're sitting at the keyboard and already know how to
solve the problem. Unfortunately I've never actually done constraint programming, and nothing in the garden proved helpful. So I was still trying to figure out how to explain the problem to a
computer when I noticed it had gotten dark, and I had only planted a few of the flowers.
Real programmers can build constraint solvers out of dirt and weeds. I, on the other hand, got to plant my flowers in the dark.
1 comment:
1. Constraint programming isn't difficult or complicated. It's basically combinatoric search with branch pruning. Most of the cleverness comes in your choice of pruning. If you've ever competed in
any programming contests, or solved many of the typical problems they pose, then you'll be familiar with the general pattern:
* A set of elements from which you can generate combinations and permutations
* A boolean function that determines success; may be existential, such as minimizing or maximizing some scalar function
Combinations and permutations are, for most problems, easily produced with one or two recursive routines, though some more complicated producers are easier to write in a language like C# - for
example, consider generating all possible binary tree shapes for a given node count. This is slightly easier to reason about when an iterator-like construct is available than without. However,
reducing tree operations to RPN operations (RPN = postorder traversal) usually makes such tree permutations easier to create.
When maximizing or minimizing, an obvious pruning suggests itself: keep track of score of current min or max, and when hit (assuming monotonically increasing or decreasing fitness function),
discard that subtree of the search space, as appropriate.
For other conditions, careful analysis of the boolean function often reveals a certain order of pruning to excise larger subsets first; or some logical inferences may lead to similar early
And finally, caching function calls (aka memoizing, aka dynamic programming) can be useful for search trees that reduce to trivially similar sub-problems. | {"url":"http://arcanesentiment.blogspot.com/2008/06/you-know-youre-programmer-when.html","timestamp":"2024-11-08T08:38:37Z","content_type":"text/html","content_length":"71810","record_id":"<urn:uuid:db74050c-c0de-446a-8106-b26020ca1cac>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00465.warc.gz"} |
How Do You Find the Length of a Leg of a Right Triangle?
Finding the missing length of a side of a right triangle? If you have the other two side lengths, you can use the Pythagorean theorem to solve! Check out this tutorial and see how to use this really
helpful theorem to find that missing side measurement! | {"url":"https://virtualnerd.com/common-core/grade-8/8_G-geometry/B/7/leg-length-right-triangle-solution","timestamp":"2024-11-02T08:16:19Z","content_type":"text/html","content_length":"29562","record_id":"<urn:uuid:73ca660f-53e8-44e8-ae06-4f4a9f875ac4>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00735.warc.gz"} |
Shiny apps – Systems Forecasting
The Minsky model was developed by Steve Keen as a simple macroeconomic model that illustrates some of the insights of Hyman Minsky. The model takes as its starting point Goodwin’s growth cycle model
(Goodwin, 1967), which can be expressed as differential equations in the employment rate and the wage share of output.
The equations for Goodwin’s model are determined by assuming simple linear relationships between the variables (see Keen’s paper for details). Changes in real wages are linked to the employment rate
via the Phillips curve. Output is assumed to be a linear function of capital stock, investment is equal to profit, and the rate of change of capital stock equals investment minus depreciation.
The equations in employment rate and wage share of output turn out to be none other than the Lotka–Volterra equations which are used in biology to model predator-prey interaction. As employment rises
from a low level (like a rabbit population), wages begin to climb (foxes), until wages becomes too high, at which point employment declines, followed by wages, and so on in a repeating limit cycle.
Limit cycle for wage share (w_fn) and employment (lambda) in Goodwin model
In order to incorporate Minsky’s insights concerning the role of debt finance, a first step is to note that when profit’s share of output is high, firms will borrow to invest. Therefore the
assumption that investment equals profit is replaced by a nonlinear function for investment. Similarly the linear Phillips curve is replaced by a more realistic nonlinear relation (in both cases, a
generalised exponential curve is used). The equation for profit is modified to include interest payments on the debt. Finally the rate of change of debt is set equal to investment minus profit.
Together, these changes mean that the simple limit cycle behavior of the Goodwin model becomes much more complex, and capable of modelling the kind of nonlinear, debt-fueled behavior that (as Minsky
showed) characterises the real economy. A separate equation also accounts for the adjustment of the price level, which converges to a markup over the monetary cost of production.
So how realistic is this model? The idea that employment level and wages are in a simple (linear or nonlinear) kind of predator-prey relation seems problematic, especially given that in recent
decades real wages in many countries have hardly budged, regardless of employment level. Similarly the notion of a constant linear “accelerator” relating output to capital stock seems a little
simplistic. Of course, as in systems biology, any systems dynamics model of the economy has to make such compromises, because otherwise the model becomes impossible to parameterise. As always, the
model is best seen as a patch which captures some aspects of the underlying dynamics.
In order to experiment with the model, I coded it up as a Shiny app. The model has rate equations for the following variables: capital K, population N, productivity a, wage rate w, debt D, and price
level P. (The model can also be run using Keen’s Minsky software.) Keen also has a version that includes an explicit monetary sector (see reference), which adds a few more equations and more
complexity. At that point though I might be tempted to look at simpler models of particular subsets of the economy.
Screenshot of Minsky app
Goodwin, Richard, 1967. A growth cycle. In: Feinstein, C.H. (Ed.), Socialism, Capitalism and Economic Growth. Cambridge University Press, Cambridge, 54–58.
Keen, S. (2013). A monetary Minsky model of the Great Moderation and the Great Recession. Journal of Economic Behavior and Organization, 86, 221-235.
The CellCycler
Tumour modelling has been an active field of research for some decades, and a number of approaches have been taken, ranging from simple models of an idealised spherical tumour, to highly complex
models which attempt to account for everything from cellular chemistry to mechanical stresses. Some models use ordinary differential equations, while others use an agent-based approach to track
individual cells.
A disadvantage of the more complex models is that they involve a large number of parameters, which can only be roughly estimated from available data. If the aim is to predict, rather than to
describe, then this leads to the problem of overfitting: the model is very flexible and can be tuned to fit available data, but is less useful for predicting for example the effect of a new drug.
Indeed, there is a rarely acknowledged tension in mathematical modelling between realism, in the sense of including lots of apparently relevant features, and predictive accuracy. When it comes to the
latter, simple models often out-perform complex models. Yet in most areas there is a strong tendency for researchers to develop increasingly intricate models. The reason appears to have less to do
with science, than with institutional effects. As one survey of business models notes (and these points would apply equally to cancer modelling) complex models are preferred in large part because: “
(1) researchers are rewarded for publishing in highly ranked journals, which favor complexity; (2) forecasters can use complex methods to provide forecasts that support decision-makers’ plans; and
(3) forecasters’ clients may be reassured by incomprehensibility.”
Being immune to all such pressures (this is just a blog post after all!) we decided to develop the CellCycler – a parsimonius “toy” model of a cancer tumour that attempts to capture the basic growth
and drug-response dynamics using only a minimal number of parameters and assumptions. The model uses circa 100 ordinary differential equations (ODEs) to simulate cells as they pass through the
phases of the cell cycle; however the equations are simple and the model only uses parameters that can be observed or reasonably well approximated. It is available online as a Shiny app.
Screenshot of the Cells page of the CellCycler. The plot shows how a cell population is affected by two different drugs.
The CellCycler model divides the cell cycle into a number of discrete compartments, and is therefore similar in spirit to other models that for example treat each phase G1, S, G2, and mitosis as a
separate compartment, with damaged cells being shunted to their own compartment (see for example the model by Checkley et al. here). Each compartment has its own set of ordinary differential
equations which govern how its volume changes with time due to growth, apoptosis, or damage from drugs. There are additional compartments for damaged cells, which may be repaired or lost to
apoptosis. Drugs are simulated using standard PK models, along with a simple description of phase-dependent drug action on cells. For the tumour growth, we use a linear model, based like the Checkley
et al. paper on the assumption of a thin growing layer (see also our post on The exponential growth effect).
The advantages of compartmentalising
Dividing the cell cycle into separate compartments has an interesting and useful side effect, which is that it introduces a degree of uncertainty into the calculation. For example, if a drug causes
damage and delays progress in a particular phase, then that drug will tend to synchronize the cell population in that state. However there is an obvious difference between cells that are affected
when they are at the start of the phase, and those that are already near the end of the phase. If the compartments are too large, that precise information about the state of cells is lost.
The only way to restore precision would be to use a very large number of compartments. But in reality, individual cells will not all have exactly the same doubling time. We therefore want to have a
degree of uncertainty. And this can be controlled by adjusting the number of compartments.
This effect is illustrated by the figure below, which shows how a perturbation at time zero in one compartment tends to blur out over time, for models with 25, 50, and 100 compartments, and a
doubling time of 24 hours. In each case a perturbation is made to compartment 1 at the beginning of the cell cycle (the magnitude is scaled to the number of compartments so the total size of the
perturbation is the same in terms of total volume). For the case with 50 compartments, the curve after one 24 hours is closely approximated by a normal distribution with standard deviation of 3.4
hours or about 14 percent. In general, the standard deviation can be shown to be approximately equal to the doubling time divided by the square root of N.
The solid lines show volume in compartment 1 following a perturbation to that compartment alone, after one cell doubling period of 24 hours. The cases shown are with N=25, 50, and 100 compartments.
Dashed lines are the corresponding normal distributions.
A unique feature of the CellCycler is that it exploits this property as a way of adjusting the variability of doubling time in the cell population. The model can therefore provide a first-order
approximation to the more complex heterogeneity that can be simulated using agent-based models. While we don’t usually have exact data on the spread of doubling times in the growing layer, the
default level of 50 compartments gives what appears to be a reasonable degree of spread (about 14 percent). Using 25 compartments gives 20 percent, while using 100 compartments decreases this to 10
Using the CellCycler
The starting point for the Shiny web application is the Cells page, which is used to model the dynamics of a growing cell population. The key parameters are the average cell doubling time, and the
fraction spent in each phase. The number of model compartments can be adjusted in the Advanced page: note that, along with doubling time spread, the choice also affects both the simulation time (more
compartments is slower), and the discretisation of the cell cycle. For example with 50 compartments the proportional phase times will be rounded off to the nearest 1/50=0.02.
The next pages, PK1 and PK2, are used to parameterise the PK models and drug effects. The program has a choice of standard PK models, with adjustable parameters such as Dose/Volume. In addition the
phase of action (choices are G1, S, G2, M, or all), and rates for death, damage, and repair can be adjusted. Finally, the Tumor page (shown below) uses the model simulation to generate a plot of
tumor radius, given an initial radius and growing layer. Plots can be overlaid with experimental data.
Screenshot of the Tumor page, showing tumor volume (black line) compared to control (grey). Cell death due to apoptosis by either drug (red and blue) and damage (green) are also shown.
We hope the CellCycler can be a useful tool for research or for exploring the dynamics of tumour growth. As mentioned above it is only a “toy” model of a tumour. However, all our models of complex
organic systems – be they of a tumor, the economy, or the global climate system – are toys compared to the real things. And of course there is nothing to stop users from extending the model to
incorporate additional effects. Though whether this will lead to improved predictive accuracy is another question.
Try the CellCycler web app here.
Stephen Checkley, Linda MacCallum, James Yates, Paul Jasper, Haobin Luo, John Tolsma, Claus Bendtsen. “Bridging the gap between in vitro and in vivo: Dose and schedule predictions for the ATR
inhibitor AZD6738,” Scientific Reports.2015;5(3)13545.
Green, Kesten C. & Armstrong, J. Scott, 2015. “Simple versus complex forecasting: The evidence,” Journal of Business Research, Elsevier, vol. 68(8), pages 1678-1685.
Application of survival analysis to P2P Lending Club loans data
Peer to peer lending is an option people are increasingly turning to, both for obtaining loans and for investment. The principle idea is that investors can decide who they give loans to, based on
information provided by the loaner, and the loaner can decide what interest rate they are willing to pay. This new lending environment can give investors higher returns than traditional savings
accounts, and loaners better interest rates than those available from commercial lenders.
Given the open nature of peer to peer lending, information is becoming readily available on who loans are given to and what the outcome of that loan was in terms of profitability for the investor.
Available information includes the loaner’s credit rating, loan amount, interest rate, annual income, amount received etc. The open-source nature of this data has clearly led to an increased
interest in analysing and modelling the data to come up with strategies for the investor which maximises their return. In this blog entry we will look at developing a model of this kind using an
approach routinely used in healthcare, survival analysis. We will provide motivation as to why this approach is useful and demonstrate how a simple strategy can lead to significant returns when
applied to data from the Lending Club.
In healthcare survival analysis is routinely used to predict the probability of survival of a patient for a given length of time based on information about that patient e.g. what diseases they have,
what treatment is given etc. It is routinely used within the healthcare sector to make decisions both at the patient level, for example what treatment to give, and at the institutional level (e.g.
health care providers), for example what new healthcare policies will decrease death associated with lung cancer. In most survival analysis studies the data-sets usually contain a significant
proportion of patients who have yet to experience the event of interest by the time the study has finished. These patients clearly do not have an event time and so are described as being
right-censored. An analysis can be conducted without these patients but this is clearly ignoring vital information and can lead to misleading and biased inferences. This could have rather large
consequences were the resultant model applied prospectively. A key part of all survival analysis tools that have been developed is therefore that they do not ignore patients who are right censored.
So what does this have to do with peer to peer lending?
The data on the loans available through sites such as the Lending Club contain loans that are current and most modelling methods described in other blogs have simply ignored these loans when building
models to maximise investor’s returns. These loans described as being current are the same as our patients in survival analysis who have yet to experience an event at the time the data was
collected. Applying a survival analysis approach will allow us to keep people whose loans are described as being current in our model development and thus utilise all information available. How can
we apply survival analysis methods to loan data though, as we are interested in maximising profit and not how quickly a loan is paid back?
We need to select relevant dependent and independent variables first before starting the analysis. The dependent variable in this case is whether a loan has finished (fully repaid, defaulted etc.)
or not (current). The independent variable chosen here is the relative return (RR) on that loan, this is basically the amount repaid divided by the amount loaned. Therefore if a loan has a RR value
less than 1 it is loss making and greater than 1 it is profit making. Clearly loans that have yet to have finished are quite likely to have an RR value less than 1 however they have not finished and
so within the survival analysis approach this is accounted for by treating that loan as being right-censored. A plot showing the survival curve of the lending club data can be seen in the below
The black line shows the fraction of loans as a function of RR. We’ve marked the break-even line in red. Crosses represent loans that are right censored. We can already see from this plot that there
are approximately 17-18% loans that are loss making, to the left of the red line. The remaining loans to the right of the red line are profit making. How do we model this data?
Having established what the independent and dependent variables are, we can now perform a survival analysis exercise on the data. There are numerous modelling options in survival analysis. We have
chosen one of the easiest options, Cox-regression/proportional hazards, to highlight the approach which may not be the optimal one. So now we have decided on the modelling approach we need to think
about what covariates we will consider.
A previous blog entry at yhat.com already highlighted certain covariates that could be useful, all of which are actually quite intuitive. We found that one of the covariates FICO range high
(essentially is a credit score) had an interesting relationship to RR, see below.
Each circle represents a loan. It’s strikingly obvious that once the last FICO Range High score exceeds ~ 700 the number of loss making loans, ones below the red line decreases quite dramatically. So
a simple risk adverse strategy would be just to invest in loans whose FICO Range High score exceeds 700, however there are still profitable loans which have a FICO Range High value less than 700. In
our survival analysis we can stratify for loans below and above this 700 FICO Range High score value.
We then performed a rather routine survival analysis. Using FICO Range High as a stratification marker we looked at a series of covariates previously identified in a univariate analysis. We ranked
each of the covariates based on the concordance probability. The concordance probability gives us information on how good a covariate is at ranking loans, a value of 0.5 suggests that covariate is no
better than tossing a coin whereas a value of 1 is perfect, which never happens! We are using concordance probability rather than p-values, which is often done, because the data-set is very large and
so many covariates come out as being “statistically significant” even though they have little effect on the concordance probability. This is a classic problem of Big Data and one option, of many, is
to focus model building on another metric to counter this issue. If we use a step-wise building approach and use a simple criterion that to include a covariate the concordance probability must
increase by at least 0.01 units, then we end up with a rather simple model: interest rate + term of loan. This model gave a concordance probability value of 0.81 in FICO Range High >700 and 0.63 for
a FICO Range High value <700. Therefore, it does a really good job once we have screened out the bad loans and not so great when we have a lot of bad loans but we have a strategy that removes those.
This final model is available online here and can be found on the web-apps section of the website. When playing with the model you will find that if interest rates are high and the term of loan is
low then regardless of the FICO Range High value all loans are profitable, however those with FICO Range High values >700 provide a higher return, see figure below.
The above plot was created by using an interest rate of 20% for a 36 month loan. The plot shows two curves, the one in red represents a loan whose FICO Range High value <700 and the black one a loan
with FICO Range High value >700. The curves describe your probability of attaining a certain amount of profit or loss. You can see that for the input values used here, the probability of making a
loss is similar regardless of the FICO Range High Value; however the amount of return is better for loans with FICO Range High value >700.
Using survival analysis techniques we have shown that you can create a relatively simple model that lends itself well for interpretation, i.e. probability curves. Performance of the model could be
improved using random survival forests – the gain may not be as large as you might expect but every percentage point counts. In a future blog we will provide an example of applying survival analysis
to actual survival data.
Rent or buy
Suppose you were offered the choice between investing in one of two assets. The first, asset A, has a long term real price history (i.e. with inflation stripped out) which looks like this:
Long term historical return of asset A, after inflation, log scale.
It seems that the real price of the asset hasn’t gone anywhere in the last 125 years, with an average compounded growth rate of about half a percent. The asset also appears to be needlessly volatile
for such a poor performance.
Asset B is shown in the next plot by the red line, with asset A shown again for comparison (but with a different vertical scale):
Long term historical return of assets A (blue line) and B (red line), after inflation, log scale.
Note again this is a log scale, so asset B has increased in price by more than a factor of a thousand, after inflation, since 1890. The average compounded growth rate, after inflation, is 6.6
percent – an improvement of over 6 percent compared to asset A.
On the face of it, it would appear that asset B – the steeply climbing red line – would be the better bet. But suppose that everyone around you believed that asset A was the correct way to build
wealth. Not only were people investing their life savings in asset A, but they were taking out highly leveraged positions in order to buy as much of it as possible. Parents were lending their
offspring the money to make a down payment on a loan so that they wouldn’t be deprived. Other buyers (without rich parents) were borrowing the down payment from secondary lenders at high interest
rates. Foreigners were using asset A as a safe store of wealth, one which seemed to be mysteriously exempt from anti-money laundering regulations. In fact, asset A had become so systemically
important that a major fraction of the country’s economy was involved in either building it, selling it, or financing it.
You may have already guessed that the blue line is the US housing market (based on the Case-Shiller index), and the red line is the S&P 500 stock market index, with dividends reinvested. The housing
index ignores factors such as the improvement in housing stock, so really measures the value of residential land. The stock market index (again based on Case-Shiller data) is what you might get from
a hypothetical index fund. In either case, things like management and transaction fees have been ignored.
So why does everyone think housing is a better investment than the stock market?
The RentOrBuyer
Of course, the comparison isn’t quite fair. For one thing, you can live in a house – an important dividend in itself – while a stock market portfolio is just numbers in an account. But the vast
discrepancy between the two means that we have to ask, is housing a good place to park your money, or is it better in financial terms to rent and invest your savings?
As an example, I was recently offered the opportunity to buy a house in the Toronto area before it went on the market. The price was $999,000, which is about average for Toronto. It was being rented
out at $2600 per month. Was it a good deal?
Usually real estate decisions are based on two factors – what similar properties are selling for, and what the rate of appreciation appears to be. In this case I was told that the houses on the
street were selling for about that amount, and furthermore were going up by about a $100K per year (the Toronto market is very hot right now). But both of these factors depend on what other people
are doing and thinking about the market – and group dynamics are not always the best measure of value (think the Dutch tulip bulb crisis).
A potentially more useful piece of information is the current rent earned by the property. This gives a sense of how much the house is worth as a provider of housing services, rather than as a
speculative investment, and therefore plays a similar role as the earnings of a company. And it offers a benchmark to which we can compare the price of the house.
Consider two different scenarios, Buy and Rent. In the Buy scenario, the costs include the initial downpayment, mortgage payments, and monthly maintenance fees (including regular repairs, utilities,
property taxes, and accrued expenses for e.g. major renovations). Once the mortgage period is complete the person ends up with a fully-paid house.
For the Rent scenario, we assume identical initial and monthly outflows. However the housing costs in this case only involve rent and utilities. The initial downpayment is therefore invested, as are
any monthly savings compared to the Buy scenario. The Rent scenario therefore has the same costs as the Buy scenario, but the person ends up with an investment portfolio instead of a house. By
showing which of these is worth more, we can see whether in financial terms it is better to buy or rent.
This is the idea behind our latest web app: the RentOrBuyer. By supplying values for price, mortgage rates, expected investment returns, etc., the user can compare the total cost of buying or renting
a property and decide whether that house is worth buying. (See also this Globe and Mail article, which also suggests useful estimates for things like maintenance costs.)
The RentOrBuyer app allows the user to compare the overall cumulative cost of buying or renting a home.
The Rent page gives details about the rent scenario including a plot of cumulative costs.
For the $999,000 house, and some perfectly reasonable assumptions for the parameters, I estimate savings by renting of about … a million dollars. Which is certainly enough to give one pause. Give it
a try yourself before you buy that beat up shack!
Of course, there are many uncertainties involved in the calculation. Numbers like interest rates and returns on investment are liable to change. We also don’t take into account factors such as
taxation, which may have an effect, depending on where you live. However, it is still possible to make reasonable assumptions. For example, an investment portfolio can be expected to earn more over a
long time period than a house (a house might be nice, but it’s not going to be the next Apple). The stock market is prone to crashes, but then so is the property market as shown by the first figure.
Mortgage rates are at historic lows and are likely to rise.
While the RentOrBuyer can only provide an estimate of the likely outcome, the answers it produces tend to be reasonably robust to changes in the assumptions, with a fair ratio of house price to rent
typically working out in the region of 200-220. Perhaps unsurprisingly, this is not far off the historical average. Institutions such as the IMF and central banks use this ratio along with other
metrics such as the ratio of average prices to earnings to detect housing bubbles. As an example, according to Moody’s Analytics, the average ratio for metro areas in the US was near its long-term
average of about 180 in 2000, reached nearly 300 in 2006 with the housing bubble, and was back to 180 in 2010.
House prices in many urban areas – in Canada, Toronto and especially Vancouver come to mind – have seen a remarkable run-up in price in recent years (see my World Finance article). However this is
probably due to a number of factors such as ultra-low interest rates following the financial crash, inflows of (possibly laundered) foreign cash, not to mention a general enthusiasm for housing which
borders on mania. The RentOrBuyer app should help give some perspective, and a reminder that the purpose of a house is to provide a place to live, not a vehicle for gambling.
Try the RentOrBuyer app here.
Housing in Crisis: When Will Metro Markets Recover? Mark Zandi, Celia Chen, Cristian deRitis, Andres Carbacho-Burgos, Moody’s Economy.com, February 2009.
The BayesianOpionionator
I recently finished Robert Matthew’s excellent book Chancing it: The laws of chance – and what they mean for you. One of the themes of the book is that reliance on conventional statistical methods,
such as the p-value for measuring statistical significance, can lead to misleading results.
An example provided by Matthews is a UK study (known as The Grampian Region Early Anistreplase Trial, aka GREAT) from the early 1990s of clot-fighting drugs for heart attack patients, which appeared
to show that administering the drugs before they reached hospital reduced the risk of death by as much as 77 percent. The range of the effect was large, but was still deemed statistically significant
according to the usual definition. However subsequent studies showed that the effect of the drug was much smaller.
Pocock and Spiegelhalter (1992) had already argued that prior studies suggested a smaller effect. They used a Bayesian approach in which a prior belief is combined with the new data to arrive at a
posterior result. The impact of a particular study depends not just on its apparent size, but also on factors such as the spread. Their calculations showed that the posterior distribution for the
GREAT study was much closer to the (less exciting) prior than to the experimental results. The reason was that the experimental spread was large, which reduced its impact in the calculation.
Given the much-remarked low degree of reproducibility of clinical studies (in the US alone it has been estimated that approximately US$28,000,000,000 is spent on preclinical research that is not
reproducible) it seems that a Bayesian approach could prove useful in many cases. To that end, we introduce the BayesianOpinionator, a web app for incorporating the effect of prior beliefs when
determining the impact of a statistical study.
Screenshot of the BayesianOpinionator
The data for the BayesianOpinionator app is assumed to be in the form of a comparison between two cases, denoted null and treated. For example in a clinical trial the treated case could correspond to
a patient population who are treated with a particular drug, and the null case would be a comparison group that are untreated. As mentioned already, a common problem with such studies is that they
produce results which appear to be statistically significant, but later turn out to be caused by a fluke. In this case the BayesianOpinionator will help to determine how seriously the results should
be taken, by taking prior beiefs and data into account. The method works by representing data in terms of binomial distributions, which as seen below lead to a simple and intuitive way of applying
weights to different effects in order to gauge their impact.
The New Data page is used to input the trial results, which can be in a number of different forms. The first is a binary table, with the two options denoted Pos and Neg – for example these could
represent fatalities versus non-fatalities. The next is a probability distribution, where the user specifies the mean and the standard deviation of the probability p of the event taking place for
each case. Finally, studies are sometimes reported as a range of the odds ratio (OR). The odds for a probability p is defined as p/1-p, so is the ratio of the chance of an event happening to the
chance of it not happening. The OR is the odds of the treated case, divided by the odds of the null case. An OR of 1 represents no change, and an OR range of 0.6 to 1 would imply up to 40 percent
improvement. Once the odds range is specified, the program searches for a virtual trial which gives the correct range. (The user is also asked to specify a null mean, otherwise the result is
In all cases, the result is a binomial distribution for the treated and null cases, with a probability p that matches the average chance of a positive event taking place. Note that the problems
studied need not be limited to binary events. For example, the data could correspond to diameter growth of a tumor with or without treatment, from a scale of 0 to 1. Alternatively, when data is input
using the probability range option, a range can be chosen to scale p between any two end points, which could represent the minimum and maximum of a particular variable. In other words, while the
binomial distribution is based on a sequence of binary outcomes, it generalises to continuous cases while retaining its convenient features.
In the next page Prior, the user inputs the same type of information to represent their prior beliefs about a trial. Again, this information is used to generate binomial distributions for the prior
case. Finally, the two sets are pooled together in order to give the posterior result on the next page. The posterior is therefore literally the sum of the prior and the new data.
The next page, Odds, shows how the new results compare with the prior in terms of impact on the posterior. The main plot shows the log-OR distribution, which is approximately normal. A feature of the
odds ratio is that it allows for a simplified representation within the Bayesian framework. The posterior distribution can be calculated as the weighted sum of the prior and the new data. The weights
are given in table form, and are represented graphically by the bubble plot in the sidebar. The size of the bubbles represents spread of log-OR, while vertical position represents weight of the data,
with heavy at the bottom.
As shown by Matthews (see this paper), the log-OR plot allows one to determine a critical prior interval (CPI) which can be viewed as the minimum necessary in order for the new result to be deemed
statistically significant (i.e. has a 95 percent chance of excluding the possibility of no effect). If the CPI is more extreme than the result, this implies that the posterior result will not be
significant unless one already considers the CPI to be realistic. For clinical trials, which for ethical reasons assume no clear advantage between the null and treated cases, the CPI acts as a
reality check on new results, because if the results are very striking it shows how flexible the prior needs to be in order to see them as meaningful.
The BayesianOpinionator Shiny app can be accessed here. | {"url":"https://systemsforecasting.com/category/shiny-apps/","timestamp":"2024-11-08T02:33:32Z","content_type":"text/html","content_length":"90802","record_id":"<urn:uuid:469b91b2-8767-4228-a7de-22d28ae171bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00171.warc.gz"} |
Lesson 12
Saree Silk Stories: Friendship Bracelets
Warm-up: True or False: Place Value Comparisons (10 minutes)
The purpose of this True or False is to elicit strategies and understandings students have for using place value to compare numbers and determine if an equation is true. When students share how they
know an equation is true or false based on looking at the total number of tens or total number of ones on each side, they look for and make use of the base-ten structure of numbers and the properties
of operations (MP7).
• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time.
• Share and record answers and strategy.
• Repeat with each statement.
Student Facing
Decide if each statement is true or false. Be prepared to explain your reasoning.
• \(24 = 10 + 14\)
• \(15 + 12 = 27\)
• \(26 = 10 + 6 + 10\)
• \(58 = 20 + 20 + 8\)
Activity Synthesis
• “How can you explain your answer for the last equation using what you know about tens and ones?” (There are only 4 tens on the right.)
Activity 1: Share Ribbon with Friends (20 minutes)
The purpose of this activity is for students to interpret and solve two-step problems involving length. After reading each story problem, students consider what questions could be asked and what
information will be needed in the second part of the problem. Students read each story with a partner, and then solve each story problem independently and compare their solutions.
Students begin the activity by looking at the first problem displayed, rather than in their books. Students represent the problem in a way that makes sense to them and share different representations
during the synthesis, explaining how these representations helped solve the problem (MP1, MP2, MP3).
This activity uses MLR6 Three Reads. Advances: reading, listening, representing
Representation: Access for Perception. Offer strips of colored paper to demonstrate the length of the ribbons, as well as the tape diagrams students create. Encourage students to cut the “ribbon” and
label the parts to represent the story. Reiterate the context and connect to the idea of subtraction.
Supports accessibility for: Conceptual Processing, Organization, Memory
• Groups of 2
• Give each group access to base-ten blocks.
• “The students in Priya’s class are sharing ribbons to make necklaces and bracelets for their friends and family members.”
MLR6 Three Reads
• Display both parts of the story, but only the problem stems, without revealing the questions.
• “We are going to read this problem 3 times.”
• 1st Read: “Lin found a piece of ribbon that is 92 cm long. She gave Noah a piece that is 35 cm. Then, Lin gave Jada 28 cm of ribbon.”
• “What is this story about?”
• 1 minute: partner discussion
• Listen for and clarify any questions about the context.
• 2nd Read: “Lin found a piece of ribbon that is 92 cm long. She gave Noah a piece that is 35 cm. Then, Lin cut off 28 cm of ribbon for Jada.”
• “Which lengths of ribbon are important to pay attention to in the story?” (length of ribbon Lin started with, length of ribbon given to Noah, length of ribbon given to Jada, length of ribbon Lin
has in the end)
• 30 seconds: quiet think time
• 1–2 minutes: partner discussion
• Share and record all quantities.
• Reveal the questions.
• 3rd Read: Read the entire problem, including the questions, aloud.
• “What are different ways we could represent this problem?” (tape diagram, equations)
• 30 seconds: quiet think time
• 1–2 minutes: partner discussion
• “Read the story again with your partner. Then decide how to represent and solve it on your own.”
• “When you have both answered the questions, compare to see if you agree.”
• 10 minutes: partner work time
• Monitor for students who represent each part of the story with:
□ tape diagrams
□ base-ten diagrams
□ other clearly-labeled drawings or diagrams
□ equations
Student Facing
1. Solve. Show your thinking. Use a diagram if it helps. Don’t forget the units.
1. Lin found a piece of ribbon that is 92 cm long. She cut a piece for Noah that is 35 cm. How much ribbon does Lin have left?
2. Then, Lin cut off 28 cm of ribbon for Jada. How much ribbon does Lin have left now?
Activity Synthesis
• Invite 2–3 previously identified students to display their work side-by-side for all to see.
• “How does each representation help you understand the problem?” (In the diagrams, they used labels to show what each part means. I can see how they used Lin's length of ribbon in both parts. In
the equations, I can see the same numbers, but it's a little harder to tell what each part means. I can make sense of them by looking at the other diagrams people made.)
Activity 2: Friendship Bracelets and Gifts (15 minutes)
The purpose of this activity is for students to represent and solve two-step story problems. The story problems are presented in parts, and students are encouraged to represent each part in a way
that makes sense to them. In the synthesis, students compare different ways they represent and solve the problem.
• Groups of 2
• Give students access to base-ten blocks.
• “Read each problem with your partner and solve it on your own. Show your thinking using diagrams, equations, or words.”
• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who use tape diagrams, base-ten diagrams, and equations to represent each part of the last problem.
Student Facing
1. Solve. Show your thinking. Don’t forget the units. Use a diagram if it helps.
1. Han has 82 inches of ribbon. He only needs 48 inches. How much should he cut off?
2. Han gives the ribbon he doesn’t need to Clare. Clare uses it to make her ribbon longer. Her ribbon was 27 inches. How long is Clare’s ribbon now?
2. Solve. Show your thinking. Don’t forget the units. Use a diagram if it helps.
1. Andre’s ribbon is too short. He has 28 inches of ribbon, but he needs it to be 50 inches long. How much more ribbon does he need?
2. Andre got the ribbon he needed from Mai. Mai now has 49 inches of ribbon left. How much ribbon did Mai start with?
Advancing Student Thinking
Students may represent and solve the first part of each problem accurately, but see the second part as a problem with two unknowns. Consider asking:
• “What new information does the second part of the problem give you? What do you need to figure out? What do you already know?”
• “What happened in the first part of the story? What did you figure out? How could you use that in the second part of the problem?”
Activity Synthesis
• Invite previously identified students to share their diagrams and equations for each part of the problem.
• “How did _____ represent the problem? How does each representation show the story problem?”
Lesson Synthesis
“Today you solved different kinds of story problems that had two parts.”
“How did you represent your thinking and keep track of your calculations? How did you keep up with the lengths you knew and what you needed to find out?” (I used diagrams to make sense of the story.
I drew base-ten diagrams to help me solve. I put a circle around my answer so I could use it for the next problem.)
“What ideas for solving story problems have you learned from others?”
Share and record responses.
Cool-down: Sharing Saree Silk Ribbon (5 minutes)
Student Facing
In this section of the unit, we learned more about standard length units. We measured using inches and feet—two length units from the the U.S. customary system. We also solved two-step story problems
about length and interpreted diagrams that represent taking a part away. This diagram shows that we know the length of the ribbon and how much was cut. The question mark represents the length of
ribbon that is left.
Han had a piece of ribbon that was 64 inches long. He cut off 28 inches to make a necklace for his sister. How much ribbon is left? | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-2/unit-3/lesson-12/lesson.html","timestamp":"2024-11-11T06:57:02Z","content_type":"text/html","content_length":"92179","record_id":"<urn:uuid:9601ae39-f69a-43bd-9cb8-fdf984f59d94>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00025.warc.gz"} |
Notations and Opinions
01/23/2008, 02:50 PM (This post was last modified: 01/23/2008, 07:58 PM by GFR.)
Ivars Wrote:........................................
Sometimes I feel that with pentation .......................................
I can not see math as not being possible to close logically while allowing it to develop because of undetermined symbols like infinity, etc..
We are still working to clarify what happens up to the s=4 hyper-operation rank, i.e. at the tetration level. In particular, concerning the "priority to the right" business, we have:
(s=1, addition): a+(a+(a+a)) = a x 4 [bracketing not necessary, add. commutable, associable]
(s=2, multipl.): a.(a.(a.a)) = a ^ 4 [bracketing not necessary, mult. commutable, associable]
(s=3, expon.): a^(a^(a^a)) = a # 4 [bracketing indispensable, exp. non-commutable, non-assoc.]
(s=4, tetrat.): a#(a#(a#a)) = a § 4 [bracketing indispensable, tetr. non-commutable, non assoc.]
Symbol § is provisionally used here for "pentation". However, concerning the same priority business, for s=5 we must also have:
(s=5, pent.): a§(a§(a§a)) = a ç 4 [bracketing indispensable, tetr. non-commutable and non-associable]. Symbol ç means here "hexation". And ... so on.
The problem is that we have not fully analyzed yet the fourth rank level. We need to show that "sexp", "slog" and sroot" are smooth uniquely defined (1- or 2-valued) "functions", before proceeding
further. Therefore, we are not ready to "sublime" up to the fifth rank.
Objects like "undeterminations", "infinitesimals" and "infinities" may strongly intervene in the analysis process, but I don't see them as really serius obstacles, if we pay the necessary attention,
at this stage. I agree that the study of the hyper-operations, starting from the tetration rank might give new ideas also for a more complete approach to undeterminate and infinite magnitudes. We
shall see!
03/25/2008, 07:58 PM (This post was last modified: 03/26/2008, 04:23 PM by James Knight.)
AND OPINION. ENJOY!
///The notation won't display right because the original post was deleted.. give me some time to fix this.... sorry!
Here is the notation I find easy to work with in my notebook. Tell me what you think.
Regular Operations
x[attachment=268]y = z
Left Inverse (Stick on the Left)
x = z[attachment=267]y
Right Inverse (stick to the right)
y = z[attachment=270]x
The Minus One Law in My Notation
(x[attachment=269]b) [attachment=270] (x) = x [attachment=269](b-1)
Logarithmic Notation
Also I am in the midst of developing a fractional notation as well as horizontal notation for logarithms.
log x (base 10) looks like this
or x~10
where the ~ is like a fraction line. I find the current/traditional notation clumsy.
for example
((a~b)~c) = log (log a)
c b
a~(b~c) = log a
log b
Anyway, I find the combination of the fractional and horizontal easier to manipulate.
03/26/2008, 02:26 PM (This post was last modified: 03/26/2008, 02:31 PM by bo198214.)
What we really need is not a new notation, but a notation that can be/is widely agreed on and that widely can be used.
Basicly, as the overwhelming majority of mathematical articles is written in (La)TeX (which is also an input option on this forum), we need an ASCII notation and a TeX notation.
A very good example is the by Gianfranco (not even officially) introduced ASCII notation [n] for the nth hyperoperation. Its very intuitive, good readable, and everyone is immediately going to accept
However there are not yet that convincing proposals for the log-type and root-type inverse functions.
I think we commonly agree on the use of slog_b for the inverse of x->b[4]x. And for ssqrt as the inverse of x->x[4]2.
One generalization would be lg[n]_b for the inverse of x->b[n]x and rt[n]^k for the inverse of x->x[n]k.
From an algebraic standpoint however our considerations fall into the category of a quasigroup. That is a set with an operation * that has unique left and right inverses. There the left and right
inverses are written as \ and / respectively. This is very mnemonic as one cancels always on the corresponding side:
(a*x)/x=a and (a/x)*x=a
x\(x*a)=a and x*(x\a)=a
applied to our problem we should choose
b \[n] y as the logarithm type inverse, i.e. b \[n] ( b[n]x )=x and
y /[n] k as the root type inverse, i.e. (b [n] k) /[n] k =b.
then is slog_b(x)=b\[4]x and ssqrt(x)=x/[4]2. This is even a bit mnemonic as \ reminds slightly of an l (for logarithm) and / reminds slightly of an r (for root). x/[3]k also reminds slightly of x^(1
03/29/2008, 03:58 PM (This post was last modified: 03/30/2008, 12:53 AM by GFR.)
bo198214 Wrote:A very good example is the by Gianfranco (not even officially) introduced ASCII notation [n] for the nth hyperoperation. Its very intuitive, good readable, and everyone is
immediately going to accept it.
Thank you, Henryk, for your kind appreciation and comments. As a matter of fact, the first idea of a new and, possibly, clear graphical notation was born during my cooperation with Konstantin
Rubtsov, in
As you see in the annexes of the two threads, we agreed on a Box Notation of direct and inverse hyperoperations (of both the log and root types) using, unfortunately, symbols not belonging to the
ASCII set. We called it the "KAR-GFR Box Notation". The square-bracketing notation of the "direct operations" was used by me for the first time in this Forum for simplifying the writing, without
loosing information. I should like to confirm it here officially, transforming this fact in a kind of official proposal.
bo198214 Wrote:...............
(a*x)/x=a and (a/x)*x=a
x\(x*a)=a and x*(x\a)=a
applied to our problem we should choose
b \[n] y as the logarithm type inverse, i.e. b \[n] ( b[n]x )=x and
y /[n] k as the root type inverse, i.e. (b [n] k) /[n] k =b.
then is slog_b(x)=b\[4]x and ssqrt(x)=x/[4]2.
As you know, the KAR-GFR Box Notation included two half-boxes for indicating the hyperroot (similar to a capital Gamma) and the hyperlog (similar to a capital L), both superscripted (or
underscripted) by the hyper-operation rank and accompanied by the appropriate bases or exponents.
In a simplifyed version of them, valid only for rank 4, I "unofficially" used, in this Forum, the following notations:
[k/]srt(x) : the k-th super-root of x
[b\]slog(x) : the superlog, base b, of x
Now, your proposal, which takes into consideration various other facts and proposals, brings to:
ssqrt(x) = [2/]srt(x) = x/[4]2
slog_b(x) = [b\]slog(x) = b\[4]x.
As a first reaction, I should like to introduce a slight modification in your interesting proposals (to be identified as ... the GFR-BO, or BO-GFR simplified ASCII notation):
ssqrt(x) = [2/]srt(x) = x/[4]2
slog_b(x) = [b\]slog(x) = b[4]\x. (modification)
In this case, for a direct hyperop such as y = b [n] k, we shall have:
b [n]\ y as the log-type inverse, i.e. ---> b [n] \ ( b [n] k ) = k and
y /[n] k as the root-type inverse, i.e. --> (b [n] k) /[n] k = b.
The remaining problem is that this simplified ASCII notation would give:
b [1] k = b + k = y --> b = y /[1] k = y - k, k = b [1] \ y = y - b
b [2] k = b * k = y --> b = y /[2] k = y / k, k = b [2] \ y = y / b
b [3] k = b ^ k = y --> b = y /[3] k = k-srt y, k = b [3] \ y = b-slog y.
In other words, for rank 3 and > 3, we are in contrast with the traditional prefixed notation of the inverse operations.
We could then try a more schematical approximated notatiuon, such as:
b [1] k = b + k = y --> b = y /1| k = y - k, k = b |1\ y = y - b
b [2] k = b * k = y --> b = y /2| k = y / k, k = b |2\ y = y / b
b [3] k = b ^ k = y --> b = y /3| k = k-srt y, k = b |3\ y = b-slog y.
The advantage of this schematical notation is that we could admit an upside-down mirror inversion of the operation symbols, in their inverse sequence, like (see the third line):
b [3] k = b ^ k = y --> b = k \3| y = k-srt y, k = y /3| b = b-slog y.
A compromise of the straight and mirror schematical notation, for always showing a "prefixed" inversing operator (acting on y at its right) could be:
b [3] k = b ^ k = y -> b = k \3| y = k-srt y, k = b |3\ y = b-slog y.
In general, for:
y = b [n] k, we might have:
b = k \n| y = y /n| k, the root-type left-inverse, and
k = b |n\ y = y |n/ b, the log-type left-inverse.
This is my "official" additional proposal, hoping not to have created more noise that it is absolutely indispensable.
Please tell me what you (BO, and ... all of you) think of it.
03/29/2008, 05:15 PM (This post was last modified: 03/29/2008, 05:17 PM by bo198214.)
Quote:b [3] k = b ^ k = y --> b = y /[3] k = k-srt y, k = b [3] \ y = b-slog y.
In other words, for rank 3 and > 3, we are in contrast with the traditional prefixed notation of the inverse operations.
I dont see this contrast, we have:
\( y /[3] k=y^{1/k} \) and \( b [3]\backslash y = \log_b y \)
(And by this you can easily remember that / corresponds to the root type.)
I mean the side is anyway arbitrary, you also have \( {^n x} \) but you write x[4]n.
GFR Wrote:The advantage of this schematical notation is that we could admit an upside-down mirror inversion of the operation symbols, in their inverse
sequence, like (see the third line):
In general, for:
y = b [n] k, we might have:
b = k \n| y = y /n| k, the root-type left-inverse, and
k = b |n\ y = y |n/ b, the log-type left-inverse.
But Gianfranco thats confusing! \ and / for the same operation depending on which side. I dont want to first think a minute what is meant by the current symbol! There is also no mnemonics attached.
Neither is a both side variant really needed nor is it usual to have it. There is no opposite side variant for -, / or ^.
So if you really desperately need the both-side variants then keep the same operation symbol! E.g. /n| and |n/ as root-type inversion, but I dont see usage for them. And you have to burden your
memory with another rule to decide on which side is the base/exponent, i.e. on the side which is not |.
However I see a bit a problem with /n|, as when you use it without spaces it can be confusing, for example
|a/n|x| = | a /n| x |
I placed some attention to this problem when I was deciding for /[n] because you can not misread the / as a division (because it is followed by an open bracket). This ruled out the other variant that
I had in mind: /n/.
However your idea to put [n]\ instead of \[n] is a better one as you can better memorize the rule
b[n]k /[n] k = b and b [n]\ ( b[n] k ) = k
as "The thing to be reduced is on the (reducing) operation side (i.e. the side with the / or \ attached)"
PS: We can call this the BO-GFR simplified ASCII notation, however by such discussions there always comes up the image of a commitee designing conventions (by long and intense democratic discussions)
which dont fit real needs of the using people. While really useful things are made without commitees! However I hope its not the case here.
03/30/2008, 12:51 AM
Of course, I was joking. I spent my entire life in committess and commissions and I am not interested in participating in new ones, until the complete stop of the engines. After that, we shall see! I
meant only to find a provisional "code" for easy future reference, in this Forum. The Future is in the hands of the gods.
03/30/2008, 05:12 AM
That is interesting, I didn't even realize I was using GFR notation the last time I used [n], so it must be easy! Also, bo198214, I am amazed at how simple, your inverse hyperop notation is, I had
heard of quasigroups before, but it didn't occur to me that they represented hyperops, but they clearly do. When it comes to exact notation, however, I would agree with GFR, in that the root notation
"y /[n] k" should be left alone, and is acceptable as is, but the log notation "b [n]\ y" is something I would perfer, because it indicates that "b [n]\" is the function because the "\" clearly
separates the "function" part from the "argument" part, and "y" is usually considered the primary argument. I vote for these two for the ASCII notation of inverse hyperops.
Andrew Robbins
03/30/2008, 06:11 AM
GFR Wrote:Please tell me what you (BO, and ... all of you) think of it.
Dear Gianfranco (and all) -
the proposals are nice.
However, noting, that we find a route for tetrational notation now, I feel I should explicitely re-remark, that for iterated exponentiation and decremented iterated exponantiation, with which I'm
involved, I always need three parameters, one additional for the top-(beginning)-exponent x for y= b^..^b^x, height h (= b's in number of h, or more generally: h being the iteration-count) and I
should step away a bit and use different symbols.
So I better announce (the ascii-notation)
y = x{4,b}h for iterated exponentiation beginning at x: b^...^b^b^x
(and from earlier discussions)
y = x{3,b}h for iterated multiplication beginning at x: x*b^h
y = x{2,b}h for iterated addition beginning at x: x+b*h
for my needs for the time being,
the "height"-function
h = hgh(x,b)
x = 1 {4,b} h
= b [4] h // related to the tetrational notation
just to prevent confusion between these two concepts and the reading of my notes in context of the tetration-discussion.
Gottfried Helms, Kassel
03/30/2008, 09:24 AM
Gottfried Wrote:...
y = x{4,b}h for iterated exponentiation beginning at x: b^...^b^b^x
(and from earlier discussions)
y = x{3,b}h for iterated multiplication beginning at x: x*b^h
y = x{2,b}h for iterated addition beginning at x: x+b*h
for my needs for the time being,
the "height"-function
h = hgh(x,b)
x = 1 {4,b} h
= b [4] h // related to the tetrational notation
I personally think that the Arrow-Iteration-Section notations discussed in my first post cover most of these use cases, but U-tetration is different enough to require a special notation. Here are my
\text{ASCII} & \text{Arrow-Iteration-Section} & \text{Special} \\
\mathtt{x\^\^y(a)}\text{ or }\mathtt{(x\^)\^y(a)}
& (x {\uparrow})^y(a)
& \exp^y_x(a)
& (y \mapsto (x {\uparrow})^y(a))^{-1}(z)
& \text{slog}_x(z) - \text{slog}_x(a)
& (x \mapsto (x {\uparrow})^y(a))^{-1}(z)
& \
\mathtt{x\^-\^y(a)}\text{ or }\mathtt{(x\^-)\^y(a)}
& (t \mapsto x^t - 1)^y(a)
& DE^y_x(a)
& (y \mapsto (t \mapsto x^t - 1)^y(a))^{-1}(z)
& \
& (x \mapsto (t \mapsto x^t - 1)^y(a))^{-1}(z)
& \
but I've seen other notations elsewhere. The one I've seen used the most is x^^y@a, although I had also used y`x`a in the past. Also, GFR uses x$y*a or something like that, which I find confusing.
Thats all about iter-exp.
Starting from scratch using Arrow-Iteration-Section notation, we find that the natural expression in ASCII is (x^)^y(a) which could be shortened to x^^y(a) which means the corresponding notation for
iterated decremented exponentials is (x^-)^y(a) which could be shortened to x^-^y(a), what do you think? About iter-dec-exp/U-tetration, this would mean that your "height" function is h = hgh(x, b,
a) = b^-^\x(a) and h = hgh(x, b) = b^-^\x which I would've called the "super-decremented-logarithm" or something.
We might even go so far as to use similar notations for superroot and superlog, so srt_n = (/^^n) and slog_b = (b^^\).
While I'm at it, I might as well summarize the other suggestions (based on BO's):
\text{ASCII} & \text{Arrow-Iteration-Section} & \text{Special} \\
& x {\uparrow}{\uparrow} y
& {}^{y}{x}
& (x {\uparrow}{\uparrow})^{-1}(z)
& \text{slog}_x(z)
& ({\uparrow}{\uparrow} y)^{-1}(z)
& \
& x {\uparrow}^{n-2} y
& x \begin{tabular}{|c|}\hline n \\\hline\end{tabular} y
& (x {\uparrow}^{n-2})^{-1}(z)
& {}^{n}_{x}\begin{tabular}{|c}z \\\hline\end{tabular}
& ({\uparrow}^{n-2} y)^{-1}(z)
& {}_{n}^{y}\begin{tabular}{|c}\hline z \\\end{tabular}
& (x {\uparrow}^{n-2})^y(a)
& \
& (y \mapsto (x {\uparrow}^{n-2})^y(a))^{-1}(z)
& \
& (x \mapsto (x {\uparrow}^{n-2})^y(a))^{-1}(z)
& \
I must say, the slash notation is by far the most expressive tetration notation I've ever seen. It allows full expression of practically anything I can think of that is hyperop/tetration related. As
you can see, it covers many topics that do not have a specialized notation yet.
Andrew Robbins
03/30/2008, 03:13 PM
andydude Wrote:I must say, the slash notation is by far the most expressive tetration notation I've ever seen. It allows full expression of practically anything I can think of that is hyperop/
tetration related. As you can see, it covers many topics that do not have a specialized notation yet.
Andrew Robbins
Yes, this looks nice - maybe because we are already a bit used to it.
However, the collection of slashes and symbols for iterated exponentiation (IE, T) and especially for decremented iterated exponentiation (DIE, U) still looks a bit obfuscating to me - again: perhaps
this is a matter of usage and experience.
While resorting to "hgh()" as function giving the height of a powertower I should add, that another -even more- natural function occurs with the notation x {4,b} h=y, since x already has a height;
"h" is here actually the height-difference of the towers x and y in terms of "bricks" or "stones"; so what in tetration is the superroot (and subequently the srt-function) may be here the stone- or
brick-function "stn" or "brk", where the latter had even some resemblance to a notation of base using b...
So until I'm getting used to some slash-notation for IE (T()) and DIE (U()) I should propose the "brk"-function indicating the value of the base, which is needed to get from x to y if h times
iterated... (well, I've no use for it so far, but...)
Gottfried Helms, Kassel | {"url":"https://tetrationforum.org/showthread.php?tid=114&pid=1692","timestamp":"2024-11-04T23:27:31Z","content_type":"application/xhtml+xml","content_length":"62670","record_id":"<urn:uuid:25cb0f86-3275-4925-9ef2-5492a63c7811>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00184.warc.gz"} |
As a planar optical wavefront propagates through a fluid medium with varying index-of-refraction, such as turbulent shear layers and atmospheric boundary layers, the wavefront becomes distorted (some
parts advance farther than others). For airborne laser systems, the aberrating flow field is conventionally divided into two regimes.
The first regime is the aperture's near field where the optical aberration arises from unsteady flow structures. The optical distortion in this regime is termed "aero-optics problem". For example,
laser communication devices mounted on high speed airborne aircraft can induce sparated turbulent flows around the aperture, and these flows degrade the beam's ability to focus in the far field.
The second regime is the Earth's atmosphere, which affects the long-range propagation of a laser beam. Optical distortions in this regime are referred to as "atmospheric propagation problem". They
usually involve very large flow structures at very low frequencies, and can be corrected by static optical components or adaptive optical (AO) systems.
Aberrations in the first regime are of high frequency and are difficult to correct. In the early days of laser technology, the aero-optical aberrations were of small magnitude ($\sim$ 1$\mu$m)
compared with the long optical wavelength ($\sim$ 10$\mu$m) and are found to be unimportant. However, since the 80's, breakthroughs have been made in laser technology, and lasers with much shorter
wavelengths ($\sim$ 1$\mu$m) become popular. For this reason, the increased relative optical aberrations has become a serious problem.
Optical aberrations originate from variations of the refractive index that is directly related to air density through the Gladstone-Dale relation, $$ n-1 = K_{GD}\rho, $$ where $n$ is the refractive
index, and $K_{GD}$ is the Gladstone-Dale constant. For light with wavelengths between 1$\mu$m and 10$\mu$m traveling in the air at standard condition ($P_{atm}=1.01325\times10^{5}$Pa, $T_{atm}=
288.15$K, $\rho_{atm}=1.225kg/m^3$), $K_{GD}$ is approximately $2.27\times10^{-4}m^{3}/kg$. The nondimensional form of the Gladstone-Dale relation is $$ n-1 = K_{GD}^{*}\rho^{*}, $$ where $K_{GD}^{*}
=K_{GD}\cdot\rho_{ref}$ is the nondimensional Gladstone-Dale constant, $\rho_{ref}$ is the reference density, and $\rho^{*}=\rho/\rho_{ref}$ is the nondimensional density. If the standard atmospheric
density $\rho_{atm}$ is taken as the reference value, the nondimensional Gladstone-Dale constant is $K_{GD}^{*}=2.8\times10^{-4}$.
A wavefront is a surface with constant phase value, and is equivalent to a surface with constant optical path length (OPL), $$ OPL(t,x,y)=\int_{z_0}^{z_1}{n(t,x,y,z)}\,dz, $$ where $z$ is the beam
propagation direction; $(x,y)$ define the plane of the initial planar wavefront that is perpendicular to the beam direction. Here we have made an assumption that the integration path is short such
that there is no noticeable change in the beam direction, i.e., the $\theta$ angle in the figure below is small. This assumption is valid for most aero-optics problems.
In practice, the difference between a distorted wavefront and its spatial mean position is more important than the wavefront itself, and this difference is defined as the optical path difference
(OPD), $$ OPD(t,x,y)=OPL(t,x,y)-\langle OPL(t,x,y)\rangle, $$ where the angle bracket means spatial average over the aperture. Conventionally, "OPD" is used interchangeably with "wavefront". This is
because that OPD is in fact the conjugate of a wavefront's variation part and has the same characteristics as a wavefront.
The overall effect of aberrations over the whole aperture in the farfield is measured by the Strehl ratio $$ SR(t)=\frac{I(t)}{I_0}, $$ where $I(t)$ is the on-axis irradiance and $I_0$ is the
diffraction-limited on-axis irradiance. Based on the large aperture approximation (LAA), the Strehl ratio can be estimated as $$ SR(t)=e^{-(2\pi OPD_{rms}(t)/\lambda)^2}, $$ where $\lambda$ is the
wavelength and $OPD_{rms}(t)$ is the root mean square of $OPD(t,x,y)$ over the aperture. The time-averaged Strehl ratio is used more often to measure the far-field optical quality, $$ \overline{SR}=e
^{-(2\pi\overline{OPD}_{rms}/\lambda)^2}, $$ which naturally makes $\overline{OPD}_{rms}$ one of the most important parameters in aero-optics.
As previously discussed, $OPD(t,x,y)$ is the conjugate of a wavefront, thus a wavefront can be written as $$ W(t,x,y)=-OPD(t,x,y). $$ Performing a Taylor series expansion on the wavefront, one can
get $$ W(t,x,y)=A + B_1(t) \cdot x + B_2(t) \cdot y + W_{HighOrder}(t,x,y), $$ where $A$ is a constant referred to as steady (or piston) component; $B_1(t)$ and $B_2(t)$ are the instantaneous tip and
tilt components, respectively; $W_{HighOrder}(t,x,y)$ is the high-order term with small amplitude but high frequency. The steady, tip, and tilt components can be effortlessly corrected using AO
systems, but the high-order component cannot be easily corrected, making it the most detrimental to optical systems. It is therefore customary to focus on the high-order wavefront (i.e., with steady,
tip, and tilt components removed) in aero-optics study. | {"url":"https://binzhang.org/research/optics","timestamp":"2024-11-08T18:19:22Z","content_type":"text/html","content_length":"8109","record_id":"<urn:uuid:6b21b25f-d24c-4841-833b-f254e20287ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00682.warc.gz"} |
IntroductionNumerical modeling of the lightning channelThe transmission line modelThe current pulseModeling of composite materialEquivalent conductivityThin layer techniqueTime domain simulations with a frequency domain solverSimulation examplesField of the lightning channelStructure close to the lightning channelDirect lightning strikeConclusionsAcknowledgementsReferences
Despite the constantly ongoing progress in the development and application of numerical methods in electromagnetics it turns out that the computation of lightning-related effects in the framework of
Electromagnetic Compatibility (EMC) still constitutes a highly challenging task. This is due to a number of difficulties that can be characterized as follows : First, the modeling of an actual
lightning channel as electromagnetic source requires to turn a physically complicated and geometrically extended excitation into a numerical model. Second, the rather long duration of a lightning
electromagnetic pulse (LEMP), together with its associated low-frequency spectrum, requires both in time and frequency domain stable and efficient numerical algorithms. Third, in actual applications
it is often necessary to calculate LEMP transfer functions of complex systems, such as aircraft, for example. This, in turn, often involves to model advanced materials and to deal with a high degree
of complexity.
In this contribution it is outlined how to numerically calculate LEMP transfer functions by the method of moments (MoM), taking into account the main difficulties mentioned above. To this end, in
Sect. 2 it is outlined, based on early work , how to take into account the influence of the lightning channel in a realistic way. The modeling of composite materials as an example for an advanced
material is explained in Sect. 3. It is also mentioned that low-frequency shielding can be modeled by a thin layer technique . Then, in Sect. 4, it is summarized how to process frequency domain
results obtained by the MoM in order to obtain time domain results that are useful for lightning analysis. Simulation examples that implement these methods are given in Sect. 5, followed by a short
summary in Sect. 6.
There are several engineering approaches for the modeling of the electric current of a lightning channel . They relate the current at the channel base to the current at any position on the channel,
where the channel itself can be modeled as a wire of length of several kilometers. Most models are categorized to be either of the Transmission Line (TL) or of the Traveling Current Source (TCS)
type. The TL type is used here, therefore it shortly is described in the following subsection. More details on its implementation within the MoM can be found in .
In the TL type model a current wave is traveling upwards with propagation speed vf, starting at the channel base, as depicted in Fig. . At positions above z′=vf⋅t the current is zero. The current at
any position z′ along the channel and at time t is given by i(z′,t)=i0(t-z′vf)if z′≤vf⋅t0if z′>vf⋅t, where i0(t) is the current pulse at the channel base and vf is the propagation speed, which
typically is chosen to be one third of the vacuum speed of light vf=13⋅c.
Illustration of the upward moving current front in the lightning channel.
For the current i0(t) the current pulse according to , which can be found in the standard, is used. The slope of this curve is continuous, which has proven to be consistent with the results of
lightning measurements. The entire pulse can be divided into three components that are multiplied by each other and determine the amplitude, the smooth rising edge, and the exponential decay,
respectively. The corresponding equation is given by i0(t)=imaxη⋅tTrisen1+tTrisen⋅exp-tThold, where imax is the maximum value of the current, η is the amplitude correction factor, which ensures that
the pulse reaches the value imax, Trise is the rise time, and Thold the decay time of the pulse. For the following investigations the parameters imax=100kA, η=0.986, Trise=1.82µs, and Thold=285µs
are used. This choice refers to a negative first stroke of threat level “severe”. A pulse with these parameter values is shown in Fig. .
Lightning current pulse according to Heidler.
Composite materials are widely used in aircraft design for weight reduction and improvement of mechanical strength. In this publication a type of carbon fiber composite is considered which consists
of several layers of carbon fibers with different orientations, enclosed in resin.
Instead of modeling single fibers one may assume an anisotropic conductivity for each layer , where the direction of the fibers in the layer corresponds to the direction of the highest conductivity
in the conductivity tensor.
A further simplification can be made by replacing the various anisotropically conducting layers by a single layer with the same overall thickness and an equivalent isotropic conductivity σeq. An
analytical investigation of the shielding effectiveness of a plane shield with several anisotropically conducting layers has shown that this simplification is valid up to several tens of MHz . Since
lightning is a low frequency phenomenon, the equivalent conductivity can be used to model the shielding properties of the composite material with respect to lightning effects. As a first step to
calculate σeq the diagonal entries of the average conductivity tensor are calculated by σxxσyy=1d∑i=1Lσξσψσψσξ⋅cos2(αi)sin2(αi)⋅di, where L is the number of layers. Here it is assumed that each
layer of fibers has the conductivity σξ in fiber direction and σψ in cross-fiber direction. The orientation of the fibers is specified by the angle αi. The thickness of layer i is denoted by di and
the total thickness of the multilayer material is denoted by d. Then the equivalent conductivity is chosen to be the smaller of the two conductivity values, σeq=min(σxx,σyy), to give a valid value
for a worst case approximation. In the following, the conductivities are chosen to be σξ=10kSm-1 and σξ=100Sm-1 and each layer has thickness di=d/L. For a symmetric layer pattern, e.g., a
multiple of four layers with a relative rotation angle of 45∘ between the fibers of adjacent layers, the equivalent isotropic conductivity is given by σeq=5050Sm-1.
Local coordinate system for one layer of carbon fibers, where the ξ-direction corresponds to the orientation of the fibers.
There are several methods to numerically model thin layers. The surface impedance boundary condition method , for example, is applicable to scattering problems, but it is not suitable for the
calculation of shielding effectiveness, since the coupling between inside and outside region of the shielding geometry has to accurately be taken into account. It might also come to mind to apply a
Green's function of layered media , but this typically requires some kind of canonical symmetry of the geometry considered. Therefore it is not immediate to apply this method to arbitrarily shaped
three dimensional structures. The thin layer technique that is applied here to efficiently model thin layers of finite size in conjunction with the MoM is based on an analytically formulated coupling
matrix. In this case, the layer has to fulfill the requirement to be thin compared to the overall dimension of the body to be modeled and it has to provide a sufficiently high conductivity. If the
conductivity is large enough then the wave propagation inside the layer is perpendicular to its surface and can be described by an analytical solution in the form of a coupling matrix which also
correctly incorporates all effects related to the wave propagation in a lossy medium. As a consequence, two regions that are separated by a two-dimensional layer, compare Fig. , can be treated by the
MoM, where the coupling through the layer is taken into account by a coupling matrix which relates the tangential fields at both sides of the layer to each other. This hybrid-technique, which
combines the MoM with an analytical solution, has proven to be stable down to frequencies in the kHz range .
Two MoM regions that are separated by a thin finitely conducting layer. Its electromagnetic properties can be described by an analytical formulation.
A priori, the lightning current is formulated in time domain while the MoM and the layer technique are formulated in frequency domain. In this section it is explained how to relate both formulations
to calculate the impulse response of a lightning current.
The spectrum I0(ω) of the current i0(t) at the channel base can be determined analytically, as described in . The spatial distribution of the current at any position z′ of the channel is a time
shifted version of i0(t), therefore the shift property of the Fourier transformation can be used to write the corresponding spectrum as I(z′,ω)=I0(ω)⋅exp-jωz′vf, which clearly states that the
magnitude of the spectrum does not depend on the position and only the phase is influenced by z′. Now the system response G(ω) to a unit excitation, in this case an impressed current with a constant
amplitude with respect to frequency, which flows spatially distributed on the channel, can be calculated. Then both spectra can be multiplied in frequency domain to find the spectrum of the pulse
response R(ω), R(ω)=G(ω)⋅I0(ω), which could, e.g., be a voltage or a field value. The function R(ω) can be transformed to time domain via an inverse Fourier transformation to yield the pulse response
The calculated spectrum consists of values that refer to discrete frequencies. This leads to a periodic extension of the pulse response in time domain. The time tex, when the periodic extension
starts, is related to the frequency difference Δf between the discrete frequencies of the spectrum by tex=1Δf. To ensure meaningful simulation results, Δf has to be small enough such that the impulse
response has decayed before the periodic extension starts. This implies that a long excitation pulse requires a small frequency step width, which sets up a limit for this method, due to the fact that
the MoM becomes unstable at very low frequencies. Hence, excitation signals with a duration of several milliseconds cannot accurately be modeled by this method. For the pulse in Fig. a frequency
step width of Δf=600Hz, which corresponds to the time tex=1.666ms, is used for the simulations in the next section.
Another important point is the maximum frequency that has to be considered to reproduce the steep rise of the pulse. In case of a lightning pulse with a rise time of a few microseconds a maximum
frequency in the range of a few MHz is sufficient.
Finally, it should be mentioned that this is a linear formulation, hence non-linear effects, which could result from matter interaction at very high field magnitudes, are not taken into account.
In this section the proposed formalism is illustrated by means of several examples. In all cases the pulse introduced in Sect. is used as excitation. In the case of a nearby lightning the channel
starts at (x,y,z)=(0,0,0) on a perfectly conducting ground and ends at (0, 0, l), where in our case the channel length l is chosen to be of length 3km. As depicted in Fig. , the current pulse starts
at the channel base an moves upwards with speed vf.
The first configuration to be considered is a single lightning channel without a neighboring structure. Then a transmission line structure, which is loaded by 50Ω resistors, is placed close to the
lightning channel. In a next step, this transmission line structure is located inside a cylindrical cavity. The geometrical details of the resulting setup are shown in Fig. . The cylinder is chosen
either as a closed one with conductivity σ=5kSm-1, which is a realistic equivalent conductivity for a carbon fiber composite, or a perfectly electrically conducting (PEC) cylinder with 12 apertures
on both sides, representing a fuselage of a passenger aircraft. As excitation a lightning channel close to the structure or a lightning channel directly attached to the structure is assumed. The
resulting four different cases are summarized in Fig. , where only short sections of the lightning channels are shown. In the case of a direct strike the impressed current flows from the top side of
the structure upwards along the lightning channel. To model the discharge along the fuselage and towards the ground a second wire connects the bottom side of the cylinder to the PEC ground. The
current on this second wire is not an impressed one, as the one on the long wire that models the lightning channel, it rather results from the MoM simulation.
Dimensions and positions of the transmission line structure and its surrounding cylinder, representing a simple model of an aircraft fuselage.
The considered simulation models where the cylindrical cavity is positioned at a height of 100m above perfectly conducting ground. Four cases are considered: closed finitely conducting cylinder 10m
distant from the lightning channel (a), closed finitely conducting cylinder subject to a direct lightning strike (b), PEC cylinder with apertures 10m distant from the lightning channel (c), and PEC
cylinder with apertures subject to a direct lightning strike (d).
Electric field of the channel at different distances from the channel at z=0, calculated by the numerical method presented in this paper and a semi-analytical formula taken from .
Magnetic field of the channel at different distances from the channel at z=0, calculated by the numerical method presented in this paper and a semi-analytical formula taken from .
Electric field of the channel at different distances from the channel at z=0, calculated for a longer time interval.
Electric field at the center of the transmission line without (a) and with (b) finitely conducting cylinder. The different time and field scales should be noted for comparison.
As a prerequisite, in this subsection the electric and magnetic fields of the lightning channel without a neighboring structure are investigated and compared to the semi-analytical formulas derived
in . With these formulas the fields at any position on the ground at z=0 can be calculated as a function of the horizontal distance to the channel.
The corresponding curves for the electric and magnetic fields are shown in Figs. and , respectively. The numerical results obtained from the MoM formalism are in excellent agreement with the
semi-analytical results, as exemplified by three observation points at distances 25, 50 and 100m. For these distances the effect of the finite length of the lightning channel in the used model is
In Fig. the same curves for the electric field as in Fig. are show up to a time of t=1ms. It can be observed that even for this longer time interval the results of the MoM formalism are still in
very good agreement with the results of the semi-analytical method.
The electric field at the center of the transmission line structure, which corresponds to the point (25, 0, 100)m, with and without the finitely conducting cylinder as shield is illustrated in
Fig. . In both cases the y component of the field is zero. As expected, the amplitudes of the electric field in the presence of the shield are much smaller if compared to the situation without
Magnetic field at the center of the transmission line, 10m distant from the lightning channel, with and without conducting cylinder.
Frequency spectrum of the system response for a lightning channel close to the structure without cylinder, with finitely conducting cylinder, and with PEC cylinder with apertures.
Voltage at the termination resistor of the transmission line with and without conducting cylinder (a) and with PEC cylinder with apertures (b) for the structures being close to the lightning channel.
The magnetic fields for these two cases are plotted in Fig. . The rise time of the magnetic field inside the finitely conducting cylinder is much larger if compared to the rise time of the magnetic
field without the cylinder. This is due to the low pass behavior of the shield with respect to the magnetic field, i.e., the high frequency components of the field are attenuated and therefore the
rise time is increased.
The related transfer function G(ω), as defined by the ratio between the voltage U(ω) at the termination resistance and an impressed current of constant amplitude of 1A per frequency, is shown in
Fig. . This transfer function includes the following effects: the creation of the fields of the lightning channel, the coupling into the cylindrical cavity by diffusion or aperture coupling, field
distribution inside the cavity, and coupling into the transmission line structure. At low frequencies the curves for the case of the transmission line alone and the transmission line inside the
finitely conducting cylinder are on top of each other. This shows that in this frequency region the presence of the conducting cylinder is hardly relevant. The magnitude of the transfer function for
the case of the PEC cylinder with apertures is lower compared to the other two cases because the magnetic field may only couple through the apertures into the fuselage. This effect is small for the
considered wavelengths which are much larger than the aperture dimensions.
Frequency spectrum of the system response for the cases of a closed conductive cylinder and a PEC cylinder with apertures if a direct lightning strike is applied.
Voltage at the termination resistor of the wire loop inside the conductive cylinder and the PEC cylinder with apertures in case of a direct lightning strike.
Finally, the time-domain voltage at the terminating resistor is shown in Fig. as a response to the current pulse according to Fig. . A delay time of 1µs can be observed, which corresponds to the
time the pulse needs to travel from the ground to the height of 100m where the structure is located. The maximum voltage for the cases of no cylinder, finitely conducting cylinder, and PEC cylinder
with apertures are 17kV, 800V, and 1.5V, respectively. The main contribution to the voltage is due to the magnetic field. Due to the low pass characteristic of the finitely conducting cylinder the
voltage has a decreased rise time, similar to the magnetic field itself.
In this subsection cases are considered where the lightning channel is directly attached to the structure, compare Fig. b and d, such that the lightning current flows directly on the surface of the
cylindrical structure.
In Fig. the spectrum of the transfer functions for both the finitely conducting cylinder and the PEC cylinder with apertures is shown. Two maximum values can be identified in the considered
frequency range. The first maximum occurs at a frequency of 372kHz. This frequency is too low for being a resonance frequency of the cylinder. Analysis shows that this frequency is related to the
wire that connects the cylinder to the ground. More precisely, the frequency turns out to be the antiresonance frequency of the wire , i.e., the imaginary part of the impedance of the wire as seen
from the current source is zero and the real part of the impedance exhibits a maximum. The second maximum of the transfer function occurs at the frequency 1.6MHz, where the length of the wire from
the cylinder to the ground equals one half of a wavelength.
The time domain response of the voltages are shown in Fig. . Oscillations are clearly visible, where the time period of the oscillation corresponds to the frequency where the first maximum of the
transfer function occurs. For the case of the finitely conducting cylinder the voltage is considerably higher if compared to the case of the PEC cylinder. Therefore, in this example, the diffusion
coupling is larger if compared to the aperture coupling, at least in the considered frequency range which is relevant for lightning analysis. | {"url":"https://ars.copernicus.org/articles/14/107/2016/ars-14-107-2016.xml","timestamp":"2024-11-08T04:36:51Z","content_type":"application/xml","content_length":"64869","record_id":"<urn:uuid:5fcf1407-1acb-47fc-bfda-833dfca4a922>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00561.warc.gz"} |
5. Find x2+y2 if x+y=−14 and xy=84
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Question Text 5. Find if and
Updated On Jul 12, 2023
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 96
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How To Draw The Infinity Sign
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runs infinitely. In this tutorial, we're going to learn how to quickly create an infinity symbol in adobe illustrator.
Drawings infinity symbol drawing Infinity symbol sketch — Stock
This symbol is also called a lemniscate, [1] after the lemniscate curves of a similar shape studied in algebraic geometry , [2] or lazy eight, in the terminology of livestock branding. Web the
infinity symbol.
Infinity Symbol Drawing at Explore collection of
Web 3d infinity symbol pattern. Web 165k views 8 years ago adobe illustrator tutorials. Web how to create an infinity symbol in powerpoint. Hi everyone it's rocky, and i just decided to post a quick.
How to create an infinity symbol in Illustrator (Tutorial) YouTube
Web how to create an infinity symbol in powerpoint. Web to type the infinity symbol in linux, just use the keyboard shortcut ctrl + shift + u, 221e. Draw a curved line that curves towards.
Infinity Symbol Drawing at Explore collection of
Type 2 2 1 e. The infinity symbol is a mathematical symbol that represents an infinitely large number. This symbol is also called a lemniscate, [1] after the lemniscate curves of a similar shape
How To Draw Infinity Symbol, Step by Step, Drawing Guide, by Dawn
Satisfying spiral drawing line illusion from my series daily art therapy ( #268) created with a sharpie on 110lb cardstock. Use the pen tool (p) to draw a base layer of the infinity sign and.
Satisfying spiral drawing line illusion from my series daily art therapy ( #268) created with a sharpie on 110lb cardstock. Draw a curved line that curves towards the right and meets the straight
line at.
How To Draw 4 Infinity Tattoo Design Style Amazing YouTube
The infinity symbol (∞) represents a line that never ends. Use the pen tool (p) to draw a base layer of the infinity sign and name it simply as ‘base’. How to type the infinity.
How to draw an Infinity sign Art, Drawing ShowMe
Web 165k views 8 years ago adobe illustrator tutorials. Web how to create an infinity symbol in powerpoint. This tutorial shows the sketching and drawing steps from start to finish. Activate the
layer by left.
How to Draw an Infinity Sign Easy Drawing Tutorial For Kids
The image below was drawn by me, but when you zoom in. Fill in the anchor, and draw tiny curved rectangles inside the infinity sign. Add a simple anchor on the side of the infinity.
Draw a simple infinity symbol (Illustrator Tutorial) — abcinformatic
Learn how to draw infinity simply by following the steps outlined in our video lessons. Press the ctrl, shift and u keys at the same time. An infinity loop or endless loop could be used.
How To Draw The Infinity Sign Web draw the initial shape of the infinity sign and cut out 2 hollow shapes in the middle. Web i'd like to know how to draw these smooth lines, especially the tapering
of the line at the bottom of the center if you draw it with a pen, it will be abrupt at the nodes. The common sign for infinity, ∞, was first time used by wallis in the mid 1650s. Draw a curved line
that curves towards the right and meets the straight line at a point. Type 2 2 1 e.
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A 4 dimensional random walk!
You start at the point $(x,y,z) = (1,1,1)$ in a Cartesian coordinate system, at time $t=0$ . If we consider time to be a fourth dimension, we start at coordinate $(x,y,z,t) = (1,1,1,0)$
Every turn you take you move randomly in one of eight (yes 8!) possible directions, six of them are the six directions available to you on the Cartesian coordinate system ( $\pm \widehat{x}$ , $\pm \
widehat{y}$ , $\pm \widehat{z}$ ), and the remaining two are to move backward or forward in time by one second.
What is the expected value for the number of moves (either in space or time or both) that it will take for you to land on a point $(3a, 3b, 3c, 3d$ ) where $a$ , $b$ , $c$ and $d$ are integers . $
(a,b,c)$ is your $(x,y,z)$ location, and $d$ is how many seconds from when you started. Remember, time only moves (forward or backward) when you travel in time, otherwise it remains still.
Clarification: On any move you can go one unit in either direction on any of the three axes, or you can go backward or forward in time by one second each with probability $1/8$ .
Note: If you end up moving in space, the move is instantaneous (i.e. time freezes), but if you move in time, you remain at a fixed location in space for that move.
Image credit: www.bbcamerica.com
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DMOPC '21 Contest 2 P4 - Water Mechanics
Submit solution
Points: 15 (partial)
Time limit: 2.0s
Memory limit: 256M
Keenan recently bought a new game, and he wants to show it to you!
Keenan's game consists of cells in a row, of varying height. Keenan also has many water buckets, and he wants to fill every cell with water.
However, as Keenan explains, water mechanics in this game are a bit weird. There is an integer chosen by the game, where water can flow at most units on equal-height cells. Specifically, if Keenan
pours his bucket into a cell , the water can flow in both directions, filling all cells until cells and with water before it loses momentum. However, if the water descends from a higher elevation to
a lower one, this limit is reset to , and the water can continue flowing as if it had just been poured into the lower cell. Water can never flow from a lower elevation to a higher one.
Keenan wants to fill all his cells with water, but he doesn't want to use that many buckets. Can you tell him the minimum number of cells he has to pour his bucket into so that all cells are filled
with water?
Subtask 1 [30%]
Subtask 2 [70%]
No additional constraints.
Input Specification
The first line will contain two space-separated integers, and .
The next line will contain space-separated integers, , the heights of the cells, in order.
Output Specification
Output the minimum number of cells Keenan has to pour into in order to fill the entire row of cells with water.
Sample Input
Sample Output
If we pour into the second and sixth cells, we can fill all the cells in moves, which we can prove to be minimal.
Note that if we pour water into the third cell, it would not reach the first cell.
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Are all reliability efforts worth the investment?
Reliability engineering involves selecting designs, procedures, plans and methods based on time and economic restrictions. The need for engineering economy comes from the fact that engineers and
managers do not work in an economic vacuum. Their decisions are influenced by the allocation of scarce resources used in production and distribution.
Managerial decision making is usually discharged within a framework of:
• Defining the alternatives clearly
• Identifying the aspects common to all alternatives
• Establishing appropriate viewpoints and decision criteria
• Considering the consequences of actions taken
Given the frequent conflicts among requirements and the potential impact on costs, project schedules and plant performance, the reliability alternatives and their consequences should be studied in
detail and be well understood.
Economics of reliability improvement
Reliability improvement is an investment. Like anything in which we invest money and resources, we expect to receive benefits that are greater than our investment. The following financial overview
provides information on various analytical tools and the data financial experts need to provide assistance.
View more content on PlantServices.com
Making reliability investment tradeoffs requires considering the time value of money. Whether the organization is for-profit or not-for-profit, resources cost money. The three dimensions of payback
analysis are:
• Cash flow required to improve reliablity
• Period over which the cash flow occurs
• Cost of money expected during the period
Reliability analysis usually requires either a simple “yes/no” decision, or selecting one of several alternatives. The time value of invested money and the future returns generated by a reliability
improvement must be quantified clearly. The time value of money simply means that a dollar in your pocket today is worth more than one a year from now. Another consideration is that forecasting
potential outcomes is more accurate in the short term than it is in the long term.
Decision-making methods include:
• Payback
• Percent rate of return (PRR)
• Average return on investment (ROI)
• Cost/benefit ratio (CBR)
The corporate controller often determines the financial rules used in justifying capital projects. Companies have rules like “Return on investment must be at least 20 percent before we will consider
a project” or “Any proposal must pay back within 18 months.” Reliability evaluations should normally use the same set of rules for consistency and to help obtain management support. It is also
important to realize that the political or financial motives behind these rules may not be entirely logical for your decision level.
Payback simply determines the number of years required to recover the original investment. Thus, if you pay $50,000 for a test instrument that saves downtime and increases production by $25,000 a
year, the payback is:
$50,000/$25,000 = 2 years
This is easy to understand. Unfortunately, it disregards the fact that the $25,000 gained the second year needs to be adjusted for the time value of money. It also assumes a uniform payback stream.
Finally, it ignores any returns after the two years. Why two years instead of some other number? There may be no good reason except “The controller says so.” If simple payback is negative, then you
probably do not want to make the investment.
Percent rate of return
Rate of return, as a percentage, is the reciprocal of the payback period. In our case above:
$25,000/$50,000 =.50 = 50 percent rate of return
This is often called the naive rate of return because it ignores the cost of money and a finite payback period.
Return on investment
Return on investment is a step better since it considers depreciation, salvage value and all benefit periods. If we acquire a test instrument for $80,000 that has a five-year life and a $5,000
salvage value, then the cost calculation, excluding depreciation, is:
($80,000-$5,000)/5 years = $15,000 per year
If the economic benefit is $135,000 for the same period, the average increment is:
($135,000-$75,000) = $60,000/5 years =$12,000 per year
The average annual ROI is:
$75,000/$135,000 = .55 = 55 percent
Ask your accountant how they handle depreciation, since it can make a major difference in the calculation.
Cost-benefit ratio
CBR takes the present value (initial project cost plus net present value) divided by the initial project cost. For example, if the project will cost $250,000 and the net present value (NPV) is
$350,000, then:
($250,000 + $350,000)/$250,000 = 2.4
It may appear that the CBR is merely a mirror image of the NPV. However, CBR considers the size of the financial investment required. For example, two competing projects could have the same NPV. But,
if one required $1 million and the other $250,000, the absolute amount might influence the choice. Compare the example above with this $1 million project:
($1,000,000 + 350,000)/$1,000,000 = 1.35
There should be little question that you would take the $250,000 project (a 2.4 return) instead of the $1 million one (1.35 return). This calculation requires that we make a management judgment on
the inflation/interest rate for the payback period and what the payback pattern will be. For example, if we spend $5,000 today to modify a machine to reduce breakdowns, the payback will come from
future improved production revenues and reduced maintenance costs.
Measuring reliability
Reliability is defined in many ways, but the most widely accepted version states that it is the ability or capacity of a plant’s production system to perform the specified function in the designated
environment for a minimum length of time or number of cycles.
The “life” of an individual system or one of its components cannot be determined except by operating it for the desired time or until it fails. Obviously, you cannot wear out the system to prove that
it will meet specifications; therefore, just as in quality assurance, you must rely on data generated by testing. As a result, reliability is measured as the probability that the system and its
components will function normally for the required time interval.
Thus, reliability can be calculated by using one or more of the following:
• Mean-time-to-failure (MTTF)
• Mean-time-between-failures (MTBF)
• Mean-time-between-maintenance action (MTBMA)
• Mean-time-before-repair (MTBR)
As a reliability measure, the MTTF of critical production systems must be greater than the planned production interval defined in the annual business plan. If it is less, the probability of
production interruptions and a resultant loss of capacity must be considered. Typically, MTTF is determined by using failure modes and effects analysis (FMEA), which calculates the failure
probability of each component that makes up a production system. In many cases, it’s provided by the system’s vendor, but you can calculate it at any point during the equipment’s life cycle.
MTBF is the historical average of actual failure intervals. Whereas MTTF is the probability of failure intervals for a specific class of machine or system, MTBF is based on the actual history in a
particular plant or application.
MTBMA defines the preventive maintenance level required to meet the MTBF or MTTF criteria. In some cases, statistical data is available for specific machinery or production systems, but more often,
it’s a measurement of actual, in-plant maintenance activities.
MTBR criteria are similar to MTBMA, but is limited to corrective maintenance, such as rebuilds, replacement of major-wear parts, and other non-preventive activities. In most cases, this is a
measurement of the actual, in-plant history, but many vendors provide statistical MTBR data for their equipment.
Practical measurement of equipment reliability is dependent on accurate record keeping. While these criteria can be calculated as theoretical values, the only true measurement is actual history.
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Price Time Series: Fractionally Differentiated Series with Prophet
In our previous post in this series we noticed that Facebook´s Prophet time series statistics tool can potentially generate decent predictions for carefully researched stocks under certain
circumstances. Before launching ourselves in a wide array search for highly seasonal, quasi-stationary financial instruments we can force a stationary series out of the price of any instrument, in
this case, for the sake of simplicity, SPY.
We will use MLFInlab fractional differentiation module (as we did here) to obtain the "best" fractional series and apply Prophet fitting and prediction to it. We will keep it to a 5 day prediction
for the time being, ideally, and after forcing the machine to do a lot of work, the best prediction windows for each season could be found, with the risk, of course, of local overfitting to past
No changes to the initial part of our process with respect to our previous post on FB Prophet:
import matplotlib.dates as mdates
import matplotlib.pyplot as plt
import numpy as np
self = QuantBook()
spy = self.AddEquity("SPY")
history = self.History(self.Securities.Keys, 365*5, Resolution.Daily)
A little improvement on the time series generation function in Prophet format, so that the output fits right away into the model:
def create_time_series(df, column='close', freq='d'):
df = df.reset_index()
time_series = df[[column]].set_index(df['time']).asfreq(freq).ffill()
time_series.columns = ['ds', 'y']
return time_series
With this, just create a time series for the current symbol, still SPY:
time_series = create_time_series(history)
With the time series created, the best fractional difference can be found, that is, the lowest value of d that gets the series through the augmented Dickey–Fuller test:
from mlfinlab import fracdiff
from statsmodels.tsa.stattools import adfuller
fracdiff_values = np.linspace(0,1,11)
symbol_list = ['SPY']
fracdiff_series_dictionary = {}
fracdiff_series_adf_tests = {}
for frac in fracdiff_values:
for symbol in symbol_list:
frac = round(frac,1)
dic_index = (symbol, frac)
fracdiff_series_dictionary[dic_index] = fracdiff.frac_diff(time_series[['y']], frac, thresh=0.1)
fracdiff_series_adf_tests[dic_index] = adfuller(fracdiff_series_dictionary[dic_index])
The dictionary series are there to be used, if needed, in the future. We only need the "best" differentiated time series:
stationary_fracdiffs = [key for key in fracdiff_series_adf_tests.keys() if fracdiff_series_adf_tests[key][1] < 0.05]
first_stationary_fracdiff = {}
diff_series = {}
for symbol, d in stationary_fracdiffs:
ds = [kvp for kvp in stationary_fracdiffs if kvp[0] == symbol]
first_stationary_fracdiff[symbol] = sorted(ds, key=lambda x: x[1], reverse=False)[0][1]
diff_series[symbol] = fracdiff_series_dictionary[(symbol, first_stationary_fracdiff[symbol])]
diff_series[symbol].columns = ['diff_close_' + str(symbol)]
for symbol in symbol_list:
print("The lowest stationary fractionally differentiated series for ", symbol, " is for d=",
first_stationary_fracdiff[symbol], ".")
The message at the end of the block tells us that in this case the best fractional differential series occurs for d=0.3. In the backtest algorithm we will let this be calculated every prediction
cycle, so it will be worth it to remove the dictionary storage and keep only the first series that fulfills the ADF test criterion. The resulting differentiated series has to be reformatted into a
the Prophet 'y'-'ds' format again:
diff_time_series = diff_series['SPY'].join(time_series['ds'])
diff_time_series.columns = ['y', 'ds']
The time series "indexes" and the differential values are aligned in this case. The weight thresholding function used to obtain the fractional differential series "uses up" some of the information
that does not make it into the final series. The points that are lost are shown in the series table:
In the date table above, as we have requested 1825 points of data, the first value for 'ds' should be 2013-08-15. A threshold of 0.1 suppresses the first 36 points of data. This is not a problem for
time series for which we have enough data, for shorter time series it may reduce the amount of data points critically to the point that no model can reliably be fitted with so little data.
The effect of changing the weight thresholds are interesting, shorter time series are obtained while extremely weighted points are dropped. In the future we will explore this effect, for the time
being the explanation is in Marcos Lopez de Prado´s Advances in Financial Machine Learning book (page 80 and on). In any case the shape, the familiar shape, of a memory-optimal stationary series is
We can now generate the Prophet model by feeding it the differentiated time series, no changes here, no options whatsoever in an attempt to check how it can work with this new series out of the box,
we also predict 5 days, no more thought given to it:
from fbprophet import Prophet
price_model = Prophet()
price_forecast = price_model.make_future_dataframe(periods=5, freq='d')
price_forecast = price_model.predict(price_forecast)
And the aspect of the fit:
_=price_model.plot(price_forecast, xlabel = 'Date', ylabel = 'Price', figsize=(12, 6))
The aspect is still good. The know past SPY behaviours are reflected in the time series and it is also stationary. Note the apparently strong mean reversion effect in the shorter term being
maintained for most of the series until COVID19 crisis volatility creates a completely wild cloud of points out of the maximum and minimum bands predicted by the Prophet fit. This may be worth
further investigation, if finally Prophet does not produce and accurate price movement prediction always, it may be able to do so reliably when trading happens at extremes out of the minimum and
maximum prediction bands. In any we can use this model now to produce a 5 day SPY prediction using a dynamic best fractional differential selection. The results are:
A little bit worse returns than for the raw price data model, returns are in any case more consistent now, with a 54% correct direction prediction. The model is not very intelligent and lacks any
risk management capabilities, still the results can be considered not bad as the machine is not ruining itself. Also not good, as a SPY buy and hold strategy netted 86% return during the same period.
Is a 5-day prediction too long of a prediction? Do we have a too large of a horizon in this model? Too short? Note that we are not calculating the fractional integral to obtain a price level again,
we are just comparing the differential direction to obtain a price direction. This is true for almost all cases, the fractional differential change will have the same direction as the price change,
this breaks down for small differential price changes below the threshold that generate erroneous signals. With additional work the model could be modified to not act, to not take the shot, if the
predicted change for the fractional differential value is below a certain threshold.
We can check the predictive capacity for 2 alternate prediction times, 3 days ahead, and 15 days ahead to learn something about the shape of a very short term prediction and a medium term prediction,
first for the short term, with a 3 ahead directional prediction:
Both generate worse predictions, probably these periods lie in between the weekly and the monthly "seasonal" effects Prophet is trying to fit to with little success. The directionality is also not
good in these models, with 46%-48% correct directionality. There seems to be no low hanging fruit from Prophet and fractional time series.
We can still try a final easy test, let's add a few more tickets to our universe, instead of focusing on predicting SPY, and accepting the full risk, we will pick the top 10 stocks by traded volume
every prediction cycle, for a 5 day window, and see if it provides a better risk control. The full fractional differential and the full Prophet fit will happen every prediction cycle, so the model
takes quite a while to complete the test: more than 24 hours as it differentiates and trains Prophet models every 5 test days:
At 51% correct direction predictions and at a 40% loss these wild, multiple Prophet models do not appear to work well together out of the box. Another approach is necessary, even if Prophet gives a
prediction edge, the model is too naive in terms of the portfolio it tries to build. The market risk is apparently all there to trap us in our seasonal preferences.
As a final wild guess; what if we fit a shorter training model into cryptocurrencies? Let's see the same Prophet out-of-the-box model fitted to BTC USD history. We will start in 2018 has there is no
reliable data to fit the model before that:
It seems, by looking at all these simple models, that Facebook Prophet could be used to support price direction predictions, it will require a deeper dive into the calculation code and the
algorithmic selection of instruments that exhibit certain characteristics. Out of the box application and popular instruments together do not offer good results. We have to keep on searching.
Remember that information in ostirion.net does not constitute financial advice, we do not hold positions in any of the companies or assets that we mention in our posts at the time of posting. If you
are in need of algorithmic model development, deployment, verification or validation do not hesitate and contact us. We will be also glad to help you with your predictive machine learning or
artificial intelligence challenges. | {"url":"https://www.ostirion.net/post/price-time-series-fractionally-differentiated-series-with-prophet","timestamp":"2024-11-11T13:51:14Z","content_type":"text/html","content_length":"1050581","record_id":"<urn:uuid:2368bdea-8c76-42e6-87e8-2aca34f9a0e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00100.warc.gz"} |
Building and Implementing Binary Trees in Python - Adventures in Machine Learning
Introduction to Binary Trees
Binary trees are one of the fundamental concepts of computer science. They are used extensively in various algorithms like sorting, searching, and indexing.
Understanding binary trees is essential for any programmer, and this article will provide an in-depth understanding of binary trees.
Definition of a Binary Tree
A binary tree is a tree data structure where each node contains a maximum of two child nodes, known as left and right child nodes or the left-subtree and right-subtree. The parent object of these two
child nodes is known as the root node of the tree.
The child nodes only have one parent node, and the root node does not have any parent object. In summary, a binary tree is made up of nodes connected by edges, where each node has two child nodes,
except for leaf nodes that do not have child nodes.
Implementation of a Binary Tree in Python
In Python, we can implement a binary tree using a class. The class contains a constructor with references to the left and right child nodes, and it also stores the data value of the node.
We can traverse the tree using different methods such as preorder, inorder, and postorder. These traversal methods determine the order in which you visit each node.
Basic Terminologies in Binary Trees
Root Node
The root node is the topmost node in a binary tree. It is the first node in the tree and does not have any parent nodes.
All other nodes in the binary tree are children of the root node.
Parent Node
The parent node is a node that has at least one child node. Each node in the binary tree has only one parent node except the root node that does not have any parent object.
The leftChild and rightChild properties of a node point to its two child nodes.
Child Node
Child nodes of a parent node are the nodes that are connected to the parent node by an edge. A parent node can have zero to two child nodes.
An edge is the link between two nodes, connecting the parent node to its child node. The edge direction is from parent to child.
Internal Nodes
A node that has at least one child node is an internal node. These nodes are used to create a binary tree structure.
Leaf Node or External Nodes
A leaf node, also known as an external node, is a node that does not have any child nodes. These nodes are found at the end of the tree and do not have any further sub-trees.
Binary trees are an instrumental data structure for any programmer that deals with algorithms, especially for sorting, searching, and indexing. In summary, each node in a binary tree can have a
maximum of two child nodes, known as left and right sub-trees.
It is easy to implement in Python, and the tree traversal methods determine the order of visiting each node in the tree. The properly described terminologies associated with binary trees are also
crucial in building the correct tree structure and operations.
Implementing a Binary Tree in Python
Binary trees are a widely used data structure in computer science and programming. They are primarily used for efficient searching and sorting algorithms.
In this article, we will discuss how to create and implement a binary tree in Python, including creating node objects, adding children to nodes, and printing data in nodes and children.
Creating Node Objects
The first step in building a binary tree is to create a node object, which represents a single item in the tree. Each node contains a data field that stores the object or value being stored.
In Python, we can create BinaryTreeNode objects for each node in the tree. The BinaryTreeNode class would have a constructor function that takes in a parameter for the data being stored in the node,
and the left and right child references.
Here is an example of creating a BinaryTreeNode class in Python:
class BinaryTreeNode:
def __init__(self, data):
self.data = data
self.leftChild = None
self.rightChild = None
In the above code, we have created a class called BinaryTreeNode that contains a constructor function that initializes the data field to the value passed in as the argument. The leftChild and
rightChild references are also initialized to None in this example.
Adding Children to Nodes
Once we have created a node object, we can add child nodes to it. In a binary tree, each node can have a maximum of two child nodes – a left child and a right child.
We can add these child nodes by updating the leftChild and rightChild references of a node object using the setLeftChild() and setRightChild() methods. Here is an example of how to create a binary
tree by adding child nodes to a root node:
root = BinaryTreeNode(1)
root.leftChild = BinaryTreeNode(2)
root.rightChild = BinaryTreeNode(3)
In the above example, we first create a root node with a value of 1.
Then we add child nodes to the root by assigning new BinaryTreeNode objects to the leftChild and rightChild properties of the root node.
Printing Data in Nodes and Children
After we have created a binary tree, we may want to print the data stored in each node or the children of any node in the tree. We can do this using the print() function.
Here is an example of how to print data in a node and its children:
In the above example, we print the data stored in the root node, as well as the data stored in its left and right child nodes.
In summary, implementing a binary tree in Python involves creating node objects for each item in the tree, adding child nodes to each node object using the leftChild and rightChild references, and
printing the data stored in nodes and their children. By understanding these topics, you will be able to create and work with binary trees in Python efficiently.
In conclusion, implementing binary trees in Python is important for programmers as they are an essential data structure used in various algorithms. The process involves creating node objects with a
constructor function, adding child nodes, and printing data in nodes and children.
The binary tree structure consists of a root node, parent nodes, child nodes, edges, internal nodes, and leaf nodes. By understanding these concepts, programmers can efficiently create and use binary
trees in their projects.
The importance of binary trees cannot be overemphasized, as they enable programmers to easily sort, search, and index data. | {"url":"https://www.adventuresinmachinelearning.com/building-and-implementing-binary-trees-in-python/","timestamp":"2024-11-06T23:57:31Z","content_type":"text/html","content_length":"72436","record_id":"<urn:uuid:8116167e-5505-4a45-8f0a-73a21bc39073>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00882.warc.gz"} |
SweetFX Settings Database
Ok so i want to use SweetFX on Battlefield 3 but pbsvc.exe (PunkBuster) is in the same folder and if i restart my PC the SweetFX plugin is on becuase pbsvc.exe is autostarting and if i shut it down i
can't play multiplayer. Is there anyway i can make SweetFX only inject to BF3.exe instead of both? | {"url":"https://sfx.k.thelazy.net/forum/sweetfx/216/","timestamp":"2024-11-11T01:57:07Z","content_type":"text/html","content_length":"5735","record_id":"<urn:uuid:01c63541-d87a-4c4b-9976-a5158debce71>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00234.warc.gz"} |
Need a Bit Info
October 25, 2005 - 18:33
Need a Bit Info
How is AP calc?
I have not even taken pre-cal but my teacher wants me to take AP Cal next year so i am gonna take it but i was hoping for a heads up. What excatly you learn about and is there any sites you know that
i can use?
for the people taking cal already, what textbooks do you use and whats your opinion for that class?
Like my Signature Picture? Then Click On it!!
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[B]E-mail Addresses[/
October 27, 2005 - 14:09
(Reply to #1) #2
Have you taken Trigonometry? Trig and Precal are the same class in different orders.
If you haven't had either one, go to one of the teachers and get a book as soon as you can. You MUST know your trigonometric identities. Without these, you will soon be lost and confused. It is
helpful to see how they are derived and how to convert one trig function into another. The identities should be listed in an appendix in your calc book also. The unit circle is also very important.
You can always take time to draw it, but it is much easier and faster if you memorize it.
I'm not sure who wrote the calc book that I used, but it was blue and red and white on the cover. And all the definitions are in yellow boxes. I found the class challenging most of the time. That is
the sign of a good class. If you learn how to differentiate and integrate very well, you've got most of what you need to know for all of AP Calculus.
If you're a math nerd, beware. This class will make you see functions and applications everywhere. Especially if you are taking Physics at the same time. I'm don't know any specific sites, but I know
that many of the solutions to previous AP problems can be found on the internet by googling the problem in quotes.
I wish you luck in your class!
[=RoyalBlue][=Comic Sans MS]
"I refuse to prove that I exist," says God, "for proof denies faith, and without faith I am nothing."
"But," say Man, "the Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It
October 29, 2005 - 15:31
(Reply to #2) #3
Well, personally, I hate calculus. It's a bit deceiving, I think. I've always been really really good at math (96-99 averages per marking pd) But now, my grades in calc have ranged from 0 (yes, zero
on a quiz that I did the work for) to a 100. The concept of caclulus isn't so hard, but the algebra that goes along with it is often very long and quite complicated. It's hard to tell how far you
need to simplify,especially what the AP graders are going to be looking for.
On the plus side though, I feel good taking a class that challenges me. I've also never been much good at sciences, but calculus has helped me in physics.
Calc is very time consuming. I spend probably 30 mins before school doing calc, 40 mins in class, an hour during lunch and studyhall and an hour after school, plus anhour or two at home.
My textbook is Calculus of a Single Variable... blue in color.
Good luck in calc next year. I'm rather pessimistic about the whole deal, but it will probably prove to be a worthwhile experience. My one piece of advice... don't be afraid to ask questions! Go in
for help if you need it, otherwise... you'll probably get behind...
October 30, 2005 - 01:16
(Reply to #3) #4
Stick with it Cocunutcremepie. I know what you mean. You must have one of the good teachers. In my experience, the AP Graders want the simplest, most exact answers from you, just like the physics
ones. Unless of course, it's a free-response. Then you should show as much detail as possible...those partial credit points really help out.
Keep rising to the challenge and work your hardest. If you ever want another view or explanation, you can always ask me. I have learned that there are many ways to reason a problem.
Good Luck and keep asking questions!
[=RoyalBlue][=Comic Sans MS]
"I refuse to prove that I exist," says God, "for proof denies faith, and without faith I am nothing."
"But," say Man, "the Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It
November 12, 2005 - 02:23
(Reply to #4) #5
Hrm... Did you like the distinct, concrete ideas of algebra? To tell you the truth, though my precalc teacher was pretty bad, I think I needed it; precalc is the bridge between your algebra and your
calculus. Algebra and trigonometry is where you learn, believe it or not, all your "basic math"- how to find x, how to graph, how to find area and volume and theta and all of that good stuff.
Precalculus is essentially a good introduction to all the theories of calculus while keeping your feet firmly aground. Then... calculus explores what's after that- it's pure application, and overall
theoretical math. Calculus itself is based on the idea of instantaneous points and limits, which in themselves are theoretical, and a bit illogical. Related rates, for example, ask you how fast
something is filling or emptying, how fast the tip of your shadow changes- all at one specific point in time. Sounds weird, but it'll make sense when you get there.
I'm a junior taking calculus AB at school, using the Larson, Hostetler and Edwards book for Calculus, fifth edition. But to tell you the truth, it's really a college level math course. Calculus is so
theoretical that for a lot of people, it takes three weeks to get their heads around the concept of limits- during which time you will be learning your derivatives. Really, I honestly do not
recommend taking calculus without having taken precalculus- you'll be swimming alone in sharky waters. Another thing I don't recommend is studying things like the quotient rule and chain rule by
yourself- it takes a good teacher to explain it correctly. I don't know if this helps... If you have any further questions, AIM me at: MurasakiYuna. Good luck!
Arandis Elearean
I'm a liberal Independent. So flame me.
November 26, 2005 - 19:18
(Reply to #5) #6
[dont listen to me- i dont know what I'm talking about and have no authority on this site]
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[B]E-mail Addresses[/
December 4, 2005 - 20:57
(Reply to #6) #7
Arandis is right. You absolutely need the algebra and trig basis of precalculus. I have a free period in which I sit in the back of the calculus class and do homework (i've already taken the class),
and from what i've noticed, those who struggle the most are the ones who did not have good teachers for precalculus, all the kids who came up from normal class compared to the honors class. there is
a huge difference. luckily, usually the first chapter of most calculus books will review all of the precalculus needed for the calculus course. i'd say obtain a copy of your calculus text and also a
good precalculus text ASAP and go through the first chapter of the calculus book, making sure you understand what they're telling you, and supplementing yourself if need be with the precalculus book.
LEARN EVERYTHING YOU CAN ABOUT TRIG! it is invaluable for doing well in calculus.
December 9, 2005 - 18:42
(Reply to #7) #8
I personally LOVED Calculus... of course, I also love math... I think that if you can enter Calculus with even a BASIC understanding of algebra and trig from Precalc or even Alg II & Trig, as long as
you have an open mind, STUDY!!!, and stay on top of things you will be fine. Even if you don't learn some algebra or trig skill before you get to Calc, you will pick it up in no time. I also would
stress that if you have a question, ASK IT!!! The moment you get behind in understand the theoretical concepts or applications to Calculus, you WILL BE BEHIND FOR A WHILE... if your teacher can't
explain it, ask someone who has a high A in the class.... (They won't mind...) From what I have seen in my past Calculus classes, I would say that the people who do well in Calc are those who PAY
ATTENTION in class, PARTICIPATE in class, do their HOMEWORK, and keep an open mind.
Best of luck, | {"url":"https://course-notes.org/comment/1015620","timestamp":"2024-11-02T21:34:50Z","content_type":"text/html","content_length":"73260","record_id":"<urn:uuid:df076973-17ca-4c40-ae35-e45ff6f90ced>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00143.warc.gz"} |
propositions p and q
The reason that sometimes form can break down is that if we look carefully some of our Ps and Qs are not, formally, propositions at all; this is often the case with arguments people use about God
a proposition as you know is formally 'a statement that can clearly be classified as true or false'. e.g. 3=1+6 is a propostion since it is clearly false whereas 'God exists' is NOT as it cannot
be clearly identified as T or F without further info, logically speaking.
i think the second argument is justified because it is equivalent in form to
~P v Q by definition of P---------> Q.
in other words
P--> Q
I - P is invalid
is something i can accept.
in sentential logic, validity concerns itself with form not content. also we are not dealing with bi-conditionals but just conditionals alone. so the following two forms is what the discussion is
p --> q,
I- q is valid.
p --> q
I- p
is inavlid.
this is the issue. so what do you say about where antecedent is justified from consequent as above in the second formulation?
thus we see that logically such arguments hold no water....
if Ala Hazrat has said the Earth is stationary then the Earth must be stationary.
P = Ala Hazrat said the Earth is stationary.
Q = The Earth is stationary
We need (see above) to find the truth value of not P or Q [~P or Q] to analyse this so.
P is true (T), therefore not P is false and Q is false therefore we have a False and a False which is False. Therefore this argument is invalid. (A disjunction i.e. 2 propostions linked by "or"
is true if either P or Q is true but if both are false it is false).
ah...formal propositional logic...i teach it in one of my maths courses:
P ---> Q = if P then Q which, mathematically, has the same truth value (T or F) as
the disjunction "Not P or Q" [in symbols ~P v Q] which means it is true (i.e. a valid argument) if either proposition ~P is true (ie. P is false) OR if Q is true...where P and Q are propositions.
It is much easier to use P and Q with symbols actually to analyse a validity of an argument...
The mathematical equivalent of the logical argument "if and only if" or "equivalent to" is mathematically equivalent to the truth value given by [~P v Q] ^ [~Q v P] i.e .it is only valid if the
proposition 'if P then Q' AND (^) its converse (if Q then P) are BOTH
Last edited: Nov 7, 2008
there are three kinds of evidences: demonstrative, dialectical and rhetorical.
the first is about 'forms' hence our logical forms to make a point. the first form is an example of formal validity and second form is an example of invalid form in logic. prove to me that the
second form is valid with reference to logic. please stick to rhetoricals. that was merely to test whether we can have a reductionist discussion but hazrat proved otherwise and instead would like
to do lenghthy khitaabs.
abu Hasan Administrator
from another thread:
thee first statements is reformatted for clarity thus:
1. if p, then q.
2. p.
3. therefore q.
1. if p, then q.
2. q.
3. therefore not-p
illustrates what? if we have to prove q is not-p, why show any dependency at all? we common folk cannot easily grasp symbols; so let us use words.
1. if i have money (p) then i am rich (q)
2. i am rich (q)
3. therefore i do not have money (not-p).
Last edited: Nov 6, 2008 | {"url":"https://sunniport.com/index.php?threads/propositions-p-and-q.5775/","timestamp":"2024-11-09T12:11:07Z","content_type":"text/html","content_length":"47823","record_id":"<urn:uuid:7b4c9f6e-68bf-4196-b776-8cad313eeb0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00312.warc.gz"} |
Which Kind of (ETF) Momentum Is Best?
When implemented via exchange-traded funds (ETF), does an equity sector momentum strategy beat an equity style momentum strategy? How do these approaches compare to a geographic equity momentum
strategy? In his paper entitled “Optimal Momentum”, runner-up for the 2011 Wagner Award presented by the National Association of Active Investment Managers, Gary Antonacci uses ETFs to compare style,
sector and geographic momentum strategies. He uses a six-month ranking period to select the top two of six iShares value-growth-size ETFs, the top three of nine SPDR sector ETFs and the top two of
four iShares region/country ETFs each month, with a 0.2% per fund switching friction. In addition, he experiments with adding short-term and intermediate-term Treasury ETFs and then gold to the
geographic momentum ranking process. His benchmarks are the Russell 1000 ETF (IWB), the AQR Momentum Index (adjusted by debiting an estimated annual trading friction of 0.7%) and equally weighted
portfolios of the ETF groups (rebalanced monthly). Using eight years of monthly ETF prices (2003 through 2010) and 34 years of related monthly index levels, he concludes that:
• Average annual trading frictions vary from 0.39% to 0.53% across momentum strategies.
• The geographic momentum strategy generates the highest Sharpe ratio among the basic ETF momentum strategies and benchmarks. The style and sector momentum strategies underperform their
equal-weighted counterparts (see the first chart below).
• Adding short-term and intermediate-term Treasury ETFs to the geographic momentum strategy boosts Sharpe ratio from 0.64 to 1.12 and dramatically reduces maximum drawdown.
• Further adding gold to the geographic-Treasury ETF momentum strategy boost the Sharpe ratio to 1.31 (see the second chart below).
• A robustness test applied to related indexes with no trading frictions over the period 1977-2010, and subperiods 1977-1993 and 1994-2010, indicates that a geographic-Treasuries-gold momentum
strategy substantially outperforms a corresponding equal-weighted strategy in terms of Sharpe ratio.
The following chart, constructed from data in the paper, summarizes annual Sharpe ratios for the various ETF momentum and benchmark strategies as specified over the 2003-2010 sample period. Results
suggest that:
• The adjusted AQR Momentum Index does not outperform buying and holding IWB.
• Sector momentum beats style momentum.
• The style and sector momentum strategies offer little or no advantage relative to buying and holding IWB, and both underperform corresponding equal weighting strategies.
• The geographic momentum strategy performs best (but, not shown, experiences the largest maximum drawdown).
The next chart, taken from the paper, translates the Sharpe ratios for the six ETF momentum strategies specified above (with “Industry” meaning sector) into cumulative performance trajectories of
$100,000 initial investments over the 2003-2010 sample period. Results show that adding Treasury ETFs and gold to the geographic momentum strategy considerably dampens volatility.
In summary, evidence from a short recent sample suggests that a geographic equity momentum strategy beats equity style and sector momentum strategies, and that adding Treasury ETFs and gold to the
geographic momentum mix substantially boosts performance.
Cautions regarding conclusions include:
• As summarized above, the “industry” momentum strategy described in the paper is a sector momentum strategy. Industry segmentation of firms is typically much finer than sector segmentation.
• The basic sample period is very short for reliable inference, consisting of only 16 independent six-month ETF ranking intervals.
• Experimentation with a variety of strategies on the same/overlapping/correlated data introduces data snooping bias, thereby likely overstating the performance of the best strategy. This bias
grows with the number of alternatives considered and is especially pernicious for a short sample period.
• Results may include borrowed data snooping bias from selection of a six-month ranking interval. “Simple Sector ETF Momentum Strategy Robustness/Sensitivity Tests” finds that sector ETF momentum
strategy performance is inconsistent for different momentum ranking intervals, with a six-month interval perhaps lucky.
• The indexes used in the 34-year robustness test represent idealized (frictionless) market environments. The frictions associated with implementing indexes as portfolios vary considerably over
time and across indexes and may be very large over parts of the sample period, undermining confidence that findings for indexes translate to real trading.
• The statistical interpretations assume that asset return distributions are tame (such as the Gaussian or normal distribution). To the extent that actual distributions are wild, these
interpretations lose meaning.
Compare results with those presented in “Simple Sector ETF Momentum Strategy Performance” (and associated robustness/refinement testing) and “Doing Momentum with Style (ETFs)”. These analyses involve
somewhat longer sample periods and pick one sector/style winner each month. The former finds that adding a long-term simple moving average signal substantially enhances returns. The latter study
finds that style momentum outperforms sector momentum. | {"url":"https://www.cxoadvisory.com/momentum-investing/which-kind-of-etf-momentum-is-best/","timestamp":"2024-11-13T11:17:02Z","content_type":"application/xhtml+xml","content_length":"146092","record_id":"<urn:uuid:4e08a421-e95e-4c45-aebc-01469fe70ae1>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00339.warc.gz"} |
GMAT & MBA Admissions | gmat math
GMAT & MBA Admissions Blog
Tags: GMAT tutor, GMAT resources, GMAT Blog, MBA Admissions, gmat test prep, gmat study skills, improving your gmat score, gmat math, gmat tutorial, gmat free tutorial, gmat calculation, gmat
It is common for GMAT students looking for a 700+ score to have many questions about GMAT combinatorics problems. These are the GMAT questions that ask you to count up all the possible arrangements
of individuals and groups in a variety of situations: How many ways can 5 men and 5 women be ordered in a line? How many high fives occur in a group of 15 people on a basketball team? GMAT tutors
often find themselves spending an inordinate amount of time helping students improve their ability to answer these types of GMAT questions.
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Tags: GMAT quant, GMAT tutor, GMAT, MBA Admissions, gmat test prep, gmat math, MBA prep, MBA degree, online gmat, gmat tutorial
This post is the second in our series on using strategies to answer specific questions from the 2018 Official Guide. Here, one of our most experienced GMAT tutors, John Easter, analyzes a question
about prime numbers using problem solving skills.
Problem #44 of the 2018 Official Guide to the GMAT states that if n is a prime number greater than 3, what is the remainder when n^2 is divided by 12?
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Tags: GMAT quant, GMAT tutor, GMAT prep, GMAT Blog, MBA Admissions, online gmat tutor, gmat study skills, gmat math
in this series, one of our most experienced GMAT tutors, John Easter, applies useful strategies to answer questions from the 2018 Official Guide.
Problem #167 of the 2018 Official Guide to the GMAT states that four extra-large sandwiches of exactly the same size were ordered for m students where m>4. Three of the sandwiches were evenly divided
among the students. Since 4 students did not want any of the fourth sandwich, it was evenly divided among the remaining students. If Carol ate one piece from each of the four sandwiches, the amount
of sandwich that she ate would be what fraction of a whole extra-large sandwich?
Read More
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Verizon 1:46 AM 85% < Kinematic equations: numerical calcula A car is traveling to the right...
Answer #1
Speed 27m/s CLee LS 2-
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s. What was the acceleration of the car as it sped up? Answer using a coordinate system where rightward is positive. Round the answer to two significant digits.
• A car is traveling to the right with a speed of 29m/s when the rider...
A car is traveling to the right with a speed of 29m/s when the rider slams on the accelerator to pass another car. The car passes in 110m with constant acceleration and reaches a speed of 34m/
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A DeepLearning Framework for Dynamic Estimation of Origin-Destination Sequence (2024)
Zheli Xiong2,Defu Lian2*, Enhong Chen2, Gang Chen3 and Xiaomin Cheng23 {zlxiong}@mail.ustc.edu.cn*Defu Lian is the corresponding author.2School of Data Science
University of Science and Technology of China, Hefei, China
Email: {liandefu,cheneh,wh5606}@ustc.edu.cn3Yangtze River Delta Information Intelligence Innovation Research Institute, China
Email: cheng@ustc.win
OD matrix estimation is a critical problem in the transportation domain. The principle method uses the traffic sensor measured information such as traffic counts to estimate the traffic demand
represented by the OD matrix. The problem is divided into two categories: static OD matrix estimation and dynamic OD matrices sequence(OD sequence for short) estimation. The above two face the
underdetermination problem caused by abundant estimated parameters and insufficient constraint information. In addition, OD sequence estimation also faces the lag challenge: due to different traffic
conditions such as congestion, identical vehicle will appear on different road sections during the same observation period, resulting in identical OD demands correspond to different trips. To this
end, this paper proposes an integrated method, which uses deep learning methods to infer the structure of OD sequence and uses structural constraints to guide traditional numerical optimization. Our
experiments show that the neural network(NN) can effectively infer the structure of the OD sequence and provide practical constraints for numerical optimization to obtain better results. Moreover,
the experiments show that provided structural information contains not only constraints on the spatial structure of OD matrices but also provides constraints on the temporal structure of OD sequence,
which solve the effect of the lagging problem well.
Index Terms:
OD matrix estimation, deep learning, neural network
I Introduction
With the development of big traffic data, a large amount of traffic data has been widely used in traffic applications such as route planning, flow prediction and traffic light control. Traffic demand
describes the trips between the divided areas, commonly referred to as the OD (Origin-Destination) matrix. The OD matrix has significant value for various traffic tasks such as traffic flow prediction[
1], trajectory prediction[2] and location recommendation[3]. On the one hand, the change in traffic demand between regions will affect the flow of the road sections, leading to a change in the
optimal vehicle path. On the other hand, OD estimation will lead to more effective traffic regulation since the already known OD demants can provide rationality for management strategies. However,
traffic demand is the data that sensors cannot directly observe, and it must be obtained through other traffic data, such as the traffic flow, to estimate the matrix.
OD matrix estimation is mainly divided into two categories: static OD matrix estimation[4] and dynamic OD sequence estimation[5]. Static OD estimation uses integrated traffic counts to estimate the
total traffic demand in the period. In comparison, dynamic OD sequence estimation uses time-varying traffic counts in a sequence of intervals to estimate the corresponding OD matrix sequence.The
bi-level framework is commonly used for OD estimation[6], which is divided into upper and lower levels. The upper level adjusts the OD matrices by minimizing the numerical gap between real and
estimated traffic counts by solving a least squares problem. The optimizing method is mainly gradient-based[7] and optimizes the OD matrices by the steepest descent method. The lower level assigns
traffic demand to road sections by analysis[8] or simulation[4] method. The upper and lower layers updates iteratively and converge to an optimum point.
Due to the limited number of road sections and abundant parameters to be estimated in the OD matrices, the optimization problem is heavily underdetermined[9]. Some researchers have alleviated this
problem by adding other observable information, such as travel speed[10], cellular probe[11], and bluetooth probe[4]to the estimator. In addition to the above challenge, dynamic OD sequence
estimation faces another lag challenge: due to different traffic conditions such as congestion, identical vehicle will appear on different road sections during the same observation period, resulting
in identical OD demands correspond to different trips[12], so current OD matrix will refer to different time-varying traffic counts according to different traffic conditions. Furthermore, traffic
conditions are caused by OD marices before and after the current OD matrix, which causes a temporal relationship between these OD matrices. To this end, some studies propose fixed maximum lags[13],
which assume that vehicles can complete their trip within a fixed maximum interval amount to eliminate the influence of observation intervals amount. And others like [12] propose using the temporal
overlap in consecutive estimation intervals to alleviate the influence between OD matrices, as shown in Fig. 1.
To address these challenges above, we use deep learning to fit the mapping relationship between the time-varying traffic counts and the structure of the OD sequence to learn the impact of lagged
traffic counts on the OD structure and the relationship between OD sequences.
Consider that numerical space makes the learning space too large for deep learning. However, distribution can constrain the learning space’s size and reflect the structural information. Furthermore,
simply using NN to infer numerical matrices will lose important information about assignments presented in the bi-level framework. Therefore, we propose a method that integrates deep learning and
numerical optimization, which merges the inferred structural information into the upper-level estimator of the bi-level framework. Providing the spatial and temporal constraints for optimization can
effectively avoid to fall into local optimum in advance and help to optimize to a better result.
In this paper, we further deliver the following contributions.
• •
We proposed a deep learning model to infer OD sequence structure which extracts the spatio-temporal constraints from time-varying traffic counts effectively.
• •
We novelly integrates deep learning method and bi-level framework to solve OD sequences estimation.
• •
Through our experiments, it is verified that the structural knowledge inferred by deep learning can provide great help for numerical optimization.
II Related Works
OD matrix estimation can be mainly divided into static OD matrix estimation and dynamic OD matrices estimation.
II-A static OD matrix estimation
For solving the static OD matrix estimation, gravity mode adopts a ”gravitational behavior” for trip demand and builds a linear or nonlinear regression model[14]. The maximum likelihood technique[15]
estimates the OD matrix over the sampled matrix on the presumption that OD pairs follow independent Poisson distributions. The entropy maximizing/information minimization method[16] seeks to select
an estimated OD matrix that adds as little information as possible from traffic counts in order to match the underdetermined problem. The Bayesian method[17] additionally resolves OD matrices by
maximizing the posterior probability, which utilizes a mix of the prior OD matrix and the observations. The maximizing/information minimization approach is a particular instance of Bayesian method
when the prior is given only a minimal amount of confidence. A method that explicitly considers both observed flow biases and the target OD matrix is built using generalized least squares (GLS)[18].
All of the aforementioned techniques need to use the prior OD matrix, which may be out of date and lead to estimation bias. Additionally, travel time[19], travel speed[10], and turning proportions[20
] are also employed directly in OD estimates due to the availability of massive traffic data and traffic simulation.
II-B dynamic OD matrices estimation
Static OD matrix estimation can only estimate OD for a specific period. While a period is divided into multiple intervals, researchers propose dynamic OD matrix estimation, which aims to estimate the
OD matrices in the corresponding time intervals and estimate the entire OD sequence further by using road traffic and other information. However, as mentioned above, the lagged flow may vary in
successive intervals according to different traffic conditions, so the estimation by simply applying the static OD matrix is no longer applicable. Dynamic OD matrix estimation is divided into two
types: off-line case and on-line case[6].
II-B1 Off-line case
The off-line method mainly discusses the direct estimation of the OD sequence when only given the corresponding observation sequence. [5] discusses off-line estimation methods, such as the
simultaneous and sequential method, and further[21] proposes a correction method based on the average OD matrix. The simultaneous method focuses on establishing an estimator to optimize all OD matrix
slices simultaneously, its idea is similar to the static OD matrix estimation. Compared to the simultaneous method, the sequential method only estimates one matrix slice at a time and infers the next
OD matrix slice based on the historical OD matrices that have been estimated. The average-based correction method estimates an average OD over the observation period, then estimates coefficients,
which are multiplied by the average OD to obtain the final OD sequence. Since each interval corresponds to an OD matrix slice, there are large amounts of parameters with non-negative constraints to
be estimated. The main optimization methods proposed by researchers are the gradient projection method[22] and the Simultaneous Perturbation Stochastic Approximation(SPSA)[23]. Similar to static OD
estimation, some researchers have integrated various traffic observations, such as vehicle transit time, travel speed, etc., into the estimators[24] to improve the accuracy.
II-B2 On-line case
The on-line method has been widely studied. Unlike the off-line method, it requires the historical periodic OD sequence and the current observation. Traditional modeling methods include Kalman filter
CF[25][26], LSQR algorithm[27], Canonical Polyadic (CP) decomposition[28]. [29] used principal component analysis (PCA) combine several machine learning models, and [30][13] utilize Dynamic Mode
Decomposition (DMD) based method. At the same time, given the excellent performance of deep learning in prediction, some researchers use deep learning methods for the dynamic prediction of OD
sequences. For example, [31] used a Graph Neural Network(GNN) to capture the spatial topology information of the graph structure and combined it with the traditional CF algorithm to improve the
accuracy of the OD matrix prediction. [32][33] used a Recurrent Neural Network(RNN) to capture the temporal features of prior OD sequence evolution to predict the current OD matrix. The primary
relationship is that the off-line methods can be used to provide a better initialization for on-line methods[34].
$n_{od}$ the number of OD nodes
$n_{sec}$ the number of road sections
$I$ the number of estimation intervals
$o$ the number of observation intervals
$\boldsymbol{\epsilon}_{\tau}$ a vector of traffic counts during observation interval $\tau$, $\boldsymbol{\epsilon}_{\tau}\in\mathbb{R}^{n_{sec}}$
$\boldsymbol{E}$ a tensor composed of $\boldsymbol{\epsilon}_{1}$ to $\boldsymbol{\epsilon}_{o}$, $\boldsymbol{E}\in\mathbb{R}^{o\times n_{sec}}$
$\boldsymbol{n}_{i}$ a vector represents the OD node $i$, $\boldsymbol{n}_{i}\in\{-1,0,1\}^{n_{sec}}$
$\boldsymbol{N}$ a tensor composed of $\boldsymbol{n}_{1}$ to $\boldsymbol{n}_{n_{od}}$, $\boldsymbol{N}\in\{-1,0,1\}^{n_{od}\times n_{sec}}$
$M_{ijt}$ the number of traffic trips from $\boldsymbol{n}_{i}$ to $\boldsymbol{n}_{j}$ during estimation interval $t$
$\boldsymbol{T}$ a tensor transformed from OD sequence, $\boldsymbol{T}\in\mathbb{R}^{I\times n_{od}^{2}}$
$\tilde{\boldsymbol{T}}$ a initial OD sequence is as a starting point of optimization.
$\hat{\boldsymbol{T}}$ an optimized OD sequence during optimization phase.
$\tilde{\boldsymbol{p}}_{t}$ production flow, a vector of trips leaving from each node during estimation interval $t$, $\tilde{\boldsymbol{p}}\in\mathbb{R}^{n_{od}}$
$\tilde{\boldsymbol{a}}_{t}$ attraction flow, a vector of trips arriving at each node during estimation interval $t$, $\tilde{\boldsymbol{a}}\in\mathbb{R}^{n_{od}}$
$\boldsymbol{p}$ global production flow, a vector concatenated from $\tilde{\boldsymbol{p}}_{1}$ to $\tilde{\boldsymbol{p}}_{I}$, $\boldsymbol{p}\in\mathbb{R}^{I\cdot n_{od}}$
$\boldsymbol{a}$ global attraction flow, a vector concatenated from $\tilde{\boldsymbol{a}}_{1}$ to $\tilde{\boldsymbol{a}}_{I}$, $\boldsymbol{a}\in\mathbb{R}^{I\cdot n_{od}}$
$\boldsymbol{D}_{E}$ a tensor of the distribution of traffic counts by normalizing $\boldsymbol{E}$
$\boldsymbol{d}_{p\ or\ a}$ a vector of the distribution by normalizing $\boldsymbol{p}$ or $\boldsymbol{a}$
$\bar{\boldsymbol{d}}_{p\ or\ a}$ the inferred distribution of production flow or attraction flow from deep learning model
$\boldsymbol{d}_{p\ or\ a}^{*}$ the inferred distribution of production flow or attraction flow from the best trained deep learning model during inference phase
$\hat{\boldsymbol{d}}_{p\ or\ a}$ the optimized distribution of production flow or attraction flow from bi-level framewrok during optimization phase
III Preliminary
III-A Definitions
As shown in Table 1, an OD node is a cluster created by grouping the intersections of road sections in the city network, and the roads connect the same OD pair are aggregated to one road section (see
Fig. 3). Considering a city network consists of $n_{od}$ OD nodes and $n_{sec}$ road sections. We use a vector of length $n_{sec}$ to represent OD node $\boldsymbol{n}_{i}$, with $n_{ij}=1$ if road
section $j$ enters node $\boldsymbol{n}_{i}$, -1 if it exits from $\boldsymbol{n}_{i}$, and 0 if it is not connected to $\boldsymbol{n}_{i}$.
During an estimation period divided into $I$ equal intervals $t=1,2,3,...,I$, and an obsevation period divided into $o$ equal intervals $\tau=1,2,3,...,o$. $\boldsymbol{\epsilon}_{\tau}$ denotes the
traffic counts of all road sections during observation interval $\tau$. The traffic trips from $\boldsymbol{n}_{i}$ to $\boldsymbol{n}_{j}$ during estimation interval $t$ is denoted by $M_{ijt}$. In
addition, we transform the OD sequence into a tensor denoted by $\boldsymbol{T}\in\mathbb{R}^{1\times I\times n_{od}^{2}\times 1}$, and $\boldsymbol{\varepsilon}$ denotes a vector concatenated from $
\boldsymbol{\epsilon}_{1}$ to $\boldsymbol{\epsilon}_{o}$. For $\varepsilon_{i}\in\boldsymbol{\varepsilon}$, we normalize its traffic count on each road section with respect to all observation
intervals as $D_{E_{ij}}=\frac{E_{ij}}{\sum\limits_{i}^{o}\sum\limits_{j}^{n_{sec}}E_{ij}}$.
Production flows $\tilde{\boldsymbol{p}}_{t}$ is used to represent a vector records the number of trips leaving each node during estimation interval $t$ as $\tilde{\boldsymbol{p}}_{t}=\sum\limits_{j}
^{n_{od}}M_{ijt}$. And attraction flows $\tilde{\boldsymbol{a}}_{t}$ is to used to represent a vector records the number of trips that arrive at each node during estimation interval $t$ as $\tilde{\
Additionally, the global production flows $\boldsymbol{p}$ is a vector concatenated from $\tilde{\boldsymbol{p}}_{1}$ to $\tilde{\boldsymbol{p}}_{I}$, and the global attraction flows $\boldsymbol{a}$
is a vector concatenated from $\tilde{\boldsymbol{a}}_{1}$ to $\tilde{\boldsymbol{a}}_{I}$, which are used to denotes the production flows and attraction flows of OD sequence, respectively. Moreover,
we normalize their flow on each OD node with respect to all OD nodes and all estimation intervals as $d_{pi}=\frac{p_{i}}{\sum\limits_{k}^{I\times n_{od}}{p}_{k}}$ and $d_{ai}=\frac{a_{i}}{\sum\
limits_{k}^{I\times n_{od}}{a}_{k}}$, respectively.
In the inference phase, $\bar{\boldsymbol{d}}_{p\ or\ a}$ denotes the inferred distribution of production flow or attraction flow from deep learning model when being trained. And the best inferred
distribution is denoted by $\boldsymbol{d}_{p\ or\ a}^{*}$.
In the optimization phase, $\hat{\boldsymbol{T}}$ denotes the tensor of optimized OD sequence from bi-level framewrok at each iteration, and correspondingly, $\hat{\boldsymbol{d}}_{p\ or\ a}$ denotes
the optimized distribution of production flows or attraction flows.
III-B Bi-level framework
In the bi-level framework, the estimation will start from an initialized OD matrix $\tilde{\boldsymbol{T}}$. In the lower level, for each observation interval, trips in every OD pair are allocated to
the road sections in an analytical or simulative way, and then an allocation tensor $\boldsymbol{P}$ is obtained. $\boldsymbol{P}_{\tau t}$ represents a matrix of proportion that OD matrix $\
boldsymbol{T}_{t}$ allocated to road sections during the observation interval $\tau$, and $\boldsymbol{P}_{\tau t}\boldsymbol{T}_{t}$ indicates the corresponding traffic counts. In the upper layer,
the traffic counts assigned by $\boldsymbol{T}_{1},\boldsymbol{T}_{1},...,\boldsymbol{T}_{t}$ are summed to get the traffic counts $\boldsymbol{\epsilon}_{\tau}$ as shown in Fig 1. Optimizing the
least squares estimator reduces the gap between optimized traffic counts $\hat{\boldsymbol{\epsilon}}_{\tau}$ and real traffic counts $\boldsymbol{\epsilon}_{\tau}$ throughout the whole observation
period $\tau=1,2,...,o$ to find a better estimated OD sequence. By iteration repeats the alternation of upper and lower levels, the final estimated OD matrix sequence is obtained when converges. The
least squares estimator is formulated as follows:
$\displaystyle Z(\hat{\boldsymbol{T}})=\min_{\hat{\boldsymbol{T}}_{1}\geq 0,...%,\hat{\boldsymbol{T}}_{I}\geq{0}}\sum_{\tau=1}^{o}\frac{1}{2}(\boldsymbol{%\epsilon}_{\tau}-\hat{\boldsymbol{\
$\displaystyle where\ \hat{\boldsymbol{\epsilon}}_{\tau}=\sum_{k=1}^{t}%\boldsymbol{P}_{\tau k}\hat{\boldsymbol{T}}_{k}$
The assignment tensor $\boldsymbol{P}\in\mathbb{R}^{o\times I\times n_{sec}\times n_{od}^{2}}$ , can be derived by analysis, for example by taking into account stochastic user equilibrium on traffic
counts or by simulation using a simulator like SUMO[35]. It displays the ratio of each OD trip to each road section during estimation intervals. Similar to[4], our assignment tensor P is computed
using a back-calculation technique based on traffic counts generated by the simulator during each iteration:
{array}[]{ccc}p_{11}&\ldots p_{1\ n_{od}%^{2}}\\p_{21}&\ldots p_{2\ n_{od}^{2}}\\\vdots&\vdots\\p_{n_{sec}\ 1}&\ldots p_{n_{sec}\ n_{od}^{2}}\\\end{array}\right)_{oI}\end{array}\right)$
III-C Optimization
Considering Eq(1) is a problem with abundant non-negativity constraints on $\boldsymbol{T}$, the gradient projection method is commonly used[22]. The idea is to evaluate the directional derivatives
of the objective function at the current point, and to obtain a descent direction by making a projection on the non-negativity constraints.
It is worth noting that we adjust the update step $\boldsymbol{T}^{k+1}:=\boldsymbol{T}^{k}\oplus(\lambda^{k}\odot\boldsymbol{d}^%{k})$ to $\boldsymbol{T}^{k+1}:=\boldsymbol{T}^{k}\odot(\boldsymbol
{e}+\lambda^{k}\odot%\boldsymbol{d}^{k})$. It has been proved that compared with the ordinary update step, this adjustment significantly improves the optimization speed of OD matrices[7], since it
proposed that update steps for larger variables should be greater. For the upper bound of step size $\lambda^{k}_{max}$ at itertion $k$, we give the corresponding adjustment $\lambda^{k}_{max}=\
mathop{\min}\{\frac{-1}{\boldsymbol{d}^{k}_{i}}|\forall i:%\boldsymbol{d}^{k}_{i}<0\}$. Since at the $(k+1)^{th}$ iteration, $\mathbf{A}_{2}\boldsymbol{T}^{k}>0$ and ensure $\mathbf{A}_{2}\boldsymbol
{T}^{k+1}>0$, let $\mathbf{A}_{2}\boldsymbol{T}^{k}\odot(\boldsymbol{e}+\lambda^{k}\odot%\boldsymbol{d}^{k})>0$, which implies $\boldsymbol{e}+\lambda^{k}\odot\boldsymbol{d}^{k}>0$. Therefore, $\
lambda^{k}<\frac{-1}{\boldsymbol{d}^{k}_{i}},\forall i:\boldsymbol{d}^{k}_{i}<0$, and then we have $\lambda^{k}_{max}=\min\{\frac{-1}{\boldsymbol{d}^{k}_{i}}\},\forall i:%\boldsymbol{d}^{k}_{i}<0$.
Finally, we search for the optimal step size $\lambda^{*k}$ at iteration $k$ based on Eq(2) and then determine the executable step size $\lambda^{k}$ according to $\lambda^{*k}$ and $\lambda^{k}_
{max}$, that is, if $\lambda^{*k}<\lambda^{k}_{max}$, set $\lambda^{k}=\lambda^{*k}$; otherwies, set $\lambda^{k}=\lambda^{k}_{max}$.
$\min_{\lambda^{k}}Z(\hat{\boldsymbol{T}}^{k}\odot(\boldsymbol{e}+\lambda^{k}%\odot\boldsymbol{d}^{k}))$ (2)
Where $\odot$ denotes the element-wise product and $\boldsymbol{e}$ is a tensor of 1s with the same dimension as $\hat{\boldsymbol{T}}^{k}$.
step 0: Giving an initial point that satisfies the constraints $\boldsymbol{T}^{0}$, expand $\boldsymbol{T}^{0}$ into a vector as $F(\boldsymbol{T}^{0})\in\mathbb{R}^{I\cdot n_{od}^{2}}$, let $k=0$,
threshold $\epsilon>0$; step 1: Construct search direction at $\boldsymbol{T}^{k}$. Let $\mathbf{A}=\left[\begin{matrix}\mathbf{A}_{1}\\\mathbf{A}_{2}\end{matrix}\right]$, $b=\left[\begin{matrix}0\\0
\end{matrix}\right]$, $\mathbf{A}_{1}F(\boldsymbol{T}^{k})=0,\mathbf{A}_{2}F(\boldsymbol{T}^{k})>0$. step 2: Let $\mathbf{M}=\left[\begin{matrix}\mathbf{A}_{1}\\\mathbf{E}\end{matrix}\right]$, Let $\
mathbf{P_{M}}=\mathbf{I}$ if $\mathbf{M}$ is empty, and $\mathbf{P_{M}}=\mathbf{I}-\mathbf{M}^{\mathrm{T}}(\mathbf{M}\mathbf{M}^{%\mathrm{T}})^{-1}\mathbf{M}$ otherwise. step 3: Calculate $\
boldsymbol{d}^{k}=-\mathbf{P_{M}}abla Z(\boldsymbol{T}^{k})$. If $\|\boldsymbol{d}^{k}\|eq 0$, to $step5$; otherwise goto $step4$. step 4: Calculate $\left[\begin{matrix}\lambda\\\mu\end{matrix}\
right]=(\mathbf{MM^{\mathrm{T}}})^{-1}\mathbf{M}abla Z(%\boldsymbol{T}^{k})$. If $u\geq 0$ stop and $\boldsymbol{T}^{k}$ is KKT point. Otherwise let $u_{i0}=\mathop{\min}\{u_{i}\}$, and remove the
row corresponding to $u_{i0}$ from $\mathbf{M}$ and goto $step2$. step 5: Determine the step size. Let $\lambda^{k}_{max}=\mathop{\min}\{\frac{-1}{\boldsymbol{d}^{k}_{i}}|\forall i:%\boldsymbol{d}^
{k}_{i}<0\}$, and determine $\lambda^{*k}$ based on Eq(2). If $\lambda^{*k}<\lambda^{k}_{max}$, let $\lambda^{k}=\lambda^{*k}$; otherwies, let $\lambda^{k}=\lambda^{k}_{max}$. step 6: Let $\
boldsymbol{T}^{k+1}:=\boldsymbol{T}^{k}\odot(\boldsymbol{e}+\lambda^{k}\odot%\boldsymbol{d}^{k}),k:=k+1$, goto $step2$.
IV method
The pipline of our proposed method will be described in detail in this section, including sampling the probe flow to compose datasets, training and inference of NN models, and combining inferred
spatial-temporal structural distributions into numerical optimization.
IV-A Probe Traffic Sampling
Firstly, as presented in our previous work on static OD estimation, most of the important trips are in a small part of OD pairs, leat to the values of other OD pairs are relatively small. So the
production and attraction flows of an OD matrix will be uneven in the reality, and this property implies the structural information of the OD matrix. Moreover, as we mentioned in Part 1, we infer
distributions rather than real numbers to reflect the structure, so the exact value is not a concern. Therefore, it is feasible to set a sparse matrix with limited values of non-zero elements to
reconstruct the specific structure information of an real OD matrix. To this end, we set $m$ to represent the maximum value of OD pairs and make it relatively small to speed up the sampling process.
Secondly, since the traffic congestion will cause lag problem, we need some probe vehicles to explore the traffic congestion. We form a original matrices sequence(OMS) by collecting $m$ vehicles from
these important OD pairs (with the number of trips $>m$) of the real OD sequence as shown in the left part of Fig. 2, these trips can be a combination of various data, such as car-hailing service
data and GPS data since these vehicles can all be seen as probe vehicles. Then, we resample each OD paris on a scale of 0.0-1.0 from OMS to obtain a dataset composed of generated OD sequences, and
calculate the corresponding global distributions $\boldsymbol{d}_{p}$, $\boldsymbol{d}_{a}$ and $\boldsymbol{D}_{E}$. The advantage of doing so is that, although we sample from a small number $m$ of
vehicles, it also can reconstruct the relationship between various traffic counts distribution and its corresponding golobal distributions under the real traffic conditions.
Finally, $\boldsymbol{D}_{E}$ are used as inputs of NN, $\boldsymbol{d}_{p}$ or $\boldsymbol{d}_{a}$ are used as labels, we form the dataset $(\boldsymbol{D}_{E},\boldsymbol{d}_{p})$ and $(\
boldsymbol{D}_{E},\boldsymbol{d}_{a})$ and train the two models separately.
IV-B Dynamic Distribution Inference
The spatio-temporal evolution of traffic counts can effectively characterize the OD sequence. We utilize traffic counts of observation interval $t$ to $t\times k+\delta$ (for $t\in[0,I-1]$, $k=\frac
{o}{I}$, $\delta>0$) as input to characterize the OD matrix of estimation interval $t$. If $\delta>k$, the observations overlap between two estimation intervals as shown in Fig 1, it indicates that
the OD matrix of the current estimation interval $t$ will affect the traffic counts of the following $t\times k+\delta$ observation intervals due to the lag problem.
In order to obtain the global distributions, we need to consider the mutual influence relationship of each OD node in spatial and temporal. For example, if the trips of node $n_{i}$ at observation
interval $t\times k+\delta_{1}$ and node $n_{j}$ at $t\times k+\delta_{2}$ both need to pass through road section $e$ at $t\times k+\delta$ when the traffic count has been given($\delta>\delta{1}$; $
\delta>\delta_{2}$), so there will be pairwise spatial-temporal dependencies between OD nodes.
IV-B1 DCGRU
In deep learning field, many studies have shown that the GNN+RNN based method can extract the spatio-temporal features well[36]. Therefore, we choose the DCGRU model as the feature extractor of each
OD matrix. It combines Diffusion Convolutional Network(DCN, a spatial-based GNN model) and Gated Recurrent Unit(GRU,an improved RNN model) and has been studied to have an outstanding performance in
capturing the long term spatio-temporal evolution characteristics of traffic flow[1] .
DCN can be adapted to deal with dependency between objects in non-Euclidean spaces according to node features $\boldsymbol{\chi}$ and the adjacency matrix $\boldsymbol{W}$. The $K$-step graph
diffusion convolution is calculated to extract the upstream and downstream spatial dependencies between their surrounding $K$-order neighbor nodes and form an integrated graph signal, which is
formulated as below:
$\displaystyle\boldsymbol{\chi}_{:,e\star g}=\sum_{k=0}^{K-1}(\theta_{k,1}(%\boldsymbol{D}_{o}^{-1}\boldsymbol{W})^{k}+\theta_{k,2}(\boldsymbol{D}_{I}^{-1%}\boldsymbol{W}^{T})^{k})\boldsymbol{\
$\displaystyle for\ e\ \in\{1,...,n_{sec}\}$ (3)
$\boldsymbol{\chi}_{\star g}$ is the graph signal obtained after each OD node fuses the $K$ order neighbors in every dimension $e$. $\boldsymbol{\theta}\in\mathbb{R}^{K\times 2}$ are learnable
parameters for the filter. $\boldsymbol{D}_{o}$ represents the diagonal matrix of the out-degree matrix of graph $g$, and $\boldsymbol{D}_{I}$ represents the diagonal matrix of the in-degree matrix,
$\boldsymbol{D}_{o}^{-1}\boldsymbol{W}$, $\boldsymbol{D}_{I}^{-1}\boldsymbol{W}^{T}$ represent the transition matrices of the diffusion process and the reverse one respectively.
GRU sends the integrated graph signal into the cell orderly to capture the temporal dependencies. The update process of feature in the GRU cell is as follows:
$\displaystyle\boldsymbol{r}^{(\tau)}=\sigma(\boldsymbol{\Theta}_{r\star g}[%\boldsymbol{\chi}^{(\tau)},\boldsymbol{H}^{(\tau-1)}]+\boldsymbol{b}_{r})$ (4)
$\displaystyle\boldsymbol{u}^{(\tau)}=\sigma(\boldsymbol{\Theta}_{u\star g}[%\boldsymbol{\chi}^{(\tau)},\boldsymbol{H}^{(\tau-1)}]+\boldsymbol{b}_{u})$ (5)
$\displaystyle\boldsymbol{C}^{(\tau)}=tanh(\boldsymbol{\Theta}_{C\star g}[%\boldsymbol{\chi}^{(\tau)},(\boldsymbol{r}^{(\tau)}\odot\boldsymbol{H}^{(\tau-%1)}]+\boldsymbol{b}_{c})$ (6)
$\displaystyle\boldsymbol{H}^{(\tau)}=\boldsymbol{u}^{(\tau)}\odot\boldsymbol{H%}^{(\tau-1)}+(1-\boldsymbol{u}^{(\tau)})\odot\boldsymbol{C}^{(\tau)}$ (7)
where $\boldsymbol{\chi}^{(\tau)}$, $\boldsymbol{H}^{(\tau)}$ denote the input and output at observation interval $\tau$, $\boldsymbol{r}^{(\tau)}$, $\boldsymbol{u}^{(\tau)}$ are reset gate and
update gate at $\tau$ respectively. $\star g$ denotes the diffusion convolution defined in Eq(3) and $\boldsymbol{\Theta}_{r}$, $\boldsymbol{\Theta}_{u}$, $\boldsymbol{\Theta}_{C}$ are learnable
parameters for the corresponding filters.
IV-B2 Multihead Self-Attention(MSA)
Standard $\boldsymbol{qkv}$ self-attention(SA) compute the attention weight $A$ over all value of elements $\boldsymbol{v}$. $A$ is based on the of query $\boldsymbol{q}$ and key $\boldsymbol{k}$ of
elements , and calculate the pairwise dependency between two elements of input sequence $\boldsymbol{\zeta}\in\mathbb{R}^{(n_{od}^{2}+1)\times d}$.
Self-attention (SA)[37] computes the attention weight A overall value of elements $\boldsymbol{v}$ to calculate the pairwise dependency between two elements of input sequence $\boldsymbol{\zeta}\in\
mathbb{R}^{(I\cdot n_{od}^{2})\times d}$ where $A$ is based on the query $\boldsymbol{q}$ and key $\boldsymbol{k}$ of elements.
$[\boldsymbol{q},\boldsymbol{k},\boldsymbol{v}]=\boldsymbol{\zeta}\boldsymbol{U%}_{qkv},\ \boldsymbol{U}_{qkv}\in\mathbb{R}^{d\times 3d_{h}}$ (8)
$A=softmax(\boldsymbol{qk}^{\mathrm{T}}/\sqrt{d_{h}})$ (9)
$SA(\boldsymbol{\zeta})=A\boldsymbol{v}$ (10)
We projected concatenated outputs from MSA, which runs $h$ SA procedures concurrently. The dimensions are kept constant by setting $d_{h}$ to $d/h$, where $h$ is the number of heads.
$\displaystyle MSA(\boldsymbol{\zeta})=[SA_{1}(\boldsymbol{\zeta});SA_{2}(%\boldsymbol{\zeta});...;SA_{h}(\boldsymbol{\zeta})]\boldsymbol{U}_{msa},$
$\displaystyle\boldsymbol{U}_{msa}\in\mathbb{R}^{h\cdot d_{h}\times d}$ (11)
$\boldsymbol{U}_{qkv}$ and $\boldsymbol{U}_{msa}$ above are learnable parameters
IV-B3 Distribution Learner
We element-wise multiply the node vector $\boldsymbol{n}_{i}$ with the distribution of traffic counts $\boldsymbol{\epsilon}_{\tau}$ to obtain the feature of OD node $i$ at observation $\tau$. In
order to get all OD nodes fetures $\boldsymbol{\chi}$ during all the $o$ observation intervals. We expand dimention of $\boldsymbol{N}$ to $n_{od}\times 1\times n_{sec}$ and $\boldsymbol{D_{E}}$ to
$1\times o\times n_{sec}$, respectively, then do the broadcast operation $\otimes$ on these two tensor as shown in Eq(12) to get the shape of $\boldsymbol{\chi}$ as $(o,n_{od},n_{sec})$. Then, divide
by the dimension $o$ of the tensor, and take the $t\times k$ to $t\times k+\delta$ (for $t\in[0,I-1]$, $k=\frac{o}{I}$, $\delta>0$) each time to obtain the input tensor $(\delta,n_{od},n_{sec})$. In
our case, we estimate an OD sequence of 12 hours, with an estimation interval every hour and an observation interval every 10 minutes. So we have $I=12,o=72,k=6$.
$\boldsymbol{\chi}=\boldsymbol{N}\otimes\boldsymbol{D}_{E}$ (12)
Subsequently, taking $\{\boldsymbol{\chi}^{(t\times k)},...,\boldsymbol{\chi}^{(t\times k+\delta)}\ %|\ t\in[0,I-1]\}$ as the input of the DCGRU module orderly, a hidden tensor $\boldsymbol{H}=\{\
boldsymbol{H}^{(t\times k+\delta)}\ |\ t\in[0,I-1]\}$ is as the output with its shape is $(I,n_{od},n_{sec})$. Then expanded $\boldsymbol{H}$ by the dimension $I$ to obtain $(I\times n_{od},n_{sec})$
as the input of the Transformer encoder. With a shared position embedding parameters added before and after Transformer encoder, we refer to the output as mutual vectors, there are $I\times n_{od}$
mutal vectors for each represents the spatio-temporal mutual information of corresponding OD node. Lastly, we operate element-wise addtion $\oplus$ on all these mutual vectors to one vector *
containing the global infromation, and do inner production between each mutual vector and the global information vector * to give a scalar for each node, then perform the softmax operations to obtain
the inferred global distribution $\bar{\boldsymbol{d}}_{p\ or\ a}$.
For model training, we choose Jensen-Shannon Divergence(JSD) as the loss function as Eq(13), which measures the distance between two distributions symmetrically.
$\displaystyle Loss_{p\ or\ a}=JS(\bar{\boldsymbol{d}}_{p\ or\ a}||\boldsymbol{%d}_{p\ or\ a})$ (13)
IV-C Estimator
Like other studies in static OD estimation and the off-line case of dynamic OD sequence estimation, we adopt the bi-level framework. The difference is the least squares approach at the upper level
merely seeks to reduce the gap between observed and simulated traffic counts, which only facilitates numerical similarity between estimated and real OD sequence. Therefore, we incorporate the
optimization with the best inference global distributions $\boldsymbol{d}_{p}^{*}$ and $\boldsymbol{d}_{a}^{*}$ and choose KLD[38] as the objective function as following.
$\displaystyle R(\hat{\boldsymbol{T}})=\min_{\hat{\boldsymbol{T}}_{1}\geq 0,...%,\hat{\boldsymbol{T}}_{I}\geq{0}}\alpha N(\hat{\boldsymbol{T}})+(1-\alpha)S(%\hat{\boldsymbol{T}})$ (14)
$\displaystyle N(\hat{\boldsymbol{T}})=\sum_{\tau=1}^{o}\frac{1}{2}(\boldsymbol%{\epsilon}_{\tau}-\hat{\boldsymbol{\epsilon}}_{\tau})^{\mathrm{T}}(\boldsymbol%{\epsilon}_{\tau}-\hat{\
$\displaystyle S(\hat{\boldsymbol{T}})=KL(\hat{\boldsymbol{d}}_{p}||\boldsymbol%{d}^{*}_{p})+KL(\hat{\boldsymbol{d}}_{a}||\boldsymbol{D}^{*}_{a})$
$\displaystyle where\ \hat{\boldsymbol{\epsilon}}_{\tau}=\sum_{k=1}^{t}%\boldsymbol{P}_{\tau k}\hat{\boldsymbol{T}}_{k}$
Our optimization process is shown in Algorithm 2. It is worth noting that since the optimization is alone the approximate distributions rather than the real distributions, the structure should not be
optimal when $S(\hat{\boldsymbol{T}})$ converges. Therefore, we then slack the structure constraint (set $\alpha$=1), which further does only numerical optimization and leads to a better point.
Input: Observed traffic counts tensor $\boldsymbol{E}$
The best inferred OD distribution $\boldsymbol{d}^{*}_{p}$ and $\boldsymbol{d}^{*}_{a}$ from pre-trained Distribution Learner, separately.
Output: Estimated OD sequence
1 Initialize Balance factor $\alpha$, Initialized OD sequence $\tilde{\boldsymbol{T}}$, set $\hat{\boldsymbol{T}}^{k}=\tilde{\boldsymbol{T}}$ and $k=0^{th}$ iteration;
2 repeat
3Lower level: Simulate $\hat{\boldsymbol{T}}^{k}$ with the simulator and obtain simulated traffic counts observations $\boldsymbol{\epsilon}_{\tau},\tau=1,2,...o$. Assignment matrix $\boldsymbol{P}$
is calculated by using a back-calculation procedure;
4Upper level: $\hat{\boldsymbol{T}}^{k+1}=\min_{\hat{\boldsymbol{T}}^{k}}R(\hat{\boldsymbol{T%}}^{k})$ based on Eq(14);
5if$S(\hat{\boldsymbol{T}}^{k})$ convergentthen
6 set $\alpha$=1
8until$R(\hat{\boldsymbol{T}}^{k})$ convergent;
We test our method on a large-scale real city network with a synthetic dataset. The effectiveness of NN for distributional inference is first validated. Then, a comparative experiment is used to
demonstrate the advantage of our optimization method compared with traditional numerical optimization. The project uses Python programming and relies on the TensorFlow system[39] for gradient
calculation and NN modeling. Sklearn library[40] and Scipy[41] are used for clustering and optimization program, respectively.
V-A Study Network
We selected the $400km^{2}$ area around Cologne, Germany, as our study network and used SUMO as the simulator program to test on a large-scale city network[42]. 71368 road sections and 31584
intersections make up the network (Fig. 3(a)). We employ the K-means[43] technique to gather OD nodes on the network according to Euclidean distance. In this case, we select $n_{od}$=15 (Fig. 3(b)),
and we aggregate the directed road sections from 782 to 64.
OD matrix(o’clock) $6-7$ $7-8$ $8-9$ $9-10$ $10-11$ $11-12$ $12-13$ $13-14$ $14-15$ $15-16$ $16-17$ $17-18$
total travels 76972 86580 40131 31971 34211 41065 50015 48899 48124 74760 95042 79205
density($>m$) 0.3689 0.3644 0.2844 0.2489 0.2533 0.2667 0.3111 0.3200 0.3022 0.3378 0.3644 0.3511
the largest OD pair 9097 11048 4918 3395 3145 4523 5664 4643 4314 7937 10315 8816
V-B Dataset
The synthetic data closely comparable to the actual conditions of urban traffic serves as the ground truth. Refer to [42] for more information.
As shown in Table 2, we took the ground truth OD matrix for 12 hours from 6:00 am to 18:00 pm from a 24-hours traffic simulation. Set the OD estimation interval to 1 hour and the traffic counts
observation interval to 10 minutes. For the input of the DCGRU module, we set $\delta$=4 to indicate that the traffic counts of the four flowing observation intervals from the current OD estimation
interval are used as input to characterize the current OD matrix. Therefore, the whole length of observation intervals is 76 (12$\times$6 since each OD estimation interval is with six observation
We set $m$=50 and sample the OMS from the ground truth OD matrix. Then resample from the original matrices sequence to get various OD sequences, and generate the corresponding observation traffic
counts $\boldsymbol{\epsilon}$ to form a data set $(\boldsymbol{D}_{E},\boldsymbol{d}_{p})$ and $(\boldsymbol{D}_{E},\boldsymbol{d}_{a})$, the size of each is 10k sample pairs.
In an 8:2 ratio, we split the dataset for training and validation. A model is trained using the training data, and its generalization performance is assessed using the validated data.
V-C OD estimating evaluation
Referring to[4], we use the numerical value indicators $RMSN(\hat{\boldsymbol{T}}_{t},\boldsymbol{T}_{t})$[44] and structural indicators $\rho(\hat{\boldsymbol{T}}_{t},\boldsymbol{T}_{t})$[45] to
measure the gap between the estimated OD matrix $\hat{\boldsymbol{T}}_{t}$ and the real OD matrix $\boldsymbol{T}_{t}$.
$RMSN(\hat{\boldsymbol{T}}_{t},\boldsymbol{T}_{t})=\frac{\sqrt{n_{od}^{2}\sum%\limits^{n_{od}^{2}}_{i}(\boldsymbol{T}_{ti}-\hat{\boldsymbol{T}}_{ti})^{2}}}{%\sum\limits^{n_{od}^{2}}_{i}\ (15a)
$\rho(\hat{\boldsymbol{T}}_{t},\boldsymbol{T}_{t})=\frac{(\boldsymbol{T}_{t}-%\boldsymbol{\mu})^{T}(\hat{\boldsymbol{T}}_{t}-\boldsymbol{\hat{\mu}})}{\sqrt{%(\boldsymbol{T}_{t}-\boldsymbol{\ (15b)
where $\boldsymbol{\mu}\in\mathbb{R}^{n_{od}^{2}}_{\geq{0}}$ is a vector with each element value equal to the mean of $\boldsymbol{T}_{t}$, and $\hat{\boldsymbol{\mu}}$, $\tilde{\boldsymbol{\mu}}$
corresponds to $\hat{\boldsymbol{T}}_{t}$ and $\tilde{\boldsymbol{T}}_{t}$, respectively.
V-D Parameter settings
head number $h$ 6
encoder layer $N$ 2
learning rate $r$ 1E-4
dimention $d$ 128
diffusion convolution step $K$ 2
maximum trips value $m$ 50
OD estimation intervals $I$ 12
observation intervals $o$ 12 and 72
sequence length $\delta$ 4
V-E Results and analyze
V-E1 Training
Firstly, As shown in Fig. 4. We set the real OD sequence as our test set, the test set curve converges indicates our sampled training data is effective for model training. Moreover, the results of
the test set is not as good as training and validation set since the distribution of real data and sampled data is not perfectly consistent. Which is a common problem in deep learning and implies it
can be further alleviated through more appropriate sampling methods.
Secondly, we tested the impact of different observation intervals length $\delta$ on the inferred results. Our experiment in Fig. 5 shows that, when $\delta=4$ the inferenced results by NN model is
the best. It means our model does not completely utilize all the observation intervals information during one estimation interval(should be $\delta=6$), and when there is an overlap ($\delta>6$), the
results get worse. It indicates that our current model has not effectively extract all the information introduced by longer observation intervals, more advanced NN model could further improve the
inference results.
V-E2 Distribution Inference
Firstly, Fig. 6(a) shows the best inference results of our NN model on the global production distribution $\boldsymbol{d}_{p}^{*}$, it reflects the ability of our model to give the global
spatio-temporal distribution of OD sequence. Secondly, we pick an OD matrix at 12-13 o’clock from the global distriution, as shown in Fig. 6(b), which shows that the model has a good performance on
spatial distribution inference, it clearly reflects the proportional between different OD nodes in the same OD matrix. Finally, we picked the distributions of four OD nodes during the whole
estimation period, as shown in Fig. 7, the model infers the temporal evolution trend of specific OD node also very well, moreover in Fig. 7 we demonstrate with different size of proportions(from le-2
to le-5) and it shows our model has a high sensitivity that, except 1e-5, since these proportions are too small to cause the model to infer the node has a proportions of 0.
V-E3 Optimization
Firstly, we compare two traditional optimization methods. These methods only use numerical optimization with two different observation interval sets: one hour and 10 minutes, respectively. The one
hour setting indicates there are 12 observation intervals since we have 12 hours during the whole estimation period, so we refer to this method as Traditional($o$=12). And another is Traditional($o$=
72) since there are 72 observation intervals for 10 minutes setting. Although 10 minutes setting has the smallest scale and can provides more constraints to alleviate the underdetermined problem, as
mentioned in[5], it also introduces more noise in the optimization process, which may leading to a poor optimization result. As seen in Table 3, the final optimization result obtained by the 10
minutes setting is worse than the one hour setting, which indicates that the noise in the optimization process offset the benefits of excessive observation intervals. Secondly, our method namd Ours
with $\boldsymbol{d}^{*}$ can mining significant information from these small sacle observation data to infer accurate global distributions $\boldsymbol{d}_{p}^{*}$ and $\boldsymbol{d}_{a}^{*}$ and
then guide optimization process to find a better result. In addition, we also provide the results obtained from Ours with $\boldsymbol{d}$, which utilizing the real global distribution $\boldsymbol
{d}_{p}$ and $\boldsymbol{d}_{a}$ as guide, and provides the upper bound of our method, which is shown in Fig. 8. It can indicate that our method has a potent extendibility, such as through better
data sampling methods or better models to infer more accurate distributions and get better optimization results.
OD matrix(o’clock) $6-7$ $7-8$ $8-9$ $9-10$ $10-11$ $11-12$ $12-13$ $13-14$ $14-15$ $15-16$ $16-17$ $17-18$ Average
Traditional($o$=12) 1.77 1.88 1.76 1.81 1.96 2.02 2.00 1.90 1.82 2.11 2.17 1.60 1.90
Ours with $\boldsymbol{d}^{*}$ 1.68 1.92 1.27 1.21 1.32 1.55 1.60 1.46 1.43 2.04 2.30 1.57 1.61
Traditional($o$=72) 2.09 2.06 2.49 1.62 1.84 2.20 1.81 1.25 2.52 2.39 1.40 1.45 1.93
Ours with $\boldsymbol{d}$ 1.33 1.51 1.01 1.51 1.43 1.41 1.41 1.35 1.15 1.44 1.51 1.18 1.35
Traditional($o$=12) 0.8470 0.8358 0.8149 0.7729 0.7266 0.7502 0.7681 0.7539 0.7573 0.7465 0.7615 0.8583 0.7828
Ours with $\boldsymbol{d}^{*}$ 0.8567 0.8167 0.9096 0.9066 0.8851 0.8614 0.8517 0.8585 0.8560 0.7618 0.7182 0.8610 0.8453
Traditional($o$=72) 0.7496 0.8677 0.5921 0.8245 0.7661 0.6735 0.8020 0.9255 0.5535 0.6182 0.9055 0.8999 0.7649
Ours with $\boldsymbol{d}$ 0.9067 0.8824 0.9435 0.8477 0.8579 0.8764 0.8818 0.8763 0.9064 0.8745 0.8685 0.9188 0.8867
Inference Inferred Optimization Traditional($o$= Ours with $\boldsymbol{d}^{*} Traditional($o$= Ours with $\boldsymbol{d}
12) $ 72) $
$KL(\boldsymbol{d}_{p}^{*}||\boldsymbol{d}_ 0.0711 $KL(\hat{\boldsymbol{d}}_{p}||\boldsymbol{d}_ 0.0756 0.0560 0.0966 0.0105
{p})$ {p})$
$KL(\boldsymbol{d}_{a}^{*}||\boldsymbol{d}_ 0.0826 $KL(\hat{\boldsymbol{d}}_{a}||\boldsymbol{d}_ 0.0772 0.0591 0.0684 0.0102
{a})$ {a})$
In Fig. 9, we demonstrate the estimation results of four OD pairs along time series with four different orders of magnitude(1e1 to 1e4), respectively, to illustrate that our method has better
performance on OD squence estimation tasks of different orders of magnitude. In there we only choose Traditional($o$=12) and Ours with $\boldsymbol{d}^{*}$ since Traditional($o$=12) has the better
performance in numerical methods, and Ours with $\boldsymbol{d}^{*}$ is an only prctical way to use approximate distributions $\boldsymbol{d}^{*}_{p}$ and $\boldsymbol{d}^{*}_{a}$ from our
distribution learner. And in Fig. 10 we show x-y plots of the estimation results for four different OD matrices.
As shown in Table 4, it is worth noting that the KLD of the best distribution $\boldsymbol{d}_{p}^{*}$ inferred by the our NN model is 0.0711. Our actual results from the final optimization is $KL(\
hat{\boldsymbol{d}}_{p}||\boldsymbol{d}_{p})$=0.0560. And KLD of the best distribution $\boldsymbol{d}_{p}^{*}$ inferred by the our NN model is 0.0826, and our actual results from the final
optimization is $KL(\hat{\boldsymbol{d}}_{a}||\boldsymbol{d}_{a})$=0.0591, which are much smaller than the approximate distributions be given since the approximate distributions are only used as a
guide in the optimization process. Moreover, our convergence speed is also faster than traditional methods. It indicates that using the approximate distribution as a guide can help the optimization
converge to a better point and faster.
In this paper, we propose a deep learning method to learn the relationship between traffic counts and OD structure information through mining information from a small amount of mixed observational
traffic data. Then we use this structure information to constrain the traditional least squares numerical optimization method based on the bi-level framework. We validate that our method outperforms
traditional numerical-only optimization methods on 12 hours of synthetic data from a large-scale city. Moreover, we present the space for future improvement of our method by improving the sampling
method and deep learning models.
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• [45]T.Djukic, S.Hoogendoorn, and H.VanLint, “Reliability assessment of dynamicod estimation methods based on structural similarity index,” Tech. Rep.,2013. | {"url":"https://savagelily.com/article/a-deeplearning-framework-for-dynamic-estimation-of-origin-destination-sequence","timestamp":"2024-11-10T20:26:38Z","content_type":"text/html","content_length":"658926","record_id":"<urn:uuid:c02befab-6838-41c8-b4d0-e2362c846283>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00643.warc.gz"} |
Linsys find
Linsys find --- Introduction ---
Linsys find is an exercise that asks you to establish a linear system according to texts of various styles, in particular systems having infinitely many solutions.
Note that you can also use the exercise Equaffine for establishing systems defining affine subspaces (line, plane, hyperplane, etc).
The exercises will be taken randomly in the list of your choice. If your choice is empty, all the available exercises will be considered.
The most recent version
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: establish a linear system according to a word problem. interactive exercises, online calculators and plotters, mathematical recreation and games
• Keywords: interactive mathematics, interactive math, server side interactivity, linear_algebra, linear_system, matrix, polynomials, variables,modelling | {"url":"https://wims.divingeek.com/wims/wims.cgi?lang=en&+module=U1%2Falgebra%2Fsysfind.en","timestamp":"2024-11-07T12:44:26Z","content_type":"text/html","content_length":"7897","record_id":"<urn:uuid:fb97abe4-1b3d-4e4c-916f-833f7ca21beb>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00582.warc.gz"} |
l_1 Trend Filtering
S.-J. Kim, K. Koh, S. Boyd, and D. Gorinevsky
SIAM Review, problems and techniques section, 51(2):339–360, May 2009.
The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on Hodrick-Prescott (H-P) filtering, a widely used method for
trend estimation. The proposed l1 trend filtering method} substitutes a sum of absolute values (i.e., an l1-norm) for the sum of squares used in H-P filtering to penalize variations in the estimated
trend. The l1 trend filtering method produces trend estimates that are piecewise linear, and therefore is well suited to analyzing time series with an underlying piecewise linear trend. The kinks,
knots, or changes in slope of the estimated trend can be interpreted as abrupt changes or events in the underlying dynamics of the time series. Using specialized interior-point methods, l1 trend
filtering can be carried out with not much more effort than H-P filtering; in particular, the number of arithmetic operations required grows linearly with the number of data points. We describe the
method and some of its basic properties, and give some illustrative examples. We show how the method is related to l1 regularization based methods in sparse signal recovery and feature selection, and
list some extensions of the basic method. | {"url":"https://stanford.edu/~boyd/papers/l1_trend_filter.html","timestamp":"2024-11-05T04:13:44Z","content_type":"application/xhtml+xml","content_length":"4661","record_id":"<urn:uuid:97b728a0-c2d9-44f3-bf60-def319af8881>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00274.warc.gz"} |
Solver that computes states and outputs for simulation
Model Configuration Pane: Solver
The Solver parameter specifies the solver that computes the states of the model during simulation and in generated code. In the process of solving the set of ordinary differential equations that
represent the system the model implements, the solver also determines the next time step for the simulation.
When you enable the Use local solver when referencing model parameter for a referenced model, the Solver parameter for the referenced model specifies the solver to use as a local solver. The local
solver solves the referenced model as a separate set of differential equations. While the top solver can be a fixed-step or variable-step solver, the local solver must be a fixed-step solver. For
more information, see Use Local Solvers in Referenced Models.
The software provides several types of fixed-step and variable-step solvers.
Variable-Step Solvers
A variable-step solver adjusts the interval between simulation time hits where the solver computes states and outputs based on the system dynamics and specified error tolerance. For most
variable-step solvers, you can configure additional parameters, including:
Fixed-Step Solvers
In general, fixed-step solvers except for ode14x and ode1be calculate the next step using this formula:
X(n+1) = X(n) + h dX(n)
where X is the state, h is the step size, and dX is the state derivative. dX(n) is calculated by a particular algorithm using one or more derivative evaluations depending on the order of the method.
For most fixed-step solvers, you can configure additional parameters, including:
auto (Automatic solver selection) (default) | discrete (no continuous states) | ...
In general, the automatic solver selection chooses an appropriate solver for each model. The choice of solver depends on several factors, including the system dynamics and the stability of the
solution. For more information, see Choose a Solver.
The options available for this parameter depend on the value you select for the solver Type.
Variable-Step Solvers
auto (Automatic solver selection)
The software selects the variable-step solver to compute the model states based on the model dynamics. For most models, the software selects an appropriate solver.
discrete (no continuous states)
Use this solver only for models that contain no states or only discrete states. The discrete variable-step solver computes the next simulation time hit by adding a step size that depends on the
rate of change for states in the model.
ode45 (Dormand-Prince)
When you manually select a solver, ode45 is an appropriate first choice for most systems. The ode45 solver computes model states using an explicit Runge-Kutta (4,5) formula, also known as the
Dormand-Prince pair, for numerical integration.
ode23 (Bogacki-Shampine)
The ode23 solver computes the model states using an explicit Runge-Kutta (2,3) formula, also known as the Bogacki-Shampine pair, for numerical integration.
For crude tolerance and in the presence of mild stiffness, the ode23 solver is more efficient than the ode45 solver.
ode113 (Adams)
The ode113 solver computes the model states using a variable-order Adams-Bashforth-Moulton PECE numerical integration technique.
The ode113 solver uses the solutions from several preceding time points to compute the current solution.
The ode113 solver can be more efficient than ode45 for stringent tolerances.
ode15s (stiff/NDF)
The ode15s solver computes the model states using variable-order numerical differentiation formulas (NDFs). NDFs are related to but more efficient than the backward differentiation formulas
(BDFs), also known as Gear's method.
The ode15s solver uses the solutions from several preceding time points to compute the current solution.
The ode15s solver is efficient for stiff problems. Try this solver if the ode45 solver fails or is inefficient.
ode23s (stiff/Mod. Rosenbrock)
The ode23s solver computes the model states using a modified second-order Rosenbrock formula.
The ode23s solver uses only the solution from the preceding time step to compute the solution for the current time step.
The ode23s solver is more efficient than the ode15s solver for crude tolerances and can solve stiff problems for which ode15s is ineffective.
ode23t (mod. stiff/Trapezoidal)
The ode23t solver computes the model states using an implementation of the trapezoidal rule.
The ode23t solver uses only the solution from the preceding time step to compute the solution for the current time step.
Use the ode23t solver for problems that are moderately stiff and require a solution with no numerical damping.
ode23tb (stiff/TR-BDF2)
The ode23tbsolver computes the model states using a multistep implementation of TR-BDF2, an implicit Runge-Kutta formula with a trapezoidal rule first stage and a second stage that consists of a
second-order backward differentiation formula. By construction, the same iteration matrix is used in both stages.
The ode23tb solver is more efficient than ode15s for crude tolerances and can solve stiff problems for which the ode15s solver is ineffective.
odeN (Nonadaptive)
The odeN solver uses a nonadaptive fixed-step integration formula to compute the model states as an explicit function of the current value of the state and the state derivatives approximated at
intermediate points.
The nonadaptive odeN solver does not adjust the simulation step size to satisfy error constraints but does reduce step size in some cases for zero-crossing detection and discrete sample times.
daessc (DAE solver for Simscape™)
The daessc solver computes the model states by solving systems of differential algebraic equations modeled using Simscape. The daessc solver provides robust algorithms specifically designed to
simulate differential algebraic equations that arise from modeling physical systems.
The daessc solver is available only with Simscape products.
Fixed-Step Solvers
auto (Automatic solver selection)
The software selects a fixed-step solver to compute the model states based on the model dynamics. For most models, the software selects an appropriate solver.
discrete (no continuous states)
Use this solver only for models that contain no states or only discrete states. The discrete fixed-step solver relies on blocks in the model to update discrete states
The discrete fixed-step solver does not support zero-crossing detection.
ode8 (Dormand-Prince)
The ode8 solver uses the eighth-order Dormand-Prince formula to compute the model state as an explicit function of the current value of the state and the state derivatives approximated at
intermediate points.
ode5 (Dormand-Prince)
The ode5 solver uses the fifth-order Dormand-Prince formula to compute the model state as an explicit function of the current value of the state and the state derivatives approximated at
intermediate points.
ode4 (Runge-Kutta)
The ode4 solver uses the fourth-order Runge-Kutta (RK4) formula to compute the model state as an explicit function of the current value of the state and the state derivatives.
ode3 (Bogacki-Shampine)
The ode3 solver computes the state of the model as an explicit function of the current value of the state and the state derivatives. The solver uses the Bogacki-Shampine Formula integration
technique to compute the state derivatives.
ode2 (Heun)
The ode2 solver uses the Heun integration method to compute the model state as an explicit function of the current value of the state and the state derivatives.
ode1 (Euler)
The ode1 solver uses the Euler integration method to compute the model state as an explicit function of the current value of the state and the state derivatives. This solver requires fewer
computations than a higher order solver but provides comparatively less accuracy.
ode14x (extrapolation)
The ode14x solver uses a combination of Newton's method and extrapolation from the current value to compute the model state as an implicit function of the state and the state derivative at the
next time step. In this example, X is the state, dX is the state derivative, and h is the step size:
X(n+1) - X(n)- h dX(n+1) = 0
This solver requires more computation per step than an explicit solver but is more accurate for a given step size.
ode1be (Backward Euler)
The ode1be solver is a Backward Euler type solver that uses a fixed number of Newton iterations and incurs a fixed computational cost. You can use the ode1be solver as a computationally efficient
fixed-step alternative to the ode14x solver.
Specify Solver Parameters Using Values Selected by Software
Open the model vdp.
mdl = "vdp";
To allow the software to select the solver to use for the model, specify the Type parameter as Fixed-step or Variable-step, and set the Solver parameter to auto. For this example, configure the
software to select a variable-step solver for the model.
1. To open the Configuration Parameters dialog box, on the Modeling tab, click Model Settings.
2. On the Solver pane, set the solver Type to Variable-step and the Solver parameter to auto (Automatic solver selection).
3. Click OK.
Alternatively, use the set_param function to set the parameter values programmatically.
set_param(mdl,"SolverType","Variable-step", ...
Simulate the model. On the Simulation tab, click Run. Alternatively, use the sim function.
As part of initializing the simulation, the software analyzes the model to select the solver. The status bar on the bottom of the Simulink Editor indicates the selected solver on the right. For this
model, the software selects the ode45 solver.
To view more information about the selected solver parameters, click the text in the status bar that indicates the selected solver. The Solver Information menu shows the selected solver and the
selected value for the Max step size parameter. For this simulation, the solver uses a maximum step size of 0.4.
If you want to lock down the solver selection and maximum step size, explicitly specify the solver parameter values. In the Solver information menu, click Accept suggested settings .
Alternatively, you can use the set_param function to specify the parameter values programmatically.
After you explicitly specify the parameter values, the solver information in the status bar and Solver information menu no longer indicate that the parameter values are automatically selected.
• The optimal solver balances acceptable accuracy with the shortest simulation time. Identifying the optimal solver for a model requires experimentation. For more information, see Choose a Solver.
• When you use fast restart, you can change the solver for the simulation without having to recompile the model. (since R2021b)
• The software uses a discrete solver for models that have no states or have only discrete states, even if you specify a continuous solver.
Recommended Settings
The table summarizes recommended values for this parameter based on considerations related to code generation.
Application Setting
Debugging No impact
Traceability No impact
Efficiency No impact
Safety precaution Discrete (no continuous states)
Programmatic Use
Parameter: SolverName or Solver
Type: string | character vector
Variable-Step Solver Values: "VariableStepAuto" | "VariableStepDiscrete" | "ode45" | "ode23" | "ode113" | "ode15s" | "ode23s" | "ode23t" | "ode23tb" | "odeN" | "daessc"
Fixed-Step Solver Values: "FixedStepAuto" | "FixedStepDiscrete" | "ode8" | "ode5" | "ode4" | "ode3" | "ode2" | "ode1" | "ode14x" | "ode1be"
Default: "VariableStepAuto"
Version History
Introduced before R2006a | {"url":"https://au.mathworks.com/help/simulink/gui/solver.html","timestamp":"2024-11-03T15:34:46Z","content_type":"text/html","content_length":"97606","record_id":"<urn:uuid:759be0e5-52ca-4cd7-b3c3-9e0d81195155>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00385.warc.gz"} |
PleistoDist Overview | David JX Tan
PleistoDist Overview
Island shape and distance metrics, normalised over Pleistocene time. A complete ground-up rebuild of PleistoDist for use with R.
PleistoDist is a tool for visualising and quantifying the effects of Pleistocene-era sea level change on islands over time. This tool comes packaged as series of R functions, for generating maps of
island extents for different Pleistocene-era sea levels, and calculating various inter-island and intra island distance and geomorphological metrics over time. This R package is a complete ground-up
rebuild of the original PleistoDist, which was written as an ArcMap plugin.
This package requires at least R v4.0.5 to function, and will automatically load the following dependencies:
To install and load this package in R, use the following commands:
You will need the following inputs in order to run PleistoDist:
• Bathymetry raster [.ASC] : PleistoDist requires an input bathymetry raster file in ASCII (.ASC) format to generate its outputs. Although PleistoDist should theoretically be able to use any type
of ASCII-formatted bathymetric map as input, this tool has been tested specifically with data from the General Bathymetric Chart of the Oceans (GEBCO: https://www.gebco.net). Locality-specific
bathymetric maps can be downloaded from https://download.gebco.net/.
• Source points [.SHP]: You will need to prepare a shapefile (.SHP format) of reference points that PleistoDist will use as sampling localities to calculate island shape parameters and inter-island
distance matrices. The shapefile should have a column titled ‘Name’ with unique identifiers for each point, otherwise PleistoDist will default to using the FID identifiers of each point. The
shapefile can be formatted in any map projection, since PleistoDist will reproject the shapefile to the user-specified projection for this analysis (see next point). Note, however, that the
reprojection process might result in points close to the shoreline ending up in the sea, so do be mindful of that.
• Map projection [EPSG code]: Because of the Earth’s spherical shape, we need to apply a map projection to accurately calculate straight-line distances between points. Users should specify a
projected coordinate system appropriate to the area being analysed using the projection’s associated EPSG code (https://epsg.org/home.html). Geographic coordinate system projections are not
recommended as those will result in distance matrices calculated in decimal degrees rather than distance units.
• Time cutoff [kya]: PleistoDist calculates the average distance between islands over a specific period of time. Users will have to specify an upper time bound (in thousands of years [kya]) for
their PleistoDist analysis, which can range from 0.1 kya to 3000 kya (i.e. 100 to 3,000,000 years ago). The lower time bound is fixed at the present day (0 kya). See the “How it works” section of
the README file for more details.
• Binning Mode and number of intervals: PleistoDist simplifies the distance over time calculation by binning either time or sea levels into a number of equal user-specified intervals. This allows
users to specify the coarseness of their analysis, with more intervals resulting in a more accurate and finer-grained analysis, although that will generate more output files and require a longer
computational time. Binning by time is recommended for beginner users since it is more intuitive and more accurate as well. See the “General PleistoDist Workflow” section of the README file for
more information on the difference between binning by time or by sea level.
General PleistoDist workflow
PleistoDist works by simplifying Pleistocene-era sea level change into discrete intervals, generating maps of island extents for each interval, calculating inter-island distances and metrics of
island shape for each interval, and performing a weighted average for each metric across all intervals. This section provides a brief overview of the PleistoDist workflow, and how each metric is
• Generating the interval file: The main role of the getintervals_time() and getintervals_sealvl() functions is to simplify reconstructed Pleistocene sea levels (by default from Bintanja and van de
Wal, 2008) into a series of discrete interval bins. These intervals can be calculated in two different ways: either by binning over time, or by binning over sea level, as is illustrated in Figure
1. Theoretically, both methods should be equally accurate when the number of intervals is very high, but for intermediate numbers of intervals, binning by time is likely to be a better measure
since it samples only one continuous section of the sea level curve per interval, and thus makes fewer assumptions about the shape of the curve for each interval. However, whether to bin
intervals by time or sea level will likely vary depending on spatial and temporal scale, as well as the spatial context of the analysis, and as such both binning modes are made available in
PleistoDist. The results of the binning process will be saved as an interval file in the output folder (see Figure 1), which will be used by subsequent modules for generating map outputs and
calculating distance matrices. Each row in the interval file corresponds to one interval, and the first interval is always set at present day, with a mean sea level of 0 m and a time interval of
0.1 kya.
Figure 1: PleistoDist provides two different methods for discretising sea level change, either by time (A) or by sea level (B). Both methods should yield similar results when the number of intervals
is very high, but will differ significantly for lower numbers of intervals. As this figure suggests, for a time cutoff of 200,000 years, 2 intervals not enough to capture the true variability of sea
level change over this time period. The results of the binning process will be written as a table to the interval file in the output folder.
• Generating maps of island extents: The makemaps() function is responsible for generating map outputs based on the interval bins contained in the interval file. This module reprojects the input
bathymetry raster into the user specified map projection using a bilinear resampling method, and generates raster and shapefile outputs of land extents based on the mean sea levels specified in
the MeanDepth column of the interval file. This function generates maps in three formats: ESRI shapefile format, a flat raster format (with no topography), and a topographic raster format that
preserves the original bathymetric elevations of each island pixel.
• Calculating island-to-island distances: PleistoDist provides 3 methods for calculating distances between islands. The pleistodist_centroid() function calculates the centroid-to-centroid distance,
while pleistodist_leastshore() calculates the shortest shore-to-shore distance between islands. The last distance metric, pleistodist_meanshore() is slightly more complicated, and estimates the
mean distance of every point on the source island shoreline facing the receiving island, to the receiving island shoreline. Because directionality matters in the case of the mean shore-to-shore
distance, the resultant distance matrix is asymmetrical, and pairwise distances from large to small islands will generally be greater than from small to large islands. The mean shore-to-shore
distance estimation method allows us to account for differences in shoreline availability between pairs of islands. For each pairwise combination of source points provided by the user,
PleistoDist selects the landmasses corresponding to the pair of points, and calculates the two distance measures between the landmasses. If both points are on the same island for that particular
interval, a distance of 0 is returned. And if one or both points are underwater during that particular interval, a value of NA is returned. Distance matrix outputs from this module are stored in
the output folder.
Figure 2: PleistoDist calculates three different distance measures between islands: the centroid-to-centroid distance, the least shore-to-shore distance, and the mean shore-to-shore distance
illustrated here with the Fijian islands of Viti Levu (left) and Gau (right). Note how the inter-island distances are asymmetric for the mean shore-to-shore distance.
• Calculating point-to-point distances: Unlike island-to-island distance metrics, which considers entire islands as the fundamental unit for calculating distances, the point-to-point distance
functions pleistodist_euclidean() and pleistodist_leastcost() calculate pairwise distances between the source points provided by the user (Figure 3). The function pleistodist_euclidean()
calculates the Euclidean distance between points (which is invariant over time and therefore only calculated once), while the pleistodist_leastcost() function calculates the least cost distance
between points for each interval (Figure 3). The resistance surface used for calculating the least cost distance is essentially a rasterised version of the shapefile for that interval, with areas
above sea level assigned a resistance value of 1, and areas underwater assigned a resistance value of 999,999. The least cost distance should thus minimise the amount of overwater movement
between points.
Figure 3: PleistoDist calculates two different distance measures between source points: the Euclidean distance between points (as the crow flies, invariant across all intervals), and the least cost
distance (which minimises overwater movement), illustrated here with the Fijian islands of Viti Levu and Gau.
• Calculating island area, perimeter, and surface area: In addition to calculating inter-island distances, PleistoDist can also calculate the shapes of islands over Pleistocene time. The three
functions, pleistoshape_area(), pleistoshape_perimeter(), and pleistoshape_surfacearea() are pretty self-explanatory, and they calculate the shape parameters of islands as identified by the
user-provided points shapefile, for each time/sea level interval defined in the interval file, as well as the weighted mean of each parameter based on the duration of each sea level/time
interval. PleistoDist will return a value of NA if the island is below sea level for that particular interval. Users can also use the pleistoshape_all() function to calculate all three shape
parameters simultaneously instead of running each subsidiary function in succession, which should save a fair bit of computational time.
• Estimating net migration between islands: MacArthur & Wilson (1967) describe a simple model for estimating the number of propagules dispersing between two islands, as given by the equation: $$ n_
{(1\rightarrow2)} = \frac{2\tan^{-1}(w_2/2d)}{360}\times \alpha A_{1}e^{(-d/\lambda)} $$ where $n_{1\rightarrow2}$ is the number of propagules of a particular species successfully dispersing from
island 1 to island 2, w[2] is the width of island 2 relative to island 1, d is the distance between the two islands, α is the population density of the propagule population on island 1, A[1] is
the area of island 1, and λ is the mean overwater dispersal distance of the propagule. While this equation is difficult to solve since it requires estimates for both α and λ, if we take the ratio
of $n_{1\rightarrow2}$ to $n_{2\rightarrow1}$ – the net migration between islands 1 and 2 – we can cancel out both the α and the exponential terms, and reduce the relationship to $$\frac{n_{(1\
rightarrow2)}}{n_{(2\rightarrow1)}}=\frac{\tan^{-1}(w_{2}/2d)A_{1}}{\tan^{-1}(w_{1}/2d)A_{2}}$$ which can easily be solved using the outputs generated by PleistoDist. As such, if there is a net
movement of propagules from island 1 to island 2, $\frac{n_{(1\rightarrow2)}}{n_{(2\rightarrow1)}}>1$, whereas if there is a net movement of propagules from island 2 to island 1, then $\frac{n_
{(1\rightarrow2)}}{n_{(2\rightarrow1)}}<1$. The net inter-island migation ratio can be estimated using the function pleistodist_netmig(), which automatically checks to see if the appropriate
distance and shape matrices exist, and calculates the net inter-island migration for each time/sea level interval as specified in the interval file, and calculate a weighted mean net migration
ratio across all intervals. Note that since the mean shore-to-shore distance matrix is asymmetrical, and we assume that the inter-island distance is symmetrical for this particular calculation,
PleistoDist will use as d the mean of $d_{(1\rightarrow2)}$ and $d_{(2\rightarrow1)}$.
• Estimating island visibility: Inter-island dispersal can also be affected by the visibility of a destination island relative to an observer on an origin island. PleistoDist therefore provides a
rudimentary way of estimating inter-island visibility using the function pleistodist_visibility(). This function works by first estimating the horizon distance of the observer point, taking into
account both the ground elevation of the observer as well as the height of the observer above the ground, using the equation: $$\text{horizon distance}=R_{e} \times \sec^{-1}(1+\frac{h_{\text
{total}}}{R_{e}})$$, where $h_{\text{total}}$ is the sum of the ground elevation and the height of the observer above the ground, and $R_{e}$ is the Earth’s radius for that particular
inter-island great circle arc, accounting for sea level change, which can be approximated by the equation $$R_{e}=\text{sea level relative to present day}+\frac{a(1-e^{2}\cos^{2}\phi\cos^{2}\
theta)^{\frac{3}{2}}}{(1-e^{2}[1-\sin^{2}\phi\cos^{2}\theta])^{\frac{1}{2}}}$$, where $a$ is Earth’s equatorial radius, $e^{2}$ is the squared eccentricity of Earth. $\phi$ the approximate
azimuthal angle between the line connecting the observer and the destiation island and Earth’s North-South axis, and $\theta$ the parametric latitude of the observer. If the destination island
falls within the radius of the horizon distance, PleistoDist calculates whether the horizon line intersects with the destination island (Figure 4). To account for the occluding effects of
mountains and other geographical features, the function then performs a viewshed analysis to estimate the area of the island that is within direct line of sight from the observer (Figure 4). Do
note, however, that due to the sampling method applied for the viewshed analysis, the calculated visible island area is likely to vary across multiple runs, so it’s best to treat the outputs of
this function as an estimate rather than an absolute value. Note also that this function is unable to account for visibility-reducing weather effects such as fog, haze, or mist.
Figure 4: PleistoDist estimates the visibility of a destination island relative to an observer on an origin island by calculating the horizon line (which defines the theoretical maximum viewing
distance relative to the observer), and performing a viewshed analysis to estimate the visible non-occluded area of the destination island.
PleistoDist assumes that the bathymetry of the area of interest is constant throughout the time period being modelled. This is an important assumption to bear in mind since bathymetry can be affected
by tectonic and volcanic activity. In the provided example of the Fijian archipelago, for instance, the island of Taveuni (see Figure 2) is likely to have emerged around 700,000 years ago (Cronin &
Neall, 2001), so analyses of Taveuni with a cutoff time close to and exceeding 700 kya are unlikely to provide meaningful results. In addition, PleistoDist is unable to account for the effect of
proximate ice sheets on the bathymetry and sea level of the area of interest, and is thus likely to be less accurate at very high or very low latitudes. It is also possible that the default global
sea level reconstruction used in the vanilla version of PleistoDist may not be accurate for particular areas of interest, in which case users are advised to use a more accurate sea level
reconstruction specific to the area of interest, bearing in mind to be aware of the column names in the new sealvl.csv file.
Further modifications/extensions
Advanced users should be able to modify the PleistoDist source code to meet their specific needs. Here are some suggestions:
• Sea level reconstruction: By default, PleistoDist uses the Pleistocene sea level reconstruction of Bintanja & van de Wal (2008), which is based on an inverse model using the ratio of marine
Oxygen-18 to Oxygen-16 isotopes. This sea level reconstruction is stored as a pre-loaded R variable, and can be replaced with your preferred sea level reconstruction (e.g. from Spratt and
Lisiecki, 2016). If you do swap out the sea level reconstruction, be sure to check and modify the getintervals.R file to make sure that this doesn’t break PleistoDist.
• Time lower bound: Vanilla PleistoDist fixes the lower time bound at the present day. Setting a different lower time bound should be relatively simple and can be achieved by modifying the
getintervals.R file.
• Bintanja, R., & van de Wal, R. S. W. (2008). North American ice-sheet dynamics and the onset of 100,000-year glacial cycles. Nature, 454(7206), 869–872. https://doi.org/10.1038/nature07158
• Cronin, S. J., & Neall, V. E. (2001). Holocene volcanic geology, volcanic hazard, and risk on Taveuni, Fiji. New Zealand Journal of Geology and Geophysics, 44(3), 417–437. https://doi.org/10.1080
• Darwell, C. T., Fischer, G., Sarnat, E. M., Friedman, N. R., Liu, C., Baiao, G., Mikheyev, A. S., & Economo, E. P. (2020). Genomic and phenomic analysis of island ant community assembly.
Molecular Ecology, 29(9), 1611–1627. https://doi.org/10.1111/mec.15326
• MacArthur R. H., & Wilson E. O. (1967). The Theory of Island Biogeography. Princeton, N.J.: Princeton University Press, 203 p.
• Spratt, R. M., & Lisiecki, L. E. (2016). A Late Pleistocene sea level stack. Climate of the Past, 12(4), 1079–1092. https://doi.org/10.5194/cp-12-1079-2016 | {"url":"https://davidbirdtan.com/pleistodist/","timestamp":"2024-11-07T01:22:01Z","content_type":"text/html","content_length":"35972","record_id":"<urn:uuid:147bdf4d-0ce1-4312-b799-db7f22d52fda>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00815.warc.gz"} |
Dirac spin liquid as an "unnecessary" q
SciPost Submission Page
Dirac spin liquid as an "unnecessary" quantum critical point on square lattice antiferromagnets
by Yunchao Zhang, Xue-Yang Song, T. Senthil
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users): Xue-Yang Song · Yunchao Zhang
Submission information
Preprint Link: https://arxiv.org/abs/2404.11654v2 (pdf)
Date submitted: 2024-06-11 16:52
Submitted by: Song, Xue-Yang
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
• Condensed Matter Physics - Theory
Specialties: • High-Energy Physics - Theory
Approach: Theoretical
Quantum spin liquids are exotic phases of quantum matter especially pertinent to many modern condensed matter systems. Dirac spin liquids (DSLs) are a class of gapless quantum spin liquids that do
not have a quasi-particle description and are potentially realized in a wide variety of spin $1/2$ magnetic systems on $2d$ lattices. In particular, the DSL in square lattice spin-$1/2$ magnets is
described at low energies by $(2+1)d$ quantum electrodynamics with $N_f=4$ flavors of massless Dirac fermions minimally coupled to an emergent $U(1)$ gauge field. The existence of a relevant,
symmetry-allowed monopole perturbation renders the DSL on the square lattice intrinsically unstable. We argue that the DSL describes a stable continuous phase transition within the familiar Neel
phase (or within the Valence Bond Solid (VBS) phase). In other words, the DSL is an "unnecessary" quantum critical point within a single phase of matter. Our result offers a novel view of the square
lattice DSL in that the critical spin liquid can exist within either the Neel or VBS state itself, and does not require leaving these conventional states.
Current status:
Has been resubmitted
Reports on this Submission
This paper proposes the existence of quantum critical points in the phase diagram of lattice anti-ferromagnets described by $N_f = 4$ quantum electrodynamics in $2+1$ dimensions. An interesting
aspect of this proposal is that the quantum critical points are between two identical phases of matter, a situation the authors refer to as that of an "unnecessary quantum critical points." I believe
this is an interesting proposal, and that the paper should be published, but I find various parts of it a bit hard to follow in the current form. I have some concrete suggestions for how to improve
it below.
Requested changes
1. In I.B, first paragraph: it would be helpful to spell out what $G_{UV}$ is explicitly, in particular to define ${\cal T}$ and to list the generators of the square lattice space group. I assume
these generators are listed in Table I in Appendix A, but a brief explanation of what these generators correspond to (lattice translations, rotations, etc.) would be very helpful. Maybe a drawing of
the lattice would be useful.
2. Paragraph containing eq. (5): it would be helpful to include a brief argument why the operator $\bar \psi_i \psi_j \Phi$ has dimension $\Delta_1 + 2 \sqrt{2}$ in the large $N_f$ limit. I assume it
is because the lowest excitation of $\psi$ in the unit charge monopole background on the sphere has energy $\sqrt{2}$ (in units of the inverse radius of the sphere), but it would be good to see this
explained more clearly.
3. It would be helpful to connect the discussion at the end of Section II (paragraphs 4, 5, 6) about $q=0$ operators to tables II and III in Appendix A. In particular, which operators in these tables
are fermion bilinears, which operators are quartic in the fermions, etc.?
4. In the second paragraph of Section III, what does it mean for the two sets of 5 monopole operators to "couple" to four fermion terms or to fermion bilinear mass terms? Does the word "couple" mean
that there's a term in the Lagrangian that contains products of monopole operators and two-fermion or four-fermion operaators? This doesn't make much sense, so the authors must mean something else by
the word "couple". It would be great to clarify this point.
5. Section III, end of 2nd paragraph: In the sentence "We label these operators as $n^a$ with $a = 1, \ldots , 5$", what do the words "these operators" refer to? Do they refer to the monopole
operators or to the fermion bilinears?
6. If the $n^a$ are fermion bilinears (as it is suggested in the 4th paragraph of Section III), which $SO(6)$ representation are the operators $n^a$ part of when $\lambda = 0$? Are they part of the $
{\bf 15}$, which under the decomposition $SO(6) \to SO(5)$ becomes ${\bf 10} + {\bf 5}$, with the ${\bf 5}$ being the $n^a$? It would be great to clarify this point.
7. Section IV, second paragraph: it would be nice to give more details supporting the idea that, in the absence of monopoles, the fermions of QED$_3$ are the $2 \pi$ vortices of an order parameter
carrying charge-1 under $U(1)_\text{top}$. Where does this come from? Why are such vortices not gauge-invariant? Why do they transform in a projective representation of $SO(6)$ (namely the ${\bf 4} +
\bar {\bf 4}$)? I think more explanation is needed.
8. When writing $\bar \psi M \psi$ in the 2nd paragraph of Section IV, do the authors mean that $M$ is an $\mathfrak{su}(4)$ matrix instead of an $SU(4)$ matrix (i.e.~Lie algebra vs.~group element)?
This should be made clear.
9. In the second paragraph of Section IV, regarding the second to last sentence, if I'm understanding it correctly: why is it that if the leading operator that breaks a symmetry has a lower scaling
dimension than the leading operator that doesn't break the symmetry, then one should expect the symmetry to be broken? It would be great to have an explanation, or to clarify the statement if I
misunderstood it.
10. Is there a difference between ${\bf n}$ in eq. (8) and $\hat n$ in Section IV.A? If there is, the authors should explain the difference; if not, it would be better I think to use the same
notation everywhere.
11. After eq. (10) it is mentioned that the operator in (10) is part of the symmetric traceless representation of SO(6). Which symmetric traceless representation? The rank-two one, namely the ${\bf
20}'$? It would be great to make this explicit.
12. Shouldn't eq. (10) have $2(n_1^2 + n_2^2 + n_3^2) - 3 (n_4^2 + n_5^2)$ instead of $n_1^2 + n_2^2 + n_3^2 - n_4^2 - n_5^2$ so that it is a traceless polynomial?
13. Third paragraph of Section IV.A: what is the evidence that the RG flow diagram is that in Figure 2? This fact is just stated with no evidence provided as far as I can tell, so I think the authors
should improve the explanation here.
14. Throughout the paper, the authors should change the notation for the rank-two symmetric traceless tensor representation of $SO(6)$ from ${\bf 20}$ to the more common notation ${\bf 20}'$. (The
irrep the authors refer to is usually called the ${\bf 20}'$; the ${\bf 20}$ is usually a different irrep of $SU(4)$.)
15. Appendix A: It would be good to explain in more detail where the assumptions for the triangular and Kagome lattices come from. As an alternative, one can remove the mention of these lattices from
the Appendix since it does not seem immediately relevant to the present paper.
16. It would be good to explain in more detail how Table I was derived. It seems very mysterious in its current form.
17. Appendix A, 2nd paragraph, regarding ''$({\bf 20}, 0)$ and $({\bf 84}, 0)$ are singlets in any lattice QED$_3$ simulation.'' Do the authors mean "are singlets" or "contain singlets"?
18. The allowed operators in the third columns of Tables II, III, IV do not obey the right symmetry and tracelessness conditions. For example, in Table II, the operator $O_1^\dagger O_1 + O_3^\dagger
O_3$ is not traceless; instead, the traceless operator should be $O_1^\dagger O_1 + O_3^\dagger O_3 - \frac 12 (O_2^\dagger O_2 + O_4^\dagger O_4 + O_5^\dagger O_5 + O_6^\dagger O_6)$.
19. If I understand correctly, I think the "allowed" operators in the third columns of Tables II, III, IV are not all the allowed operators. For instance, in the bottom right box of Table II, it
seems to me that we can also have $O_2^\dagger O_2 - \text{trace}$, $O_4^\dagger O_4 - \text{trace}$, etc. It would be great if the authors could comment on this point (or correct me if I'm wrong).
20. At the end of the Appendix, various estimates for scaling dimensions are given. The authors should give a reference or explain how they are derived.
21. A general comment: the authors propose that $N_f = 4$ QED$_3$ can be found in the phase diagram of square lattice anti-ferromagnets. But as far as I can tell, the arguments in Sections III and IV
involve continuum field theory, in particular QED$_3$ and its deformations. Do I understand correctly that the continuum field theory picture is a good approximation only if the parameters $\lambda$
and $\kappa$ are close to the origin in Figure 2? If so, is it obvious that this range of parameters can be accessed from the lattice Hamiltonian? If not, where is the continuum approximation valid?
The manuscript proposed Dirac spin liquid (DSL) as an unnecessary quantum critical point in square lattice antiferromagnets. The main idea is based on the conjecture that, deforming DSL with the
relevant $2\pi$ monopole will trigger a RG flow towards the SO(5) DQCP, which will further flow to an ordered phase due to some dangerously irrelevant operator. Therefore, DSL can in principle appear
as a critical point, inside the same ordered phase, i.e. Neel order or valence bond solid.
The proposal is interesting, and would be helpful for future experimental or numerical search of DSL inside the more conventional ordered phase. I am happy to recommend this paper.
Requested changes
1. In the second paragraph of Sec. IV, the authors wrote "Then the fermions of QED3 can be viewed as the basic $2\pi$ vortices...". I am not sure I understand this statement, maybe it is good to
elaborate it a bit.
2. Page 6, first paragraph, there is a typo, "The QED3 CFT will then describe a sexond..." should be second.
Publish (easily meets expectations and criteria for this Journal; among top 50%)
This paper deals argues that QED$_3$ with $N_f=4$ fermions can be realized as an "unnecessary" critical point sitting within the Neel phase of 2+1 D quantum antiferromagnets on the square lattice
(and also within the VBS phase). "Unnecessary" here means that on both sides of the codimension one critical manifold we find the same phase. Their argument starts by identifying the relevant
operator which has to be tuned at criticality (charge-1 monopole). Then they argue that for any size of the perturbing monopole coupling the RG flow terminates in the same theory, SO(5) sigma model
with a WZW term, Eq. (8). Then they discuss how this picture changes in presence of (dangerously) irrelevant perturbations arising from the microscopic symmetry.
This is an interesting paper and I think it should be published. I have however a few questions and requests.
- Consider the RG flows (6) with the irrelevant terms ... set to zero. Is the RG flow with $\lambda$ positive and negative supposed to be identical for all distances or only the IR fixed point (8) is
supposed to be identical? The first possibility would be realized e.g. when perturbing the Ising model by the magnetic field perturbation, as is trivial to show since the sign of the coupling is
flipped by a Z2. The second possibility is much more nontrivial. Whichever it is, it would be worth pointing out explicitly.
- p.3 last but one paragraph "hence must be irrelevant". "Must be" sounds confusing here. Can the authors rephrase it more explicitly, e.g. that they believe it to be irrelevant based on this
evidence and you will assume it so? (This is in line with the description of $\Delta_2$ above)
- p.4 second paragraph. "One set of 5 will couple to four fermion terms" "orthogonal set of 5 operators that couple to the adjoint" I don't understand what the authors mean by these phrases. Could be
some jargon that I'm not aware of. I'd be grateful for more details here.
- p.5 first paragraph "has slower correlations" Can the authors explain what "slower" means here and why this implies the expectation that flavor symmetry is broken?
- second column "supported by searches using the conformal bootstrap". Here and elsewhere the authors demonstrate familiarity with the conformal bootstrap results, which is great. Can they provide
some references here, and they do in other places of the paper?
- p.6 sexond->second
- Appendix A. Are results in tables I,II,III,IV,V new? A citation or , alternatively, at least some details on the derivation would be needed.
- Ref. [52] - "to a tricritical point" Several recent papers providing evidence for tricritical point include 2405.06607, 2405.04470, 2307.05307. I believe a reference these and other relevant works
here would be appropriate, to properly balance the cited evidence for the complexed fixed points. | {"url":"https://scipost.org/submissions/2404.11654v2/","timestamp":"2024-11-12T22:52:50Z","content_type":"text/html","content_length":"47556","record_id":"<urn:uuid:3d849f19-3a8b-4dc2-82cc-a001fd7cc595>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00542.warc.gz"} |
A Plus B (Hard)
Submit solution
Points: 12
Time limit: 2.5s
Memory limit: 64M
Allowed languages
ALGOL 68, Assembly, Brain****, C, C++, COBOL, Forth, Fortran, Java, Lua, Text, Turing
's teacher realized that he was cheating, and was using the code you wrote to save his marks. So, the math teacher decided that he will mess up your program by using numbers larger than . In fact, he
will give problems involving addition of -digit numbers as punishment. However, is once again on top — he has promised you a "reward" if you help him again. You suspect it might not be anything more
than 12 points, but you still have your hopes up...
Input Specification
The first line will contain an integer , the number of addition problems needs to do. The next lines will each contain two space-separated integers with up to digits in decimal, the two integers
needs to add. will never be greater than .
Warning: the test cases are a lot harder than the sample.
Output Specification
Output lines, the solutions to the addition problems in order.
Sample Input
226077045628835347875 -572260769919042128358
-803119834418378628674 236083700054616110639
-435599336891761067707 451767479989987922363
Sample Output
□ The goal is to implement your own version of the biginteger class.
• I'm pretty sure half the top solutions are just bigint classes copied and pasted from other sources. Like, if it was written just for this problem, why would they implement abs, and never use it?
Or multiply?
□ commented on Jan. 30, 2021, 2:52 a.m. ← edited
String ans = "This is " + "slow";
• Why is BF not an allowed language for this question?
The only practical way to solve even the original "A Plus B" in BF would require you to implement your own algorithm for addition of arbitrary-length integers.
• Not sure why I'm WA on two cases. On case #2, the input is 7808787 -2084742, and it WA because it's outputting 5403285 (wrong), but when I run the same input locally, it outputs 5724045 (right).
I have no idea why this is happening. I thought it was because some variables weren't being reset, but all of the variables used in calculations are local. Any ideas?
□ commented on Oct. 29, 2017, 1:47 p.m. ← edited
Unfortunately, using a double is not sufficient enough (nor accurate enough) to store a digit number. Try finding another way you can store the digits (it may involve creating your own data
☆ cough cough string cough cough
• Can someone tell me what I'm doing wrong? Or at least give a case to test?
□ commented on Oct. 22, 2017, 9:25 p.m. ← edited
-99 99
99 -99
-100 99
100 -99
• Can the two integers have leading zeros?
• commented on March 30, 2017, 3:16 p.m. ← edit 2
□ commented on March 30, 2017, 5:01 p.m. ← edit 2
The difficulty of this in Python is the same as the regular aplusb (i.e. not worth 12 points).
• commented on Jan. 21, 2017, 11:15 p.m. ← edited
□ commented on Jan. 22, 2017, 12:00 a.m. ← edited
Because copy/pasting code from Java's BigInteger implementation was never intended as a correct solution to this problem. BigInteger is explicitly disallowed, so why would copypasta of it be
any different?
• commented on Aug. 28, 2016, 9:36 p.m. ← edited
Java is now allowed for this problem, though use of the BigInteger and BigDecimal classes is disallowed.
□ commented on Oct. 8, 2017, 9:39 a.m. ← edited | {"url":"https://dmoj.ca/problem/aplusb2","timestamp":"2024-11-05T13:06:04Z","content_type":"text/html","content_length":"97429","record_id":"<urn:uuid:0889d539-2bcb-4bc0-a37d-36f7302e216b>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00516.warc.gz"} |
Degree Sequence
Out Degree Sequence And In Degree Sequence
Out-Degrees and In-Degrees of a Vertex
Definition: For a directed graph $G = (V(G), E(G))$ and a vertex $x_1 \in V(G)$, the Out-Degree of $x_1$ refers to the number of arcs incident from $x_1$. That is, the number of arcs directed away
from the vertex $x_1$. The In-Degree of $x_1$ refers to the number of arcs incident to $x_1$. That is, the number of arcs directed towards the vertex $x_1$.
We denote the out-degree of a vertex $x_1$ by the notation $\mathrm{outdeg}(x_1)$ while we denote the in-degree of the same vertex by the notation $\mathrm{indeg}(x_1)$.
For example, let's look at the following directed graph.
In the following graph above, the out-degrees of each vertex are in blue, while the in-degrees of each vertex are in red.
Out-Degree Sequence and In-Degree Sequence of a Graph
Definition: For a directed graph $G = (V(G), E(G))$, the Out-Degree Sequence is a sequence obtained by ordering the out-degrees of all vertices in $V(G)$ in increasing order. The In-Degree Sequence
is a sequence obtained by ordering the in-degrees of all vertices in $V(G)$ in increasing order.
From the graph earlier, the out-degree sequence (blue degrees) is $(0, 1, 1, 1, 2, 3)$, while the in-degree sequence (red degrees) is $(0, 1, 1, 2, 2, 2)$.
Remark: The degree sequence of a graph $G$ that is NOT directed has the degrees of each vertex ordered in decreasing order, while the degree sequence of a graph $G$ that is directed has the degrees
of each vertex ordered in increasing order. | {"url":"http://mathonline.wikidot.com/out-degree-sequence-and-in-degree-sequence","timestamp":"2024-11-06T08:30:10Z","content_type":"application/xhtml+xml","content_length":"15693","record_id":"<urn:uuid:6392e1e4-35f3-4276-81f9-05a79d9a9101>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00335.warc.gz"} |
Euler Ancestral
Euler Ancestral is a sampler in text-to-image diffusion models that is based on the Euler method for solving differential equations. It is a fast and efficient sampler that can often generate good
outputs in 20-30 steps.
Euler Ancestral works by sampling noise from the diffusion process and then subtracting it from the image. However, unlike other samplers, Euler Ancestral also adds some random noise back to the
image at each step. This helps to prevent the image from becoming too blurry and to generate more diverse images.
Euler Ancestral is a good choice for applications where speed and efficiency are important. It is also a good choice for applications where diversity is important, such as generating creative images
or generating images that are different from each other.
Here are some examples of how Euler Ancestral can be used to generate different types of images:
• A simple image, such as a black cat sitting on a red couch: Euler Ancestral can generate this type of image in a few dozen steps.
• A more complex image, such as a realistic portrait of Albert Einstein: Euler Ancestral may require more steps to generate this type of image, but it can still generate good results in a
reasonable amount of time.
• A creative image, such as a painting of a cat in the style of Pablo Picasso: Euler Ancestral is a good choice for generating creative images, as it can produce a variety of different outputs from
the same text prompt.
Overall, Euler Ancestral is a versatile and powerful sampler for text-to-image diffusion models. It is a good choice for applications where speed, efficiency, and diversity are important. | {"url":"https://feedsee.com/aiw/Euler_Ancestral/","timestamp":"2024-11-09T06:06:54Z","content_type":"text/html","content_length":"3753","record_id":"<urn:uuid:b4343ac3-042d-480b-96e0-30a7ef0b378e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00731.warc.gz"} |
Видеотека: I. Halacheva, The periplectic Lie superalgebra from a diagrammatic perspective
Аннотация: One approach to the study of the representation theory of the general linear Lie algebra is to look at the endomorphism algebras of tensor powers of the vector representation and of bigger
tensor products with more general representations. The classical and higher Schur-Weyl dualities relate these algebras to the symmetric group and the degenerate affine Hecke algebra respectively, and
have been subsequently generalized to other types, such as the symplectic and special orthogonal Lie algebras. We study the periplectic Lie superalgebra p(n) and define diagrammatically the affine VW
supercategory sVW, which generalizes the Brauer supercategory and maps to the category of p(n)-representations. This functor allows us to obtain a basis for the morphism spaces of sVW as well as
study the representation theory of p(n).
Язык доклада: английский | {"url":"https://m.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=19120","timestamp":"2024-11-12T00:36:52Z","content_type":"text/html","content_length":"7228","record_id":"<urn:uuid:c7b49371-c378-450a-b5ea-9143ce358c2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00886.warc.gz"} |
10.12.19 14.11.19 Kathlén Kohn (KTH Royal Institute of Technology, Stockholm)
The geometry of neural networks
A fundamental goal in the theory of deep learning is to explain why the optimization of the loss function of a neural network does not seem to be affected by the presence of non-global local minima.
Even in the case of linear networks, the existing literature paints a purely analytical picture of the loss, and provides no explanation as to *why* such architectures exhibit no bad local minima. We
explain the intrinsic geometric reasons for this behavior of linear networks. For neural networks in general, we discuss the neuromanifold, i.e., the space of functions parameterized by a network
with a fixed architecture. For instance, the neuromanifold of a linear network is a determinantal variety, a classical object of study in algebraic geometry. We introduce a natural distinction
between pure critical points, which only depend on the neuromanifold, and spurious critical points, which arise from the parameterization. This talk is based on joint work with Matthew Trager and
Joan Bruna.
23.10.19 Věra Kůrková (Institute of Computer Science, Czech Academy of Sciences, Czech Republic)
Lower Bounds on Complexity of Shallow Networks
Although originally biologically inspired neural networks were introduced as multilayer computational models, shallow networks have been dominant in applications till the recent renewal of interest
in deep architectures. Experimental evidence and successful applications of deep networks pose theoretical questions asking: When and why are deep networks better than shallow ones? This lecture will
present some probabilistic and constructive results showing limitations of shallow networks. It will show how geometrical properties of high-dimensional spaces imply probabilistic lower bounds on
network complexity. Bounds depending on covering numbers of dictionaries of computational units will be derived and combined with estimates of sizes of some common dictionaries used in
neurocomputing. Probabilistic results will be complemented by constructive ones built using Hadamard matrices and pseudo-noise sequences. The results will be illustrated by an example of a class of
functions which can be computed by two-hidden-layer perceptron networks of considerably smaller model complexities than by networks with only one hidden layer. Connections with the No Free Lunch
Theorem and the central paradox of coding theory will be discussed.
18.09.19 Vladimir Temlyakov (University of South Carolina)
Supervised learning and sampling error of integral norms in function classes
In this talk we will discuss how known results from two areas of research -- supervised learning theory and numerical integration -- can be used in sampling discretization of the square norm on
different function classes.
28.05.19 Nicolas Garcia Trillos (Department of Statistics, University of Wisconsin-Madison, USA)
The use of geometry to learn from data, and the learning of geometry from data.
In this talk I will explore from different perspectives the duality between geometry and data, and mostly focus on two of them. First, ideas from geometry and analysis have inspired a variety of
popular procedures for supervised and semi supervised learning tasks, an example of which is the so called spectral clustering algorithm, where for a given data set $\mathit{X}=\{x_1, \dots, x_n\}$
with a weighted graph structure $\Gamma= (\mathit{X},W)$ on $\mathit{X}$, one uses the spectrum of an associated graph Laplacian to find a meaningful partition of the data set. Conversely, one can
ask whether such ML algorithms are more concretely linked with the geometric ideas that inspired them: If the data set consists of samples from a distribution supported on a manifold (or at least
approximately so), and the weights depend inversely on the distance between the points, do these procedures converge as the number of samples goes to infinity towards analogous procedures at the
continuum level?. I will explore this question mathematically using ideas from the calculus of variations, PDE theory, and optimal transport. Then, I will discuss a second perspective: what is the
impact of the "choice of geometry" in learning, and how, in the presence of labeled data, this can be related to the inverse problem of learning geometry from data?
10.04.19 Gabriel Peyré (CNRS and Ecole Normale Supérieure, Paris, France)
Computational Optimal Transport for Data Sciences
Optimal transport (OT) has become a fundamental mathematical tool at the interface between calculus of variations, partial differential equations and probability. It took however much more time for
this notion to become mainstream in numerical applications. This situation is in large part due to the high computational cost of the underlying optimization problems. There is a recent wave of
activity on the use of OT-related methods in fields as diverse as computer vision, computer graphics, statistical inference, machine learning and image processing. In this talk, I will review an
emerging class of numerical approaches for the approximate resolution of OT-based optimization problems. This offers a new perspective for the application of OT in imaging sciences (to perform color
transfer or shape and texture morphing) and machine learning (to perform clustering, classification and generative models in deep learning). More information and references can be found on the
website of our book "Computational Optimal Transport" https://optimaltransport.github.io/
07.03.19 Stefania Petra (Universität Heidelberg)
Compressed Sensing - From Theory To Practice
Compressed sensing (CS) is a new sampling theory and has become a major research direction in applied mathematics in the last 10 years. The key idea of CS for addressing the big data problem is to
avoid sampling data that can be recovered afterwards. However, mathematical recovery guarantees depend on assumptions that are often too strong in practice. The extension of the mathematical theory
as well as the development of new applications in various fields are the subject of many current research activities in the field. The talk will highlight some of the challenges of bridging the gap
between theory and practicality of CS.
14.02.19 Felix Krahmer (Technische Universität München)
Blind deconvolution with randomness - convex geometry and algorithmic approaches
Blind deconvolution problems are ubiquitous in many areas of imaging and technology and have been the object of study for several decades. Recently, inspired by the theory of compressed sensing, a
new viewpoint has been introduced, motivated by applications in wireless application, where a signal is transmitted through an unknown channel. Namely, the idea is to randomly embed the signal into a
higher dimensional space before transmission. Due to the resulting redundancy, one can hope to recover both the signal and the channel parameters. In this talk we will discuss recovery guarantees for
this problem. In this talk, we will focus on convex approaches based on lifting as they have first been studied by Ahmed et al. (2014). We show that one encounters a fundamentally different geometric
behavior as compared to generic bilinear measurements. In addition, we will review recent progress on the study of efficient nonconvex recovery methods. This talk is based on joint work with Dominik
Stöger (TUM).
28.01.19 Nils Bertschinger (Frankfurt Institute for Advanced Studies - FIAS, Germany)
A geometric structure underlying stock correlations
Estimating covariances between financial assets plays an important role in risk management and portfolio allocation. Here, we show that from a Bayesian perspective market factor models, such as the
famous CAPM, can be understood as linear latent space embeddings. Based on this insight, we consider extensions allowing for non-linear embeddings via Gaussian processes and discuss some
applications. In general, all these models are unidentified as the choice of coordinate frame for the latent space is arbitrary. To remove this symmetry we reparameterize the factor loadings in terms
of an orthogonal frame and its singular values and provide an efficient implementation based on Householder transformations. Finally, relying on results from random matrix theory we derive the
parameter distribution corresponding to a Gaussian prior on the factor loadings. | {"url":"https://www.mis.mpg.de/de/events/series/mathematics-of-data-seminar","timestamp":"2024-11-08T08:25:45Z","content_type":"text/html","content_length":"253304","record_id":"<urn:uuid:2b4c4c2a-f865-437f-b8bc-88f77e2bae40>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00619.warc.gz"} |
What Is The Classification Of Radon?
To what level does radon fall under the categorization system?
Classification: Radon is a noble gas and a nonmetal
Color: colorless
Atomic weight: (222), no stable isotopes
State: gas
Melting point: -71 o C, 202 K
Radon is a chemical element with the symbol Rn and the atomic number 86, and it is found in the atmosphere. Radon, which is classified as a noble gas, is a gas that exists at normal temperature.
What is the atomic number of radon?
Radon is a chemical element with the atomic symbol Rn, the atomic number 86, and the atomic weight 222.0. Radon is a radioactive gas that occurs naturally in the environment and has no odor or
flavor. It is created as a result of the radioactive decay of uranium in the environment. Uranium may be found in trace levels in nearly all rocks and soils.
What is radon (Rn)?
• Radon is a naturally radioactive element with the atomic symbol Rn and the atomic number 86.
• It is found in the environment.
• It is a member of the noble gas family that may be found in soil and is emitted as a byproduct of the decay of radioactive elements.
• Radon is a chemical element with the atomic symbol Rn, the atomic number 86, and the atomic weight 222.0.
• Radon is a radioactive gas that occurs naturally in the environment and has no odor or flavor.
What is radon and how is it formed?
• Radon is a radioactive gas that occurs naturally in the environment and has no odor or flavor.
• It is created as a result of the radioactive decay of uranium in the environment.
• Uranium may be found in trace levels in nearly all rocks and soils.
• It progressively degrades into other compounds, such as radium, which degrades further into radon, which is a radioactive gas.
• Radon is a radioactive gas that undergoes radioactive decay as well.
What is the best radon level measurement?
• The Becquerels per cubic meter (Bq/m3) unit of measurement for radon is the recommended unit of measurement.
• In radioactive disintegration, one Becquerel equals one radioactive disintegration per second.
• Radon levels that are safe.
• A radon level of zero is the most accurate measurement.
• Unfortunately, it is not an option at this time.
• The average worldwide outdoor radon concentration ranges between 5 and 15 Bq/m3, which is equivalent to 0.135 and 0.405 pCi/L.
What is radon classified as?
Radon (Rn) is a chemical element that is a heavy radioactive gas that belongs to Group 18 (noble gases) of the periodic table. It is produced by the radioactive decay of radium and is found in the
Is radon a nonmetal or metal?
The chemical element radon is classified as a noble gas and a nonmetal, according to the periodic table. Fredrich E. Dorn made the discovery in the year 1900. Data Zone is a place where you may store
Classification: Radon is a noble gas and a nonmetal
Color: colorless
Atomic weight: (222), no stable isotopes
State: gas
Melting point: -71 oC, 202 K
Is radon a pure substance?
It is possible to remove gases from the gas in the tube by removing oxygen, nitrogen, water vapor, carbon dioxide, and other gases. The radon gas that remains is completely pure.
Is radium a metal or nonmetal?
Radium (Ra) is a radioactive chemical element that is the heaviest of the alkaline-earth metals in Group 2 (IIa) of the periodic table. Radium is the heaviest of the alkaline-earth metals in the
periodic table. Radium is a silvery-white metal that does not occur naturally as a free-floating element.
What are radon gases?
• Radon is a radioactive noble gas that is produced by the disintegration of radium in the soil and is found in the atmosphere.
• Radium is also a daughter or offspring nuclide of Uranium, which is a radioactive isotope (Uranium decay).
• Radon is a colorless, odorless, and invisible gas that can only be identified with the use of specialized equipment and protocols.
• Radon is a carcinogen that can cause cancer.
Is radon an isotope?
Isotopes. Radon does not have any stable isotopes. Nine radioactive isotopes with atomic weights ranging from 193 to 231 have been identified and described. The most stable isotope is 222Rn, which is
a decay product of 226Ra, which is a decay product of 238U. It is also the most abundant isotope in nature.
What are the properties of radon?
Radon has the following properties: a melting point of -71°C, a boiling point of -61.8°C, a gas density of 9.73 g/l, a specific gravity of the liquid state of 4.4 at -62°C, and a specific gravity of
the solid state of 4, with a valence of 0 in most cases. Radon has the following properties: (it does form some compounds, however, such as radon fluoride).
Is sulfur a metal nonmetal or metalloid?
Nonmetallic chemical element Sulfur (S), sometimes written sulfur, that belongs to the oxygen group of the periodic table (Group 16). It is one of the most reactive elements known, and it is also one
of the most toxic.
Is radon a primary or secondary pollutant?
Among the contaminants that can be found are radon, perfumes, and offensive scents. Dust, smoke particles, nitrogen, carbon, and other such substances are considered primary. Secondary pollutants are
formed when main pollutants react or interact with one another, or with the fundamental elements, to form new pollutants.
Is radon a molecule?
A radon atom is represented by the symbol 222. Radon is a radioactive gas that occurs naturally in the environment and has no odor or flavor. It is created as a result of the radioactive decay of
uranium in the environment. 4.3.3 Related Element
Element Name Radon
Element Symbol Rn
Atomic Number 86
What are 5 facts about radon?
1. Here are a few facts regarding radon, as well as what you can do to keep your family healthy and safe while living in your home. Radon is a radioactive substance. Radon is a radioactive gas that
occurs naturally in the environment.
2. Radon is a cancer-causing gas
3. there are no acute signs.
4. You must do a radon test.
5. In both indoor and outdoor environments, radon may accumulate
6. radon can accumulate in any building.
What type of radiation is radium?
The majority of them are caused by gamma radiation, which has the ability to travel a great distance through air. Even being in close proximity to high quantities of radium is hazardous. Radium is a
chemical that has been linked to cancer in the past. It has been shown that high amounts of radium exposure can increase the likelihood of developing bone, liver, and breast cancer.
Can I buy radium?
The vast majority of individuals are unable to purchase radium. As a result, in order to receive radium, we propose that only qualified purchasers be contacted. Radium is not often available for
purchase by the general public.
Where is radium from?
The majority of the radium is derived from uranium mines in the Democratic Republic of the Congo and Canada, respectively. According to Chemistry Explained, radium is recovered from uranium ores
today in a manner similar to that used by Marie and Pierre Curie in the late 1890s and early 1900s, with the exception of the use of radioactive elements.
What is radon gas?
IDENTIFICATION: Radon, also known as Rn atomic and number 86 on the periodic table, is a colorless radioactive gas that occurs naturally in the environment. It has no discernible odor or flavor. It
has a high solubility in water. Radon belongs to a family of gases known as rare, inert, or noble gases, which indicates that it is chemically inert. Radon is also a radioactive gas.
Is RADON a radioactive element?
Radon is a naturally radioactive element with the atomic symbol Rn and the atomic number 86. It is found in the environment. It is a member of the noble gas family that may be found in soil and is
emitted as a byproduct of the decay of radioactive elements. Radon is a chemical element with the atomic symbol Rn, the atomic number 86, and the atomic weight 222.0.
What is the chemical formula for radon?
• In earthquake prediction, the study of air transport, and in the exploration for petroleum and uranium, radon is utilized to make predictions.
• Radon is a chemical element with the atomic symbol Rn, the atomic number 86, and the atomic weight 222.0.
• Radon (0) is a monoatomic radon with an oxidation state of zero, making it the purest form of the gas.
• radon Lexichem TK 2.7.0 was used to compute this result (PubChem release 2021.05.07) InChI=1S/Rn
What is radon and how does it affect you?
• Radon is a radioactive gas that occurs naturally and has the potential to cause lung cancer.
• Radon gas is a radioactive gas that is inert, colorless, and odorless.
• Radon can be found in tiny concentrations in the atmosphere as a result of natural processes.
• Radon dissipates quickly in the open air and, in most cases, does not pose a health risk.
• The majority of radon exposure happens inside, in settings like homes, schools, and businesses. | {"url":"https://kanyakumari-info.com/tips/what-is-the-classification-of-radon.html","timestamp":"2024-11-03T21:52:31Z","content_type":"text/html","content_length":"29459","record_id":"<urn:uuid:9a47599e-72ce-4b25-8312-8138e747b043>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00307.warc.gz"} |
Alan Turing
Alan M. Turing, 1912-06-22/3? - 1954-06-07. A British mathematician, inventor of the Turing Machine. Turing also proposed the Turing test. Turing's work was fundamental in the theoretical foundations
of computer science.
Turing was a student and fellow of
King's College Cambridge
and was a graduate student at
Princeton University
from 1936 to 1938. While at Princeton Turing published "On Computable Numbers", a paper in which he conceived an
abstract machine
, now called a
Turing Machine
. Turing returned to England in 1938 and during World War II, he worked in the British Foreign Office. He masterminded operations at
Bletchley Park
, UK which were highly successful in cracking the Nazis "Enigma" codes during World War II. Some of his early advances in computer design were inspired by the need to perform many repetitive symbolic
manipulations quickly. Before the building of the
computer this work was done by a roomful of women. In 1945 he joined the
National Physical Laboratory
in London and worked on the design and construction of a large computer, named
Automatic Computing Engine
(ACE). In 1949 Turing became deputy director of the Computing Laboratory at Manchester where the
Manchester Automatic Digital Machine
, the worlds largest memory computer, was being built. He also worked on theories of
artificial intelligence
, and on the application of mathematical theory to biological forms. In 1952 he published the first part of his theoretical study of morphogenesis, the development of pattern and form in living
organisms. Turing was gay, and died rather young under mysterious circumstances. He was arrested for violation of British homosexuality statutes in 1952. He died of potassium cyanide poisoning while
conducting electrolysis experiments. An inquest concluded that it was self-administered but it is now thought by some to have been an accident. There is an excellent biography of Turing by Andrew
Hodges, subtitled "The Enigma of Intelligence" and a play based on it called "Breaking the Code". There was also a popular summary of his work in Douglas Hofstadter's book "Gödel, Escher, Bach".
Last updated: 2001-10-09
Nearby terms:
Alan Kay ♦ Alan M. Turing ♦ Alan Shugart ♦ Alan Turing ♦ ALARP ♦ A-law ♦ ALC
Try this search on Wikipedia, Wiktionary, Google, OneLook. | {"url":"https://foldoc.org/Alan+Turing","timestamp":"2024-11-11T05:27:01Z","content_type":"text/html","content_length":"11283","record_id":"<urn:uuid:cba7cdc9-6ec3-4067-92cf-12122fba2678>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00156.warc.gz"} |
Boosting the margin: A new explanation for the effectiveness of voting methods
One of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large, and often
is observed to decrease even after the training error reaches zero. In this paper, we show that this phenomenon is related to the distribution of margins of the training examples with respect to the
generated voting classification rule, where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label. We
show that techniques used in the analysis of Vapnik's support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the
test error. We also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples. Finally, we compare our explanation to those based
on the bias-variance decomposition.
All Science Journal Classification (ASJC) codes
• Statistics and Probability
• Statistics, Probability and Uncertainty
• Bagging
• Boosting
• Decision trees
• Ensemble methods
• Error-correcting
• Markov chain
• Monte Carlo
• Neural networks
• Output coding
Dive into the research topics of 'Boosting the margin: A new explanation for the effectiveness of voting methods'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/boosting-the-margin-a-new-explanation-for-the-effectiveness-of-vo","timestamp":"2024-11-02T21:33:03Z","content_type":"text/html","content_length":"46243","record_id":"<urn:uuid:b1760b71-b50e-4db5-bceb-df6ef22f4c0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00243.warc.gz"} |
Target-Mediated Drug Disposition: New Derivation of the Michaelis-Menten Model, and Why It Is Often Sufficient for Description of Drugs with TMDD
Leonid Gibiansky, Ekaterina Gibiansky
QuantPharm LLC, MD,USA
Purpose: To derive the Irreversible Binding (IB) and Michaelis-Menten (MM) approximations of the Target-Mediated Drug Disposition (TMDD) equations; to investigate parameter ranges where these
approximations can be used for description of TMDD data and for estimation of target production rate and free target suppression.
Methods: The IB approximation was derived assuming that (i) the drug-target binding is irreversible and (ii) the free target concentration is in quasi-steady-state. Further, the MM approximation was
derived assuming that the free target concentration is much smaller than the drug concentration. A population PK dataset (3355 observations from 224 subjects) was simulated using the full TMDD model.
The MM approximation was used to describe the simulated data. Predicted drug concentrations were compared with the true values. Bias and precision of the parameter estimates were investigated.
Results: The IB equations for a drug that is described by a two-compartment model and administered as intravenous bolus (D[2]), intravenous infusion (In(t)) and subcutaneous (D[1]) doses are
presented below:
dC[dif]/dt =[In(t)+k[a]A[d]]/V - (k[el]+k[pt])C-k[syn]C/(K[IB]+C)+k[tp] A[T]/V,
dA[T]/dt=k[pt ]C V-k[tp ]A[T],
C = 0.5 {C[dif ]-K[IB]+[ (C[dif ]+K[IB])^2+4 R[0] K[IB]]^1/2},
Here C[dif]=C - R; C and R are the concentrations of the free (unbound) drug and the target in the central compartment, k[el] is the linear elimination rate, k[on], k[deg], k[int], k[syn] are the
binding, degradation, internalization, and the target production rate; V is the central compartment volume; R[0]=k[syn]/k[deg] is the baseline target concentration.
The IB approximation is valid for high-affinity (large k[on]) drugs in cases where the drug-target dissociation rate k[off] is either small or much smaller than k[int]. This is typical for the
therapeutic monoclonal antibodies with membrane-bound targets. If R[0] is much smaller than C then C[dif]=C and the irreversible binding equations are equivalent to the model with the
Michaelis-Menten elimination (V[max]=k[syn], K[M]=K[IB],R[0]=0). The discrepancy between the true and MM solutions does not exceed R[0]. In the simulation study for a system with R[0] significantly
smaller than C, the MM model precisely estimated all relevant TMDD parameters and provided unbiased population and individual predictions of the unbound drug concentrations C and the target
production rate k[syn].
Conclusions: The new IB and MM approximations of the TMDD equations were derived. The simulated examples demonstrated validity of these approximations and their ability to estimate the TMDD
parameters. The results extend the parameter range where the Michaelis-Menten model describes the TMDD data.
Reference: PAGE 19 (2010) Abstr 1728 [www.page-meeting.org/?abstract=1728]
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Charge-based modeling of ultra narrow junctionless cylindrical nanowire FETs
With the current trend of increasing complexity of industrial systems, the construction and monitoring of health indicators becomes even more challenging. Given that health indicators are commonly
employed to predict the end of life, a crucial criterion fo ...
Research Publishing | {"url":"https://graphsearch.epfl.ch/en/publication/289549","timestamp":"2024-11-07T10:23:20Z","content_type":"text/html","content_length":"103560","record_id":"<urn:uuid:f5886b62-8bce-4c8e-85cd-d9d42f3e1212>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00743.warc.gz"} |
Voltage Mode
On this page
Torque control using voltage
This torque control approach allows you to run the BLDC and Stepper motor as it is simple DC motor, where you set the target voltage \(U_q\) to be set to the motor and the FOC algorithm calculates
the necessary phase voltages \(u_a\) ,\(u_b\) and \(u_c\) for smooth operation. This mode is enabled by:
// voltage torque control mode
motor.torque_controller = TorqueControlType::voltage;
There are three different ways to control the torque of your motor using voltage requiring different knowledge about your motor mechanical parameters:
Block diagrams of the three torque control techniques based on voltage control can be represented as:
Choose the motor type:
Choose the voltage control type:
Voltage control + Phase resistance + KV rating + Phase inductance
Pure voltage control
The voltage control algorithm reads the angle \(a\) from the position sensor and the gets target \(U_q\) voltage value from the user and using the FOC algorithm sets the appropriate \(u_a\), \(u_b\)
and \(u_c\) voltages to the motor. FOC algorithm ensures that these voltages generate the magnetic force in the motor rotor exactly with 90 degree offset from its permanent magnetic field, which
guarantees maximal torque, this is called commutation.
The assumption of the pure voltage control is that the torque generated (which is proportional to the current \(I = k \tau\)) in the motor is proportional the voltage as \(U_q\) set by the user.
Maximal torque corresponds to the maximal \(U_q\) which is conditioned by the power supply voltage available, and the minimal torque is of course for \(U_q= 0\).
\[U_q \approx I = k\tau\]
⚠️ Practical limitations
This torque control approach is the fastest and most simple one to setup by it does not limit the current in any way!
Expected motor behavior
If the user sets the desired voltage \(U_q\) of 0 Amps, the motor should not move and have some resistance, not too much, but more than when the motor is disconnected from the driver.
If you set a certain desired voltage \(U_g\) your motor should start moving and the velocity reached should be proportional to the voltage set \(U_q\). The behavior should be very similar to the DC
motor controlled by changing the voltage on its wires.
Voltage control with current estimation
Block diagram of this torque control strategy is as follows
If the user provides the phase resistance \(R\) value of the motor, the user can set the desired current \(I_d\) (that generates the desired torque \(I_d = k\tau_d\)) and the library will
automatically calculate the appropriate voltage \(U_q\).
\[U_q = I_d R = (k\tau_d) R\]
User can specify the phase resistance of the motor either through the constructor for example
// BLDCMotor(pole pair number, phase resistance)
BLDCMotor motor = BLDCMotor( 11, 2.5 );
// StepperMotor(pole pair number, phase resistance)
StepperMotor motor = StepperMotor( 50, 1.0 );
or just setting the parameter:
motor.phase_resistance = 1.5; // ex. 1.5 Ohms
⚠️ Practical limitations
The resulting current in the motor can, in some cases, be higher than the desired current \(I_d\) but the order of the magnitude should be preserved. The better you know the phase resistance
value \(R\) the better the current estimation will work. However, as the current \(I_d\) depends also of the back-emf voltage, not only the voltage \(U_q\), this current estimation strategy is
very limited. The relationship \(U_q=I_dR\) is true only if the motor is not moving (no Back-EMF voltage generated). If the motor is moving the back-emf voltage generated will reduce the voltage
set to the motor \( I_dR = U_q - U_{bemf}\). The practical limitation of this approach then will be that the desired current \(I_d\) is only set to the motor when it is static, and as soon as the
motor moves the actual current set to the motor decreases.
Expected motor behavior
If the user sets the desired current of 0 Amps, the motor should not move and have some resistance, not too much, but more than when the motor is disconnected from the driver.
If you set a certain desired current \(I_d\) your motor should start moving and the velocity reached should be proportional to the current set \(I_d\).
For current \(I_d > 0\) motor does not move
If your desired current is set to some value that is not 0, but your motor does not move, your phase resistance value \(R\) is probably too low. Try increasing it.
Voltage control with current estimation and Back-EMF compensation
Block diagram of this torque control strategy is as follows
If the user provides the phase resistance \(R\) value and the motor’s \(KV\) rating of the motor, the user can set the desired current \(I_d\) (that generates the desired torque \(I_d = k\tau_d\))
directly. The library will automatically calculate the appropriate voltage \(U_q\) while compensating for the generated Back EMF voltage by keeping track of the motor velocity \(v\).
\[U_q = I_d R + U_{bemf}= (k\tau_d) R + \frac{v}{KV}\]
User can specify the phase resistance and the KV rating of the motor either through the constructor for example
// BLDCMotor(pole pair number, phase resistance [Ohms], KV rating [rpm/V])
BLDCMotor motor = BLDCMotor( 11, 2.5, 120 );
// StepperMotor(pole pair number, phase resistance [Ohms], KV rating [rpm/V])
StepperMotor motor = StepperMotor( 50, 1.5, 20 );
RULE OF THUMB: KV value
KV rating of the motor is defined as speed of the motor in rpm with the set voltage \(U_q\) of 1 Volt. If you do not know your motor's KV rating you can easily measure it using the library. Run
your motor int the voltage mode and set the target voltage to one 1V and read the motor velocity. SimpleFOClibrary shows that velocity in the rad/s so in order to convert it to the rpm you just
need to multiply it by \(30/\pi \approx 10\).
As explained above as the Back-EMF constant of the motor is always a bit smaller than the inverse of the KV rating ( \(k_{bemf}<1/KV\) ), the rule of thumb is to set the KV rating 10-20% higher
than the one given in the datasheet, or the one determined experimentally.
With the \(R\) and \(KV\) information the SimpleFOClibrary is able to estimate current set to the motor and the user will be able to control the motor torque, provided the motor parameters are
correct (enough 😄).
⚠️ Practical limitations
Back-EMF voltage is defined as \(U_{bemf} = k_{bemf}v\) and calculating it based on the motor \(KV\) rating of the motor is just an approximation because the motor BEMF constant \(k_{bemf}\) is
not exacly \(k_{bemf}=1/KV\). It can be shown that the back-emf constant is always somewhat smaller than the inverse of the KV rating: \[k_{bemf}<\frac{1}{KV}\]
Expected motor behavior
If the user sets the desired current of 0 Amps, the motor should have very low resistance, much lower than in the two torque control strategies above. The motor should feel like it is almost
For current 0 motor moves
If your desired current is set to 0, but when you move your motor with your hand it continues moving on its own and does not come back to a complete stop, your \(KV\) value is too high. Try
reducing it.
If you set a certain desired current \(I_d\) your motor should accelerate to its maximum velocity. The acceleration value is proportional to the motor torque and will be proportional to the current \
(I_d\). So for larger currents your motor will accelerate faster and for the lower currents it will accelerate slower. But for the motor without load, regardless of set target current \(I_d\) the
motor should reach its max velocity.
For current \(I_d > 0\) motor does not move
If your desired current is set to some value that is not 0, but your motor does not move, your phase resistance value \(R\) is probably too low. Try increasing it.
Voltage control using current estimation with Back-EMF and lag compensation
Block diagram of this torque control strategy is as follows
If the user provides the phase resistance \(R\) value and the motor’s \(KV\) rating of the motor, the user can set the desired current \(I_d\) (that generates the desired torque \(I_d = k\tau_d\))
directly. The library will automatically calculate the appropriate voltage \(U_q\) while compensating for the generated Back EMF voltage by keeping track of the motor velocity \(v\).
\[U_q = I_d R + U_{bemf}= (k\tau_d) R + \frac{v}{KV}\]
Additionally if the user sets the phase inductance value \(L\), the library will be able to compensate for the lag of the torque vector by calculating an appropriate d-axis voltage \(U_d\)
\[U_d = -I_d L v n_{pp} = -(k\tau_d)L v n_{pp}\]
where \(n_{pp}\) is the number of motor’s pole pairs. By compensating the lag of the torque vector due to the motor rotation velocity \(v\), the motor will be able to spin with higher max velocity.
Therefore the lag compensation will have the most effect if application requires going to the maximal motor velocity.
User can specify the phase resistance and the KV rating of the motor either through the constructor for example
// BLDCMotor(pole pair number, phase resistance [Ohms], KV rating [rpm/V], phase inductance [H])
BLDCMotor motor = BLDCMotor( 11, 2.5, 120, 0.01 );
// StepperMotor(pole pair number, phase resistance [Ohms], KV rating [rpm/V], phase inductance [H])
StepperMotor motor = StepperMotor( 50, 1.5, 20, 0.01 );
RULE OF THUMB: KV value
KV rating of the motor is defined as speed of the motor in rpm with the set voltage \(U_q\) of 1 Volt. If you do not know your motor's KV rating you can easily measure it using the library. Run
your motor int the voltage mode and set the target voltage to one 1V and read the motor velocity. SimpleFOClibrary shows that velocity in the rad/s so in order to convert it to the rpm you just
need to multiply it by \(30/\pi \approx 10\).
As explained above as the Back-EMF constant of the motor is always a bit smaller than the inverse of the KV rating ( \(k_{bemf}<1/KV\) ), the rule of thumb is to set the KV rating 10-20% higher
than the one given in the datasheet, or the one determined experimentally.
With the \(R\) and \(KV\) information the SimpleFOClibrary is able to estimate current set to the motor and the user will be able to control the motor torque, provided the motor parameters are
correct (enough 😄).
⚠️ Practical limitations
Back-EMF voltage is defined as \(U_{bemf} = k_{bemf}v\) and calculating it based on the motor \(KV\) rating of the motor is just an approximation because the motor BEMF constant \(k_{bemf}\) is
not exacly \(k_{bemf}=1/KV\). It can be shown that the back-emf constant is always somewhat smaller than the inverse of the KV rating: \[k_{bemf}<\frac{1}{KV}\]
Expected motor behavior
If the user sets the desired current of 0 Amps, the motor should have very low resistance, much lower than in the two torque control strategies above. The motor should feel like it is almost
For current 0 motor moves
If your desired current is set to 0, but when you move your motor with your hand it continues moving on its own and does not come back to a complete stop, your \(KV\) value is too high. Try
reducing it.
If you set a certain desired current \(I_d\) your motor should accelerate to its maximum velocity. The acceleration value is proportional to the motor torque and will be proportional to the current \
(I_d\). So for larger currents your motor will accelerate faster and for the lower currents it will accelerate slower. But for the motor without load, regardless of set target current \(I_d\) the
motor should reach its max velocity.
For current \(I_d > 0\) motor does not move
If your desired current is set to some value that is not 0, but your motor does not move, your phase resistance value \(R\) is probably too low. Try increasing it.
For different values of the motor phase inductance \(L\) motor will be able to reach different maximal velocities. The higher the inductance value the higher the maximal velocity. However, after
certain inductance value the motor maximal velocity will stop being affected as it will reach its absolute max velocity.
How to find the phase inductance \(L\) value
Start with low value, such as 0.1mH and set your target current \(I_d\) to certain value to allow the motor to accelerate to it's max velocity. Then use the Commander interface to change the
inductance and see see how the motor's velocity evolves. By raising the \(L\) value, the velocity should increase. After certain \(L\) value the velocity will stop increasing and if you continue
it might even decrease. So use the minimal \(L\) value that reaches the max velocity.
For more info about the theory of the torque control check the section Digging deeper section or go directly to torque control theory.
Torque control example code
A simple example of the voltage based torque control and setting the target current by serial command interface.
#include <SimpleFOC.h>
// BLDC motor & driver instance
BLDCMotor motor = BLDCMotor(11);
BLDCDriver3PWM driver = BLDCDriver3PWM(9, 5, 6, 8);
// encoder instance
Encoder encoder = Encoder(2, 3, 500);
// channel A and B callbacks
void doA(){encoder.handleA();}
void doB(){encoder.handleB();}
// instantiate the commander
Commander command = Commander(Serial);
void doMotor(char* cmd) { command.motor(&motor, cmd); }
void setup() {
// initialize encoder sensor hardware
encoder.enableInterrupts(doA, doB);
// link the motor to the sensor
// driver config
// power supply voltage [V]
driver.voltage_power_supply = 12;
// link driver
// set the torque control type
motor.phase_resistance = 12.5; // 12.5 Ohms
motor.torque_controller = TorqueControlType::voltage;
// set motion control loop to be used
motor.controller = MotionControlType::torque;
// use monitoring with serial
// comment out if not needed
// initialize motor
// align sensor and start FOC
// add target command M
command.add('M', doMotor, "motor");
Serial.println(F("Motor ready."));
Serial.println(F("Set the target current using serial terminal:"));
void loop() {
// main FOC algorithm function
// Motion control function
// user communication
#include <SimpleFOC.h>
// Stepper motor & driver instance
StepperMotor motor = StepperMotor(50); // nema17 200 steps per revolution
StepperDriver2PWM driver = StepperDriver2PWM(9, 5, 6, 8);
// encoder instance
Encoder encoder = Encoder(2, 3, 500);
// channel A and B callbacks
void doA(){encoder.handleA();}
void doB(){encoder.handleB();}
// instantiate the commander
Commander command = Commander(Serial);
void doMotor(char* cmd) { command.motor(&motor, cmd); }
void setup() {
// initialize encoder sensor hardware
encoder.enableInterrupts(doA, doB);
// link the motor to the sensor
// driver config
// power supply voltage [V]
driver.voltage_power_supply = 12;
// link driver
// set the torque control type
motor.phase_resistance = 1.5; // 1.5 Ohms
motor.torque_controller = TorqueControlType::voltage;
// set motion control loop to be used
motor.controller = MotionControlType::torque;
// use monitoring with serial
// comment out if not needed
// initialize motor
// align sensor and start FOC
// add target command M
command.add('M', doMotor, "motor");
Serial.println(F("Motor ready."));
Serial.println(F("Set the target current using serial terminal:"));
void loop() {
// main FOC algorithm function
// Motion control function
// user communication | {"url":"https://docs.simplefoc.com/voltage_torque_mode","timestamp":"2024-11-14T22:30:20Z","content_type":"text/html","content_length":"76462","record_id":"<urn:uuid:0f39546b-078b-457b-a118-c5db35e22796>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00196.warc.gz"} |
Continuity Definition Calculus | Hire Someone To Do Calculus Exam For Me
Continuity Definition Calculus A calculus of variations is an elementary consequence of a few standard notions of calculus (Lebesgue) and of differentiation (Gorossi) along with some helpful, and not
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Compatible class groups are precisely $G$-submodules of $\mathbb Z^\times$ and are considered countably many. $G: \mathbb Z^\times \to \mathbb Z$ is open under commutative group action such that $G \
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“chain” for commutative ${\mathbb Z}$-actions is a chain diagram with homeworks from $\mathbb Z$ that consists of squares. We have a family of $*$-homomorphisms $\{x_n : x_n \cdot g \in G\}$, where
$x_n \cdot g = x_n^{-1} \cdot g$, for $g \in G$, defined as follows. We set $x_n \cdot g := \sum_a^b x_a^{-1} g$. The why not try here $C_G: \mathbb Z \to \mathbb Z$ is a chain complex. An
$*$-homomorphism $x \to \sqsubseteq \mathbb Z$ can be extended to an $*$-homomorphism $\tau \mapsto \sum_n x_n^{-1} \tau$. Let $C = \{ 1Continuity Definition Calculus Möbius, Hilborn and Aihara hold
that if the value of the normal operator in the Cauchy–Riemann representation $Q$ can be assumed as an even fraction with high probability, there exists a nondegenerate, weakly nondegenerate,
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$$\mathfrak{H}(d, \nu) = \{\{\nu\}\mid\; \nu \textrm{ continuous and of class} \; \nu \, \textrm{ infinite)} \\ \mathbb{Z}^d \rightarrow \mathbb{Z}^d$$ is in general not obvious. We will also observe
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replaced by the local weak condition $t \rightarrow \infty$ at $t=0$. Acknowledgments {#ack.no.conf} =============== This work has been supported by the funds from FAPES (Centro de Craciones Estados
Unidos, Chile) and RFBR grant 01-02-00812-a (GIS). [99]{} S. DeMott, *From a quantum approximation*, Second (1995) ed. D. A. Birkedal and B. Moekel’son . JContinuity Definition Calculus – by showing
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formula without changing the definition of the relation in question (see Theorem 6.2). In other words, in terms of the equation of V, we have : [2, 1, 2] – 3 which thus gives the equation that you
and I found out quickly. It is still in this form, in fact, but we can consider it for a moment as a rule that will clarify us to what you need to write out. Therefore (6) gives us In fact, the
formula in Theorem 6.3 says that that. In the equation of V, we shall have : Now, the formula in Theorem 6.4: or [2, 1, 2] – 2 Your words are hard to follow, but we’ll take one property that you have
used that many times in this book: the property of decreasing the right argument of. (Theorem 6.
Taking Online Classes For Someone Else
5 and 6.6 are here.) | {"url":"https://hirecalculusexam.com/continuity-definition-calculus","timestamp":"2024-11-08T21:35:04Z","content_type":"text/html","content_length":"104866","record_id":"<urn:uuid:718b8fab-49c6-4d48-b595-9c0ef1025cbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00246.warc.gz"} |
seminars - On the Cauchy problem for quasilinear dispersive PDEs III
(ID: 402 031 2420)
Quasilinear dispersive PDEs often arise in fluid dynamics and plasma physics as effective models. The goal of this lecture series is to provide an introduction to the theory of local well-/
ill-posedness of the Cauchy problem for such equations. In the first part, I will cover classical concepts that are relevant to the wellposedness theory of quasilinear evolution equations, such as
hyperbolicity, energy estimate, and the continuity of the solution map. In the second part, I will discuss illposedness mechanisms in the dispersive case, and techniques for proving wellposedness in
the absence of such obstructions. An emphasis will be given on the phenomenon of degenerate dispersion, which is a strong instability mechanism for conservative quasilinear dispersive PDE. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=speaker&order_type=desc&l=en&page=27&document_srl=816786","timestamp":"2024-11-14T07:27:48Z","content_type":"text/html","content_length":"45981","record_id":"<urn:uuid:f27895db-1933-45c7-ba87-8778c6bb5d8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00054.warc.gz"} |
Is it OK to use formula after 4 weeks of opening?
Category: medical health cold and flu
Formula isn't sterile. You can't guarantee that your baby won't get sick if you keep your formula for more than 4 weeks after opening.
Likewise, people ask, does formula really go bad after 1 month?
Prepared formula can be stored in the refrigerator for up to 24 hours before using. Do not freeze prepared formula. Powdered formula can be used for one month after it has been opened. We don't
recommend using the formula after the 30 days of opening because the nutrients start to degrade.
Furthermore, does opened formula go bad?
Baby formula lasts for 24 hours once opened. Nutrients deteriorate with time and need to be at a certain level for proper infant development. So, DO NOT use baby formula after the manufacturer
recommended use by date.
Also Know, how long can you keep a tin of formula once opened?
A prepared (but untouched) bottle of formula can be stored in the back of the fridge for 24 hours. Opened containers of ready-to-feed and liquid concentrate formulas are good for 48 hours. Powdered
formula should be used within one month of opening the can or tub. Your fridge should be at 4°C or 40° F.
What do you do with opened baby formula?
Place any opened ready-to-feed liquid formula in a sealed container and discard any unused portions after 48 hours. Store liquid concentrate formula. Store opened liquid concentrate formula in a
covered container and refrigerator immediately after opening.
What happens if baby drinks expired formula?
Expired baby formula.
Unopened cans of formula generally last a year—and this is the one food you don't want to serve past its use by date. It has nothing to do with food safety, but beyond that use by date, the nutrients
in the formula will start to degrade, Chapman says.
Why is Formula only good for 1 hour?
Formula that's been prepared should be consumed or stored in the refrigerator within 1 hour. If it has been at room temperature for more than 1 hour, throw it away. Formula may be prepared ahead of
time (for up to 24 hours) if you store it in the refrigerator to prevent the formation of bacteria.
Will old formula make baby sick?
The germs can live in dry foods, such as powdered infant formula, powdered milk, herbal teas, and starches. Anybody can get sick from Cronobacter, but infection occurs most often in infants.
Do you have to throw out formula after an hour?
Formula that is sitting out at room temperature must be thrown away after 1 hour. If it is in the refrigerator, pre-mixed formula (also called ready-to-feed formula) that is opened must be thrown out
after 48 hours. If your baby does not drink all the formula within one hour, throw it out.
Is it OK to use both powder and ready to feed formula?
No, there's nothing wrong with switching from ready-to-feed formula to the powdered variety. Remember that milk made from powder may look different from what you're used to. When you're away from
home, keep the powder separate from the water and mix them just before feeding.
What is the ratio of water to formula?
Concentrated formulas are mixed 1:1 with water. Ready-to-feed formulas do not need any added water. Powdered formulas are mixed 2 ounces (60 mL) of water per each level scoop of powder. Never add
extra water because dilute formula can cause a seizure.
Why do you have to put water in before formula?
Preparing formula with boiled water
Excessive boiling can increase the concentration of impurities. Let the water cool to room temperature before adding to formula. Making formula with boiling water can cause clumping and decrease the
nutritional value.
How do you measure formulas?
For powdered formula:
1. Determine the amount of formula you want to prepare, following instructions on the package.
2. Use a measuring cup to measure the amount of water needed and add the water to the bottle.
3. Use the scoop that came with the formula container.
4. Pour the scoop or scoops into the bottle.
How long is powdered formula good for after mixing?
Yes, as long as your baby doesn't drink from the bottle. An unused bottle of formula mixed from powder can last up to 24 hours in the fridge. That's why many parents opt to make a larger batch of
formula in the morning and portion out into bottles — or pour into bottles as needed — for use throughout the day.
How do you store formula milk for night feeds?
It is ok to store baby bottles with mixed baby formula in the fridge for up to 24 hours, you may then heat them when you need them. This means you e.g. can make all bottles needed for nightly feeds
before you go to bed, keep them refridgerated and heat baby bottle in seconds in the microwave.
How long should a can of formula last?
Some formula will be wasted because you have to dump the bottle an hour after the feeding starts. The open cans of powder are good for one month, so if you are feeding formula only part-time make
sure you do not buy such a big can that it won't be used up in a month.
Why should you use formula within 4 weeks of opening?
Formula isn't sterile. You will probably find that there is bacteria in it. Also every time you open the tin it's exposed to air therefore moisture. You can't guarantee that your baby won't get sick
if you keep your formula for more than 4 weeks after opening.
How do you store ready to feed formula?
Ready-to-use formula: Once you've opened ready-to-use (premixed) liquid formula, store it in closed bottles or tightly cover the container and refrigerate immediately. After 48 hours, discard any
that's left over, because bacteria may have formed.
Can you reheat formula?
Unfortunately, you can't reheat it. Formula should be used immediately and never be reheated. You should discard whatever formula is left. Note: Babies don't actually require warm milk (whether it's
formula or breast milk).
How long does Enfamil ready to use formula last?
Once prepared, Enfamil^® powder formulas can be kept in the refrigerator (35-40° F or 2-4° C), covered, for up to 24 hours and Enfamil liquid formulas up to 48 hours. A prepared bottle can be kept at
room temperature for up to a total of two hours.
Can you microwave formula?
Milk that's "baby-ready" should feel lukewarm. Heating breast milk or infant formula in the microwave is not recommended. Studies have shown that microwaves heat baby's milk and formula unevenly.
This results in "hot spots" that can scald a baby's mouth and throat. | {"url":"https://findanyanswer.com/is-it-ok-to-use-formula-after-4-weeks-of-opening","timestamp":"2024-11-13T06:39:50Z","content_type":"text/html","content_length":"32565","record_id":"<urn:uuid:fa5671cc-3b2c-4a54-aa8b-a7bbb5b61878>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00334.warc.gz"} |
Stephane Boucher ●
September 21, 2023
In case you forget or didn't already know, registering for the 2023 DSP Online Conference automatically gives you 10 months of unlimited access to all sessions from previous editions of the
conference. So for the price of an engineering book, you not only get access to the upcoming 2023 DSP Online Conference but also to hours upon hours of on-demand DSP gold from some of the best
experts in the field.
The value you get for your small investment is simply huge. Many of the...
In this article, I present a simple approach for designing interpolators that takes the guesswork out of determining the stopbands.
This blog presents a most computationally-efficient guaranteed-stable real-time sliding discrete Fourier transform (SDFT) algorithm. The phrase “real-time” means the network computes one spectral
output sample, equal to a single-bin output of an N‑point discrete Fourier transform (DFT), for each input signal sample.
Proposed Guaranteed Stable SDFT
My proposed guaranteed stable SDFT, whose development is given in [1], is shown in Figure 1(a). The output sequence Xk(n) is an N-point...
Jason Sachs ●
May 28, 2023
About a decade ago, I wrote two articles:
Each of these are about delta-sigma modulation, but they’re short and sweet, and not very in-depth. And the 2013 article was really more about analog-to-digital converters. So we’re going to revisit
the subject, this time with a lot more technical depth — in fact, I’ve had to split this...
Although I generally agree that money can't buy happiness, I recently made a purchase that has brought me countless hours of pure joy. In this blog post, I want to share my excitement with the
DSPRelated community, because I know there are many audio and music enthusiasts here, and also because I suspect there is a lot of DSP magic behind this product. And I would love to hear your
opinions and experiences if you have also bought or tried the Sonos ERA 300 wireless speaker, or any other...
Neil Robertson ●
April 19, 2023
There are many software applications that allow modeling LC filters in the frequency domain. But sometimes it is useful to have a time domain model, such as when you need to analyze a mixed analog
and DSP system. For example, the system in Figure 1 includes an LC filter as well as a DSP portion. The LC filter could be an anti-alias filter, a channel filter, or some other LC network. For a
design using undersampling, the filter would be bandpass [1]. By modeling...
The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1]. So, if we can find H(k) by whatever method,
it should be identical to the DFT of h(n). In this article, we’ll find H(k) by using complex exponentials, and we’ll see that it is indeed identical to the DFT of h(n).
Consider the four-tap FIR filter in Figure 1, where each block labeled Ts represents a delay of one...
Most signal processing intensive applications on FPGA are still implemented relying on integer or fixed-point arithmetic. It is not easy to find the key ideas on quantization, fixed-point and integer
arithmetic. In a series of articles, I aim to clarify some concepts and add examples on how things are done in real life. The ideas covered are the result of my professional experience and hands-on
In this article I will present the most fundamental question you...
Cedron Dawg ●
December 10, 2022
This article is a summary of all the articles I've written here at DspRelated. The main focus has always been an increased understanding of the Discrete Fourier Transform (DFT). The references are
grouped by topic and ordered in a reasonable reading order. All the articles are meant to teach math, or give examples of math, in context within a specific application. Many of the articles also
have sample programs which demonstrate the equations derived in the articles. My...
In this part, I’ll show how to design a Hilbert Transformer using the coefficients of a half-band filter as a starting point, which turns out to be remarkably simple. I’ll also show how a half-band
filter can be synthesized using the Matlab function firpm, which employs the Parks-McClellan algorithm.
A half-band filter is a type of lowpass, even-symmetric FIR filter having an odd number of taps, with the even-numbered taps (except for the main tap) equal to zero. This...
The problem of "spectral inversion" comes up fairly frequently in the context of signal processing for communication systems. In short, "spectral inversion" is the reversal of the orientation of the
signal bandwidth with respect to the carrier frequency. Rick Lyons' article on "Spectral Flipping" at http://www.dsprelated.com/showarticle/37.php discusses methods of handling the inversion (as
shown in Figure 1a and 1b) at the signal center frequency. Since most communication systems process...
Jason Sachs ●
December 4, 2013
Happy Thanksgiving! Maybe the memory of eating too much turkey is fresh in your mind. If so, this would be a good time to talk about overflow.
In the world of floating-point arithmetic, overflow is possible but not particularly common. You can get it when numbers become too large; IEEE double-precision floating-point numbers support a range
of just under 21024, and if you go beyond that you have problems:
for k in [10, 100, 1000, 1020, 1023, 1023.9, 1023.9999, 1024]: try: ...
This article shows how to implement a Butterworth IIR lowpass filter as a cascade of second-order IIR filters, or biquads. We’ll derive how to calculate the coefficients of the biquads and do some
examples using a Matlab function biquad_synth provided in the Appendix. Although we’ll be designing Butterworth filters, the approach applies to any all-pole lowpass filter (Chebyshev, Bessel,
etc). As we’ll see, the cascaded-biquad design is less sensitive to coefficient...
1. Introduction
Figure 1.1 is a block diagram of a digital PLL (DPLL). The purpose of the DPLL is to lock the phase of a numerically controlled oscillator (NCO) to a reference signal. The loop includes a phase
detector to compute phase error and a loop filter to set loop dynamic performance. The output of the loop filter controls the frequency and phase of the NCO, driving the phase error to zero.
One application of the DPLL is to recover the timing in a digital...
In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT)[1] . These include windowing the signal, taking the
magnitude-squared of the DFT, and computing the vector of frequencies. The Matlab function pwelch [2] performs all these steps, and it also has the option to use DFT averaging to compute the
so-called Welch power spectral density estimate [3,4].
In this article, I’ll present some...
Evaluating the performance of communication systems, and wireless systems in particular, usually involves quantifying some performance metric as a function of Signal-to-Noise-Ratio (SNR) or some
similar measurement. Many systems require performance evaluation in multipath channels, some in Doppler conditions and other impairments related to mobility. Some have interference metrics to measure
against, but nearly all include noise power as an impairment. Not all systems are...
Power law functions are common in science and engineering. A surprising property is that the Fourier transform of a power law is also a power law. But this is only the start- there are many
interesting features that soon become apparent. This may even be the key to solving an 80-year mystery in physics.
It starts with the following Fourier transform:
The general form is tα ↔ ω-(α+1), where α is a constant. For example, t2 ↔...
Discrete-time systems are remarkable: the time response can be computed from mere difference equations, and the coefficients ai, bi of these equations are also the coefficients of H(z). Here, I try
to illustrate this remarkableness by converting a continuous-time second-order system to an approximately equivalent discrete-time system. With a discrete-time model, we can then easily compute the
time response to any input. But note that the goal here is as much to...
Jason Sachs ●
May 28, 2023
About a decade ago, I wrote two articles:
Each of these are about delta-sigma modulation, but they’re short and sweet, and not very in-depth. And the 2013 article was really more about analog-to-digital converters. So we’re going to revisit
the subject, this time with a lot more technical depth — in fact, I’ve had to split this...
This article covers interpolation basics, and provides a numerical example of interpolation of a time signal. Figure 1 illustrates what we mean by interpolation. The top plot shows a continuous
time signal, and the middle plot shows a sampled version with sample time Ts. The goal of interpolation is to increase the sample rate such that the new (interpolated) sample values are close to the
values of the continuous signal at the sample times [1]. For example, if...
Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2
fast Fourier transform (FFT).
Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two algorithms showing how to
compute the individual twiddle factors of an N-point decimation-in-frequency...
Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many
Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place.
Here I focus on envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope)...
While there are plenty of canned functions to design Butterworth IIR filters [1], it’s instructive and not that complicated to design them from scratch. You can do it in 12 lines of Matlab code. In
this article, we’ll create a Matlab function butter_synth.m to design lowpass Butterworth filters of any order. Here is an example function call for a 5th order filter:
N= 5 % Filter order fc= 10; % Hz cutoff freq fs= 100; % Hz sample freq [b,a]=...
The problem of "spectral inversion" comes up fairly frequently in the context of signal processing for communication systems. In short, "spectral inversion" is the reversal of the orientation of the
signal bandwidth with respect to the carrier frequency. Rick Lyons' article on "Spectral Flipping" at http://www.dsprelated.com/showarticle/37.php discusses methods of handling the inversion (as
shown in Figure 1a and 1b) at the signal center frequency. Since most communication systems process...
If you need to compute inverse fast Fourier transforms (inverse FFTs) but you only have forward FFT software (or forward FFT FPGA cores) available to you, below are four ways to solve your problem.
Preliminaries To define what we're thinking about here, an N-point forward FFT and an N-point inverse FFT are described by:
$$ Forward \ FFT \rightarrow X(m) = \sum_{n=0}^{N-1} x(n)e^{-j2\pi nm/N} \tag{1} $$ $$ Inverse \ FFT \rightarrow x(n) = {1 \over N} \sum_{m=0}^{N-1}...
In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT)[1] . These include windowing the signal, taking the
magnitude-squared of the DFT, and computing the vector of frequencies. The Matlab function pwelch [2] performs all these steps, and it also has the option to use DFT averaging to compute the
so-called Welch power spectral density estimate [3,4].
In this article, I’ll present some...
This blog may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete
Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.
To be specific, if we perform an N-point DFT on N real-valued time-domain samples of a discrete cosine wave, having exactly integer k cycles over N time samples, the peak magnitude of the cosine
Evaluating the performance of communication systems, and wireless systems in particular, usually involves quantifying some performance metric as a function of Signal-to-Noise-Ratio (SNR) or some
similar measurement. Many systems require performance evaluation in multipath channels, some in Doppler conditions and other impairments related to mobility. Some have interference metrics to measure
against, but nearly all include noise power as an impairment. Not all systems are...
In the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR
design II ). In this post I am going to conclude the IIR filter design review with an example.
Previous posts:
In the recent past, high data rate wireless communications is often considered synonymous to an Orthogonal Frequency Division Multiplexing (OFDM) system. OFDM is a special case of multi-carrier
communication as opposed to a conventional single-carrier system.
The concepts on which OFDM is based are so simple that almost everyone in the wireless community is a technical expert in this subject. However, I have always felt an absence of a really simple guide
on how OFDM works which can...
When the idea of live-streaming parts of Embedded World came to me, I got so excited that I knew I had to make it happen. I perceived the opportunity as a win-win-win-win.
• win #1 - Engineers who could not make it to Embedded World would be able to sample the huge event,
• win #2 - The organisation behind EW would benefit from the extra exposure
• win #3 - Lecturers and vendors who would be live-streamed would reach a (much) larger audience
• win #4 - I would get...
Stephane Boucher ●
February 21, 2019
Do you have a Twitter and/or Linkedin account?
If you do, please consider paying close attention for the next few days to the EmbeddedRelated Twitter account and to my personal Linkedin account (feel free to connect). This is where I will be
posting lots of updates about how the EmbeddedRelated.tv live streaming experience is going at Embedded World.
The most successful this live broadcasting experience will be, the better the chances that I will be able to do it...
Stephane Boucher ●
February 21, 2019
With the upcoming Embedded Word just around the corner, I am very excited to launch the EmbeddedRelated.tv platform.
This is where you will find the schedule for all the live broadcasts that I will be doing from Embedded World next week. Please note that the schedule will be evolving constantly, even during the
show, so I suggest your refresh the page often. For instance, I am still unsure if I will be able to do the 'opening of the doors' broadcast as...
Stephane Boucher ●
February 12, 2019
For those of you who won't be attending Embedded World this year, I will try to be your eyes and ears by video streaming live from the show floor.
I am not talking improvised streaming from a phone, but real, high quality HD streaming with a high-end camera and a device that will bond three internet connections (one wifi and two cellular) to
ensure a steady, and hopefully reliable, stream. All this to hopefully give those of you who cannot be there in person a virtual...
This was my first time at Sensors Expo and my second time in Silicon Valley and I must say I had a great time.
Before I share with you what I find to be, by far, my best 'highlights' video yet for a conference/trade show, let me try to entertain you with a few anecdotes from this trip. If you are not
interested by my stories or maybe don't have the extra minutes needed to read them, please feel free to skip to the end of this blog post to watch the...
This will be my first time attending this show and I must say that I am excited. I am bringing with me my cameras and other video equipment with the intention to capture as much footage as possible
and produce a (hopefully) fun to watch 'highlights' video. I will also try to film as many demos as possible and share them with you.
I enjoy going to shows like this one as it gives me the opportunity to get out of my home-office (from where I manage and run the *Related sites) and actually...
Many of you have the knowledge and talent to write technical articles that would benefit the EE community. What is missing for most of you though, and very understandably so, is the time and
motivation to do it.
But what if you could make some money to compensate for your time spent on writing the article(s)? Would some of you find the motivation and make the time?
I am thinking of implementing a system/mechanism that would allow the EE community to...
After the interview videos last week, this week I am very happy to release two more videos taken at Embedded World 2018 and that I am proud of.
For both videos, I made extensive use of my two new toys, a Zhiyun Crane Gimbal and a Sony a6300 camera.
The use of a gimbal like the Zhiyun makes a big difference in terms of making the footage look much more stable and cinematographic.
As for the Sony camera, it takes fantastic slow-motion footage and...
Stephane Boucher ●
March 21, 2018
Once again this year, I had the chance to go to Embedded World in Nuremberg Germany. And once again this year, I brought my video equipment to try and capture some of the most interesting things at
the show.
Something new this year, I asked Jacob Beningo if he would partner with me in doing interviews with a few vendors. I would operate the camera while Jacob would ask the right questions to the vendors
to make them talk about the key products/features that... | {"url":"https://www.dsprelated.com/blogs-2/mp/all/all.php","timestamp":"2024-11-07T05:27:09Z","content_type":"text/html","content_length":"85273","record_id":"<urn:uuid:3cd976d3-8e31-4528-badc-4b15651419b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00537.warc.gz"} |
Cumulative ground deformation induced by train operations
With the rapid development of railway construction, many railway projects have crossed the soft soil deposit. In order to obtain ground settlement data after train operation to ensure the safety of
railway, the train-track dynamic model and track-ground finite element model were established to predict accumulated deformation of soft soil foundation induced by dynamic train load. Compared to the
measured vibration acceleration on the sleeper and ground, the predicted vibration acceleration can match well with the measured vibration acceleration across the full range of time history shown.
The results showed that after many times of train operations, the cumulative deformations of the ground at different depth are quite different, it may affect the safety of the railway operation, the
foundation reinforcement measures can be considered during construction.
1. Introduction
With the rapid development of railway construction, many railway projects have crossed the soft soil deposit. The original survey design specifications and related manuals cannot meet the
requirements of the design standards for the evaluation and treatment of the soft soil deposit. The settlement of the ground surface, especially when it occurs in a rapid and differential manner, may
cause severe damage on human-built structures and the loss of human lives.
Recently, some scholars studied train induced ground vibration and settlement through measurement, numerical simulation, and empirical equation [1, 2]. Shih et al. [3] built a finite element model of
the track and ground to study train induced soft ground deflections. Wu et al. [4] used real-time strong-motion observation tests in a permafrost region along Qinghai-Tibet Railway to study
displacement response of embankment. Cui et al. [5] based on numerical analyses for the actual tunnel excavation to examine the settlements of ground surface. Therefore, the research on ground
deformation monitoring over soft soil deposit has undoubtedly played an important guiding role in railway design.
In order to obtain ground settlement data after train operation to ensure the safety of railway, the train-track dynamic model and track-ground finite element model were established to predict
accumulated deformation of soft soil foundation induced by dynamic train load. The results can evaluate the safety of the railway operation.
2. Method
In order to analyze the dynamic deformation mechanism caused by moving train loads, this section establishes the of train-track coupling dynamic model, fully considering the influence of track
irregularity, and calculating the wheel-rail interaction when the train runs on the track structure in the time domain. On this basis, the finite element model of track structure and ground is built
and excitation by moving train loads, and the dynamic response characteristics of the ground under moving load are obtained, and the long-term accumulated deformation of ground induced by dynamic
train loads is predicted.
2.1. Train-track dynamic model
The axial load of the train passing through the uneven track including static load and dynamic load. The static load is generated by the self-weight of the train, and the dynamic load is formed by
the contact of the wheel and rail when the train is running. In order to calculate the dynamic response between the train and the track, a train-track interaction dynamic model is established, which
takes into account the interaction between the train vehicle, the rails, the underlying underlay and the fasteners, sleepers, track beds and foundations. It includes the vehicle submodel and the
track structure submodel, as shown in Fig. 1.
Fig. 1Train-track dynamic model
A train consists of eight cars, each of which includes 10 degrees of freedom. The train is simulated to run on the track at a constant speed, and the relative displacement between the trains can be
ignored. According to the Lagrange equation of motion, the dynamic equilibrium equation of the train model can be established as Eq. (1):
where ${\left[M\right]}_{c}$, ${\left[K\right]}_{c}$ and ${\left[C\right]}_{c}$ are mass matrix, stiffness matrix and damping matrix of train, respectively. ${\left\{\stackrel{¨}{\mathbf{q}}\right\}}
_{c}$, ${\left\{\stackrel{˙}{\mathbf{q}}\right\}}_{c}$ and ${\left\{\mathbf{q}\right\}}_{c}$ are acceleration, velocity and displacement vector of train. ${\left\{F\right\}}_{c}$ is the wheel and
track interaction force [6]. The train dynamic parameters are shown in Table 1.
Table 1Train dynamic parameters
Distance between two bogies 18.0 m Wheelset mass 1900 kg
Distance between two axles of bogie 2.4 m Wheel rolling radius 0.4575 m
Car mass 29600 kg Primary suspension stiffness 8.37×10^5 kN/m
Car body inertia 2.319×10^6 kg∙m^2 Secondary suspension stiffness 4.1×10^5 kN/m
Bogie mass 1700 kg Primary suspension damping 3 ×10^5 kN∙s/m
Bogie inertia 1700 kg∙m^2 Secondary suspension damping 1.087 ×10^5 kN∙s/m
The entire track structure system simulation is based on the finite element method and is set up as a three-layer structure model including rails, sleepers, ballast and subgrade, as shown in Fig. 1.
The degree of freedom of the track system is set on the same plane as the degree of freedom of the vehicle system. In order to reduce the size of the finite element stiffness matrix, the steel rail
is assumed to be a discrete finite length Euler beam, ignoring the influence of shear force and moment of inertia, and an adjacent analysis unit is formed by the adjacent two sleepers and ballast
supporting the rail, as shown in Fig. 2. The track structure dynamic parameters are shown in Table 2.
Based on the Hamilton principle, the dynamic equation of the orbit model is as follows:
where ${\left[M\right]}_{1}$, ${\left[K\right]}_{1}$ and ${\left[C\right]}_{1}$ are mass matrix, stiffness matrix and damping matrix of track structure, respectively. ${\left\{\stackrel{¨}{\mathbf
{q}}\right\}}_{1}$, ${\left\{\stackrel{˙}{\mathbf{q}}\right\}}_{1}$ and ${\left\{\mathbf{q}\right\}}_{1}$ are acceleration, velocity and displacement vector of track structure. ${\left\{F\right\}}_
{1}$ is the wheel and track interaction force [6].
Fig. 2Track structure unit
Table 2Track structure parameters
Rail elastic modulus N/m^2 2.06×10^11 Rail mass kg/m 60.64
Rail cross-sectional area m^2 7.745×10^-3 Rail section moment of inertia m^4 3.217×10^-5
Rail density kg/m^3 7830 Rail elastic pad damping N∙s/m 5×10^4
Rail elastic pad stiffness MN/m 100 Sleeper mass kg 251
Sleeper spacing m 0.6 Ballast elastic modulus Pa 0.8×10^8
Ballast mass kg 630 Subgrade elastic modulus Pa/m 1.3×10^8
Ballast damping N∙s/m 1.6×10^5 Subgrade damping N∙s/m 6.32×10^4
The wheel-rail contact force acts on the wheel-rail contact point, and the forces acting on the wheel pair and the rail are equal in magnitude and opposite in direction. The wheel-rail contact force
expression is as follows:
mathrm{}\mathrm{}{q}_{wi}-{q}_{ri}\ge 0,& \\ 0,\mathrm{}\mathrm{}\mathrm{}{q}_{wi}-{q}_{ri}<0,& \end{array}\right\$
where ${F}_{0}$ is train static load. ${q}_{wi}$ and ${q}_{ri}$ are vertical displacement of the wheel at the wheel-rail contact and the vertical displacement of the track, respectively. $h$
represents the irregularity of the track surface at the wheel-rail contact.
Track irregularity is the main reason of dynamic response caused by train operation, and it is the excitation source of wheel-rail system. Considering that the established train-track dynamic model
only considers the vertical dynamic response, the track irregularity is based on the spectrum used by Lombaert [7]. The simulation is as follows:
where ${k}_{1,0}=$ 1 rad/s, $S\left({k}_{1,0}\right)=\mathrm{}$1×10^-8 m^3, $w=$ 3.5.
2.2. Track-ground finite element model
The track-ground model was based on finite element to build. The track structure model consists of rails, sleepers, ballasts. The track adopts 60 kg/m, U75V hot-rolled steel rail, and the distance
between two rails is 1.435 m. The relevant physical property parameters are shown in Table 3.
Table 3Physical and mechanical parameters of track structure finite element model
Model Elastic modulus (MPa) Poisson’s ratio Density (kg/m^3)
Rail 210000 0.25 7850
Sleeper 30000 0.2 2400
Ballast 300 0.35 1800
For the ground model, according to the detailed geological investigation, the physical and mechanical parameters of the main soil layers are shown in the table below.
Table 4Physical and mechanical parameters of ground model
Soil layer Thickness (m) Density (g/cm^3) Elasticity modulus (MPa) Poisson’s ratio $u$ ${V}_{p}$ (m/s) ${V}_{s}$[](m/s)
Artificial fill 2 1.85 200 0.30 390.5 210.0
Mucky soil 8 1.83 180 0.32 374.6 180.4
Fine sand 2.5 1.87 250 0.25 412.1 254.3
Medium-coarse sand 15 1.90 300 0.25 435.1 262.3
For the train load, this paper simulates moving train loads by applying wheel-rail contact force which obtain from train-track dynamic model to the rail surface. The train speed is 20 km/h. For the
boundary condition, by coupling the finite element and the infinite element to simulate infinite ground. It has wide applicability in the problem of simulating the infinite domain without losing the
accuracy of calculation. The track-ground model is shown in Fig. 3.
Fig. 3Track-ground finite element model
2.3. Accumulated deformation of soft soil foundation induced by dynamic train load
The cumulative deformation of the ground under the moving train dynamic load can be calculated based on the empirical formula which proposed by Chai and Miura [8]. The formula was established by the
soil cumulative plastic strain results based on the indoor dynamic triaxial test, which is defined as:
${\epsilon }_{p}=a{\left(\frac{{q}_{d}}{{q}_{f}}\right)}^{m}{\left(1+\frac{{q}_{s}}{{q}_{f}}\right)}^{n}{N}^{b},$
where ${\epsilon }_{p}$ is the cumulative plastic strain, $N$ is the number of cyclic loadings, ${q}_{d}$ is the dynamic deviatoric stress, ${q}_{s}$ is the initial static deviatoric stress, and ${q}
_{f}$ is the static strength of the soil. When the soil is undrained, ${q}_{f}=\text{2}{C}_{u}$, ${C}_{u}$ can be obtained by shear test.
The specific analysis process is: a) dynamic analysis of the three-dimensional track-ground finite element model. The dynamic moving load is the wheel-rail contact force obtained by the train-track
dynamic model. b) The dynamic analysis is carried out to obtain the dynamic response of the soil after one single train pass-by. The maximum dynamic deviator stress ${q}_{d}$ of the soil can be
obtained. c) The static analysis of the model can obtain the initial static deviator stress ${q}_{s}$ of the soil. d) Use the empirical formula Eq. (5) to calculate the plastic cumulative strain of
the ground under any vehicle load times, and calculate the cumulative deformation of the ground.
3. Results
In order to verify the validity of the models, the calculated vibration accelerations on the sleeper and ground (10 m from the track) were compared with the measurement data, as shown in Fig. 4.
Predicted vibration acceleration for both on the sleeper and ground compare well the measured vibration acceleration across the full range of time history shown.
Fig. 4Comparison of measured and predicted acceleration
b) 10 m away from the track on the ground
Fig. 5Ground settlement changed with different train pass-by numbers
The cumulative plasticity in the soil is larger in the surrounding area of the train operation and is smaller away from the train running area. In the soil below the train running, the cumulative
plastic strain is the largest. The integral deformation of the soil can be obtained by integrating the plastic strain in the depth direction. It can be seen from Fig. 5 that the cumulative
deformation of the ground increases with the increase of the number of times of loading. The deformation rate is the largest in the early stage of loading, and as the number of loads increases, the
growth rate of cumulative deformation gradually decreases, and 500,000 times is the inflection point of the curve. Before 500,000 times of train operations, the cumulative deformations of the soil at
the ground surface, 1 m under the ground surface and 3 m under the ground surface are close, and when the number of loading increases to 10,000,000 times and 20,000,000 times, the cumulative
deformations of the ground at different depth are quite different. It may affect the safety of the railway operation, the foundation reinforcement measures can be considered during construction.
4. Conclusions
The train-track coupling dynamic model and track-ground finite element model were stablished to predicted accumulated deformation of soft soil ground induced by dynamic train load. The conclusions
can be drawn as below:
1) Compared to the measured vibration acceleration on the sleeper and ground, the predicted vibration acceleration can match well with the measured vibration acceleration across the full range of
time history shown.
2) Train operations may affect the safety of the railway operation, the foundation reinforcement measures can be considered during construction.
• Pakbaz M. S., Imanzadeh S., Bagherinia K. H. Characteristics of diaphragm wall lateral deformations and ground surface settlements: case study in Iran-Ahwaz metro. Tunnelling and Underground
Space Technology, Vol. 35, 2013, p. 109-121.
• Wang F., Gou B., Qin Y. Modeling tunneling-induced ground surface settlement development using a wavelet smooth relevance vector machine. Computers and Geotechnics, Vol. 54, 2013, p. 125-132.
• Shih J. Y., Thompson D. J., Zervos A. The influence of soil nonlinear properties on the track/ground vibration induced by trains running on soft ground. Transportation Geotechnics, Vol. 11, 2017,
p. 1-16.
• Wu Z., Chen T., Zhao T., Wang L. Dynamic response analysis of railway embankments under train loads in permafrost regions of the Qinghai-Tibet Plateau. Soil Dynamics and Earthquake Engineering,
Vol. 112, 2018, p. 1-7.
• Cui Y., Kishida K., Kimura M. Prevention of the ground subsidence by using the foot reinforcement side pile during the shallow overburden tunnel excavation in unconsolidated ground. Tunnelling
and Underground Space Technology, Vol. 63, 2017, p. 194-204.
• Zhang J., Gao Q., Tan S. J., Zhong W. X. A precise integration method for solving coupled vehicle-track dynamics with nonlinear wheel-rail contact. Journal of Sound and Vibration, Vol. 331, Issue
21, 2012, p. 4763-4773.
• Lombaert G., Degrande G. Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains. Journal of Sound and Vibration, Vol. 319, Issue 3, 2009, p. 1036-1066.
About this article
Vibration in transportation engineering
train operation
vibration acceleration
Copyright © 2018 Sangui Wang, et al.
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/20374","timestamp":"2024-11-05T13:28:14Z","content_type":"text/html","content_length":"120636","record_id":"<urn:uuid:f2d4b0b8-fcef-44ed-a9ca-8beb9dd18e17>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00536.warc.gz"} |
The effectiveness of ancient knowledge: Inspiring Archimedes
Placing well-known names in “top-ten” lists by profession or mentioning them in comparison to one another as to how “great” they were is not the right way to ascribe merit when it comes to
mathematicians. While “tops” lists provide an objective way to classify sport legends in terms of scores, records, and memorable matches, they are not quite adequate for saying how “great” a
mathematician is, in terms of the number of his or her discoveries. For what really counts in mathematics is fruitfulness, fertileness, and inspiring work. The main criterion for qualifying a
discovery as remarkable is the degree to which that discovery or individual influences subsequent discoveries or theories.
Archimedes is such a figure in the history of mathematics in that both his work and he as a man inspired scientists and mathematicians, and he should inspire us all. Indeed, it is not accidental at
all that his “Eureka!” became and has remained over millennia the paradigmatic cry of victory of the arduous, rational self-search for a solution to a theoretical issue. Perhaps you yourself shouted
it after struggling to solve a difficult math problem.
Archimedes of Syracuse lived in the third century B.C. in Greece and is known as a Greek mathematician, physicist, engineering inventor, and astronomer. Such distinct qualifications are specific only
for modern and contemporary time, as in that ancient time there were not such disciplinary distinctions—mathematics and natural sciences were all under the same umbrella, usually called
‘philosophy’ or ‘natural philosophy’. Differentiation of these disciplines came in the 16th-17th century, the age of birth of modern science.
Archimedes’ theoretical achievements decisively influenced contemporary mathematics and physics. His thinking focused equally on both geometry and its physical applications, like all sorts of useful
mechanical devices, astronomy, and nontraditional measurements.
As a mathematician, he discovered the method of exhaustion for computing the area of (or under) a curved shape. This method stemmed from the idea that a curved shape is the image of an infinite
aggregation and accumulation of straight segments, and implicitly, its area consists of an infinite aggregation of regular shapes such as triangles and other polygons. Although such an idea can be
expressed rigorously only in terms of present-day mathematical analysis with the notion of limit being essential, the tools of Euclidean geometry sufficed for Archimedes at that time to make his
point and effectively provide precise measurements of such areas.
In brief, the method of exhaustion assumes inscribing inside the area to be measured a sequence of polygons (properly chosen) whose areas converge to the area of that shape (or approximate the area,
in the terms of that period). The difference between the area of the initial shape and that of the n-th polygon will become arbitrarily small as n increases. Then, the possible values of the area in
question are successively eliminated with the lower areas in that sequence, by using the reductio ad absurdum logical principle.
An easy example of using the method of exhaustion is what is known as Archimedes’ quadrature of the parabola. He proved that the area determined by a parabola and a straight line intersecting it is
the infinite sum .
The idea was to dissect that parabolic segment into an infinite number of triangles (a procedure known as triangulation), as shown in Figure 1. Each of these triangles is inscribed in its own
parabolic segment in the same way that the initial triangle is inscribed in the large segment. Assuming the area of the initial triangle is 1, Archimedes proved the area of the parabolic segment to
be 4/3.
Figure 1. The triangulation of the parabolic segment.
What is remarkable is that Archimedes computed the infinite sum above (which is the limit of a geometrical series with the common ratio ¼, in modern terms), also by geometrical means. He considered a
unit square tiled with four squares of side length ½, then each of these smaller squares tiled with four squares of side length ¼ , and so on, as in Figure 2. Then he showed that the areas of all the
squares lining up with the diagonal of the initial square cover 1/3 of the area of the initial square. You could try to reconstruct this proof as an exercise; it is not hard at all.
Figure 2. The geometrical computation for the sum of the series
The method that Archimedes used for finding the area of curved shapes is simultaneously simple and complex—complex because he had to deal with infinity and argue for such concept. The concept of
infinity was present in Euclid’s fundamental concepts of his geometry (just a few decades before), where segments, lines, and shapes are sets containing an infinite number of points; yet this
infinity reflected “numerosity” and continuity, a “large” and “dense” infinity. However, the infinity that Archimedes found and used in his sequences of shapes was of another kind: it was a
“sequential” infinity, “smaller than” the geometrical infinity. It was “as large as” the infinity of natural numbers; however, it had the peculiarity (at that time) of accumulating itself near a
certain number and as such, generating numbers arbitrarily close to zero.
These concepts, although having a geometrical origin in Archimedes’ work, inspired two fundamental concepts for modern integral and differential calculus (pioneered by Leibniz, in the 17th century),
namely the infinitesimals and the idea of limit.
Infinitesimals are positive quantities so small that there is no mathematical way to measure them. They designate a concept that over the history of mathematics gave serious headaches to
mathematicians who struggled to provide axiomatic constructions for them in order to embed them as numbers among the reals. Such constructions were necessary since the naturals, rationals, and reals
benefited from rigorous systematic constructions that defined them as numbers. One of the main features of mathematics as a discipline is that any of its fundamental concepts must have its own
mathematics, including sets and numbers. Detected with the ancient work of Archimedes, infinitesimals came to be quite controversial in modern and contemporary mathematics. Several number systems
have been axiomatically designed to embed infinitesimals; however, not all mathematicians accepted them as real numbers.
Still related to curved shapes, it is worth mentioning that Archimedes provided one of the first approximations for , by placing it in the interval . He did that by inscribing and circumscribing a
circle with two similar 96-sided regular polygons.
There are plenty of contributions of Archimedes in various fields of mathematics besides geometry. One of my math professors commented that a mathematician becomes really great when his name comes to
be “adjectivized”. Well, there are many notions that are called after Archimedes: Archimedean absolute value, Archimedean circle, Archimedean field, Archimedean group, Archimedean spiral, to name
just notions from mathematics.
However, Archimedes is best known for his principle in the physics of water. The legendary tale about his discovering this principle with his “Eureka!” while taking a bath is well known (though some
historians don’t take it as true, arguing that the story appeared nowhere in the known written works of Archimedes). The tale says that in searching for a way to determine whether the crown of King
Hiero II of Syracuse was made of pure gold or of a metal surreptitiously alloyed with silver, without damaging the crown, he discovered the principle of buoyancy, which states that every object
immersed in a liquid is buoyed up by a force equal to the weight of the liquid displaced by the object. Applied to his particular crown problem, this principle allowed Archimedes to find the volume
of the crown and then its density, by dividing mass by volume. If this density were found to be lower than the usual density of gold, then less dense metals had been added. Archimedes’ principle is
nowadays a canon law of physics fundamental for hydrostatics and fluid mechanics.
If the crown tale is true, it is just another example of Archimedes’ inclination toward practical application of his knowledge. He was not content simply with his theoretical achievements, but wanted
the applications of those achievements to benefit the community. His mechanical inventions—the screw for raising water, block-and-tackle pulley systems, the odometer (measuring distances), the
planetarium—stand as beautiful evidences of his caring about people’s daily life.
Archimedes also involved himself in the military actions defending the city of Syracuse against enemy attacks, not as a soldier, but as a brilliant inventor of weapons. He designed a claw consisting
of a long arm from which a large metal grappling hook was suspended. When the claw was dropped onto a rival ship, the arm would swing upwards, lifting the ship out of the water. It is known as
Archimedes’ claw.
Yet, the most ingenious weapon seems to be Archimedes’ heat ray device. The device was a system of mirrors acting together as a parabolic reflector of the sunlight. The heated rays converged toward
the focus and set fire to the enemy’s ships. Several recent experiments have reproduced Archimedes’ heat ray device and proved it to be functional and effective.
Figure 3. Painting by Giulio Parigi (1599), depicting a mirror used by Archimedes to burn Roman ships
We know that many brilliant mathematicians are so dedicated to their theoretical work and creation that they tend to remain “suspended” in the abstract realm of their pursuits. There, they try to
exploit their minds to reach as high an intellectual plane as possible, and it is often difficult for them to get back “on the ground” to make their knowledge work for practical applicative purposes.
The ancient ages provided us with a marvelous counterexample of that tendency in the person of Archimedes, who dedicated his time and knowledge to serve his community, from providing people with
effective tools and annihilating enemy ships, to leaving them mathematical and physical principles they will use fundamentally in their future science over millennia. | {"url":"https://www.magazine.philscience.org/2022/03/21/the-effectiveness-of-ancient-knowledge-inspiring-archimedes/","timestamp":"2024-11-15T03:25:10Z","content_type":"text/html","content_length":"45422","record_id":"<urn:uuid:62af8ff6-e624-4d82-98d6-087ada04d822>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00137.warc.gz"} |
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Also Read: How to Apply for the Best Business Loan in India? - Types of Govt. Loan Schemes
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Loan Amount
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Also Read: Understanding EPF Loan - A Comprehensive Guide
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Unit 6 Radioactivity What Is the Nucleus Like
Name ______
Ch 21 · Nuclear Chemistry
1. Show how a, b, and g rays each behave when they pass through an electric field. Use the diagram below to illustrate your answer. What direction does each particle travel and what bending occurs.
Match the rays with the following ideas. Each ray may be used once, more than once, or not at all.
a) alpha b) beta c) gamma
___ 2. Two protons and two neutrons ___ 7. High speed electron
___ 3. a ___ 8. He
___ 4. Higher energy than x-rays ___ 9. b-
___ 5. Helium nucleus ___ 10. e
___ 6. Most easily stopped.
Complete the following nuclear equations:
11. Co + n ® Mn + c
12. C ® N + c
13. Mo ® c + e + g
14. (alpha decay of U-235)
15. K + c ® Ar
/ 16. What is the half life of the graphed material? ____
17. What mass of radioisotope will remain after 12.0 hours? _____
18. Plot the data for a substance with a half-life of 1.5 hours, beginning with 100 grams.
19. Lr-257 has a half life of 8 seconds. What % of a sample will remain 32 seconds after it is made?
20. Na-24 has a half life of 15 hours. What is the rate constant, k, for Na-24 (include units)?
21. A 64 gram sample of I-131 is tested after 40 days and is found to contain only 2 grams of I-131. What is the half life of I-131?
22. The sum of 2 protons, 2 neutrons, and 2 electrons is 4.0322980 amu; however, the measured mass of He is only 4.00260 amu. What happens to this mass?
Match the types of nuclear changes with the equations:
___23. Fission
(occurs in nuclear reactors & atomic bombs)
___24. Fusion
(occurs in the Sun & hydrogen bombs)
___25. Neutron Bombardment
___26. Electron Capture /
27. a) Iodine-130 has a half-life of 8.0 days. What is the value of the rate constant, k, for I-130?
b) What percentage of a sample of I-130 remains after 35 days?
c) What sort of decay would you predict for I-130? ______(alpha, beta, positron)
28. A wooden artifact has a carbon 14 activity of 24.9 counts per minute as compared with an activity of 32.5 counts per minute for a standard from 0 age. Knowing the half life of C-14 is 5715 years,
determine the age of the artifact. Show your work!!
29. Design an experiment where you use radioactive Fe-59 (a beta emitter with half life of 44.5 days) to determine the extent to which rabbits are able to convert a particular iron compound in their
diet into blood hemoglobin, which contains iron atoms. | {"url":"https://docest.com/doc/252898/unit-6-radioactivity-what-is-the-nucleus-like","timestamp":"2024-11-02T12:12:04Z","content_type":"text/html","content_length":"23824","record_id":"<urn:uuid:2100cef8-abca-41c0-99d6-ea7587e3961f>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00644.warc.gz"} |
Level 1 Math
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Days: 116
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Sylladot was developed to work for all students. Our multisensory approach helps children with learning disabilities such as autism, dyslexia, and attention-deficit disorder.
Accelerate Progress
It takes about 100 days to complete each level. You can easily finish 2 or 3 levels in a given year. Acceleration should be used to catch up rather than to get ahead. | {"url":"https://sylladot.com/products/level-1-math","timestamp":"2024-11-09T19:30:00Z","content_type":"text/html","content_length":"97256","record_id":"<urn:uuid:ebe0e610-3b48-4374-9f86-a4c37c45250a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00866.warc.gz"} |
Academic Calendar -
Admission to this module is discontinued effective September 1, 2022. Students currently enrolled in the module will be permitted to graduate upon fulfilment of the module requirements by August 31,
Completion of first-year requirements with no failures. Students must have an average of at least 70% in 3.0 principal courses, including:
0.5 course: Calculus 1000A/B, Calculus 1500A/B or the former Calculus 1100A/B;
0.5 course: (Calculus 1501A/B (recommended)) or (Calculus 1301A/B with a mark of at least 85%);
plus 2.0 additional courses, with no mark in these principal courses below 60%. Mathematics 1600A/B, and Mathematics 1120A/B, if taken in first year, will count toward the 3.0 principal courses.
Mathematics 1120A/B and Mathematics 1600A/B are recommended.Note: Mathematics 1600A/B must be completed prior to Mathematics 2120A/B.
3.5 courses: Calculus 2502A/B, Calculus 2503A/B, Mathematics 2120A/B, Mathematics 2122A/B, Mathematics 2155F/G, Mathematics 3020A/B, Mathematics 3150A/B.2.5 courses from: Actuarial Science 2553A/B,
Applied Mathematics 2402A, Applied Mathematics 2814F/G, Applied Mathematics 3811A/B, Applied Mathematics 3815A/B, Computer Science 2209A/B, Computer Science 2210A/B, Computer Science 3331A/B,
Computer Science 3340A/B, Earth Sciences 2222A/B, Economics 2210A/B, Mathematics 2124A/B, Mathematics 2156A/B, Mathematics 2251F/G, Mathematics 3124A/B, Mathematics 3152A/B, Mathematics 4158A/B/Y,
Philosophy 2250, Philosophy 2251F/G, Philosophy 2252W/X, Philosophy 2254A/B, Philosophy 3201A/B, Statistical Sciences 2857A/B, Statistical Sciences 2858A/B, the former Philosophy 3202B, the former
Philosophy 4201A/B, the former Philosophy 4202A/B. Note that some of these courses have prerequisites that are not part of the module.3.0 courses from: Actuarial Sciences, Applied Mathematics,
Computer Science, Financial Modelling, Mathematics, or Statistical Sciences courses, at the 2100 level or above.
It is strongly recommended that Mathematics 2122A/B be completed in the year of entry into the module.
Note: Students intending to pursue graduate studies in Pure Mathematics should take the Honours Specialization in Mathematics module. | {"url":"https://westerncalendar.uwo.ca/Modules.cfm?ModuleID=21061&SelectedCalendar=Live&ArchiveID=","timestamp":"2024-11-04T04:51:40Z","content_type":"text/html","content_length":"16669","record_id":"<urn:uuid:a88cb163-b672-4830-83ea-bcf9f195ae7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00844.warc.gz"} |
Sets, Relations and Functions: Introduction - Mathematics
The concepts of sets, relations and functions occupy a fundamental place in the mainstream of mathematical thinking.
The concepts of sets, relations and functions occupy a fundamental place in the mainstream of mathematical thinking. As rightly stated by the Russian mathematician Luzin the concept of functions did
not arise suddenly. It underwent profound changes in time. Galileo (1564-1642) explicitly used the dependency of one quantity on another in the study of planetary motions. Descartes (1596-1650)
clearly stated that an equation in two variables, geometrically represented by a curve, indicates dependence between variable quantities. Leibnitz (1646-1716) used the word “function”, in a 1673
manuscript, to mean any quantity varying from point to point of a curve. Dirichlet (1805-1859), a student of Gauss, was credited with the modern “formal” definition of function with notation y = f(x)
. In the 20th century, this concept was extended to include all arbitrary correspondence satisfying the uniqueness condition between sets and numerical or non-numerical values.
With the development of set theory, initiated by Cantor (1845-1918), the notion of function continued to evolve. From the notion of correspon-dence, mathematicians moved to the notion of relation.
However even now in the theory of computation, a function is not viewed as a relation but as a computational rule. The modern definition of a function is given in terms of relation so as to suit to
develop artificial intelligence.
In the previous classes, we have studied and are well versed with the real numbers and arithmetic operations on them. We also learnt about sets of real numbers, Venn diagrams, Cartesian product of
sets, basic definitions of relations and functions. For better understanding, we recall more about sets and Cartesian products of sets. In this chapter, we see a new facelift to the mathematical
notions of “Relations” and “Functions”.
Tags : Mathematics , 11th Mathematics : UNIT 1 : Sets, Relations and Functions
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th Mathematics : UNIT 1 : Sets, Relations and Functions : Sets, Relations and Functions: Introduction | Mathematics | {"url":"https://www.brainkart.com/article/Sets,-Relations-and-Functions--Introduction_33891/","timestamp":"2024-11-12T02:35:36Z","content_type":"text/html","content_length":"29310","record_id":"<urn:uuid:2c8b8dcb-9724-49ff-8adc-00a6741c4dad>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00633.warc.gz"} |
Looking Good on Course Evaluations
At the end of each semester, students provide feedback on their courses via anonymous course evaluations. However, use of student evaluations as indicators of quality of the course and teaching
effectiveness is often criticized since these measures may be reflecting biases in favor of nonteaching-related characteristics, such as the physical appearance of the instructor.
The 2005 Economics of Education Review article titled “Beauty in the Classroom: Instructors’ Pulchritude and Putative Pedagogical Productivity” finds that instructors who are viewed to be better
looking receive higher instructional ratings. This article and the accompanying data set are relatable to students since they have first-hand experience evaluating courses and professors. They also
have pre-existing notions about how some of the variables in the data set (e.g., class size, whether the professor is a native speaker, etc.) might affect their learning. Last, almost all students
are familiar with RateMyProfessors.com, an online destination for professor ratings, where students rate their professors on, among other factors, hotness.
The Data
The study uses data from end of semester student evaluations for a large sample of professors from The University of Texas at Austin. These data are merged with descriptors of the professors and the
classes. In addition, six students rate the professors’ physical appearance.
A list of the variables in the data set and their descriptions are provided in Table 1, and the complete data set can be found in the supplemental material. This is a slightly modified version of the
original data set that was released as part of the replication data for Data Analysis Using Regression and Multilevel/Hierarchical Models.
Ideas for Using the Data in the Classroom
This data set/paper combination is approachable enough to be used in an introductory statistics course and complex enough to be used in an advanced undergraduate or graduate course on multilevel/
hierarchical modeling.
The complexity of the data comes from the study design. The researchers first sampled 94 professors, and then collected data on courses they taught over a two-year period between 2000 and 2002. This
sampling scheme resulted in 463 classes, with the number of classes taught by a unique professor in the sample ranging from 1–13. Therefore, the observations in this data set (classes) are not truly
independent. While this is an interesting problem to tackle in an advanced course, we suggest treating the cases as independent observations when using this data set in more elementary courses.
However, while students in these lower-level courses may not have the tools to correctly handle the hierarchical structure of the data, it is important for them to read the original paper to evaluate
whether this simplifying assumption is actually reasonable and gain important insights into the data.
We have used this data set as part of a final project in an introductory statistics course (that covers multiple regression). Working in teams, students built models predicting course evaluation
scores and presented their final product in a poster session and research paper. To help increase variety in the final product, we kept the assignment open ended. An abbreviated version of the
assignment is given below:
• Data: Explain the data, including implications for the scope of inference.
• Simple Linear Regression: Choose one quantitative explanatory variable and do a simple linear regression to predict average course evaluations.
• Two Variable Comparisons: Explore relationships between each of the explanatory variables and average course evaluations as well as relationships between the explanatory variables.
• Multiple Regression: Decide on a “best” model for predicting course evaluations and use it to obtain a predicted course rating for this course.
• Conclusion: What have you learned about course ratings?
Teams focused on different aspects of the data and had different approaches to model selection, which promoted rich discussion during the poster session. One advantage of the poster session was that
it required teams to reveal their answers simultaneously, eliminating the answer drift we tend to see in sequential presentations.
This data set also can be used in class discussions interspersed throughout the semester or at the end of a semester as review. Below, we have provided a series of questions that can be used as
starting points for discussion. These questions do not require models that take into account the hierarchical structure of the data. For a list of discussion questions involving multilevel/
hierarchical modeling of these data, see chapters 12, 13, and 16 in Data Analysis Using Regression and Multilevel/Hierarchical Models.
Discussion Questions
Data and Study Design
1. What does each observation in this data set represent? How are the observations sampled?
2. Are the observations independent of each other? Why, or why not?
3. Is this an observational study or an experiment? The original research question posed in the paper is whether beauty leads directly to the differences in course evaluations. Given the study
design, is it possible to answer this question as it is phrased? If not, how would you rephrase the question?
4. In what analyses did the authors use the picture variables (pic_outfit, pic_color, pic_full_dept)? Should these variables be included in a model used to answer the main research question of the
Course and Professor Evaluations
5. Describe the distributions of average course and professor evaluations. Is the distribution skewed? What does that tell you about how students rate courses? Is this what you expected to see? Why
or why not?
6. How do average course and professor evaluations relate to each other? Do students tend to rate courses or professors more highly?
Professor and Class Characteristics
7. Explore bivariate relationships between various professor characteristics (rank, ethnicity, gender, language, age) as well as between course evaluations and each of these variables. Can you spot
any trends?
8. Would you expect to see a relationship between class size and how highly the course is rated? If so, in which direction would you expect this relationship to be? Check if the data appear to
support your expectation.
9. Are one-credit courses or multi-credit courses rated more highly? What reasoning do the authors give for this trend? Do you agree with their reasoning?
Beauty Scores
10. The paper states that students were asked to “use a 1 to 10 rating scale, […], to make their ratings independent of age, and to keep 5 in mind as an average.” Does it appear that the students
followed the instructions?
11. Make scatterplots of beauty scores given by each student against the others. Do the students appear to agree on the beauty scores of professors?
12. Do male and female students tend to score similarly?
13. Do beauty scores appear to be dependent on whether the picture was black&white or in color, or whether the professor wore a formal outfit or not in the picture?
14. Fit a model predicting average course evaluations from average beauty scores and interpret the slope. Is average beauty score a statistically significant predictor? Does it appear to be a
practically significant variable?
15. Should you include all beauty scores as explanatory variables in a model predicting course evaluations, a few of them, or only the average beauty score? Explain your reasoning and any selection
criteria you might use.
16. The authors use unit normalized beauty scores in their analysis. They also create a composite standardized beauty rating for each instructor and they note that this reduces measurement error.
Create this new composite standardized beauty rating variable and explain why this approach reduces measurement error.
Putting It All Together
17. Fit a multiple regression model predicting average course evaluations using an appropriate selection of the explanatory variables (excluding, at a minimum, average professor evaluation). Justify
the model selection method you use.
18. Is beauty associated with differences in course evaluations? Is the association still present after accounting for other relevant variables? Do your findings agree with the results presented in
the paper?
19. Use graphical diagnostic methods to check if conditions are met for this model. If conditions are not met, what are the implications?
20. Choose a model without any beauty variables to obtain a predicted course rating for this course and calculate the corresponding interval for this prediction. (Note that this question might
require you to share some personal information with the students [e.g., age]. If you are feeling adventurous, you might consider allowing them to leave the beauty variable(s) in the model.)
This last question can lead to a discussion about generalizability—whether a model built on data from The University of Austin at Texas should be used for predictions at other institutions. Working
with these data also gets students to think about the criteria they use when they fill out course evaluation forms; whether these variables are valid indicators of learning; and whether their own
biases about looks, gender, race, and age are coming into play.
While some of the analyses presented in this article are beyond the scope of an introductory course, or even a second course in regression, a few simplifying assumptions make these data accessible to
students at any level.
Further Reading
Hamermesh, D.S., and A. Parker. 2005. Beauty in the classroom: Instructors’ pulchritude and putative pedagogical productivity. Economics of Education Review 24:369–376.
Gelman, A., and J. Hill. 2007. Data analysis using regression and multilevel/hierarchical models. New York: Cambridge University Press. Replication data.
In Taking a Chance in the Classroom, column editors Dalene Stangl, Mine Çetinkaya-Rundel, and Kari Lock Morgan focus on pedagogical approaches to communicating the fundamental ideas of statistical
thinking in a classroom using data sets from CHANCE and elsewhere. | {"url":"https://chance.amstat.org/2013/04/looking-good/","timestamp":"2024-11-11T21:19:00Z","content_type":"application/xhtml+xml","content_length":"47579","record_id":"<urn:uuid:4d1ce001-89f3-49a4-862a-2ddd5c05bdb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00285.warc.gz"} |
Guidance Navigation and Control (GNC) in context of Aerospace Engineering
12 Oct 2024
Guidance, Navigation, and Control (GNC) in Aerospace Engineering: A Comprehensive Overview
In the realm of aerospace engineering, Guidance, Navigation, and Control (GNC) is a critical system that enables aircraft, spacecraft, and missiles to accurately reach their destinations while
ensuring safe and efficient operation. This article delves into the fundamentals of GNC, exploring its components, principles, and applications in the context of aerospace engineering.
What is Guidance, Navigation, and Control?
Guidance refers to the process of determining the desired trajectory or path for a vehicle to follow. Navigation involves measuring the vehicle’s position, velocity, and attitude (orientation)
relative to its surroundings. Control, on the other hand, is the actuation system that adjusts the vehicle’s flight parameters to stay on course.
Components of GNC
1. Guidance: This component determines the desired trajectory or path for the vehicle. Guidance systems use various algorithms, such as linear quadratic regulator (LQR) or model predictive control
(MPC), to calculate the optimal trajectory based on mission requirements and environmental conditions.
2. Navigation: Navigation systems provide the vehicle’s current state information, including position, velocity, and attitude. Common navigation techniques include:
□ Inertial Measurement Unit (IMU): uses accelerometers and gyroscopes to measure the vehicle’s acceleration and angular rate.
□ Global Positioning System (GPS): relies on satellite signals to determine the vehicle’s position and velocity.
□ Inertial Navigation System (INS): combines IMU data with GPS information to provide a more accurate navigation solution.
3. Control: The control system adjusts the vehicle’s flight parameters, such as thrust, pitch, and yaw, to stay on course. Control algorithms, like proportional-integral-derivative (PID) or sliding
mode control (SMC), are used to stabilize the vehicle and maintain its desired trajectory.
Principles of GNC
1. State Estimation: The navigation system estimates the vehicle’s current state by combining sensor data with prior knowledge of the environment.
2. Trajectory Planning: The guidance system plans the optimal trajectory based on mission requirements, environmental conditions, and the vehicle’s current state.
3. Control Loop: The control system adjusts the vehicle’s flight parameters to stay on course, using feedback from sensors and the navigation system.
Applications of GNC in Aerospace Engineering
1. Aircraft Autopilot Systems: GNC is used in commercial aircraft autopilot systems to ensure safe and efficient flight.
2. Spacecraft Navigation: GNC plays a crucial role in spacecraft navigation, enabling accurate trajectory planning and control for interplanetary missions.
3. Missile Guidance: GNC is used in missile guidance systems to ensure accurate targeting and minimize collateral damage.
Mathematical Formulas
1. Linear Quadratic Regulator (LQR): The LQR algorithm minimizes the cost function J(t) = ∫[t0, t] [x^T Q x + u^T R u] dt, where x is the state vector, u is the control input, Q and R are weighting
2. Proportional-Integral-Derivative (PID): The PID controller adjusts the control output u(t) = Kp e(t) + Ki ∫[0, t] e(τ) dτ + KD de/dt, where e(t) is the error signal, Kp, Ki, and KD are gain
3. Sliding Mode Control (SMC): The SMC algorithm uses a switching function s(t) = c^T x + d^T u to determine the control output u(t) = -λ sign(s(t)), where λ is the switching gain.
Guidance, Navigation, and Control (GNC) is a critical system in aerospace engineering that enables accurate trajectory planning and control for aircraft, spacecraft, and missiles. By understanding
the components, principles, and applications of GNC, engineers can design and develop more efficient and reliable systems for various aerospace missions.
Related articles for ‘Aerospace Engineering’ :
• Reading: Guidance Navigation and Control (GNC) in context of Aerospace Engineering
Calculators for ‘Aerospace Engineering’ | {"url":"https://blog.truegeometry.com/tutorials/education/088363f89f663943d043b6c991f0c0ad/JSON_TO_ARTCL_Guidance_Navigation_and_Control_GNC_in_context_of_Aerospace_Engi.html","timestamp":"2024-11-03T03:16:47Z","content_type":"text/html","content_length":"19008","record_id":"<urn:uuid:84089168-9412-41cc-9794-9985368cc19e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00666.warc.gz"} |
CSE 101: Design and Analysis of
CSE 101: Design and Analysis of Algorithms
Fall 2024
• First day of class is September 27th in the CENTER 105.
• Discussion sections are Monday 2-3pm in CENTER 212.
• Please remember to compete the FinAid survey on canvas by the end of the second week of class.
Discussion Notes:
Other: Course Description
: CSE 101 will covers the basics of the design and analysis of algorithms with a focus on non-numerical algorithms. We will cover general algorithmic techniques such as divide and conquer, greedy
algorithms and dynamic programming. We will also discuss some important graph algorithms as well as NP-completeness and techniques for dealing with it. In the process of doing so, we will investigate
several important algorithms such as sorting, shortest paths and longest common substrings. Time permitting we may also cover additional topics such as linear programming, number theoretic algorithms
and quantum computation. | {"url":"https://cseweb.ucsd.edu/~dakane/CSE101/","timestamp":"2024-11-07T16:47:00Z","content_type":"text/html","content_length":"4999","record_id":"<urn:uuid:5e5d242d-fad3-44e4-a80b-37afe07c37ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00574.warc.gz"} |
Basic Laws In Marine Electricity
What is Kirchhoff’s current law?
The total incoming current to a junction is equal to total outgoing current from that junction of an electric circuit or In other words the net current flowing at a junction of a circuit is zero.
Faraday’s First Law - Electromagnetic Induction
Whenever a conductor is placed in a varying magnetic field, an EMF is induced. A current is induced when the conductor circuit is closed. This current is named an induced current.
What is mutual induction?
When an electromotive force (EMF) is produced in a coil due to the change in current in a coupled coil, the effect is known as mutual inductance.
What is the Electromotive force (EMF)?
The force which motivates the electrons to flow in a circuit known as electromotive force.
Its unit is volt.
What is Kirchoff's voltage law?
According to this law, the sum of voltages in a closed circuit on a closed path is equal to zero. Σv=0
Ohm’s law
> Accordingly at constant temperature and pressure the current through a conductor is directly proportional to the voltage across the conductor.
I ∝ V
> Constructional resistance is directly proportional to length of conductor and inversely proportional to the cross-sectional area of the conductor.
Lenz law
> Accordingly whatever may be the cause of production the induced EMF opposes its initial cause
Post a Comment
>> Your Comments are always appreciated...
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Echoes of Eternity: Time's Infinite Tapestry in Nature
Written on
Chapter 1: The Concept of Half-Life
In 1906, physicist Ernest Rutherford introduced the concept of the "half-life" to quantify the duration it takes for half of the atoms in a sample of isotopes to transform into other elements. This
measure exhibits an exponential and probabilistic nature, with recorded values spanning an astonishing range—from approximately one hundred sextillionth of a second for hydrogen-5 to over 18
septillion years for xenon-124. This remarkable variance highlights the profound expanse of Nature’s timeline, stretching from the infinitesimal to the infinite, far beyond human understanding.
What intrigues me most is the inherent nature of these immense figures embedded within these elements and their fundamental particles. It appears these quantum states possess an intrinsic awareness
of time, yielding metrics that are truly mind-boggling. The vast time scales in both directions suggest that the intricacies of reality may be far deeper and broader than we can fathom. The idea of a
Multiverse isn't necessary for our minds to be astounded.
Furthermore, it raises the question: If a single xenon isotope has a half-life a trillion times greater than the age of the universe itself, what other astonishing discoveries lie ahead? What are the
extremes of time?
The finding of xenon was an unexpected outcome of an experiment aimed at detecting Dark Matter, while other efforts focused on measuring proton decay have proven fruitless. However, this does not
imply that protons do not decay. If a seemingly trivial xenon isotope boasts a half-life of 10²³ years, could it be that our calculations have been inaccurate, suggesting that fundamental particles
may take 10?? years or even longer to decay?
In contrast, the 13.8 billion years since the Big Bang now seem merely a fleeting moment. This discrepancy challenges our foundational beliefs about what we understand regarding the universe's
origins and its ultimate fate.
Section 1.1: The Fate of the Universe
Scientists actively engage in discussions about how our Universe may ultimately conclude. Since the revelation in 1998 that the cosmological constant is positive, theories about accelerated expansion
fueled by Dark Energy have gained traction. Astonishingly, approximately 97% of the observable Universe is moving away from us at speeds exceeding that of light. If we project this trend forward, it
suggests that in the blink of an eye, all will vanish.
As the Universe continues to expand exponentially, it grows colder and darker. Red dwarf stars will eventually extinguish after only a few trillion years, while supermassive black holes may linger
for more than a googol years. As space and time stretch and thin, the density of matter and energy diminishes. Should protons decay, the only remnants will be quantum fluctuations of thermal
equilibrium at temperatures nearing 1/10³? degrees Kelvin.
In this so-called De Sitter space, concepts of time and extensibility may lose their significance; clocks would become obsolete, and the ubiquitous quantum state may resemble that existing just
before the Big Bang. Theoretical physicist Roger Penrose has speculated that such a state might ignite another Big Bang, propelling the Universe through cycles of thermal demise and explosive
This inquiry naturally leads to questions about the origin of the initial cycle and the properties that define our Universe (are there infinite cycles, or just one with endless variations?).
Section 1.2: Our Place in the Cosmic Clock
Regardless of how the Universe began or will conclude, the natural world is constructed of intrinsic clocks set to astonishingly small and large intervals, echoing the concept of eternity woven into
the very fabric of reality.
Each of us is born with our own internal clocks ticking away, from our extremities to our telomeres. The average life expectancy of a human is about 72 years, which equates to roughly 2x10³?
half-lives of hydrogen-5 or 1/(6x10²?) the half-life of xenon-124.
In terms of physical dimensions, our bodies occupy a space that is precisely nestled between the Planck length limit, which defines the smallest measurable units, and the outer edges of the
observable Universe. As Nature's clocks continue to expand and contract in duration, our significance within the grand narrative appears increasingly diminished.
History has shown that every time we position ourselves and our planet at the center of the Universe, we have been proven mistaken. Our tendency to regard our existence as unique or special reflects
a misplaced arrogance. We remain suspended amid unfathomably vast and eternally inaccessible extremes, frail and finite in our nature. While belief in a higher power might provide comfort, science
often complicates that pursuit.
Just under a century ago, we were unaware that the Universe extended beyond our own Milky Way galaxy, and prior to 1670, we had no knowledge of ecosystems existing at a scale smaller than a human
As we grapple with the infinite, perhaps the next best approach is to learn how to measure time with clocks calibrated to an astounding billion billion billion billion times the age of the Universe.
Time is inexorably running out—yet, who is keeping track?
Chapter 2: The Infinite Nature of Time | {"url":"https://dayonehk.com/echoes-of-eternity-time-infinite-tapestry.html","timestamp":"2024-11-02T01:02:56Z","content_type":"text/html","content_length":"14212","record_id":"<urn:uuid:3d377e3c-ec63-4c89-a8ae-1f379617170c>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00156.warc.gz"} |
Definition of RATIONALNESS
: having reason or understanding
: relating to, based on, or agreeable to reason : reasonable
a rational explanation
rational behavior
: involving only multiplication, division, addition, and subtraction and only a finite number of times
: relating to, consisting of, or being one or more rational numbers
a rational root of an equation
Examples of rational in a Sentence
Adjective human beings are rational creatures insisted there was a rational explanation for the strange creaking noises and that there were no such things as ghosts
Recent Examples on the Web
Social media speeds up the spread of market sentiment, contributing to volatility that undermines rational pricing. —Carrie McCabe, Forbes, 17 Oct. 2024 His reflexive tendency to extend Lindsay the
benefit of the doubt seemed both credulous and entirely rational. —Eren Orbey, The New Yorker, 14 Oct. 2024
For instance, in one group, collect all rationals that, when squared, are less than 2; in the other, put all rationals whose squares are greater than 2. —Jordana Cepelewicz, Quanta Magazine, 21 June
2024 At the time the database had a little over 3 million elliptic curves over the rationals. —Lyndie Chiou, Quanta Magazine, 5 Mar. 2024 See all Example Sentences for rational
Last Updated: - Updated example sentences | {"url":"https://www.merriam-webster.com/dictionary/rationalness","timestamp":"2024-11-01T23:30:56Z","content_type":"text/html","content_length":"690523","record_id":"<urn:uuid:c0993269-3bf9-454f-a706-5e67c5c607f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00621.warc.gz"} |
Drag of rectangular planform cavity in a flat plate with a turbulent boundary layer for Mach numbers up to 3. Part I: Closed flow
ESDU 00006
Drag of rectangular planform cavity in a flat plate with a turbulent boundary layer for Mach numbers up to 3. Part I: Closed flow
ESDU 00006 develops a method for predicting the drag where the cavity length is large compared to the depth and the shear flow enters the cavity and attaches to the floor before separating to pass
over the rear wall with a stagnation point near the top of the wall (i.e. closed flow). A family of curves suggests an upper limit of cavity depth to length ratio for closed flow in terms of
free-stream Mach number and cavity width to length or width to height ratio. In supersonic flow it is possible for vortices to form as the shear flow spills over the side edges of the cavity and
their impingement on the rear wall gives rise to an increase in drag for which an estimation method is also provided. Tables give the ranges of parameters covered in the construction of the method.
The prediction of the ratio of the drag coefficient, based on floor area, to the local skin friction coefficient at the cavity mid-length station (in the absence of the cavity) is assessed to be
within 1 for low-speed flow (Mach number less than 0.1) and to within 2 for high-speed flow (Mach number between 0.5 and 3). Worked examples illustrate the use of the method. A companion ESDU
document, ESDU 00007, deals with other types of cavity flow, known as transitional and open. The third Item in the series, ESDU 10016, deals with the effect on cavity drag of a pair of doors open at
90°, including the effects of three different treatments of the door leading and trailing edges.
Software for the method is provided as ESDUpac A1607 and also with an interface as Toolbox 16007.
Indexed under:
Data Item ESDU 00006
• PDF
Format: • with software
• with interactive graphs
• Amendment (C), 01 Sep 2017
Status: • Published in Release 2023-02 (Apr 2023)
Previous Releases:
ISBN: • 978 1 86246 109 3
The Data Item document you have requested is available only to subscribers or purchasers.
• Subscribers login here.
• If you are not an ESDU subscriber you can
This Data Item is complemented by the following software:
Name Details
Title ESDU 16007: Estimation of the drag of a rectangular planform cavity, with or without doors, in a flat plate with a turbulent boundary layer
Version 1.0
Toolbox 16007 This app estimates the drag of a rectangular planform cavity, with or without doors, in a flat plate with a turbulent boundary layer for Mach numbers up to 3.
Run Details
The program combines the prediction methods of ESDU 00006 for closed flows, ESDU 00007 for open and transitional flows and the method of ESDU 10016 for estimating the effect
of doors.
Name Details
Title Computer program for the estimation of the drag of a rectangular planform cavity, with or without doors, in a flat plate with a turbulent boundary layer for Mach numbers up to 3
Version 1.0
ESDUpac A1607 (Used in Toolbox App)
This program is only available to subscribers.
The graphs listed below are available only to subscribers.
This Data Item contains 15 interactive graph(s) as listed below.
Graph Title
Figure 1 Boundaries for closed flow (closed flow likely below each curve)
Figure 2 Cavity width factor, F
Figure 3 Value of a[0]
Figure 4 Cavity width factor, G
Figure 5 Value of b[0]
Figure 6 Mach number factor on a[0]
Figure 7 Mach number factor on b[0]
Figure 8 Mach number factor on a[rw0]
Figure 9 Mach number factor on b[rw0]
Figure 10 Factor on C[Drw]/C[fm] to give drag due to side-edge vortices
Figure 11 Exponent in Equation (4.25) for C[fm]
Figure 12 Mach number factors for C[fm]
Figure 13 Local skin friction coefficient on a flat plate
Figure 14 Boundary layer thickness on a flat plate | {"url":"https://esdu.com/cgi-bin/ps.pl?sess=unlicensed_1241031021717lqf&t=doc&p=esdu_00006c&tab=graphs","timestamp":"2024-11-06T14:02:01Z","content_type":"text/html","content_length":"44966","record_id":"<urn:uuid:e803a8ce-247e-4cab-b286-f40b161dcc9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00724.warc.gz"} |
Pascal's triangle
One approach is to notice - after some trial and error - that the number of paths at each node is represented by the equivalent entry in Pascal's triangle. So in general for a NxN grid one would
construct Pascal's triangle up to height 2N and take the entry in the center position of the last row. | {"url":"https://techscreen.ec.tuwien.ac.at/taxonomy/term/10756","timestamp":"2024-11-08T14:04:08Z","content_type":"application/xhtml+xml","content_length":"17482","record_id":"<urn:uuid:a028fc2d-84db-4489-aedc-7aada083b537>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00622.warc.gz"} |
Kelly McCown
New and exciting!
This month debuted a new resource for 5th grade math, the 5th Grade Math Centers Bundle.
This 5th Grade Math Centers Bundle is a GROWING BUNDLE covering ALL 5th grade standards! Your students will enjoy completing the different math centers activities independently or with a group of
classmates. All fifth grade math topics are addressed in this bundle.
Be sure to view the PREVIEW to see everything inside!
This Bundle is 20% OFF the price of all total math centers activities.
Easy to incorporate into your math workshops. Students will love learning and independently working at each math center. All math skills are covered in the 5th grade math centers bundle.
Here's what you'll get:
• 16 Math Centers
• Centers Instructions
• Over 95 printable activities included
• Student Answer Sheets
• Answer Keys
Your students will love practicing math skills with math centers. They'll be independently working on 7 different activities in a group setting. Including math activities with practice, applications,
real life math, and in a teacher group.
Prep is quick and easy! Just print the math center and student answer sheet pages and you're ready for class! No cutting required.
• Place Value
• Multiplication
• Numerical Expressions
• Dividing By Whole Numbers
• Adding Decimals
• Subtracting Decimals
• Multiplying Decimals - Coming 9/15/22
• Dividing Decimals - Coming 9/15/22
• Adding Fractions With Unlike Denominators - Coming 9/15/22
• Subtracting Fractions With Unlike Denominators - Coming 9/22/22
• Multiplying Fractions - Coming 9/22/22
• Dividing Fractions - Coming 9/22/22
• Patterns & Graphing -Coming 9/30/22
• Convert Units of Measurement - Coming 9/30/22
• Geometry - Coming 9/30/22
• Volume - Coming 9/30/22
Ready for 8th grade math review?
Let's look at 5 different resources that can improve your math test review.
There are many ways to do math test prep review in the classroom. I've compiled a list of resources that are ready to use now. By doing at least one actionable strategy of math test prep you can
improve your students' scores by 10%. Choose the best one for your kids now.
1. Math Test Prep Lesson Plans Grade 8. These Math Test Prep Lesson Plans cover all 8th Grade middle school standards. A great tool to prepare students for the end of year test. 3 weeks of math
lesson plans done for you. Easy to use and implement in your math class.
2. End of Year Math Review Folder Grade 8. This Math Folder includes information, suggestions for testing, and review of seventh grade math skills. A great tool for students to use before the big
test. Easy to print and use with any type of folder.
3. Middle School Math Reference Sheet Activities and Quiz. These Math Reference Sheet activities and quiz cover all 6th, 7th, and 8th grade skills students need to the end of year test. A great tool
to use for students as an instructional aide or support. Easy to use before any assessment!
4. End of Year Math Review 8th Grade Booklet. Help students understand how to answer assessment questions based on the eighth grade math standards. Students also review key vocabulary words and
assess their understanding of all seventh grade standards and math skills.
5. End of Year Math Review BINGO Game Grade 8. Students will review key eighth-grade math skills while having fun playing BINGO!
Math Test Prep is important for your kids to practice specific skills and strategies. These skills and strategies will help them feel comfortable and capable of acing the standardized math test this
year. Choose one test prep and watch your students' confidence in math increase as their skills increase to meet the grade level standards.
Happy Reviewing!
Ready for 7th grade math review?
Let's look at 5 different resources that can improve your math test review.
There are many ways to do math test prep review in the classroom. I've compiled a list of resources that are ready to use now. By doing at least one actionable strategy of math test prep you can
improve your students' scores by 10%. Choose the best one for your kids now.
1. Math Test Prep Lesson Plans Grade 7. These Math Test Prep Lesson Plans cover all 7th Grade middle school standards. A great tool to prepare students for the end of year test. 3 weeks of math
lesson plans done for you. Easy to use and implement in your math class.
2. End of Year Math Review Folder Grade 7. This Math Folder includes information, suggestions for testing, and review of seventh grade math skills. A great tool for students to use before the big
test. Easy to print and use with any type of folder.
3. Middle School Math Reference Sheet Activities and Quiz. These Math Reference Sheet activities and quiz cover all 6th, 7th, and 8th grade skills students need to the end of year test. A great tool
to use for students as an instructional aide or support. Easy to use before any assessment!
4. End of Year Math Review 7th Grade Booklet. Help students understand how to answer assessment questions based on the seventh grade math standards. Students also review key vocabulary words and
assess their understanding of all seventh grade standards and math skills.
5. End of Year Math Review BINGO Game Grade 7. Students will review key seventh-grade math skills while having fun playing BINGO!
Math Test Prep is important for your kids to practice specific skills and strategies. These skills and strategies will help them feel comfortable and capable of acing the standardized math test this
year. Choose one test prep and watch your students' confidence in math increase as their skills increase to meet the grade level standards.
Happy Reviewing!
Ready for 6th grade math review?
Let's look at 6 different resources that can improve your math test review.
There are many ways to do math test prep review. I've compiled a list of resources that are ready to use now. By doing at least one actionable strategy of math test prep you can improve your
students' scores by 10%. Choose the best one for your kids now.
1. Math Test Prep Lesson Plans Grade 6. These Math Test Prep Lesson Plans cover all 6th Grade middle school standards. A great tool to prepare students for the end of year test. 3 weeks of math
lesson plans done for you. Easy to use and implement in your math class.
2. End of Year Math Review Folder Grade 6. This Math Folder includes information, suggestions for testing, and review of sixth grade math skills. A great tool for students to use before the big test.
Easy to print and use with any type of folder.
3. Middle School Math Reference Sheet Activities and Quiz. These Math Reference Sheet activities and quiz cover all 6th, 7th, and 8th grade skills students need to the end of year test. A great tool
to use for students as an instructional aide or support. Easy to use before any assessment!
4. End of Year Math Review 6th Grade Booklet. Help students understand how to answer assessment questions based on the sixth grade math standards. Students also review key vocabulary words and assess
their understanding of all sixth grade standards and math skills.
5. Test Prep Math Practice Worksheets Grade 6. A great tool to use as a Review, Test Prep, Intervention, or Classwork before the end of year exam. This test prep packet covers all math skills and
standards for sixth grade. It is an excellent overview of the 6th Grade Math Curriculum.
6. End of Year Math Review BINGO Game Grade 6. Students will review key sixth-grade math skills while having fun playing BINGO!
Math Test Prep is important for your kids to practice specific skills and strategies. These skills and strategies will help them feel comfortable and capable of acing the standardized math test this
year. Choose one test prep and watch your students' confidence in math increase as their skills increase to meet the grade level standards.
Happy Reviewing!
How do you measure a students' knowledge?
There are many different ways to measure and assess a student. The key is to look for the most important grade level skills and measure if there is mastery of skills. A great way to measure skills is
with a math test. Here are 3 Math Tests that matter and why you should be using them too.
Diagnostic Test
A diagnostic math test is the perfect assessment to see what students know and don't yet know. It's a great resource to keep in a student’s file portfolio for parent conferences, to show growth from
the beginning of the year to the end, or to show the guidance counselor results that could be used to move the student to a different class. Diagnostic math tests can be administered any time during
the first semester.
All diagnostic math tests are aligned to grade level standards and skills.
Midyear Test
A midyear math test is an excellent way to assess how students' knowledge from the first half of the school year. This midyear assessment is a great resource to keep in a student’s file portfolio for
parent conferences, to show growth from the beginning of the year to the middle of the year, or to use for math test prep before the state assessment. Midyear math tests can be administered any time
during December or January after the first half of the school year.
All midyear math tests are aligned to grade level standards for the 1st and 2nd semesters.
End of Year Test
An end of year math test is the best way to measure a school year. The end of year math assessment is a great resource to review the year’s curriculum and to show growth from the beginning of the
year. The results can be shared with parents, administration, and to show the guidance counselor results that could be used to place the student in next year’s math class. End of year math tests can
be administered any time during May or June after the second half of the school year.
All end of year math tests are aligned to grade level standards for the year.
Measuring knowledge is key in knowing if your students have grown. You can't improve if you don't know what skills have been mastered and what you need to reteach or review. Using just 3 math skills
tests can make the difference and create a foundation for your students to grow and improve upon.
Happy Teaching and Testing! | {"url":"http://www.kellymccown.com/","timestamp":"2024-11-06T11:32:48Z","content_type":"application/xhtml+xml","content_length":"218920","record_id":"<urn:uuid:9edec220-3ddb-4518-8b6e-a23807716f9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00724.warc.gz"} |
Exponential Relationships (1 of 6)
Learning Objectives
• Use an exponential model (when appropriate) to describe the relationship between two quantitative variables. Interpret the model in context.
In our first example of exponential relationships, we investigate a nonlinear model for growth in a population over time.
The Return of the Bald Eagle
During the mid-20th century, the population of bald eagles in the lower 48 states declined substantially. A highly toxic pesticide, DDT, was the main cause of the decline. DDT causes damage to bird
egg shells. By 1963, bald eagles were in danger of complete extinction. Only 417 pairs of bald eagles remained. In 1967, the bald eagle became an official endangered species. Then in 1972, the EPA
banned the use of DDT in the United States. The impact of the ban was a dramatic turnaround in the fate of the bald eagle.
Here is the data. Note that in the table, we defined t, our explanatory variable, to be Years after 1950. The response variable is the number of bald eagle pairs that are mating.
Our goal is to find an equation to model this relationship.
Here is a scatterplot of the data. We can see that the relationship appears somewhat linear, particularly for years after 1980 (t = 30). The correlation coefficient for this data set is high, r =
The least squares regression line is Predicted eagle pairs = −3,878.11 + 185.4t. Below is a scatterplot of the data with the least-squares regression line and the residual plot.
We see a clear pattern in the residuals, suggesting that a linear model does not capture patterns in the data.
Note: This is a reminder that a large r-value does not guarantee that a linear model is a good fit.
Conclusion: The data set for eagle pairs is clearly nonlinear. We need a better model.
In the scatterplot below, we fit an exponential model to the data. Notice how well this model describes the relationship between the variables. There is very little scatter about the exponential
curve. There is a strong, positive exponential relationship between these variables.
The equation of the exponential model is Predicted eagle pairs = 121 (1.083)^t.
Note: In this equation, the t-variable is an exponent. Sometimes you will see this written with the caret symbol: ^. So Predicted y = 121 (1.083)^t and Predicted y = 121(1.083) ^ t mean the same
Now we use the exponential model to make predictions about the number of bald eagle mating pairs. We also compare the predictions from the exponential model to the linear model. Because there is a
strong exponential relationship and a weaker linear relationship in the data, we expect the predictions from the exponential model to be better.
In 1963 (t = 13), there were 417 mating pairs.
• According to the linear model: Predicted eagle pairs = –3,878.11 + 185.4 (13) = (–1,468). Obviously, a negative value does not make sense for a count of eagle pairs.
• According to the exponential model: Predicted eagle pairs = 121 (1.083)^13 = 341. So the exponential model underestimates by 417 – 341 = 76 mating pairs. But this is a much better prediction than
we got from the linear model.
In 2000 (t = 50), there were 6,471 mating pairs.
• According to the linear model: Predicted eagle pairs = –3,878.11 + 185.4 (50) = 5,392. So the linear model underestimates by 6,471 – 5392 = 1,079 mating pairs.
• According to the exponential model: Predicted eagle pairs = 121 (1.083)^50 = 6,519. So the exponential model overestimates by 6,591 – 6,471 = 48 mating pairs. This is a much better prediction
than we got from the linear model. | {"url":"https://courses.lumenlearning.com/atd-herkimer-statisticssocsci/chapter/exponential-relationships-1-of-6/","timestamp":"2024-11-08T06:23:36Z","content_type":"text/html","content_length":"52982","record_id":"<urn:uuid:ddc56ee2-2fce-4cb0-a2bf-b0cb4fcd9791>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00228.warc.gz"} |
Reversal Dynamic Pivot Points Exponential Moving Average Strategy
1. Reversal Dynamic Pivot Points Exponential Moving Average Strategy
Reversal Dynamic Pivot Points Exponential Moving Average Strategy
, Date: 2023-12-08 11:37:36
The Reversal Dynamic Pivot Points Exponential Moving Average strategy combines reversal trading and dynamic support resistance levels. It uses the Stochastic oscillator to identify market reversal
points and calculates daily support/resistance based on previous day’s high, low and close prices. It goes long or short when both the reversal and pivot points strategies generate buy or sell
signals. The strategy is suitable for medium-term trading.
Strategy Logic
Reversal Strategy
The reversal strategy is based on the rationale that when markets become overbought or oversold, prices tend to revert back to the value range. Specifically, this reversal strategy follows Ulf
Jensen’s rules:
Go long when close has been higher than previous close for 2 consecutive days and 9-day Slow K line is below 50; Go short when close has been lower than previous close for 2 consecutive days and
9-day Fast K line is above 50.
Dynamic Pivot Points Strategy
The dynamic pivot points strategy calculates the current day’s support and resistance levels based on previous day’s high, low and close prices. The formulas are:
Pivot Point = (High + Low + Close) / 3
Support 1 = Pivot Point - (High - Pivot Point)
Resistance 1 = Pivot Point + (Pivot Point - Low)
It goes long when close is higher than Resistance 1 and goes short when close is lower than Support 1.
Dual Signals
This strategy combines the reversal and dynamic pivot points strategies. It only enters positions when signals from both strategies align. This helps filter out some false signals and improves
The biggest advantage of this strategy is it combines the strengths of both reversal and dynamic S/R strategies - it can benefit from major trend reversals and also identify key support and
resistance levels. Compared to individual strategies, it has better stability from filtering out some false signals.
Also, the strategy has few parameters and is easy to implement and optimize.
The strategy also has the following risks:
• Failed reversal - prices may over-extend and continue to trend despite reversal signal.
• Breach of support/resistance levels - prices may breakthrough calculated S/R levels resulting in wrong signals.
• Dual signals too conservative, missing runs. The dual signal mechanism may filter too many trades.
• Fine-tune parameters, combine other factors to confirm reversals.
• Use stop loss to control loss.
• Adjust rules to allow more trading opportunities.
Enhancement Opportunities
The strategy can be enhanced in the following areas:
1. Test different Stochastic parameters combinations to improve sensitivity in identifying reversals.
2. Test different moving averages or longer term indicators to better gauge overall trend.
3. Add other factors to determine market structure, e.g. volume indicators.
4. Optimize dual signal rules to capture more trades.
5. Incorporate stop loss to manage risks.
The Reversal Dynamic Pivot Points Exponential Moving Average strategy combines the strengths of reversal trading and dynamic support resistance analysis. It can benefit from major trend turning
points and also gauge intraday directionality against key levels. By utilizing dual-signal mechanism, it has good stability in filtering out false trades. The strategy can be further optimized by
tuning parameters, testing additional filters etc. to enhance performance.
start: 2023-11-07 00:00:00
end: 2023-12-07 00:00:00
period: 1h
basePeriod: 15m
exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}]
// Copyright by HPotter v1.0 25/03/2020
// This is combo strategies for get a cumulative signal.
// First strategy
// This System was created from the Book "How I Tripled My Money In The
// Futures Market" by Ulf Jensen, Page 183. This is reverse type of strategies.
// The strategy buys at market, if close price is higher than the previous close
// during 2 days and the meaning of 9-days Stochastic Slow Oscillator is lower than 50.
// The strategy sells at market, if close price is lower than the previous close price
// during 2 days and the meaning of 9-days Stochastic Fast Oscillator is higher than 50.
// Second strategy
// This Pivot points is calculated on the current day.
// Pivot points simply took the high, low, and closing price from the previous period and
// divided by 3 to find the pivot. From this pivot, traders would then base their
// calculations for three support, and three resistance levels. The calculation for the most
// basic flavor of pivot points, known as ‘floor-trader pivots’, along with their support and
// resistance levels.
// WARNING:
// - For purpose educate only
// - This script to change bars colors.
Reversal123(Length, KSmoothing, DLength, Level) =>
vFast = sma(stoch(close, high, low, Length), KSmoothing)
vSlow = sma(vFast, DLength)
pos = 0.0
pos := iff(close[2] < close[1] and close > close[1] and vFast < vSlow and vFast > Level, 1,
iff(close[2] > close[1] and close < close[1] and vFast > vSlow and vFast < Level, -1, nz(pos[1], 0)))
DPP() =>
pos = 0
xHigh = security(syminfo.tickerid,"D", high[1])
xLow = security(syminfo.tickerid,"D", low[1])
xClose = security(syminfo.tickerid,"D", close[1])
vPP = (xHigh+xLow+xClose) / 3
vR1 = vPP+(vPP-xLow)
vS1 = vPP-(xHigh - vPP)
pos := iff(close > vR1, 1,
iff(close < vS1, -1, nz(pos[1], 0)))
strategy(title="Combo Backtest 123 Reversal & Dynamic Pivot Point", shorttitle="Combo", overlay = true)
Length = input(14, minval=1)
KSmoothing = input(1, minval=1)
DLength = input(3, minval=1)
Level = input(50, minval=1)
reverse = input(false, title="Trade reverse")
posReversal123 = Reversal123(Length, KSmoothing, DLength, Level)
posDPP = DPP()
pos = iff(posReversal123 == 1 and posDPP == 1 , 1,
iff(posReversal123 == -1 and posDPP == -1, -1, 0))
possig = iff(reverse and pos == 1, -1,
iff(reverse and pos == -1 , 1, pos))
if (possig == 1)
strategy.entry("Long", strategy.long)
if (possig == -1)
strategy.entry("Short", strategy.short)
if (possig == 0)
barcolor(possig == -1 ? #b50404: possig == 1 ? #079605 : #0536b3 ) | {"url":"https://www.fmz.com/strategy/434678","timestamp":"2024-11-08T15:19:11Z","content_type":"text/html","content_length":"15649","record_id":"<urn:uuid:85b83f83-31ab-4691-903f-173fd0b8e257>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00010.warc.gz"} |
Miller Indices-Procedure for finding and Important Features
Miller Indices
In this lecture, we are going to learn about the Miller Indices, their definition, the Procedure for finding Miller Indices, and some important features of Miller Indices of crystal plane. So let’s
start with the introduction of Miller Indices.
Introduction of Miller Indices
In a crystal, there exists direction and planes which contain a large concentration of atoms. Therefore it is necessary to locate these directions and planes for a crystal to analyze. The problem is
how to identify the direction and to designate ( to choose) a plane in a crystal.
Here, let us discuss briefly the method of designating a plane in crystal. This method was suggested by Miller.
What is Miller Indices?
Miller introduces a set of three numbers to designate a plane in a crystal. This set of three numbers is known as the Miller indices of the concerned plane.
Also Read: CRYSTAL SYSTEMS AND BRAVAIS LATTICES
Procedure for Finding Miller Indices
The steps in the determination of Miller indices of a plane are illustrated with the aid of Figure. Consider the plane ABC which cuts 1 unit along the X-axis, 2 units along the Y-axis, and three
units along the Z-axis.
Step 1:
• Find the intercept of the plane ABC along the three axes X, Y, and Z. Let it be OA, OB, and OC. Express the intercepts in terms of multiples of axial lengths, i.e., lattice parameters. Let them
be OA = pa, OB = qb, and OC = rc where p, q, and r are the intercept numerical values along the three-axis.
p=1, q=2 and r = 3
Hence, OA : OB: OC = pa:qb: rc = 1a: 2b : 3c
• Therefore, the intercepts are 1a, 2b, and 3c along the three axes.
Step 2:
• Find the reciprocal of the numerical intercept values.
i.e., \frac{1}{p}\frac{1}{q}\frac{1}{r}
• For the example shown the reciprocal of the numerical intercept values are:
Step 3:
• Convert these reciprocals into whole numbers by multiplying each with their least common multiple (LCM). In this example, the LCM is 6. Therefore,
6\times \frac{1}{1}\;,6\times \frac{1}{2}\;, 6\times\frac{1}{3}
6 \;3\; 2
Step 4:
• Enclose these numbers in the bracket. This represents the indices of the given plane and is called the Miller Indices of the plane.
• For example, as shown, the Miller indices are (6 3 2).
• It is generally denoted by ( h k l). It can also be noticed that,
Definition -1: Thus, Miller indices may be defined as the reciprocal of the intercepts made by the plane along the three crystallographic axes which are reduced to the smallest numbers.
Definition-2: Miller Indices are the three smallest possible integers, which have the same ratio as the reciprocals of the intercepts of the plane concerned along the three axes.
Also Read: Engineering Materials | Classification Of Engineering Materials
Important Features of Miller Indices of Crystal Plane
When it comes to Miller indices, several important features come into play, providing valuable insights into crystallography. Let’s delve into these essential aspects:
1. Infinite Intercepts:
• Any plane that runs parallel to at least one coordinate axis exhibits an infinite intercept (∞). Consequently, the Miller index for that particular axis becomes zero.
2. Parallel Planes:
• Equally spaced parallel planes with a specific alignment share the same index numbers (h k l). It’s important to note that Miller indices represent not just a single plane, but rather a
combination of multiple parallel planes.
3. The ratio of Indices:
• The ratio between the indices holds significance above all else. The specific planes themselves are not of primary importance.
4. Origin and Non-Zero Intercept:
• A plane passing through the origin is defined in comparison to a parallel plane with non-zero intercepts.
5. Equally Distant Planes:
• Parallel planes that are equally spaced possess identical Miller indices. Hence, Miller indices are utilized to represent a set of parallel planes.
6. Parallelism through Ratio:
• If two planes have the same ratio of Miller indices, such as 844 and 422 or 211, they can be deemed parallel to each other.
7. Dividing the Axes:
• If the Miller indices for a plane are denoted as h k l, the plane will divide the axes into equivalent sections of a/h, b/k, and c/l.
8. Precision in Multi-Digit Indices:
• When the integers in Miller indices consist of more than one digit, it is essential to separate them by commas for clarity, for example, (3, 11, 12).
9. Crystal Directions and Planes:
• In a crystal family, the directions of the crystals are not necessarily parallel to each other. Similarly, not all planes within a family are guaranteed to be parallel.
10. Antiparallel Directions and Planes:
• By changing the signs of all indices in a crystal direction, an antiparallel or conflicting direction is obtained. Similarly, altering the signs of all indices in a plane leads to a plane
situated at an equivalent distance on the opposite side of its origin.
Also Read: Dielectric Polarization | Types of Polarization in Dielectrics
Frequently Asked Questions on Miller Indices
1. Explain Miller Indices.
The Miller indices definition can be stated as the mathematical representation of the crystallographic planes in three dimensions. Miller evolved a method to designate the orientation and
direction of the set of parallel planes with respect to the coordinate system by numbers h, k, and l (integers) known as the Miller indices. The planes represented by the h k l miller indices are
also known as the h k l planes.
2. State Any Two Rules to Determine the Miller Indices.
Following are the rules to be followed for determining the miller indices:
1. Determine the intercepts (a,b,c) of the planes along the crystallographic axes, in terms of unit cell dimensions.
2. Consider the reciprocal of the intercepts measured.
3. Clear the fractions, and reduce them to the lowest terms in the same ratio by considering the LCM.
4. If the h k l plane has a negative intercept, the negative number is denoted by a bar ( ̅) above the number.
3. Are Miller indices at all times positive in value or they can be negative as well?
According to Miller indices, two or more parallel planes can have similar indices which can ultimately be a negative value, zero, or a positive value. This completely depends on the intercept on
the axes and nothing else. This leads us to the conclusion that Miller indices do not necessarily always have to be positive.
4. What is the meaning of the line that is displayed over a number in Miller indices?
A Miller index having a 0 value simply means that the plane is in a parallel position to the corresponding axis. Negative indices, on the other hand, are indicated with a bar drawn over the
integer number. In cubic crystal systems, the Miller indices of a plane are exactly similar to those of the way vertical to the plane.
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2 (number) - Wikiwand
2 (Two; ( listen)) is a number, numeral, and glyph. It is the number after 1 (one) and the number before 3 (three). In Roman numerals, it is II.
Pronunciation of the number 2
Two has many meanings in math. For example: ${\displaystyle 1+1=2}$.^[1] An integer is even if half of it equals an integer. If the last digit of a number is even, then the number is even. This means
that if you multiply 2 times anything, it will end in 0, 2, 4, 6, or 8.
Two is the smallest, first, and only even prime number. The next prime number is three. Two and three are the only prime numbers next to each other. The even numbers above two are not prime because
they are divisible by 2.
Fractions with 2 in the bottom do not yield infinite.
Two is the framework of the binary system used in computers. The binary way is the simplest system of numbers where natural numbers (the numbers used to count) can be written.
Two also has the unique property that 2+2 = 2·2 = 2^2 and 2! + 2 = 2^2.
Powers of two are important to computer science.
The square root of two was the first known irrational number.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
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The argument f should be an object of class "fv", containing both empirical estimates \(\widehat f(r)\) and a theoretical value \(f_0(r)\) for a summary function.
The P--P version of f is the function \(g(x) = \widehat f (f_0^{-1}(x))\) where \(f_0^{-1}\) is the inverse function of \(f_0\). A plot of \(g(x)\) against \(x\) is equivalent to a plot of \(\widehat
f(r)\) against \(f_0(r)\) for all \(r\). If f is a cumulative distribution function (such as the result of Fest or Gest) then this is a P--P plot, a plot of the observed versus theoretical
probabilities for the distribution. The diagonal line \(y=x\) corresponds to perfect agreement between observed and theoretical distribution.
The Q--Q version of f is the function \(h(x) = f_0^{-1}(\widehat f(x))\). If f is a cumulative distribution function, a plot of \(h(x)\) against \(x\) is a Q--Q plot, a plot of the observed versus
theoretical quantiles of the distribution. The diagonal line \(y=x\) corresponds to perfect agreement between observed and theoretical distribution. Another straight line corresponds to the situation
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Sum and difference of cubes worksheets
sum and difference of cubes worksheets Related topics: sum multiply roots equation quadratic
formula to find square root
convert a fraction to a decimal point
common factors of 48
8th grade math formula chart
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linear to sqaure root
calculator for solving synthetic division
algebra used in everyday life
evaluate rational expression calculator
means nearly the same as "each generation builds shrines to the radicals crucified by its fathers"
negative exponents and rational functions
log ti89
linear equation graphing calculator
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I need help with topics covered in Algebra 1 class and look for help to understand everything that I need to know so I can improve my grades.
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From: Atlanta, GA
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IlbendF Posted: Tuesday 10th of May 20:27
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Dolknankey Posted: Thursday 12th of May 17:57
Hello , I am in Intermediate algebra and I purchased Algebrator a few weeks ago. It has been so much easier since then to do my algebra homework! My grades also got much
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Vild Posted: Thursday 12th of May 21:18
Algebrator is the program that I have used through several algebra classes - Algebra 2, Algebra 2 and Pre Algebra. It is a truly a great piece of math software. I remember
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CDS & AFCAT 1 2025 Exam Maths Simple & Compound Interest Class 1
In competitive exams like the Combined Defence Services (CDS) and Air Force Common Admission Test (AFCAT), questions on Simple Interest (SI) and Compound Interest (CI) form a crucial part of the
mathematics section. These topics are considered relatively straightforward but require clarity of concepts and quick calculations. Recently, a class was conducted to explain these concepts, focusing
on the basics such as principal, interest, the rate of interest, time, and the simple interest formula. In this blog, we will go over the core topics that were covered in the class and provide
strategies to help you prepare effectively for the exams.
Key Concepts in Simple Interest and Compound Interest
1. Principal:
The principal refers to the initial amount of money that is borrowed or invested. All interest calculations are based on this sum. Understanding the role of the principal is crucial since both simple
and compound interest formulas use this as the starting point.
2. Interest:
Interest is the extra amount paid by a borrower to a lender, or the return earned by someone who invests money. There are two types of interest commonly discussed in these exams: Simple Interest and
Compound Interest.
3. Rate of Interest:
This is the percentage of the principal that is paid or earned as interest over a given period, typically a year. In most questions, the rate of interest is presented annually, but it can be adjusted
for different time periods depending on the problem.
4. Time:
The time period is the duration for which the money is borrowed or invested. The interest is usually calculated for a specific period—monthly, quarterly, or annually.
Simple Interest (SI)
Simple Interest is the interest calculated on the original principal for the entire duration of the loan or investment. It remains constant, which makes calculations simple and straightforward. The
total amount of interest earned or paid is directly proportional to the principal, the rate of interest, and the time. This linear relationship makes it easy to understand and solve questions
In the class, the instructor emphasized the importance of thoroughly understanding the relationship between the principal, interest rate, time, and simple interest. Questions in CDS and AFCAT exams
typically revolve around calculating one of these variables when the other three are provided.
Types of Questions Discussed in the Class
During the class, several types of questions were discussed, focusing on both Simple Interest and Compound Interest. Here are some of the common question formats that were highlighted:
1. Basic Simple Interest Calculation:
These questions are the most straightforward and usually ask you to calculate the interest earned or paid based on the given principal, rate of interest, and time period. These problems are
simple and serve as the foundation for more complex interest-related problems.
2. Finding the Principal or Rate of Interest:
In some questions, the interest, time, and rate might be given, and you are asked to find the principal amount. Alternatively, the principal, interest, and time could be provided, and you’ll
need to calculate the rate of interest. These questions test your ability to manipulate the basic relationships between these variables.
3. Compound Interest with Different Compounding Frequencies:
Compound interest problems often ask for interest calculations when the interest is compounded annually, semi-annually, or quarterly. Understanding how to adjust for different compounding periods
is key to solving these problems quickly and correctly.
4. Difference Between Simple Interest and Compound Interest:
A common question type in CDS and AFCAT exams involves calculating the difference between simple interest and compound interest over a given time period. This type of question highlights how much
extra interest is earned through compounding, and the calculations must be precise.
Strategies to Prepare for Simple Interest and Compound Interest
While Simple and Compound Interest are considered easier topics, mastering them can give you an edge in the mathematics section of CDS and AFCAT exams. Here are some strategies discussed in the class
that will help you prepare:
1. Focus on Conceptual Clarity
Before diving into practice questions, ensure that you fully understand the underlying concepts of principal, interest, rate, and time. Learn how they are interrelated and how they affect each other
in both Simple and Compound Interest calculations.
2. Practice Different Types of Questions
Since questions can range from simple interest calculations to more complex compound interest problems, make sure to practice a variety of question types. Start with basic problems and gradually move
on to questions involving different compounding periods and mixed interest calculations.
3. Master Quick Calculation Techniques
In competitive exams, time is of the essence. Mastering shortcuts and mental math techniques can help you save valuable time during the exam. Practice calculating interest amounts quickly and
accurately, especially when working with compound interest problems where time-consuming calculations are common.
4. Use Previous Year Papers
One of the best ways to prepare for any exam is by solving previous years’ papers. Go through the SI and CI questions from previous CDS and AFCAT exams to get a sense of the types of problems that
are commonly asked. Pay attention to patterns and frequently repeated question types.
5. Stay Updated on Tricks for Compound Interest
Compound Interest problems often involve lengthy calculations. Learning a few tricks to simplify these can save you time. For instance, when interest is compounded annually, semi-annually, or
quarterly, practice adjusting the rate of interest and time period accordingly.
6. Avoid Careless Mistakes
Many students lose marks due to small errors in reading the question or misinterpreting the given data. In questions involving Compound Interest, especially with different compounding periods,
double-check your calculations to avoid such mistakes.
7. Time Management
During the exam, it’s important to allocate your time wisely. Simple Interest questions are generally quicker to solve, while Compound Interest may require more time. Practice solving these
questions under timed conditions to develop a good sense of pacing.
Simple Interest and Compound Interest are important topics in the mathematics section of the CDS and AFCAT exams. A recent class on this topic emphasized the importance of understanding basic
concepts like principal, interest, rate, and time, and how they apply to both types of interest calculations. While Simple Interest problems are straightforward, Compound Interest questions can be
more complex, especially when different compounding periods are involved.
By focusing on building a solid foundation of the concepts, practicing different types of problems, and mastering quick calculation techniques, you can confidently tackle these questions in the exam.
Remember to stay organized in your preparation and regularly solve previous year papers to familiarize yourself with the exam format. With consistent effort and effective time management, you’ll be
well-prepared to ace the Simple and Compound Interest sections of your CDS and AFCAT exams.
Good luck!
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Cumulative Return On Investment
Since , the average annual total return for the S&P , an unmanaged index of large U.S. stocks, has been about 10%. Investments that offer the potential. To calculate the total return over the period,
divide the ending value by the beginning value and then subtract one. measures how the value of an investment has changed over time. This calculation considers the fund's performance along with the
size and timing of cash flows. Total return figures take into account not only the increases (and decreases) in the prices of the shares you own but also the value of any payouts you received. ROI is
generally defined as the ratio of net profit over the total cost of the investment. ROI is most useful to your business goals when it refers to something.
An ETF also typically has ongoing fees in the form of its expense ratio, referred to in the ETF's prospectus as the total annual fund operating expenses. You. The return on investment (ROI) is return
per dollar invested. It is a measure of investment performance, as opposed to size. Cumulative return is the return on the investment in total. For instance, the money gained in the first year of an
investment would be the annualized return. Investment return and principal value of Fund shares will fluctuate so that Total Return Bond Fund (A), Eaton Vance, EBABX, , , , , That may sound like a
good return, but on an annualized basis the return is about 5% a year. To give you a better understanding of how well your investments. total change in the value of an investment over a specifc
period. Over longer time periods, the cumulative return of an investment can be a very large number. Annualized total return gives the yearly return of a fund calculated to demonstrate the rate of
return necessary to achieve a cumulative return. Return on Investment is a key business metric that measures the profitability of investments or marketing activities by weighing the size of the
upfront. return, based on their unique investment objectives and risk tolerance. cumulative total return if performance had been constant over the entire. The annualized total return is the return
that an investment earns each year for a given period. · It is useful when comparing investments with different lengths. Total return, for a given period, is defined as share price performance
including the value of all re-invested dividends. Each dividend is calculated re-.
Return on investment (ROI) is a financial ratio that measures the profit generated from investments. Since it can be difficult for companies to establish. Cumulative return refers to the aggregate
amount an investment gains or loses irrespective of time, and can be presented as either a numerical sum total or as a. An annualized total return is the return earned on an investment each year. It
is computed as a geometric average of the returns of each year earned over a. The Annualized Return Calculator computes the annualized return of an investment held for a specified number of years.
The true performance most people want to know is cumulative, how much money have I actually made. The annualized returns will be time-weighted. Total return includes change in share prices and, in
each case The investment return and principal value vary so that an investor's shares. You can calculate the return on your investment by subtracting the initial amount of money that you put in from
the final value of your financial investment. spark-servis.ru provides a FREE return on investment calculator and other ROI calculators to compare the impact of taxes on your investments. ROI is
calculated by dividing a company's net income (earnings after tax) by total investments (total invested capital) and multiplying the result by
The total amount of interest earned, after the effects of inflation have been calculated. Total future value. The total value of the investment after the. The cumulative return over both periods, R
c, is (1 + R 1)(1 + R 2) – 1 = R c. The cumulative return is sometimes referred to as the total return. In the first year of investing, you may generate returns on your initial investment, while in
the second year, you invest the capital (your initial investment). Capital Market Investment Basics The return is the total income an investor gets from his/her investment every year and is usually
quoted as a percentage of. The Total Return Investment Pool (TRIP) is an investment pool established in by the regents and is available to UC campuses and the UC Office of the.
Calculate Annualized Returns for Investments in Excel
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Gr 4_OA_PropertyPrimeNumbers_Problem_Critique | Bridging Practices Among Connecticut Mathematics Educators
Gr 4_OA_PropertyPrimeNumbers_Problem_Critique
This task is designed for fourth grade students learning about how to classify odd and prime numbers. Students are asked to critique a statement made by a student regarding odd and prime numbers. The
task addresses a common misconception that all odd numbers are also prime numbers. Students are asked to use diagrams to help support answers and also provide a counter example in order to prove a
statement is false.
Microsoft Word version: 4_OA_PropertyPrimeNumbers_Problem_Critique
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Notes - DAA HT23, Shortest paths and relaxation
Shortest paths and relaxation
What is the process of relaxing an edge $(u, v)$?
Updating the distance to $v$ if going via $u$ is cheaper.
Let $\delta(x, y)$ represent the shortest path distance between $x$ and $y$ and let $w(u, v)$ represent the weight of the edge $(u, v)$. What is the triangle inequality in the context of shortest
paths from $s$?
For any edge $(u, v) \in E$
\[\delta(s, v) \le \delta(s, u) + w(u, v)\]
Let $\delta(x, y)$ represent the shortest path distance between $x$ and $y$. What is the upper-bound property in the context of shortest paths from $s$?
For every vertex $v$ we have
\[v.d \ge \delta(s, v)\]
and once it is equal, it never changes.
Still need to do:
• Convergence property
• Path-relaxation property
• Predeccessor-subgraph property
Dijkstra’s Algorithm
Breadth-first search and Dijkstra’s algorithm are almost the same algorithm but with one small change. What data structure does Dijkstra’s use rather than a FIFO queue?
What is the running time of Dijkstra’s algorithm?
\[O((|V|+|E|)\log |V|)\]
Can you state pseudocode for Dijkstra’s algorithm on a graph $G$ and starting from $s$?
DIJKSTRA(G, s):
for every vertex v in G:
v.pi = null
v.d = infinity
s.d = 0
Q = priority queue indexed by distance
while Q is nonempty:
u = DEQUEUE(Q)
for every vertex v adjacent to u:
if u.d + w(u, v) < v.d:
v.d = u.d + w(u, v)
v.pi = u
DECREASE-KEY(Q, v, v.d)
Bellman-Ford algorithm
Dijkstra’s algorithm doesn’t work when there’s negative weights. Assuming the graph has no negative cycles, what’s the name of the algorithm that solves the shortest path problem for graphs with
negative weights?
What’s the time complexity of the Bellman-Ford algorithm?
How many times does the Bellman-Ford algorithm relax every edge?
$ \vert V \vert -1$ times.
Can you state pseudocode for the Bellman-Ford algorithm on a graph $G$ starting from $s$?
BELLMAN-FORD(G, s):
for every vertex v in G:
v.d = infinity
v.pi = null
s.d = 0
for i in 1 to |V| - 1:
for (u, v) in E:
if u.d + w(u, v) < v.d:
v.d = u.d + w(u, v)
v.pi = u
for each (u, v) in E:
if u.d + w(u, v) < v.d:
return false
return true
Shortest paths in DAGs
When the input graph is a DAG (possibly with negative weights), there’s better algorithms than Dijkstra’s or the Bellman-Ford algorithm for finding shortest paths. What’s the best time complexity you
can achieve?
\[O(|V| + |E|)\]
Can you state pseudocode for finding the shortest path in a DAG $G$ and from a start vertex $s$?
DAG-SHORTEST-PATHS(G, s):
topologically sort vertices of G
for each vertex v in G:
v.d = infinity
v.pi = null
s.d = 0
for each vertex u in the topologically sorted G:
for each vertex v adjacent to u
if u.d + w(u, v) < v.d:
v.d = u.d + w(u, v)
v.pi = u
Can you briefly justify why
DAG-SHORTEST-PATHS(G, s):
topologically sort vertices of G
for each vertex v in G:
v.d = infinity
v.pi = null
s.d = 0
for each vertex u in the topologically sorted G:
for each vertex v adjacent to u
if u.d + w(u, v) < v.d:
v.d = u.d + w(u, v)
v.pi = u
is a correct algorithm?
By considering vertices in topological order, a vertex is only considered once all of its predecessors have been processed.
Floyd-Warshall algorithm
What’s the $O( \vert V \vert ^3)$ algorithm for finding the shortest-path between every two vertices in a graph called?
The Floyd-Warshall algorithm.
What’s the time complexity of the Floyd-Warshall algorithm?::
When solving the all-pairs shortest path problem using the Floyd-Warshall algorithm, what data structure do you use for keeping track of intermediate results (that are used by the dynamic programming
\[d[i, j, k]\]
which is the shortest distance from $i$ to $j$ whose intermediate nodes are in the interval $[1, k]$.
What optimal substructure about $s \xrightarrow{p _ 1} u \xrightarrow{p _ 2} v$ is used by the Floyd-Warshall algorithm?
If $s \xrightarrow{p _ 1} u \xrightarrow{p _ 2} v$ is the shortest path from $s$ to $v$, then
• $s \xrightarrow{p _ 1} u$ is the shortest path from $s$ to $u$
• $u \xrightarrow{p _ 2} v$ is the shortest path from $u$ to $v$.
\[d[i, j, k]\]
be the shortest distance from $i$ to $j$ whose intermediate nodes are in the interval $[1, k]$. Given also that if $s \xrightarrow{p _ 1} u \xrightarrow{p _ 2} v$ is the shortest path from $s$ to
$v$, then
• $s \xrightarrow{p _ 1} u$ is the shortest path from $s$ to $u$
• $u \xrightarrow{p _ 2} v$ is the shortest path from $u$ to $v$.
then what recurrence can you use to solve the all-pairs shortest path algorithm?
\[d[i, j, k+1] = \min (d[i, j, k], d[i, k+1, k] + d[k+1, j, k] )\]
Can you give pseudocode for the Floyd-Warshall all-pairs shortest path algorithm?
FLOYD-WARSHALL(G, w):
for i = 1 to |V|:
for j = 1 to |V|:
d[i, j, 0] = infinity
for each edge (i, j) in E:
d[i, j, 0] = w(i, j)
for k = 0 to |V| - 1
for i = 1 to |V|
for j = 1 to |V|
d[i, j, k+1] = min(
d[i, j, k],
d[i, k+1, k] + d[k+1, j, k]
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One- and Two-Sided Testing, Mixture Distributions
Consider testing the hypothesis . If is the open interval , then only a one-sided alternative hypothesis is meaningful,
This is the appropriate set of hypotheses, for example, when is the variance of a G-side random effect. The positivity constraint on is required for valid conditional and marginal distributions of
the data. Verbeke and Molenberghs (2003) refer to this situation as the constrained case.
However, if one focuses on the validity of the marginal distribution alone, then negative values for might be permissible, provided that the marginal variance remains positive definite. In the
vernacular or Verbeke and Molenberghs (2003), this is the unconstrained case. The appropriate alternative hypothesis is then two-sided,
Several important issues are connected to the choice of hypotheses. The GLIMMIX procedure by default imposes constraints on some covariance parameters. For example, variances and scale parameters
have a lower bound of 0. This implies a constrained setting with one-sided alternatives. If you specify the NOBOUND option in the PROC GLIMMIX statement, or the NOBOUND option in the PARMS statement,
the boundary restrictions are lifted from the covariance parameters and the GLIMMIX procedure takes an unconstrained stance in the sense of Verbeke and Molenberghs (2003). The alternative hypotheses
for variance components are then two-sided.
When and , the value of under the null hypothesis is on the boundary of the parameter space. The distribution of the likelihood ratio test statistic is then nonstandard. In general, it is a mixture
of distributions, and in certain special cases, it is a mixture of central chi-square distributions. Important contributions to the understanding of the asymptotic behavior of the likelihood ratio
and score test statistic in this situation have been made by, for example, Self and Liang (1987); Shapiro (1988); Silvapulle and Silvapulle (1995). Stram and Lee (1994, 1995) applied the results of
Self and Liang (1987) to likelihood ratio testing in the mixed model with uncorrelated errors. Verbeke and Molenberghs (2003) compared the score and likelihood ratio tests in random effects models
with unstructured matrix and provide further results on mixture distributions.
The GLIMMIX procedure recognizes the following special cases in the computation of p-values ( denotes the realized value of the test statistic). Notice that the probabilities of general chi-square
mixture distributions do not equal linear combination of central chi-square probabilities (Davis 1977; Johnson, Kotz, and Balakrishnan 1994, Section 18.8).
1. parameters are tested, and neither parameters specified under nor nuisance parameters are on the boundary of the parameters space (Case 4 in Self and Liang 1987). The p-value is computed by the
classical result:
2. One parameter is specified under and it falls on the boundary. No other parameters are on the boundary (Case 5 in Self and Liang 1987).
Note that this implies a 50:50 mixture of a and a distribution. This is also Case 1 in Verbeke and Molenberghs (2000, p. 69).
3. Two parameters are specified under , and one falls on the boundary. No nuisance parameters are on the boundary (Case 6 in Self and Liang 1987).
A special case of this scenario is the addition of a random effect to a model with a single random effect and unstructured covariance matrix (Case 2 in Verbeke and Molenberghs 2000, p. 70).
4. Removing j random effects from uncorrelated random effects (Verbeke and Molenberghs, 2003).
Note that this case includes the case of testing a single random effects variance against zero, which leads to a 50:50 mixture of a and a as in 2.
5. Removing a random effect from an unstructured matrix (Case 3 in Verbeke and Molenberghs 2000, p. 71).
where k is the number of random effects (columns of ) in the full model. Case 5 in Self and Liang (1987) describes a special case.
When the GLIMMIX procedure determines that estimates of nuisance parameters (parameters not specified under ) fall on the boundary, no mixture results are computed.
You can request that the procedure not use mixtures with the CLASSICAL option in the COVTEST statement. If mixtures are used, the Note column of the “Likelihood Ratio Tests of Covariance Parameters”
table contains the “MI” entry. The “DF” entry is used when PROC GLIMMIX determines that the standard computation of p-values is appropriate. The “–” entry is used when the classical computation was
used because the testing and model scenario does not match one of the special cases described previously. | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_glimmix_details28.htm","timestamp":"2024-11-07T16:11:08Z","content_type":"application/xhtml+xml","content_length":"30705","record_id":"<urn:uuid:43706219-208f-4d20-96c0-ecf199ccf294>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00884.warc.gz"} |
Doppelblock rules - Xavier Badin de Montjoye
Doppelblock rules
Doppelblock rules : shade exactly two cells in each row and each column. In the remaining cells, place an integer between 1 and X-2, where X is the number of cells in each row, such that each number
appears exactly once in each row and each column. Number outside the grid indicate the sum of the numbers between the two shaded cells. Some cells may already be filled for you.
Here is an example of a Doppelblock grid and its solution
Solve online here
History of Doppelblock
According to the WPC unofficial wiki, doppelblock appeared in online qualifier for the Japan Puzzle Championship 2016 by an unknown author. It was latter independently created by Naoki Inaba (Japan). | {"url":"https://xavierbdm.fr/en/rules-doppelblock/","timestamp":"2024-11-14T11:46:55Z","content_type":"text/html","content_length":"44239","record_id":"<urn:uuid:57e307e9-31d2-4a3e-8074-7b359df609b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00117.warc.gz"} |