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Back propagation for the seriously hands-on
I have just finished the Stanford back propagation exercise, and to put it mildly, it was a ****.
So is back propagation complicated? And indeed what is it?
These are my notes so that I don’t have to go through all the pain when I do this again. I am not an expert and the agreement with Stanford is that we don’t give away the answer particularly at the
level of code. So use with care and understand that this can’t tell you everything. You need to follow some lecture notes too.
Starting from the top: What is back propagation?
Back propagation is a numerical algorithm that allows us to calculate an economical formula for predicting something.
I am going to stick to the example that Stanford uses because the world of robotics seems infinitely more useful than my customary field of psychology. Professor Ng uses an example of handwriting
recognition much as the Royal Mail must use for reading postal codes.
We scan a whole lot of digits and save each digit as a row of 1’s and 0’s representing ink being present on any one of 400 (20×20) pixels. Can you imagine it?
Other problems will always start the same way – with many cases or training examples, one to each row; and each example described by an extraordinary large number of features. Here we have 400
features or columns of X.
The second set of necessary input data is one last column labeling the row. If we are reading digits, this column will be made up of digits 0-9 (though 0 is written down as 10 for computing
reasons). The digit is still 0 in reality and if we reconstructed the digit by arranging the 400 pixels, it will still be seen to the human eye as 0.
The task is to learn a shorthand way for a computer to see a similar scan of 400 pixels and say, aha, that’s a 1, or that’s a 2 and so on.
Of course the computer will not be 100% accurate but it will get well over 95% correct as we will see.
So that is the input data: a big matrix with examples along the rows of features and with the last column being the correct value – the digit from (10, 1-9) in this case.
How does back propagation work?
Back propagation programs work iteratively without any assumptions about statistics that we are used to in psych.
The computer boffins start by taking a wild guess of the importance of each pixel for a digit, and see what the computer would predict with those weights. That is called the forward pass.
Then based on what the computer got right or wrong, they work backwards to adjust the weights or importance of each pixel for each digit.
And remembering that computers are pretty fast, the computer can buzz back and forth asking “how’s this?”.
After a set number of trials, it stops improving itself and tells us how well it can read the digits, i.e., compares its answers to the right answers in the last column of our input data.
What is a hidden layer?
Back proagation also has another neat trick. Instead of using pixels to predict digits, it works with an intermediate or hidden layer. So the pixels predict some units in the hidden layer and the
hidden layer predicts the digits. Choosing the number of units in the hidden layer is done by trying lots of versions (10 hidden units, 50 hidden units, etc) but I guess computer scientists can pick
the range of the right answer as they get experienced with real world problems.
In this example, the solution worked with 25 hidden layers. That is, 400 pixels were used to make predictions about 25 units which predict which of 10 digits made the data.
The task of the computing scientist is to calculate the weights from the pixels to the hidden layers and from the hidden layers to the digits and then report the answer with a % of “training
accuracy” – over 95%, for example.
Steps in back propagation
We have already covered the first four steps
Step 1: Training data
Get lots of training data with one example on each row and lots of features for each example in the columns.
Make sure the row is labeled correctly in the last column.
Step 2: Decide on the number of units in the hidden layer
Find out what other people have tried for similar problems and start there (that’s the limit of my knowledge so far).
Step 3: Initialize some weights
I said before, we start with wild guess. Actually we start with some tiny numbers but the numbers are random.
We need one set of weights linking each pixel to each hidden layer (25 x 400)* and another set linking each hidden layer to each digit (10 x 25)*.
The asterisk means that a bias factor might be added in raising one or the other number by 1. To keep things simple, I am not going to discuss the bias factor. I’ll just flag where it comes up. Be
careful with them though because I am tired and they might be wrong.
Step 4: Calculate the first wildly inaccurate prediction of the digits
Use the input data and the weights to calculate initial values for the hidden layer.
Our input data of training examples and features (5000 examples by 400 pixels) is crossed with the appropriate initial random weights (25 x 400) to get a new matrix of hidden layer values. Each
training example will have 25 new values (5000 x 25)*.
Then repeat again from the hidden layer to the layer of digits or output layer making another matrix of 5000 x 10.
In the very last step, the calculated value is converted into a probability with the well know sigmoid function. It would be familiar if you saw it. I’ll try to patch it in.
The values calculated at the hidden layer are converted into these probability-type values and they are used for the next step and the final answer is converted in the same way.
Now we have a probability type figure for each of 10 digits for each training example (5000 x 10)*.
Step 5: Find out how well we are doing
In this step, we first convert the correct answer (which was a 1, or 5, or 7 or whatever the digit was) into 1’s and 0’s – so we have another matrix (5000 x10).
We compare this with the one we calculated in Step 4 using simple subtraction and make yet another matrix (5000 x 10).
Step 6: The backward pass begins
So far so good. All pretty commonsensical. The fun starts when we have to find a way to adjust those guessed weights that we used at the start.
Staying at a commonsensical level, we will take error that we have in that big 5000 x 10 matrix calculated in Step 5 and partition it up so we can ‘track’ the error back to training examples and
hidden layers and then from hidden layers to pixels. And this is what the computing scientists do. T
hey take one training example at a time (one of the 5000 rows), pick out the error for digit 1, and break it up. And do it again for digit 2 up to digit 0 (which we input as 10).
Step 7: Working with one training example at a time
It might seem odd to work with one training example at a time, and I suspect that is just a convenience for noobes, but stick with the program. If you don’t, life gets so complicated, you will feel
like giving up.
So take example one, which is row 1; and do the stuff. And repeat for row 1, and so on until you are done.
In computing this is done with a loop: for 1: m where m is the number of training examples or rows (5000 in our case). The machine is happy doing the same thing 5000 times.
So we do everything we did before this step but we start by extracting our row of features: our X or training data how has 1 row and 400 features (1 x 400)*.
And we still have one label, or correct answer but remember we will turn that into a row of 1’s and 0’s. So if the right answer is 5, the row will be 0000100000 (1 x10).
And we can recalculate our error, or uplift the right row from matrix of observed values that we calculated in Step 6. The errors at the ‘output_layer’ will be a row of ten numbers (1 x 10). They
can be positive or negative and the number bit will be less than 1.
Step 8: Now we have to figure out the error in the hidden layer
So we know our starting point of pixels (those never get changed), the correct label (never gets changed) and the error that we calculated for this particular forward pass or iteration. After we
adjust the weights and make another forward pass, our errors change of course and hopefully get smaller.
We now want to work on the hidden layer, which of course is hidden. Actually it doesn’t exist. It is a mathematical convenience to set up this temporary “tab”. Nonetheless, we want to partition the
errors we saw at the output layer back to the units in the hidden layer (25 in our case)*.
Just like we had at the output layer, where we had one row of errors (1 x 10), we now want a row or column of errors for the hidden layer (1 x25 or 25 x 1)*.
We work out this error by taking the weights we used in the forward pass and multiplying by the observed error and weighting again by another probabilistic value. This wasn’t explained all that
well. I’ve seen other explanations and it makes intuitive sense. I suspect our version is something to do with computing.
So here goes. To take the error for hidden layer unit 1, we take the ten weights that we had linking that hidden unit to each digit. Or we can take the matrix of weights (10 x 25)* and match them
against the row of observed errors (1 x 10). To do this with matrix algebra, then we turn the first matrix on its side (25 x 10) and the second on its side (10 x 1) and we the computer will not only
multiply, it will add up as well giving us one column of errors (1 x25).* Actually we must weight each of these by the probabilistic type function that we called sigmoidGradient.
We put into sigmoidGradient a row for the training example that was calculated earlier on as the original data times the weights between the pixels and the hidden layer ((5000 x 400*) times (25 x
400*))– the latter is tipped on its side to perform matrix algebra and produce a matrix of 25* values for each training example (5000 x 25*).
Picking up the column of data that we calculated one paragraph up, we now have two columns (25* x1) which we multiple (in matrix algebra .* so we can do multiplication of columns like we do in
Now we have a column of errors for the hidden layer for this one particular training example (25* x1). (Our errors at the output layer for this person was in a row (1 x 10).
Step 9: Figure out how much to adjust the weights
Now we know how much error is in the output layer and the hidden layer, we can work on adjusting the weights.
Remember we have two sets of weights. Between the output and hidden layer we had (10 x 25*) and between the input layer and the hidden layer, we had (25 x 400*). We deal with each set of weights
Taking the smaller one first (for no particular reason but that we start somewhere), we weight the values of the hidden layer with the amount of error in the output layer. Disoriented? I was.
Let’s look again what we did before. Before we used the errors in the output layer to weight the weights between output and hidden layer and we weighted that with a probabilistic version of input
data times the weights coming between input and hidden layers. That seemingly complicated calculation produced a set of errors – one for each hidden layer – just for this training example because we
still working with just one row of data (see Step 8).
Now we are doing something similar but not the same at all. We take the same differences from the output layer (1 x10) and use them to weight the values of the hidden layer that we calculated on the
forward pass (1×25*). This produces (and this is important) a matrix that will have the same proportions as the weights between the hidden and output layer. So if we have 10 output possibilities
(as we do) and 25* units in the hidden layer, then at this stage we are calculating a 10 x 25* matrix.
So for each training example (original row), we have 250 little error scores, one for each combination of output and hidden units (in this case 10×25*).
Eventually we want to find the average of these little errors over all our training examples (all 5000), so we whisk this data out of the for loop into another matrix. As good programmers, we set
this up before and filled it up with zeros (before the for loop started). As we loop over training examples, we just add in the numbers and we get a total of errors over all training examples (5000)
for each of the combos of hidden unit and output unit (10 x25*).
And doing it again
We have a set of errors now for the connections between hidden and output layers. We need to do this again for the connections between the input layer and the hidden layer.
We already have the errors for the hidden layer (25* x1) (see Step 8). We use these to weight the input values (or maybe we should think of that the other way round – we use the input values to
weight the differences).
We take the errors for the hidden layer (25 x1) and multiple by the row of original data ( 1 x 400*) and we will get a matrix of (25 x 400*) – just like our table of weights! You might notice I did
not put an asterisk on the 25 x1 matrix. This is deliberate. At this point, we take out the bias factor that we put in before.
We do the same trick of storing the matrix of error codes (25 x 400*) in a blank matrix that we set up earlier and then adding the scores for the next training example, and then the next as we loop
through all 5000.
Step 10: Moving on
Now we have got what we want: two matrices, exactly the same size as the matrices for the weights ( 25 x 400* and 10 x 25*). Inside these matrices are the errors added up over all training examples
To get the average, we just have to divide by the number of training examples (5000 in this case). In matrix algebra we just say – see that matrix? Divide every cell by m (the number of training
examples). Done.
These matrices – one 25 x 400* and the other 10 x 25* are then used to calculate new tables of weights. And we rinse and repeat.
1. Forward pass : make a new set of predictions
2. Back propagation as I described above.
3. Get two matrices of errors: yay!
4. Recalculate weights.
5. Stop when we have done enough.
The next questions are how are the weights recalculated and how do we know if we have done enough?
Recalculating weights
The code for the back propagation algorithm is contained within a function that has two purposes:
• To calculate the cost of a set of weights (average error in predictions if you like)
• And the matrices that we calculated to change the weights (also called gradients).
The program works in this order
• Some random weights
• Set up the step-size for learning (little or big guesses up or down) and number of iterations (forward/back ward passes)
• Call a specialized function for ‘advanced optimization’ – we could write a kluxy one but this is the one we are using
• The advanced optimizer calls our function.
• And then performs its own magic to update the weights.
• We get called again, do our thing, rinse and repeat.
How do we know we have done enough?
Mainly the program will stop at the number of iterations we have set. Then it works out the error rate at that point – how many digits are we getting right and how many not.
Oddly, we don’t want 100% because that would probably just mean we are picking up something quirky about our data. Mine eventually ran at around 98% meaning there is still human work and management
of error to do if we are machine reading postal codes. At least that is what I am assuming.
There you have it. The outline of the back propagation. I haven’t taken into account the bias factor but I have stressed the size of the matrices all the way through, because if there is one thing
I have learned, that’s how the computing guys make sure they aren’t getting muddled up. So we should too.
So now I will go through and add an * where the bias factor would come into play.
Hope this helps. I hope it helps me when I try to do this again. Good luck!
The regularization parameter
Ah, nearly forgot – the regularization parameter. Those values – those little bits of error in the two matrices that are the same size as the weights – (25×400*) and (10×25*)?
Each cell in the matrix except for the first column in each which represents the bias factor, must be adjusted slightly by a regularization parameter before we are done and hand the matrices over to
the bigger program
The formula is pretty simple. It is just the theta value for that cell times by the learning rate (set in the main program) and divided by the number of training cases. Each of the two matrices is
adjusted separately. A relatively trivial bit of arithmetic. | {"url":"http://flowingmotion.jojordan.org/2011/11/10/back-propagation-for-the-seriously-hands-on/","timestamp":"2024-11-02T18:33:47Z","content_type":"text/html","content_length":"90656","record_id":"<urn:uuid:f3566504-1951-4ef9-b5e0-ad3bd9421fba>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00571.warc.gz"} |
Weighted hypersoft configuration model
Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions,
we are dealing with configuration models. If the degree sequence is constrained exactly, the corresponding microcanonical ensemble of random graphs with a given degree sequence is the configuration
model per se. If the degree sequence is constrained only on average, the corresponding grand-canonical ensemble of random graphs with a given expected degree sequence is the soft configuration model.
If the degree sequence is not fixed at all but randomly drawn from a fixed distribution, the corresponding hypercanonical ensemble of random graphs with a given degree distribution is the hypersoft
configuration model, a more adequate description of dynamic real-world networks in which degree sequences are never fixed but degree distributions often stay stable. Here, we introduce the hypersoft
configuration model of weighted networks. The main contribution is a particular version of the model with power-law degree and strength distributions, and superlinear scaling of strengths with
degrees, mimicking the properties of some real-world networks. As a byproduct, we generalize the notions of sparse graphons and their entropy to weighted networks.
Related publications | {"url":"https://www.networkscienceinstitute.org/publications/weighted-hypersoft-configuration-model","timestamp":"2024-11-12T00:23:30Z","content_type":"text/html","content_length":"22078","record_id":"<urn:uuid:d0c688fc-4f53-4389-b9c0-01c73cac6cbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00820.warc.gz"} |
M^3 (Making Math Meaningful)
If you have yet to read Sara VanDerWerf's post about her 5 X 5 game go here and read it now. It sounds awesome, doesn't it? I didn't read it until I was already on break, but I look forward to
playing it with my students at some point down the road.
At a very early hour this morning, I thought that this game could be modified to help students practice working with logarithms. Instead of having the numbers 1-10 in their grid, they could have
logs. They would still place numbers of equal value next to each other, but some would look like log[2]8, others like log[3]27 and others like 3. They would also still need to find the sum of squares
that match. I would have them write the sum of each set of matching expressions as a single logarithm so that they would need to use their log rules.
Clearly, I need to actually work through this, but I wanted to get it down before I forgot. I think you could also do this with trig expressions (sin π/6, cos 5π/3, etc.), but have thought that
through even less than the log idea.
If you have feedback, please let me know. Ideas to build on or telling me why this is a bad idea are both welcome :)
Today is our last day before the break. This generally means treats, music and movies throughout the school, with very little, if any, work being done. However, 11 of my students opted to rewrite the
skills portion of the quadratics test from Monday. I didn't even have 11 students show up to my grade 12 class first period! Their dedication to learning and improving is remarkable. I feel very
privileged to be their teacher.
Happy holidays to all!
Day 68
Yesterday was day 2 of the test. I don't really want to talk about it...
Day 69
We changed gears completely today and looked at the equation of a circle. This is one of those weird topics in our curriculum that doesn't really fit in. I do it as an application of the distance
formula, but we only look at circles centered at the origin so I don't even get to make connections to the transformations we have done. Well, it's not part of the curriculum, but I usually have them
explore circles that are not centered at the origin.
I started off with a little group activity. They had to plot 12 points and calculate the distance between each point and the origin. This is what it looked like:
They easily saw the pattern (all the answers were 5!) and that the points formed a circle. So then I asked them to define a circle. Here is what they said:
Pretty good, right? We went on to develop the equation of a circle.
Then they filled in a big table giving them different information (equation, radius, intercepts, graph).
Next, we looked at this question:
We hopped onto Desmos to see what this looked like. They could easily see if a point was inside, outside or on the circle, so they talked in their groups about how we could figure this out without
the picture.
In our discussions, we talked more about radius that radius squared, so that is how we looked at the example that follows.
And then class ended early so students could go clean up their lockers. They were not sad about not getting homework.
Today, students had the opportunity to work on review questions (quadratics) and ask for help. I returned yesterday's test and went over the questions where they had to solve a quadratic. We also
spent a little bit of time talking about how to determine the number of real roots of a quadratic. I like having a day in between test days as it gives students a chance to refocus and ask questions
about skills which with they are struggling.
Completely off topic, if you haven't checked out Desmos' latest activity, Marbleslides, go do that now! Here is the link to the quadratic one.
Not much to report today. Test day is always a combination of feelings. Satisfaction that some students have really understood the material, combined with sadness that some really have not. I have to
push down that voice that wants to say "But we did this!" as the evidence clearly shows that some just didn't get it. More work tomorrow to prepare for part II of the test on Wednesday.
I collected some great quadratic summaries today and asked four students if I could scan their work and post it on Edmodo. I like students to be able to see each other's work.
They spent the period working on review questions. The cycle 3 test will be over two days. The first day will cover mixture-type linear system questions, shortest distance between a point and a line
and some of the skills of quadratics (factoring, completing the square...). The second day will be all about quadratics where they have to choose the right tool to answer the question and interpret
answers as necessary.
What's left in the course? Sine and cosine law, comparing y = 2^x and y = x^2 and equations of circles centered at the origin. I think we will look at y = 2^x (and negative exponents) next Thursday
and do Penny Circle on Friday and save the rest for January.
We started today with a couple of leftover examples from yesterday.
We talked about the fact that the second answer is not double the first as the diver is speeding up.
They had done problems similar to the next one for homework, but I wanted to make sure that they were all solid on how to approach this type of question.
Then we moved on to making frames, inspired by (stolen from) Fawn Nguyen. Here is her post about it.
Each pair got one picture (I use black & white pictures as they look better photocopied) and four frames (on thicker paper) to work with as I told them their first attempt would likely be
unsuccessful, but that they should learn from it.
Eventually they wanted help. They didn't quite beg for it but they did ask for the math that would help them. That was good. I gave them this and let them have a little more time to work it out.
Then we went over the solution together.
Here is one group's finished product!
After that they worked on one more problem.
Time was tight so we looked at the solution in a Desmos kind of way. I explained how to set up the equation - that the variable was the number of decreases in price. They usually find these questions
And their homework for today:
I left it very open-ended as I want them to make something that is meaningful for them.
I struggled with the title of this lesson and blog post as all the "problems" are "math land" questions - we wrap a fake context around what we want our students to show us and ignore real world
constraints. However, here is what we did today.
We focused a lot on what we were trying to find and which tool would get us there, with stops along the way for those students that were lost and confused. Here is another one:
And a third example:
Here is today's homework.
Several times over the past few weeks we have talked about the fact that we could only find the zeros of a quadratic if we could factor it. Today we found ways of dealing with the case when it does
not factor.
As a class they were struggling with this so I opened up Desmos and asked for some values. One student gave me an equation with decimal values for 'b' and 'c' which gave zeros that would not easily
have been found algebraically. This, once again, set up the need to find another way to solve quadratics.
We worked through a couple of examples where the equation was in vertex form. I told them that one of the big challenges was knowing when to expand. In these cases, solving would be easier if we did
not expand.
This proved to be a great opportunity to review the effect of the 'a' value when graphing. We went back to our pattern of "from the vertex, go right/left 1, up 1; right/left 2, up 4; right/left 3, up
9" to graph each of these parabolas. The algebraic solution, although new, seemed to make sense.
Okay - so now given an equation in standard form, they could complete the square to get it in vertex form, then solve as we did above. I told them we could generalize the process and then I did. The
curriculum says that students should be able to follow the development of the quadratic formula, not replicate it, so it was all pencils/pens down as we worked through a case with numbers alongside
the general equation.
As is often the case, students were generally unimpressed with this "ugly" formula. I asked if they would rather complete the square then solve each time they could not factor or simply substitute
values into the formula. Some were sold.
Then we practiced with three particular questions.
It took a little questioning to get them all to see that what was under the square root was the determining factor in the number of solutions. I had a Desmos file ready to go, but felt that they
understood how the discriminant showed the number of roots so I skipped it. We did a couple of examples to be sure.
Then we started on a more interesting question. The actual calculations are not difficult but choosing what tool to use is.
Here is yesterday's homework and here is today's.
As many of my students were away on a field trip on Friday I told them to each find someone who was in class, and have them explain completing the square. They spent about 15 minutes on this. I
thought it would be a good way to bring those who had missed class up to speed, but would also be of benefit to those who were there as they had to explain the concept well enough for their
classmates to understand.
What we worked on today was really more of the same.
I illustrated the point that we could not make a square with two x^2 tiles. I asked if we could if we have four... they thought a bit and many said yes. What about three? No. Nine? Yes.
I showed them that we divide up the x^2 tiles and create a square for each one, dividing the x tiles evenly among them.
We translated this into a chart method and repeated the process algebraically, too.
We continued with more examples, relying less on the tiles each time yet always tying the process back to them - "Why are we dividing by 2? Why are we squaring?".
When we looked at the next example, using tiles or the chart became less meaningful as it's hard to think of having -3 of each square. I think they had enough experience and a solid enough
understanding of why we were doing what we were doing to move to the algebraic form.
I will post today's homework tomorrow as DropBox is not cooperating right now.
We started today by looking for patterns in perfect square trinomials.
They noticed the pattern which we consolidated:
Then we looked at it another way. In groups of four, they each answered one of these questions and then wrote the sum of the four answers in the middle box. This allows me to quickly see if they are
correct and, if they are not, they have to work together to figure out which question(s) are wrong.
We talked about how we can find the vertex of a parabola. Factor the quadratic, take the average of the zeros, then substitute that value back in the equation. But what if you can't factor the
quadratic? One student said he could always find the zeros... "Desmos!", he said :) Then the algebra tiles came out and we starting completing the square.
The idea of making a square is not difficult when they work with the tiles. We kept the 7 unit tiles off to the side and then added one positive unit tile to fill in the square. This meant that we
also needed to add one negative unit tile to ensure that we weren't changing the value of expression. Writing the equation in vertex form was quite straightforward, as was stating the vertex.
I gave them the steps - this may be useful for those students who were away today.
Then we practiced some more.
It was time to start to move toward an algebraic solution so we started by noticing what is happening with the numbers, and then repeated one of the previous examples without tiles.
We did a few more examples.
Along the way we talked about why we needed to move away from the tiles. What if the number of x-tiles was not even? But we did a simple case together with tiles - not the actual tiles though, as I
do not want them split in half!
Today's homework was to go back over any old homework that they had either not completed or done incorrectly. Next class we will look at cases where the a-value is not 1. | {"url":"https://marybourassa.blogspot.com/2015/12/","timestamp":"2024-11-14T01:59:52Z","content_type":"text/html","content_length":"171173","record_id":"<urn:uuid:8307decc-9572-4df6-a7aa-90bc647b2d65>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00476.warc.gz"} |
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accepted from students of diverse backgrounds and institutions: graduates of research universities or four year colleges, from the U.S. and worldwide. Most applicants, but not all, have undergraduate
degrees in physics; many have had significant research experience. Seeking to identify the most qualified students who show promise of excellence in research and teaching, the admissions process is
highly selective and very competitive.
Doctor of Philosophy
During the first year of the doctoral program, a student takes introductory graduate physics courses and usually serves as a teaching assistant assigned to one of the introductory or intermediate
undergraduate physics courses. Students are encouraged to explore research opportunities during their first year. Students are strongly encouraged to take the graduate diagnostic examination prior to
their first quarter in the program. The results of this examination will determine which of the introductory graduate courses the student must take to achieve candidacy. After achieving candidacy and
identifying a research sponsor, the student begins dissertation research while completing course requirements. Within a year after research begins, a PhD committee is formed with the sponsor as
chairman. The student continues research, from time to time consulting with the members of the committee, until completion of the dissertation. The average length of time for completion of the PhD
program in physics is about six years.
In addition to fulfilling University and divisional requirements, a candidate for the degree of Doctor of Philosophy in physics must:
• Achieve Candidacy.
• Fulfill the experimental physics requirement by completing PHYS 33400 Adv Experimental Physics or PHYS 33500 Adv Experimental Physics Project.
• Pass four post candidacy advanced graduate courses devoted to the broad physics research areas of (A) Condensed Matter Physics, (B) Particle Physics, (C) Large Scale Physics (i.e. Astrophysics
and/or Cosmology related), and (D) Intermediate Electives. The four courses selected must include at least one from each of the categories (A), (B), and (C).
• Pass two other advanced (40000 level) courses either in physics or in a field related to the student’s Ph.D. research. The latter requires department approval.
• Within the first year after beginning research, convene a first meeting of the Ph.D. committee to review plans for the proposed thesis research and for fulfilling the remaining Ph.D.
• Attend annual meetings with the thesis committee.
• One to two quarters prior to the defense of the dissertation, hold a pre-oral meeting at which the student and the Ph.D. committee discuss the research project.
• Defend the dissertation before the Ph.D. committee.
• Submit for publication to a refereed scientific journal the thesis which has been approved by the Ph.D. committee or a paper based on the thesis. A letter from the editor acknowledging receipt of
the thesis must be provided to the department office.
Acquire further information about our doctoral program with Zosia Krusberg, Director of Graduate Studies.
Master of Science
The graduate program of the Department of Physics is oriented toward students who intend to earn a Ph.D. degree in physics. Therefore, the department does not offer admission to students whose goal
is the Master of Science degree. However, the department does offer a master’s degree to students who are already in the physics Ph.D. program or other approved graduate programs in the University.
Normally it takes one and a half years for a student to complete the master’s program. A master’s degree is not required for continued study toward the doctorate.
In addition to fulfilling University and Divisional requirements, a candidate for the degree of Master of Science in physics must demonstrate a satisfactory level of understanding of the fundamental
principles of physics by passing nine approved courses with a minimum grade point average of 2.5. Six of the nine courses must be:
PHYS 31600 Adv Classical Mechanics 100
PHYS 33000 Math Methods Of Physics-1 100
PHYS 34100 Graduate Quantum Mechanics-1 100
PHYS 32200 Advanced Electrodynamics I 100
PHYS 35200 Statistical Mechanics 100
PHYS 33400 Adv Experimental Physics 100
PHYS 33500 Adv Experimental Physics Project 100
The experimental physics requirement can be fulfilled either through PHYS 33400 Adv Experimental Physics or PHYS 33500 Adv Experimental Physics Project.
Testing out of certain courses (PHYS 31600, 32200, 32300, 34100, 34200, and 35200) on the Graduate Diagnostic Exam can be applied toward the Master’s degree in place of taking the course. The 2.5
GPA minimum applies only to courses taken in addition to those credited by performance on the Graduate Diagnostic Exam.
The Department may approve substitutions to this list where warranted.
Teaching Opportunities
Part of the training of graduate students is dedicated to obtaining experience and facility in teaching. Most first year students are supported by teaching assistantships, which provide the
opportunity for them to engage in a variety of teaching related activities. These may include supervising undergraduate laboratory sections, conducting discussion and problem sessions, holding office
hours, and grading written work for specific courses. Fellowship holders are invited to participate in these activities at reduced levels of commitment to gain experience in the teaching of physics.
During the Autumn quarter first year graduate students attend the weekly workshop, Teaching and Learning of Physics, which is an important element in their training as teachers of physics.
Teaching Facilities
All formal class work takes place in the modern lecture halls and classrooms and instructional laboratories of the Kersten Physics Teaching Center. This building also houses special equipment and
support facilities for student experimental projects, departmental administrative offices, and meeting rooms. The center is situated on the science quadrangle near the John Crerar Science Library,
which holds over 1,000,000 volumes and provides modern literature search and data retrieval systems.
Research Facilities
Most of the experimental and theoretical research of Physics faculty and graduate students is carried out within the Enrico Fermi Institute, the James Franck Institute and the Institute for
Biophysical Dynamics. These research institutes provide close interdisciplinary contact, crossing the traditional boundaries between departments. This broad scientific endeavor is reflected in
students’ activities and contributes to their outlook toward research.
In the Enrico Fermi Institute, members of the Department of Physics carry out theoretical research in particle theory, string theory, field theory, general relativity, and theoretical astrophysics
and cosmology. There are active experimental groups in high energy physics, nuclear physics, astrophysics and space physics, infrared and optical astronomy, and microwave background observations.
Some of this research is conducted at the Fermi National Accelerator Laboratory, at Argonne National Laboratory (both of these are near Chicago), and at the European Organization for Nuclear Research
(CERN) in Geneva, Switzerland.
Physics faculty in the James Franck Institute study chemical, solid state, condensed matter, and statistical physics. Fields of interest include chaos, chemical kinetics, critical phenomena, high Tc
superconductivity, nonlinear dynamics, low temperature, disordered and amorphous systems, the dynamics of glasses, fluid dynamics, surface and interface phenomena, nonlinear and nanoscale optics,
unstable and metastable systems, laser cooling and trapping, atomic physics, and polymer physics. Much of the research utilizes specialized facilities operated by the institute, including a low
temperature laboratory, a materials preparation laboratory, x-ray diffraction and analytical chemistry laboratories, laser equipment, a scanning tunneling microscope, and extensive shop facilities.
Some members of the faculty are involved in research at Argonne National Laboratory.
The Institute for Biophysical Dynamics includes members of both the Physical Sciences and Biological Sciences Divisions, and focuses on the physical basis for molecular and cellular processes. This
interface between the physical and biological sciences is an exciting area that is developing rapidly, with a bi-directional impact. Research topics include the creation of physical materials by
biological self assembly, the molecular basis of macromolecular interactions and cellular signaling, the derivation of sequence structure function relationships by computational means, and structure
function relationships in membranes.
In the areas of chemical and atomic physics, research toward the doctorate may be done in either the physics or the chemistry department. Facilities are available for research in crystal chemistry;
molecular physics; molecular spectra from infrared to far ultraviolet, Bose Einstein condensation, and Raman spectra, both experimental and theoretical; surface physics; statistical mechanics; radio
chemistry; and quantum electronics.
Interdisciplinary research leading to a Ph.D. degree in physics may be carried out under the guidance of faculty committees including members of other departments in the Division of the Physical
Sciences, such as Astronomy & Astrophysics, Chemistry, Computer Science, Geophysical Sciences or Mathematics, or related departments in the Division of the Biological Sciences.
Admission and Student Aid
Most students entering the graduate program of the Department of Physics of the University of Chicago hold a bachelor’s or master’s degree in physics from an accredited college or university.
December 15 is the deadline for applications for admission in the following autumn quarter. The Graduate Record Examination (GRE) given by the Educational Testing Service is required of all
applicants. Applicants should submit recent scores on the verbal, quantitative, and analytic writing tests and on the advanced subject test in physics. Arrangements should be made to take the
examination no later than September in order that the results be available in time for the department’s consideration. Applicants from non-English speaking countries must provide the scores achieved
on the TOEFL or the IELTS.
All full time physics graduate students in good standing receive financial aid. Most graduate students serve as teaching assistants in their first year.
The department has instituted a small bridge-to-Ph.D. program which does not require the Graduate Record Examination. The application deadline for this program varies but is expected to be mid to
late spring.
For information including faulty research interests, application instructions, and other important program details please visit our department website http://physics.uchicago.edu/. You can also reach
out to physics@uchicago.edu with any questions or concerns regarding the admissions process.
Grading Policy
The department’s grading policy is available on the departmental website.
Course Requirements
Course requirements are available on the department’s website.
Physics Courses
PHYS 30101. Analytical Methods of Physics I. 100 Units.
This course focuses on analytical techniques used in physics. It is designed to have flexible topical coverage so that the course may be geared to the registered students. Enrollment is by instructor
approval only.
Instructor(s): D. Reed Terms Offered: Autumn
Prerequisite(s): Permission of the instructor.
PHYS 30102. Analytical Methods of Physics II. 100 Units.
Course focuses on analytical techniques used in Physics. It is designed to have flexible topical coverage so that the course may be geared to registered students. Enrollment is by instructor approval
PHYS 30103. Analytical Methods of Physics III. 100 Units.
PHYS 31600. Adv Classical Mechanics. 100 Units.
This course begins with variational formulation of classical mechanics of point particles, including discussion of the principle of least action, Poisson brackets, and Hamilton-Jacobi theory. These
concepts are generalized to continuous systems with infinite number of degrees of freedom, including a discussion of the transition to quantum mechanics.
Terms Offered: Autumn
Prerequisite(s): PHYS 18500
PHYS 31700. Symplectic Methods of Classical Dynamics. 100 Units.
This course covers advanced techniques in classical dynamics including Lagrangian mechanics on manifolds, differential forms, symplectic structures on manifolds, the Lie algebra of vector fields and
Hamiltonian functions, and symplectic geometry.
Terms Offered: Spring
PHYS 32200-32300. Advanced Electrodynamics I-II.
This two-quarter sequence covers electromagnetic properties of continuous media, gauge transformations, electromagnetic waves, radiation, relativistic electrodynamics, Lorentz theory of electrons,
and theoretical optics. There is considerable emphasis on the mathematical methods behind the development of the physics of these problems.
PHYS 32200. Advanced Electrodynamics I. 100 Units.
Terms Offered: Winter
Prerequisite(s): PHYS 22700 and 23500
PHYS 32300. Advanced Electrodynamics II. 100 Units.
Terms Offered: Spring
Prerequisite(s): PHYS 32200
PHYS 33000. Math Methods Of Physics-1. 100 Units.
Topics include complex analysis, linear algebra, differential equations, boundary value problems, and special functions.
Terms Offered: Autumn
Prerequisite(s): PHYS 22700
PHYS 33400. Adv Experimental Physics. 100 Units.
For course description contact Physics.
Terms Offered: Spring
PHYS 33500. Adv Experimental Physics Project. 100 Units.
For course description contact Physics.
PHYS 34100-34200. Advanced Quantum Mechanics I-II.
This two-quarter sequence covers wave functions and their physical content, one-dimensional systems, WKB method, operators and matrix mechanics, angular momentum and spin, two- and three-dimensional
systems, the Pauli principle, perturbation theory, Born approximation, and scattering theory.
PHYS 34100. Graduate Quantum Mechanics-1. 100 Units.
This course is a two-quarter sequence that covers wave functions and their physical content, one dimensional systems, WKB method, operators and matrix mechanics, angular momentum and spin,
two-and-three dimensional systems, with Pauli principle, perturbation theory, Born approximation, and scattering theory.
Terms Offered: Autumn
Prerequisite(s): PHYS 23500
PHYS 34200. Graduate Quantum Mechanics-2. 100 Units.
This two-quarter sequence covers wave functions and their physical content, one-dimensional systems, WKB method, operators and matrix mechanics, angular momentum and spin, two- and three-dimensional
systems, the Pauli principle, perturbation theory, Born approximation, and scattering theory
Terms Offered: Winter
Prerequisite(s): PHYS 34100
PHYS 35200. Statistical Mechanics. 100 Units.
This course covers principles of statistical mechanics and thermodynamics, as well as their applications to problems in physics and chemistry.
Terms Offered: Spring
Prerequisite(s): PHYS 19700 and 23500
PHYS 35300. Advanced Statistical Mechanics. 100 Units.
This course will cover advanced topics in collective behavior, mean field theory, fluctuations, scaling hypothesis. Perturbative renormalization group, series expansions, low-dimensional systems and
topological defects, random systems and conformal symmetry.
PHYS 36100. Solid State Physics. 100 Units.
Topics include Properties of Insulators, Electronic Properties of Solids, Thermal Properties, Optical Properties of Solids, and Transport in Metals (conductivity, Hall effect, etc.)
Terms Offered: Autumn
Prerequisite(s): PHYS 23600, 34200, 35200
PHYS 36300. Particle Physics. 100 Units.
PHYS 36400. General Relativity. 100 Units.
This is advanced-level course on general relativity treats special relativity, manifolds, curvature, gravitation, the Schwarzschild solution and black holes.
Terms Offered: Winter 2014
PHYS 36600. Adv Condensed Matter Physics. 100 Units.
Phasetransitions, Magnetism, Superconductivity, Disorder, Quantum Hall Effect, Superfluidity, Physics of Low-dimensional systems, Fermiliquid theory, and Quasi-crystals.
Terms Offered: Winter
PHYS 36700. Soft Condensed Matter Phys. 100 Units.
This course will cover topics including granular and colloidal matter, jamming, fluids, instabilities and topological shapes and transitions between them.
PHYS 37100. Introduction To Cosmology. 100 Units.
PHYS 37200. Particle Astrophysics. 100 Units.
This course treats various topics in particle astrophysics.
Terms Offered: TBD
PHYS 38500. Advanced Mathematical Methods. 100 Units.
Course description unavailable.
Terms Offered: Winter
PHYS 38520. Advanced Mathematical Methods: Topology. 100 Units.
This course covers topology. It alternates years with PHYS 38510 (group theory).
Terms Offered: Winter
PHYS 38600. Advanced Methods of Data Analysis. 100 Units.
This course covers advanced methods of data analysis including probability distributions, propagation of errors, Bayesian approaches, maximum likelihood estimators, confidence intervals, and more.
Terms Offered: Spring
PHYS 39000. PREP for Candidacy. 300.00 Units.
Registration for students who have not yet reached Ph.D. candidacy.
PHYS 39800. Research: Physics. 300.00 Units.
Registration for students performing individually arranged research projects not related to a doctoral thesis.
PHYS 39900. Prep For Candidacy Examination. 300.00 Units.
PHYS 40600. Nuclear Physics. 100 Units.
No description Available
PHYS 40700. X-ray Lasers and Applications. 100 Units.
This course will introduce the basic concepts of accelerator-based x-ray light sources (XFELs and synchrotrons) and survey contemporary x-ray applications such as nonlinear multiphoton absorption,
induced transparency/saturable absorption, and atomic x-ray lasing in systems ranging from atoms to clusters to solids.
PHYS 41000. Accelerator Physics. 100 Units.
The course begins with the historical development of accelerators and their applications. Following a brief review of special relativity, the bulk of the course will focus on acceleration methods and
phase stability, basic concepts of magnet design, and transverse linear particle motion. Basic accelerator components such as bending and focusing magnets, electrostatic deflectors, beam diagnostics
and radio frequency accelerating structures will be described. The basic concepts of magnet design will be introduced, along with a discussion of particle beam optics. An introduction to resonances,
linear coupling, space charge, magnet errors, and synchrotron radiation will also be given. Topics in longitudinal and transverse beam dynamics will be explored, including synchrotron and betatron
particle motion. Lastly, a number of additional topics will be reviewed, including synchrotron radiation sources, free electron lasers, high energy colliders, and accelerators for radiation therapy.
Several laboratory sessions will provide hands-on experience with hardware and measurement instrumentation.
Terms Offered: Autumn
PHYS 41100. Many Body Theory. 100 Units.
The course will follow roughly the new textbook by Piers Coleman "Introduction to Many-Body Physics". The topics are: Second quantization, Path integral, Quantum fields, Green functions, Feynman
diagrams, Landau Fermi Liquid theory, Phase transitions, BCS theory, more advanced topics.
PHYS 41101. Entanglement in Many-Body Systems. 100 Units.
This course starts with an introduction to quantum information theory: density operators, quantum channels and measurements, von Neumann entropy, mutual information, and entanglement. It continues
with a discussion of topological quantum computation. The course then concludes with a discussion of entanglement in many-body ground states, including the area law, topological entanglement entropy,
and entanglement in conformal field theories.
Terms Offered: Winter
Prerequisite(s): PHYS 342 or equivalent
PHYS 41200. Topological Quantum Matter. 100 Units.
PHYS 41300. Topological Phases in Condensed Matter. 100 Units.
Terms Offered: Winter
Prerequisite(s): PHYS 36100
PHYS 42100. Fractional Quantum Hall Effect. 100 Units.
PHYS 42600. Fluid Mechanics. 100 Units.
Terms Offered: Spring
PHYS 44000. Principles of Particle Detectors. 100 Units.
PHYS 44100. Advanced Particle Detectors. 100 Units.
We will explore the development of modern detector types, and examine opportunities for developing new capabilities in a variety of fields.
Terms Offered: Spring
Prerequisite(s): PHYS 32300
PHYS 44300. Quantum Field Theory I. 100 Units.
Topics include Basic Field Theory, Scattering and Feynman Rules, and One Loop Effects.
Terms Offered: Autumn
Prerequisite(s): PHYS 34200
PHYS 44400. Quantum Field Theory II. 100 Units.
Topics include Path integral formulation of QFT, Renormalization, Non-Abelian gauge theory.
Terms Offered: Winter
PHYS 44500. Quantum Field Theory-3. 100 Units.
PHYS 44800. Field Theory in Condensed Matter. 100 Units.
Course description unavailable.
Terms Offered: Autumn
PHYS 45700. Implementation of Quantum Information Processors. 100 Units.
This course emphasizes the experimental aspects of quantum information focusing on implementations rather than algorithms. Several candidate quantum information systems will be discussed including
ion traps, neutral atoms, superconducting circuits, semiconducting quantum dots, and linear optics.
PHYS 45710. Physics of Superconducting Circuits. 100 Units.
This course will give a brief introduction to superconductivity as it relates to building quantum circuits. Circuit quantization will be introduced and used to derive the Hamiltonians of several
standard circuits including sensors such as single electron transistors and superconducting quantum interference devices as well as various flavors of superconducting qubit. We will study cavity QED
and how such physics is realized with superconducting circuits. We will discuss the experiments used to characterize such quantum systems. The course will have a strong numerics component across all
Terms Offered: Spring
Prerequisite(s): PHYS 34200 or MENG 31400 or consent of Instructor
PHYS 45800. The Physics of Quantum Information. 100 Units.
PHYS 46000. Gravitational Waves. 100 Units.
This course will provide a broad overview of gravitational waves, with a focus on current results from LIGO. We will cover the basics of gravitational wave theory, compact binary coalescence and
sources of gravitational wave, ground-based gravitational wave detection, LIGO and the first detections, LIGO's black holes and how the Universe might have made them, gravitational wave astrophysics,
and the near future of gravitational wave science.
PHYS 46200. Nuclear Astrophysics. 100 Units.
Terms Offered: Autumn
PHYS 46700. Quantum Field Theory in Curved Spacetime I. 100 Units.
This course covers introductory topics in the study of quantum field theory in curved spacetime. These topics include QFT for a free scalar field and for globally hyperbolic curved spacetimes, and
the Unruh effect.
PHYS 46800. Quantum Field Theory in Curved Spacetime II. 100 Units.
This course covers advanced topics in the study of quantum field theory in curved spacetime. These topics include the Hawking effect, quantum perturbations in cosmology, black hole evaporation and
information loss, and other modern topics.
PHYS 46900. Effective Field Theories. 100 Units.
PHYS 47100. Modern Atomic Physics. 100 Units.
This course is an introduction to modern atomic physics, and focuses on phenomena revealed by new experimental techniques.
Terms Offered: Winter
PHYS 48102. Neutrino Physics. 100 Units.
This is an advanced course on neutrino phenomenology. The topics include neutrino flavor transformations, neutrino mass, sterile neutinos, non-standard interactions of neutrinos, and other topics of
modern interest.
PHYS 48103. Standard Model of Particle Physics and Beyond. 100 Units.
This course provides an overview of the Standard Model of particle physics, the problems with it, and the candidates of physics beyond the Standard model that can solve those problems.
Terms Offered: Winter
PHYS 48300. String Theory-1. 100 Units.
First quarter of a two-quarter sequence on string theory.
Terms Offered: Winter
PHYS 48400. String Theory-II. 100 Units.
Second quarter of a two-quarter sequence on string theory.
PHYS 49000. Basic Principles of Biophysics. 100 Units.
This course is designed to expose graduate students in the physical sciences to conceptual and quantitative questions about biological systems. It will cover a broad range of biological examples from
vision in flies and developing embryos to swimming bacteria and gene regulation. This course does not assume specialized biological knowledge or advanced mathematical skills.
PHYS 49100. Biological Physics. 100 Units.
Course will be structured around unifying problems and themes found across biology that benefit from a quantitative approach. No specialized biological knowledge assumed. Topics covered include:
active solution to passive problems, self-replication: the origin of life and evolution, mass, energy and growth laws, biological behaviors as stable dynamical attractors.
Terms Offered: Spring
PHYS 49900. Advanced Research: Physics. 300.00 Units.
This course is for students performing research toward their doctoral thesis.
PHYS 70000. Advanced Study: Physics. 300.00 Units.
Advanced Study: Physics | {"url":"http://graduateannouncements.uchicago.edu/graduate/departmentofphysics/","timestamp":"2024-11-08T10:40:07Z","content_type":"application/xhtml+xml","content_length":"54535","record_id":"<urn:uuid:9c9ec0c4-3f58-4802-bb14-e2f7e3f4941e>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00173.warc.gz"} |
How to calculate circumference - Easy to Calculate
Circles are a very common shape. If you think about it, we are surrounded by circular figures in our everyday life: most glassware we use to drink water, coffee mugs, bicycle and car tires,
engagement rings, roundabouts, our pupils, among many other.
Since ancient times, humans have been attracted to this simple but puzzling geometry for many different reasons: it has no straight sides, it possesses an infinite number of symmetry planes, and it
is closely related to the mathematical constant π (keep reading to learn more about this!).
An important characteristic of circles is their circumference, or the distance covered around it. Let’s learn how to calculate it without the need of flexible measuring tape, and let’s discover why
so many great mathematicians are so attracted to it.
How to calculate circumference
To calculate a circle’s circumference first find its radius, r, which equals half of its diameter. Then, apply the following equation to find its circumference, C:
Where π = 3,14159.
What is the circumference?
Circumference is defined as the perimeter of a circle. But what exactly is the perimeter? Its name comes from Greek, as many concepts in geometry do, and is formed by περί (peri), which means
“around”, and μέτρον (metron), which means “to measure”. Together, they form an expression that means “to measure around” something, in this case, around the space enclosed by geometrical figures.
Ancient Greeks loved to experiment with different shapes they called polygons. These are shapes built with straight segments called sides, which form a closed figure. You have probably seen many
polygons. For example, triangles are 3-sided polygons, while rectangles —and squares among them— are 4-sided polygons. Every polygon has a specific perimeter, which depends on how long its sides
Another figure Greeks were very fond of is the circle. In this case, the perimeter is referred to as circumference, which in turn comes from the Latin circum, which means “around”, and ferre, which
means “to carry”. This is because, as with any type of perimeter, it refers to the length of the path that outlines a shape. Since, in this case, the shape is circular, the perimeter will revolve
around it, and thus the Latin expression to “carry around” something makes perfect sense.
A circle’s circumference can be measured as any other length using a measuring tape. Nevertheless, this is not practical at times. For example, imagine you need to determine the perimeter of a
circular piece of land to install a fence of a proper length. If the land is too big, going around it to measure its perimeter in sections would simply take a lot of time and effort.
Another way to measure a circle’s circumference could be by extending its perimeter in a straight line, and then measuring it with any suitable device. This is, of course, not possible in many
scenarios, but it could look something like the following image. Here, C represents the circle’s circumference.
Finally, a more practical way to determine a circle’s circumference is by use of its enigmatic properties. Let’s see how:
A very interesting property of circles is the relationship between their perimeter and their diameter. The latter simply refers to how wide a circle is. It is alternately defined as the length of any
line that connects two points on its perimeter, and which at the same time passes through its center. The following image shows both quantities:
What is interesting about circles is that no matter its size, the ratio of its circumference and its diameter will always result in the same number: 3,14159265359… We call this number pi and
represent it with the Greek letter π. This simple but puzzling phenomenon has been known for thousands of years, and can be written as:
It is important to remember that this equation holds for any circle in the universe. Since it relates two basic characteristics of a circle, it can be used to calculate them if the other one is
known. If we solve the equation for the circumference we get:
This way, a new method to find a circle’s circumference can be extracted from the definition of π. A third important characteristic of a circle is its radius, which simply equals half of its
By replacing equation 4 in equation 3, we get equation 1, which is the most common mathematical definition of the circumference. Now, to calculate this value for any circle, you only need to find its
radius, multiply it by 2 and then by the mathematical constant π. Since π is unitless, the circumference inherits the units of length from the radius.
Equation 3 implies that, for any circle with diameter d, its circumference will have a value of pi times d. If we choose a circle of diameter equal to 1 m, its circumference will be pi meters, as the
following animation shows:
Taken from: Wikimedia Commons, John Reid, 2006.
Example 1:
Find the circumference of a circle with a diameter equal to 1,5 m.
Answer: 4,71 m
Example 2:
What is the diameter of a circle with a circumference of 10,3 m?
Answer: 3,28 m | {"url":"https://easytocalculate.com/how-to-calculate-circumference/","timestamp":"2024-11-07T20:11:02Z","content_type":"text/html","content_length":"116156","record_id":"<urn:uuid:8a8a924d-e4ec-499b-bd84-07ac73ce8537>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00669.warc.gz"} |
Artun Bayer
Oct 07, 2021
Abstract:Graph neural networks (GNNs) have achieved superior performance on node classification tasks in the last few years. Commonly, this is framed in a transductive semi-supervised learning setup
wherein the entire graph, including the target nodes to be labeled, is available for training. Driven in part by scalability, recent works have focused on the inductive case where only the labeled
portion of a graph is available for training. In this context, our current work considers a challenging inductive setting where a set of labeled graphs are available for training while the unlabeled
target graph is completely separate, i.e., there are no connections between labeled and unlabeled nodes. Under the implicit assumption that the testing and training graphs come from similar
distributions, our goal is to develop a labeling function that generalizes to unobserved connectivity structures. To that end, we employ a graph neural tangent kernel (GNTK) that corresponds to
infinitely wide GNNs to find correspondences between nodes in different graphs based on both the topology and the node features. We augment the capabilities of the GNTK with residual connections and
empirically illustrate its performance gains on standard benchmarks.
* Under review at IEEE ICASSP 2022 | {"url":"https://www.catalyzex.com/author/Artun%20Bayer","timestamp":"2024-11-08T18:05:33Z","content_type":"text/html","content_length":"66947","record_id":"<urn:uuid:359020da-4f27-492f-beb1-12b83bf5b2fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00470.warc.gz"} |
A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage,
the terms fat-tailed and heavy-tailed are sometimes synonymous; fat-tailed is sometimes also defined as a subset of heavy-tailed. Different research communities favor one or the other largely for
historical reasons, and may have differences in the precise definition of either.
Fat-tailed distributions have been empirically encountered in a variety of areas: physics, earth sciences, economics and political science. The class of fat-tailed distributions includes those whose
tails decay like a power law, which is a common point of reference in their use in the scientific literature. However, fat-tailed distributions also include other slowly-decaying distributions, such
as the log-normal.^[1]
The extreme case: a power-law distribution
The most extreme case of a fat tail is given by a distribution whose tail decays like a power law.
A variety of Cauchy distributions for various location and scale parameters. Cauchy distributions are examples of fat-tailed distributions.
That is, if the complementary cumulative distribution of a random variable X can be expressed as
${\displaystyle \Pr \left\{\ X>x\ \right\}\sim x^{-\alpha }\quad }$ as ${\displaystyle \quad x\to \infty \quad }$ for ${\displaystyle \quad \alpha >0\;,}$
then the distribution is said to have a fat tail if ${\displaystyle \alpha <2}$ . For such values the variance and the skewness of the tail are mathematically undefined (a special property of the
power-law distribution), and hence larger than any normal or exponential distribution. For values of ${\displaystyle \ \alpha >2\ ,}$ the claim of a fat tail is more ambiguous, because in this
parameter range, the variance, skewness, and kurtosis can be finite, depending on the precise value of ${\displaystyle \ \alpha \ ,}$ and thus potentially smaller than a high-variance normal or
exponential tail. This ambiguity often leads to disagreements about precisely what is, or is not, a fat-tailed distribution. For ${\displaystyle \ k>\alpha -1\ ,}$ the ${\displaystyle \ k^{\mathsf
{th}}\ }$ moment is infinite, so for every power law distribution, some moments are undefined.^[2]
Here the tilde notation "${\displaystyle \sim }$ " means that the tail of the distribution decays like a power law; more technically, it refers to the asymptotic equivalence of functions –
meaning that their ratio asymptotically tends to a constant.
Fat tails and risk estimate distortions
Lévy flight from a Cauchy distribution compared to Brownian motion (below). Central events are more common and rare events more extreme in the Cauchy distribution than in Brownian motion. A single
event may comprise 99% of total variation, hence the "undefined variance".
Lévy flight from a normal distribution (Brownian motion).
Compared to fat-tailed distributions, in the normal distribution, events that deviate from the mean by five or more standard deviations ("5-sigma events") have lower probability, meaning that in the
normal distribution extreme events are less likely than for fat-tailed distributions. Fat-tailed distributions such as the Cauchy distribution (and all other stable distributions with the exception
of the normal distribution) have "undefined sigma" (more technically, the variance is undefined).
As a consequence, when data arise from an underlying fat-tailed distribution, shoehorning in the "normal distribution" model of risk—and estimating sigma based (necessarily) on a finite sample
size—would understate the true degree of predictive difficulty (and of risk). Many—notably Benoît Mandelbrot as well as Nassim Taleb—have noted this shortcoming of the normal distribution model and
have proposed that fat-tailed distributions such as the stable distributions govern asset returns frequently found in finance.^[3]^[4]^[5]
The Black–Scholes model of option pricing is based on a normal distribution. If the distribution is actually a fat-tailed one, then the model will under-price options that are far out of the money,
since a 5- or 7-sigma event is much more likely than the normal distribution would predict.^[6]
Applications in economics
In finance, fat tails often occur but are considered undesirable because of the additional risk they imply. For example, an investment strategy may have an expected return, after one year, that is
five times its standard deviation. Assuming a normal distribution, the likelihood of its failure (negative return) is less than one in a million; in practice, it may be higher. Normal distributions
that emerge in finance generally do so because the factors influencing an asset's value or price are mathematically "well-behaved", and the central limit theorem provides for such a distribution.
However, traumatic "real-world" events (such as an oil shock, a large corporate bankruptcy, or an abrupt change in a political situation) are usually not mathematically well-behaved.
Historical examples include the Wall Street Crash of 1929, Black Monday (1987), Dot-com bubble, 2007–2008 financial crisis, 2010 flash crash, the 2020 stock market crash and the unpegging of some
Fat tails in market return distributions also have some behavioral origins (investor excessive optimism or pessimism leading to large market moves) and are therefore studied in behavioral finance.
In marketing, the familiar 80-20 rule frequently found (e.g. "20% of customers account for 80% of the revenue") is a manifestation of a fat tail distribution underlying the data.^[8]
The "fat tails" are also observed in commodity markets or in the record industry, especially in phonographic markets. The probability density function for logarithm of weekly record sales changes is
highly leptokurtic and characterized by a narrower and larger maximum, and by a fatter tail than in the normal distribution case. On the other hand, this distribution has only one fat tail associated
with an increase in sales due to promotion of the new records that enter the charts.^[9]
See also
External links
• Examples of Fat Tails in Financial Time Series
• Fat Tail Distribution - John A. Robb Archived 2017-03-17 at the Wayback Machine | {"url":"https://www.knowpia.com/knowpedia/Fat-tailed_distribution","timestamp":"2024-11-08T09:22:29Z","content_type":"text/html","content_length":"102396","record_id":"<urn:uuid:b902d664-ca22-4cde-9882-750be9a16181>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00171.warc.gz"} |
A Geometric Theory of Everything
The December issue of Scientific American is out, and it has an article by Garrett Lisi and Jim Weatherall about geometry and unification entitled A Geometric Theory of Everything. Much of the
article is about the geometry of Lie groups, fiber-bundles and connections that underpins the Standard Model as well as general relativity, and it promotes the idea of searching for a unified theory
that would involve embedding the SU(3)xSU(2)xU(1) of the Standard Model and the Spin(3,1) Lorentz group in a larger Lie group.
The similarities between (pseudo)-Riemannian geometry in the “vierbein” formalism where there is a local Spin(3,1) symmetry, and the Standard Model with its local symmetries makes the idea of trying
to somehow unify these into a single mathematical structure quite appealing. There’s a long history of such attempts and an extensive literature, sometimes under the name of “graviGUT”s. For a recent
example, see here for some recent lectures by Roberto Percacci. The Scientific American article discusses two related unification schemes of this sort, one by Nesti and Percacci that uses SO(3,11),
another by Garrett that uses E[8]. Garrett’s first article about this is here, the latest version here.
While I’m very sympathetic to the idea of trying to put these known local symmetry groups together, in a set-up close to our known formalism for quantizing theories with gauge symmetry, it still
seems to me that major obstructions to this have always been and are still there, and I’m skeptical that the ideas about unification mentioned in the Scientific American article are close to success.
I find it more likely that some major new ideas about the relationship between internal and space-time symmetry are still needed. But we’ll see, maybe the LHC will find new particles, new dimensions,
or explain electroweak symmetry breaking, leading to a clear path forward.
For a really skeptical and hostile take on why these “graviGUT” ideas can’t work, see blog postings here and here by Jacques Distler, and an article here he wrote with Skip Garibaldi. For a recent
workshop featuring Lisi, as well as many of the most active mathematicians working on representations of exceptional groups, see here. Some of the talks feature my favorite new mathematical
construction, Dirac Cohomology.
One somewhat unusual aspect of Garrett’s work on all this, and of the Scientific American article, is that his discussion of Lie groups puts their maximal torus front and center, as well as the
fascinating diagrams you get labeling the weights of various representations under the action of these maximal tori. He has a wonderful fun toy to play with that displays these things, which he calls
the Elementary Particle Explorer. I hear that t-shirts will soon be available…
Update: T-shirts are available here.
110 Responses to A Geometric Theory of Everything
1. Aaron wrote:
This is often a gauge field (gauge mediated susy breaking) or a graviton (gravity mediated supersymmetry breaking).
This is not what “gravity mediated supersymmetry breaking” means. “Gravity mediated supersymmetry breaking” means “supersymmetry breaking mediated by any set of Planck-suppressed operators.” In
some sense, the minimal version of it is what’s known as “anomaly mediation,” but it encompasses a huge range of models (and some not-quite-models, like “mSUGRA” or “minimal supergravity” which
is more of an ansatz than a model, and which unfortunately is most of what experimentalists have been setting limits on for decades).
The trouble with generic Planck-suppressed operators is the flavor problem. As John said, there are about a hundred parameters in the MSSM with soft SUSY breaking, but phenomenology imposes
strong restrictions so that really only about 20 are completely independent. If you tried to wander very far outside of this low-dimensional subspace in the 100-dimensional parameter space, you
would be in gross conflict with observations. To give an example, if selectrons and smuons are both light, they have to be almost the same mass. So gravity mediation requires extra structure to
explain these phenomenological facts, and this structure must be present at or near the Planck scale and survive running down to low energies.
As for John’s question:
Does someone know how to get a supersymmetric extension of the Standard Model where supersymmetry is broken spontaneously?
It’s important to note that particles beyond those of the MSSM are needed for supersymmetry to be broken spontaneously, which is why models always involve a hidden sector. This was realized quite
early on; the paper by Dimopoulos & Georgi that introduced the MSSM with soft SUSY breaking explained that without a hidden sector (i.e. if SUSY is broken spontaneously in the MSSM alone), the
theory would always have a scalar lighter than the up or down quark.
2. John: Thanks for this — I hope we can clarify a lot, and maybe even make some new progress.
Most importantly, particles have very distinct personalities, whereas all 248 dimensions of E8 look alike. E8 is very symmetrical, that’s why people like it. But this beautiful symmetry needs
to be severely broken for anything like real-world physics to fall out.
Excellent point. Here’s what happens. We start with an E8 principal bundle with connection (not a superconnection). The symmetry breaks when this connection gets a vacuum expectation value (VEV),
[tex]A \simeq E_0[/tex] (I’m not sure if TeX is working here in the comments, as it once was), which leaves the curvature 0. (One way this could happen spontaneously, starting with an E8
invariant action, is described in the paper with Lee and Simone, but the particular mechanism isn’t so important.) This spontaneous symmetry breaking picks out some directions in E8 as special,
allowing all other generators in E8 to be identified (and named) with respect to these, based on their Lie brackets. Since it’s key, let me describe this in more detail. If we describe the E8
(-24) Lie algebra as
e8 = spin(12,4) + 128^+_S
then the VEV of the connection is [tex]E_0 = 1/4 e_0 \phi_0[/tex], in which [tex]\phi_0[/tex] is the VEV of a Higgs multiplet that transforms as a 12 vector under a spin(11,1) subalgebra of the
spin(12,4), and [tex]e_0[/tex] is the 1-form frame field of deSitter spacetime, transforming as a 4 vector under a spin(1,3) subalgebra of spin(12,4), such that the nonzero VEV is in the
complement of spin(1,3) and spin(11,1) in the spin(12,4) of e8. It had to be deSitter spacetime if the curvature of the connection is to be 0, with cosmological constant related to the Higgs VEV.
Personally, I think this symmetry breaking mechanism — combining cosmogenisis with a Higgs model — is… awesome. I’d enjoy getting your feedback on it.
I see no way to get E8 symmetries that mix fermions and bosons in a model where the symmetry of E8 has been broken by deliberately chopping it into bosonic and fermionic parts.
The “bosonic and fermionic” parts of the connection can only mix before spontaneous symmetry breaking — which is to say, before our universe technically exists. However, if an appropriate action
has been chosen that is independent of the “fermion” parts of the E8 connection, then there is a prescription for replacing the “fermion” parts of the connection (1-forms valued in the parts e8
that we’re calling the fermion part, based on [tex]E_0[/tex]) with Grassmann fields, which are identified as fermions (or pre-fermions, if you like). Now, based on our action, and on [tex]E_0[/
tex], we could separate out the [tex]128^+_S[/tex] as the fermion part, or, as I consider preferable, we could break e8 up as
e8 = spin(4,4) + spin(8) + 8×8^+_S + 8×8^-_S + 8x8_V
and consider those last three blocks of 64 as pre-fermion Grassmann fields. This works because spin(4,4) + spin(8) is reductive in e8.
Ah, as I’m reading this, I see I have an email from you…
3. John Baez,
A recent, up to date, small (20 pages) concise review with a comparison of the various mechanisms (pros and cons) and potential string theory realizations is the following
arXiv:1006.0949 by Alwis
4. This is not what “gravity mediated supersymmetry breaking” means.
Geez. I go away for a few years and I already start forgetting things. Enh. Phenomenology was never my thing anyways.
For Garrett, you still haven’t explained whether the infinitesimal generators of your symmetry are all commuting or if some or Grassman.
And do you still claim to be able to reproduce any part of the standard model action?
I do apologize for confusing your paper with an earlier one of Lee’s which was fermion free, however. However, your paper seems more along the lines of Percacci’s earlier paper where fermions are
considered separately and not in the same multiplets as bosons.
I will probably have to bow out of this discussion now, however.
5. “This works because spin(4,4) + spin(8) is reductive in e8”
How the Standard Model gauge group sits inside of spin(4,4) + spin(8) ?
“Personally, I think this symmetry breaking mechanism — combining cosmogenisis with a Higgs model — is… awesome.”
Does that mean that you predict the cosmological constant is electro-weak scale in size?
The resulting paper is here, which I believe makes things crystal clear, as well as forming a more complete introduction to E8 Theory.
Except that you have still completely failed to answer Distler’s criticism. Allow me to quote the relevant excerpt from your paper:
Distler and Garibaldi prove that … when one embeds gravity and the Standard Model in E8, there are also mirror fermions. They then claim this prediction of mirror fermions (the existence of
“non-chiral matter”) makes E8 Theory unviable. However, since there is currently no good explanation for why any fermions have the masses they do, it is overly presumptuous to proclaim the
failure of E8 unification – since the detailed mechanism behind particle masses is unknown, and mirror fermions with large masses could exist in nature.
This is completely misleading. For one thing, the phrase “mirror fermions” is ambiguous. There are models of particle physics which impose exact parity symmetry, which requires introducing
so-called “mirror matter”. However, in this case, the extra particles are charged under a different gauge group to the visible particles, and are easy to ‘hide’. What you have is nothing like
In the standard model, no fermion mass terms are allowed, because they violate the gauge symmetry. The reason is that the left-handed fermions are in a complex representation of the gauge group.
After electroweak symmetry breaking, all fermions are vector-like with respect to the remaining gauge symmetry, and mass terms can be written down. In practice, these come from Yukawa couplings
to the Higgs field. However, in your model, the fermion content is doubled, such that the left-handed fermions now fall into a real representation of the standard model gauge group (in fact, R +
R-bar, where R is the standard model rep). Therefore there is nothing to forbid mass terms for all the fermions, and in fact these should be generated radiatively in the absence of supersymmetry.
So generically, all fermions in your theory should have masses roughly of the cut-off scale (probably the Planck scale here).
This is a serious problem, and why Distler rightly calls your model a ‘zero-generation’ model. You can’t just wave your hands about it — you have to at least provide a solution in principle, or
there is no reason to think your model is anything more than a pretty exercise in group theory.
Let me finish by explaining why this is so different to supersymmetry. Before SUSY breaking, there are no mass terms in the MSSM. For the fermions, the reason is the same as for the standard
model, and the bosons are related to the fermions by SUSY. After SUSY breaking, nothing stops us writing down mass terms for the bosons, but those for the fermions are still forbidden by
chirality. That’s why it is natural for the (unseen) scalar partners to be significantly more massive than the standard model fermions. The Higgs is different, because there is an up-type Higgs
and a down-type Higgs, and together they form a real representation, so one can write down a supersymmetric mass term.
7. Garrett suggested to sombody:
Might I recommend you choose one of the other ToE’s that have made fruitful progress over the last 30 years okay, I will — oh wait, there are none.
I suppose you have followed Alain Connes’ construction (here is a survey and links) of the standard model by a Kaluza-Klein compactification in spectral geometry. It unifies all standard model
gauge fields, gravity as well as the Higgs as components of a single spin connection. Connes finds a remarkably simple characterizaiton of the vector bundle over the compactification space such
that its sections poduce precisely the standard model particle spectrum, three chiral generations and all.
Alain Connes had computed the Higgs mass in this model under the big-desert hypothesis to a value that was in a rather remarkable chain of events experimentall ruled out shortly afterwards by the
Tevatron. But the big desert is a big assumption and people got over the shock and are making better assumptions now. We’ll see.
Apart from being a nice geometrical unification of gravity and the other forces (credits ought to go all the way back to Kaluza and Klein, but in spectral geometry their orginal idea works out
better) Connes’ model has some other striking features:
the total dimension of the compactified spacetime in the model as seen by K-theory is and has to be, as they showed, to produce exactly the standard model spectrum plus gravity: D= 4+6.
Now “as seen by K-theory” was shown by Stolz and Teichner and students to mean in a precise sense: as seen by quantum superparticles (here is some link — you can ask me for a better link). In
fact what they consider is almost exactly the spectral triples that Connes considers, with some slight variation and from a slightly different angle. For the relation see the nLab entry on
spectral triple (ask me to expand that entry…).
As also indicated at that entry: there is a decent theory of how to obtain a spectral triple as the point particle limit of a superconformal 2-dimensional CFT. Yan Soibelmal will have an article
on that in our book Precisely because Connes’ model turns out to have real K-theory dimension 4+6 does it have a chance to be the point particle limit of a critical 2D SCFT. That would even give
it the UV-completion — as they say — that would make its quantization consistent (which, remember, contains gravity).
I think there is some impressive progress here. It is not coming out of the physics departments, though, but out of the math departmens. For some reason.
8. (I’m continuing through the comments chronologically, picking up where I left off, trying not to miss anything directed to me that’s important. Peter, thanks for allowing the discussion.)
Second, even among fermions, different particles have drastically different personalities: for example their masses, and the rates at which they turn into each other, which are described by
the numbers in the Cabibbo-Kobayashi-Maskawa matrix and the Maki-Nakagawa-Sakata matrix. Similarly, in the realm of the bosons there’s the Higgs mass and other numbers. Garrett’s work has
nothing to say about these. Without these numbers we can’t do real-world physics. But the big problem is this: I don’t see any way to get these numbers into the game without further breaking
down the symmetry of E8. Why? Because again, E8 symmetry wants all particles to be alike, but these numbers describe how they’re not.
Getting the CKM and MNS matrices is the goal, and it is true, E8 Theory is not there yet — and I have been completely candid about this at every opportunity. But I do think there is hope, and my
work may soon have something to say about these. How can this possibly work? Well, it all has to start with symmetry breaking, as described in my previous comment. After that, the masses of all
other particles are determined by how they interact with the Higgs.
Third, there’s no way to pack all known fermions into E8 without positing two copies of E8 and giving every fermion a mysterious unseen partner called a “mirror fermion”. To keep these
rascals from being seen we could claim they’re more massive than the guys we see – but no method for this has been described yet, to my knowledge.
There is potentially a way to fit three generations of fermions into E8 and avoid the mirror fermion problem. The basic idea is that, using an inner-automorphism of E8 related to triality, we can
independently gauge transform the 64 mirror fermions, and 64 pre-fermions in the 8_Vx8_V of a so(4,4)+so(8) subalgebra of e8, into generations of usual fermions that will all interact differently
with the Higgs. I don’t yet know if CKM and MNS will come out of this, but I’m working on it. In the meantime, yes, it’s fair to say the model is incomplete, and the burden is on me (or some
other researcher) to figure out how this can work, but it’s premature to say it can’t work.
9. Mark:
Lisi should: a) publish some papers in journals
Did that.
b) answer john baez et al.’s skepticism
Working on it. But, I also encourage skepticism.
c) present some testable predictions
The testable predictions are that if any new particles are found that don’t fit E8, such as superparticles (which many expect to see), then this theory is wrong.
d) have them tested
In progress.
e) be labeled the next-einstein and move on to selling i-phone universe-splitter apps and t-shirts.
What people label me, whether it’s crackpot, next-einstein, or surfer-physics dude, isn’t really up to me. And I’m not selling anything. I am, however, helping friends sell things I think are
cool, and see no problem with that. Also, I did get paid to write the SciAm article, and will use the money to buy a new surfboard.
Add to this the powerful behind-the-scenes financial and media forces at play here
What powerful financial and media forces?
and that this has been going on for over three years with no new developments nor solutions to the problems from Lisi
If you look at my list of (1)-(6) issues in my comment above, you’ll see that there has been good progress on several of them.
10. Brian: Peter has addressed your points. My only addition is that I have published work on the theory, without difficulty, with coauthors. But as the Bogdanovs showed, this means little.
11. Peter:
You need people who know what they are talking about to discuss the issue as honestly and clearly as they can. In many cases things will come down to whether there’s any hope that future work
will fix known problems with some idea (and I think that’s what’s going on here, as well as in string theory). Reasonable people will differ, with those who believe problems can be overcome
going on to try and do so.
Precisely. Thank you.
12. ned:
“I’ve been spending every other day surfing or kitesurfing here in Maui.” No wonder his peers are jealous.”
My life hasn’t been all roses. But I am working on making it easier for other scientists to come spend some time in Maui — that’s what the Pacific Science Institute is about.
13. Aaron:
For Garrett, you still haven’t explained whether the infinitesimal generators of your symmetry are all commuting or if some or Grassman.
The superconnection is the formal sum of a 1-form and an anti-commuting Grassmann field, both valued in different parts of some algebra. That algebra can be a Lie algebra, such as E8, or it can
be a Lie superalgebra — the necessary restriction is that the 1-form be valued in a reductive subalgebra.
And do you still claim to be able to reproduce any part of the standard model action
Yes, see the paper with Lee and Simone.
I do apologize for confusing your paper with an earlier one of Lee’s which was fermion free, however. However, your paper seems more along the lines of Percacci’s earlier paper where fermions
are considered separately and not in the same multiplets as bosons.
True. We wanted to make this less unfamiliar and upsetting to people by keeping the bosons and fermions separate.
I will probably have to bow out of this discussion now, however.
Sorry to lose your input.
14. Wolfgang:
How the Standard Model gauge group sits inside of spin(4,4) + spin(8) ?
Excellent observation. It doesn’t. The Standard Model and gravity fits in spin(12,4), and spin(4,4)+spin(8) fits in that. There are two SM generators, W^+ and W^-, that occupy the complement.
When considering triality, one needs to use spin(4,4) and/or spin(8). I’m not yet sure whether those W’s will remain in the complement as bosons, displacing two fermion degrees of freedom, or
whether things will work some other way. A key idea that came from Banff is that the spin(4,4)+spin(8) subalgebra of e8 which relates to triality might be a different subalgebra than the one
containing the Standard Model, including having different Cartan subalgebras. This will, of course, give a mixing mess, but the question will be whether that mess is the CKM and MNS mess.
Does that mean that you predict the cosmological constant is electro-weak scale in size?
Yes! That seems terrible, but the hope is that the cosmological constant runs from this value at the unification scale down to the tiny value we see at low energies.
15. “There are two SM generators, W^+ and W^-, that occupy the complement. When considering triality, one needs to use spin(4,4) and/or spin(8). I’m not yet sure whether those W’s will remain in the
complement as bosons, displacing two fermion degrees of freedom, or whether things will work some other way.”
Didn’t you explain that all the generators of e8 are either in spin(4,4)+ spin(8) or in one of the 8×8 blocks exchanged by your triality?
So, if some of 8×8 block is bosons, where are rest of fermions? And, if Ws live in 8×8 block, does that mean they form triplet under triality? Do different generations have different gauge bosons
coupling to them?
Sorry for so many questions, but this is very confusing!
16. Rhys:
Except that you have still completely failed to answer Distler’s criticism.
No, Distler and Garibaldi’s claim is that one cannot even get one generation of fermions in E8. This paper answers that directly by showing explicitly how a generation of fermions does fit in E8.
(And it explains it via a direct identification of generators, which is quite nifty.)
This is completely misleading.
No, it is direct. It is Distler’s language that is misleading. When he says “there are no chiral generations,” what he actually means is that there is a generation and what he calls an
anti-generation and I call mirror fermions. For him and you to use this twist of mathematical language to say “there are no generations” is a lie.
For one thing, the phrase mirror fermions is ambiguous.
The top Google hits and I disagree.
There are models of particle physics which impose exact parity symmetry, which requires introducing so-called mirror matter.
If I had said “mirror matter,” then yes, it would have been ambiguous. But I did not.
fermions in your theory should have masses roughly of the cut-off scale
That’s such a fun word, “should.” With that one word, you are presuming what nature does — when the truth is that we just don’t know. It is fine if you want to say “it should not be,” it is a lie
to say “it can not be,” at least until we know what nature actually does. And in that same phrase, you are incorrectly presuming, as do Distler and Garibaldi, that my theory must have mirror
fermions in it, when I have said several times that I expect these to be gauge transformed to usual fermions.
You can’t just wave your hands about it — you have to at least provide a solution in principle, or there is no reason to think your model is anything more than a pretty exercise in group
This is a valid point. I cannot say I have a complete theory of everything, and I do not, until these problems are solved. But I can and do wave my hands about how they might be solved in
principle. And even if it ends up having been a pretty exercise in group theory, I won’t have considered it a waste of time, because I think there is a lot here — especially the dodge of the
Coleman-Mandula theorem via symmetry breaking — that is true about nature, even if we don’t yet have the full picture.
17. This may not be the proper place to bring this up. But does anyone have a view on Penrose’s ‘conformal cyclic cosmology’? I am reading his book now and but have only seen a couple of arxiv
articles mentioning it…..
18. Urs: I agree that Alain Connes’ model is fascinating and deserves more attention from physicists. But it has not been fruitful in making successfully tested new HEP predictions — nor has any
model in the past 30 years. It was not my intent to be discouraging, or particularly disparaging of other ideas — I was mostly counter-snarking Aaron.
19. Steve,
I’ve heard Penrose talk about this, but didn’t really understand the point. I look forward to reading his book (it isn’t out in the US yet), and might write about it then. But, I’m no
cosmologist, best to find a blog run by someone who is to discuss the subject.
But it has not been fruitful in making successfully tested new HEP predictions — nor has any model in the past 30 years.
It has. Within weeks even. It was experimentally verified that the big desert assumption is inconsistent in this model with experiment.
It’s an impressive model. And I didn’t quite say that “it deserves more attention from physicists”. Let them spend their time with what pleases them. Instead I mentioned this in reply to your
insinuation that there is nothing promising in fundamentall model building out there besides your idea. It occurred to me that you might actually think that’s true. And maybe because the most
impressive progress in fundamental physics these days does not quite percolate through the physics community.
21. Wolfgang (and John):
So, if some of 8×8 block is bosons, where are rest of fermions? And, if Ws live in 8×8 block, does that mean they form triplet under triality? Do different generations have different gauge
bosons coupling to them?
Here we are on the edge of what I’m working on. So I don’t yet know what the complete picture is, and my remarks here will be speculative. We do know that under the decomposition
e8(-24) = spin(12,4) + 128^+_S
that one generation of fermions can be the 64^+_S rep (part of the above 128) of a spin(11,3) subalgebra of the above spin(12,4). And in fact, the known gravitational and Standard Model bosons
can fit in a spin(5,3)+spin(6) subalgebra of spin(11,3). But, spin(5,3) and spin(7,1) don’t have triality automorphisms. However, spin(4,4) and spin(8) do, so we can decompose e8 as
e8(-24) = spin(4,4) + spin(8) + 8x8_V + 8x8_+ + 8x8_-
and consider innner automorphisms of e8, corresponding to so(4,4) and so(8) triality, that interchange those three blocks of 64. If we put gravitational spin(1,3) in the spin(4,4), and strong su
(3) in the spin(8), and the photon and the Z in both, then we’re stuck with at least the W^+ and the W^- in that 8x8_V. And you’re right that if we’re identifying those three blocks of 64 by
triality that we’re probably going to be missing at least three sets of two fermion degrees of freedom to accommodate those W’s. Maybe nature has chosen to exclude the right-handed components of
neutrinos in this way? Or, an even weirder speculation, maybe that particular spin(4,4)+spin(8) is not completely in the spin(12,4)? There would have to be a lot of mixing angles to describe the
geometry of how these spin groups are mutually related, but we want something like that to come out anyway that corresponds to CKM and MNS. The bottom line is that I don’t know how this works
yet, but it’s really fun and interesting! (And please do correct me if I’ve made any mistakes here.)
22. Urs: I didn’t mean to insinuate that there aren’t other promising models. But I don’t consider a prediction proven false to be a “fruitful” prediction in the usual sense, though these impressive
events and the positive aspects of Connes’ model are not lost on me. I do consider this E8 Theory to be even more fascinating and promising, but I’m biased. Although, it indisputably looks really
good on the new T-shirts. 🙂
23. Garrett, about
e8(-24) = spin(4,4) + spin(8) + 8x8_V + 8x8_+ + 8x8_-
where you put
gravity spin(1,3) in the spin(4,4)
color su(3) in the spin(8)
are “stuck with at least the W+ and W- in that 8x8_V”
could you find within “that 8x8_V” a spacetime base
manifold that is an 8-dim Kaluza-Klein M4 x CP2
M4 is 4-dim Minkowski spacetime
CP2 is internal symmetry space.
Then since CP2 = SU(3)/U(2)
you would have the electroweak U(2) (weak bosons and photon)
naturally included in your structure,
the added benefit of getting not only fermions and bosons,
but spacetime itself as part of your E8.
The 8-dim Kaluza-Klein idea is not mine,
but is due to N. A. Batakis
who wrote Class. Quantum Grav. 3 (1986) L99-L105 in which he showed
that “… In a standard Kaluza-Klein framework,
M4 x CP2 allows the classical unified description of an SU(3) gauge field with gravity
… an additional SU(2) x U(l) gauge field structure is uncovered …
As a result,
M4 x CP2 could conceivably accommodate the classical limit of a fully unified theory
for the fundamental interactions and matter fields …”.
Roughly, he uses the structure CP2 = SU(3)/U(2) with the local U(2) giving electroweak and the global SU(3) working for color since its global action is on CP2 which is, due to Kaluza-Klein
local with respect to M4 Minkowski spacetime.
As to why Batakis is not well known and his model fell into obscurity,
Batakis never handled fermions properly in his model.
Since he had nothing to work with but M4xCP2 Kaluza-Klein,
he was reduced to introducing fermions sort of ad hoc by hand,
and he could not show that they worked nicely with his gauge bosons.
since all your structures (spacetime, gauge bosons, fermions) would come from E8 they can be shown to work together nicely.
24. “If we put gravitational spin(1,3) in the spin(4,4), and strong su(3) in the spin(8), and the photon and the Z in both, then we’re stuck with at least the W^+ and the W^- in that 8x8_V.”
If I understand what you are saying, there are three W^+s and three W^-s (the ones in the 8x8_V and their triality partners in the other 8×8 blocks). How to reconcile that with there being only
one SU(2), whose gauge field corresponding to the diagonal generator occurs only once?
How does the SU(2) Yang Mills action look?
I understood that you wanted the physics of having three generations of fermions, but I don’t think you want three generations of W’s.
25. Tony:
Your idea sounds pretty close to constructing a Cartan geometry starting from E8. However, if we mod E8 by spin(4,4)+spin(8), we get not only the 8x8_V but also the 128 spinor as the base, which
is waaaay too big. It seems much cleaner to consider an E8 principal bundle over a 4D base. If it turns out that structure isn’t rich enough, then I’m open to re-considering an 8D base and KK
with CP2. I agree it looks pretty good, but I want to see what I can do with just a 4D base, E8, and triality first.
26. Wolfgang:
You are understanding things perfectly.
But when looking for a mixing mechanism, I think it’s probably good to have these issues in mind but not focus on them too hard. The W’s are important, but they’re not the only problem. We also
have to either get rid of or give large masses to all the X bosons somehow — the various gauge fields other than those of the SM. And, of course, we want to mix the fermions (including mirrors)
and get the CKM and MNS for them — all of this in one go, with limited options.
I think a good thing to try is going to be using the so(4,4)+so(8) decomposition to calculate a set of E8 inner automorphisms related to triality, then try applying those inner automorphisms back
in the so(12,4) + 128 decomposition to see how it can mix elements. I want to see what is learned from trying that before focusing on specifics.
27. > Also, I did get paid to write the SciAm article, and will use the money
> to buy a new surfboard.
this is the coolest statement on this blog yet 🙂
and kudos that you resisted invoking the anthropic principle … yet
28. Peter,
Penrose’s book has been available on Amazon since late October.
29. “The W’s are important, but they’re not the only problem. We also have to either get rid of or give large masses to all the X bosons somehow”
OK. But other gauge symmetries could be broken at Planck scale. Standard Model gauge symmetries are supposed to be unbroken down to low energies. That seems much more restrictive.
One more question:
In previous comments, you seemed to say that even if triality idea doesn’t work out, E8 theory is still OK.
My understanding from trying to read distler-garibaldi paper is that they show two things,
1) 128 (out of 248) fields are fermions
2) fermion spectrum is non-chiral
so (assuming no triality), fermions are at best 1 generation and 1 mirror-generation of Standard Model. From this they conclude E8 theory is not viable.
Do you say E8 theory with 1 generation and 1 mirror-generation (even if no triality) is still viable theory of Nature?
If so, could you explain how?
30. The lack of response to Connes’ theory is indeed interesting. I think the problem is that nonbody has been able to explain in a language that particle theorists can understand whether this is
indeed a new idea (and if so, what the new idea is) or whether this is just a complicated way to formulate an old idea (GUT’s or maybe Gravi-GUT’s). Where is Witten when you need him?
31. Sorry. I have one more question. About cosmological constant being electro-weak scale in size, you said:
“Yes! That seems terrible, but the hope is that the cosmological constant runs from this value at the unification scale down to the tiny value we see at low energies.”
Does that mean the relation between electro-weak scale and cosmological constant is accidental feature of your classical lagrangian? Is there a more general lagrangian where they are independent
I ask because in renormalizable theory all counterterms should appear as possible terms in classical lagrangian.
32. Didn’t Connes predict a 170 GeV Higgs? Which was the first region to be ruled out by the Tevatron?
33. Just to be clear,
my intent was not to suggest
“… mod E8 by spin(4,4)+spin(8) …”
in which case “… we get not only the 8x8_V but also the 128 spinor as the base …”
but to suggest
mod E8 by both spin(4,4)+spin(8) and also the 128 spinor
(that may be a 2-stage process)
so that
we get only the 8x8_V as the base
and then
to let the 8x8_V represent an 8-dim M4 x CP2 Kaluza-Klein.
The lack of response to Connes’ theory is indeed interesting. I think the problem is that nonbody has been able to explain in a language that particle theorists can understand whether this is
indeed a new idea (and if so, what the new idea is) or whether this is just a complicated way to formulate an old idea (GUT’s or maybe Gravi-GUT’s).
Yes. Generally my impression is that the number of theoretical physicists actively aware of or at least interested in the issues of what it means to find a conceptual or even axiomatic framework
for fundamental physics is currently much lower than it used to be. It seems to me that in the early 90s or so the situation has been very different. In fact from that time date a few articles by
string theorists who had read Connes, had understood what he is after and had tried to connect it to string theory.
Because the curious thing is: what Connes suggests is precisely the 1-dimensional version of the very idea of perturbative string theory (which is the 2d version of an even more general idea):
regard the algebraic data characterizing a d-dimensional super QFT as a stand-in for the geometric data characterizing the target space of which this QFT would be the sigma-model, if it were one.
What in Connes’s setup is a spectral triple is a vertex operator algebra for the string.
(References that discuss how to make this statement precise are at spectral triple and 2-spectral triple).
Where is Witten when you need him?
And why do you necessarily need him?
Lately Witten seems to be busy providing more evidence for the holographic principle of higher category theory (scroll down to see what i mean).
35. Wolfgang:
In previous comments, you seemed to say that even if triality idea doesn’t work out, E8 theory is still OK.
What I was saying was slightly different. I do think the triality idea is going to have to work out for E8 Theory to be a good theory. However, since I only have some rough ideas on how triality
might work out, I have been forced by critics to defend the theory without it. Without triality, the best I can say is that the theory is incomplete and unattractive, but not necessarily wrong.
The best way to look at it, in my opinion, is that we currently know exactly how gravity and the Standard Model gauge fields along with one generation of fermions can embed in E8, which is
incredibly cool. And there are some indications of how to get the other two generations, with a much tighter fit, but that is not yet clear.
My understanding from trying to read distler-garibaldi paper is that they show two things, (1) 128 (out of 248) fields are fermions. (2) fermion spectrum is non-chiral. So (assuming no
triality), fermions are at best 1 generation and 1 mirror-generation of Standard Model. From this they conclude E8 theory is not viable. Do you say E8 theory with 1 generation and 1
mirror-generation (even if no triality) is still viable theory of Nature?
This is a straw man setup. I disagree with Distler and Garibaldi at step (1) — they insisted on using this [tex]Z_2[/tex] grading of E8 even though I said the theory would rely on other options.
However, even this silly straw man is not easy to knock down, because mirror fermions have not been completely ruled out, even if they make it ugly.
Essentially, I think E8 Theory is about half way done. We’ve got gravity, the Standard Model, a generation of fermions, and a nice symmetry breaking mechanism. And the triality-related gauge
transformations I’m working with are very encouraging. It is kind of stupid to assess a half-done theory as if was supposed to be a complete theory of nature. It’s like looking at a half-built
house and saying “oh, that’s no good — it’s leaky.” Rather, one needs to assess E8 Theory as a research program moving towards a complete ToE. And, from that point of view, it’s doing pretty
36. Wolfgang:
Does that mean the relation between electro-weak scale and cosmological constant is accidental feature of your classical lagrangian? Is there a more general lagrangian where they are
independent parameters?
The relation between Higgs VEV and cosmological constant is even more fundamental than the Lagrangian. If the bosonic connection is
[tex]H = \frac{1}{2} \omega + \frac{1}{4} e \phi + A[/tex]
then its curvature is
[tex]F = \frac{1}{2}(R – \frac{1}{8}\phi^2 e e) + \frac{1}{4} ( T \phi – e D \phi) + F_A[/tex]
and the relationship between Higgs VEV and cosmological constant, [tex]\Lambda = \frac{3}{4} \phi_0^2[/tex], comes from [tex]F_0 = 0[/tex], which I consider more fundamental than the Lagrangian.
One might be able to cook up a way to change that relationship, but I wouldn’t recommend it.
37. (Hmm, that “8211;” above is a “-“. I don’t know why it did that.)
38. Tony: I think one is only allowed to mod out by subgroups.
39. “This is a straw man setup. I disagree with Distler and Garibaldi at step (1) — they insisted on using this Z_2 grading of E8 even though I said the theory would rely on other options.”
Maybe I expressed myself badly.
Fermions transform as Lorentz spinors. Your triality idea is to change how fields in the 248 transform under Lorentz group. Without triality (which remains to be worked out), fields transform
according to the “naive” transformation rule. Do you agree that naive transformation rule gives 128 fermions or is even that part wrong?
Put differently: if the triality idea doesn’t work out, do you have another way to avoid distler-garibaldi conclusion?
“The relation between Higgs VEV and cosmological constant is even more fundamental than the Lagrangian. … One might be able to cook up a way to change that relationship, but I wouldn’t recommend
If you don’t change it, how do you avoid cosmological constant of order the electro-weak symmetry breaking scale?
In earlier comment, you said “the hope is that the cosmological constant runs from this value at the unification scale down to the tiny value we see at low energies.” But if Higgs VEV and
cosmological constant are tied together as you say how can one be big (250 GeV) and the other tiny?
40. Garrett, you say “one is only allowed to mod out by subgroups”.
Maybe you and I are not using “mod out” in the same sense,
and maybe (since it is a term with which I am not very familiar)
I have been misusing it, so here is what I am trying to say in
terms of graded Lie algebras:
Consider Thomas Larsson’s 7-grading of E8 which is of the form
E8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3
with graded dimensions
E8 = 8 + 28 + 56 + (sl(8) + 1) + 56 + 28 + 8
The odd graded part of E8 has 8+56 + 56+8 = 64+64 = 128 dimensions
and corresponds to your 128 spinor.
The even graded part of E8 has 28 + 64 + 28 = 120 dimensions
and corresponds to your D8 Lie algebra so(4,12)
My first stage is to “mod out” the odd graded 128 spinor,
which leads to the next stage about the D8.
The D8 Lie algebra has a 3-grading which is of the form
D8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3
with graded dimensions
D8 = 28 + (sl(8) + 1) + 28
The odd graded part of D8 has 28 + 28 = 56 dimensions
and corresponds to your D4 + D4 Lie algebras so(4,4) and so(8)
The even graded part of D8 has 64 dimensions
and corresponds to your 8x8_V.
My second stage is to “mod out” the odd graded D4 + D4,
which leaves your 64-dimensional 8x8_V to represent
an 8-dim spacetime that can (by breaking octonionic symmetry
down to quaternionic) give you a M4 x CP2 Kaluza-Klein
with the Batakis structure giving you the U(2) from CP2= SU(3)/U(2).
Then you can construct a nice Lagrangian as follows:
Base Manifold from the 8-dim Kaluza-Klein in the 8x8_V
Gauge Boson terms from D4 + D4 “modded out” in stage 2
Fermion terms from 128 half-spinor “modded out” in stage 1.
if you look at the geometry of the octonionic/quaternionic
symmetry breaking down to 4+4 dim Kaluza-Klein you see
that Meinhard Mayer’s mechanism (Hadronic Journal 4 (1981) 108-152)
(he is physics professor emeritus at U. C. Irvine)
gives the Higgs scalar.
41. Garrett,
What about this: whenever you talk about using triality on a part of E(8), it seems you are not talking about E(8) anymore, but a semiderect product of SO(8)XE(8), SO(8) being the group that
“insert” the triality. Now, what do you think of this?
42. Hello Peter & Garrett,
Well, it seems that Peter is quite skeptical of Garrett’s theory, and that Garrett is too, if less so. The question, then, is why does it keep getting so much attention and funding?
Peter writes, ”
My understanding is that Garrett is well aware that his proposal has problems. In the Scientific American article he writes:
“All new ideas must endure a trial by fire, and this one is no exception. Many physicists are skeptical—and rightly so. The theory remains incomplete.”
I have no problem with skepticism, I’m skeptical about many of Garrett’s ideas too. If Jacques wants to make a clean technical argument showing the nature of the problems with Garrett’s proposal,
that’s great, and could be potentially worthwhile. But I don’t see any reason for the hostile, sneering tone of Jacques’s blog posting explaining these points. This is not the way to
professionally make a credible technical argument. ”
Are there not a lot of other theories out there which we can be skeptical about? So why is Garrett’s “theory” getting all the attention from television, magazines, and the press? Who is pushing/
promoting this and why?
Insights? Ideas? Thanks!
43. Wolfgang:
Fermions transform as Lorentz spinors. Your triality idea is to change how fields in the 248 transform under Lorentz group. Without triality (which remains to be worked out), fields transform
according to the “naive” transformation rule. Do you agree that naive transformation rule gives 128 fermions or is even that part wrong?
That is correct.
Put differently: if the triality idea doesn’t work out, do you have another way to avoid distler-garibaldi conclusion?
No, without triality, we’re stuck with mirror fermions. But the Distler-Garibaldi conclusion that “the theory can’t work” would still be untrue, because mirror fermions could exist. But I don’t
think they do — I think triality will work.
But if Higgs VEV and cosmological constant are tied together as you say how can one be big (250 GeV) and the other tiny?
I haven’t done the calculation, but perhaps they run independently, with the effective cosmological constant getting contributions from gravity, and the Higgs mass from Standard Model and other
44. Tony:
If one were to try and build a universe by deforming the E8 Lie group, the nicest way to do it would probably be to use Cartan geometry, by which the base spacetime is modeled on the (too large)
symmetric space obtained by moding E8 out by a subgroup.
Of course, you’re also welcome to just start with an 8D base and a principal bundle, which is less restrictive, and play with different gradings and KK schemes as you are here.
45. Daniel:
When I am talking about E8 triality I am talking about the triality outer automorphisms of the so(4,4) and so(8) subalgebras, and the corresponding inner automorphisms of E8.
46. Gregor:
Since you cannot accept that the media has been attracted to a story about an unusual physicist who has come up with an interesting new theory, the attention must be because I am so incredibly
Happy Thanksgiving!
47. “No, without triality, we’re stuck with mirror fermions. But the Distler-Garibaldi conclusion that “the theory can’t work” would still be untrue, because mirror fermions could exist.”
Sorry. That I don’t understand.
Without triality, you are stuck with one generation and one mirror-generation. I don’t see how you can say that “works” as a theory of nature. Could you explain?
” But I don’t think they do — I think triality will work.”
Maybe it will. But it faces serious obstacles (see above discussion about W bosons).
“I haven’t done the calculation, but perhaps they run independently,”
Every independently-running coupling constant corresponds to an independent term you can add to classical lagrangian. You just explained that in your theory electro-weak scale and cosmological
constant are not independently adjustable coupling constants. So how can they run independently?
48. >“No, without triality, we’re stuck with mirror fermions.
> But the Distler-Garibaldi conclusion that
> “the theory can’t work” would still be
> untrue, because mirror fermions could exist.”
> Sorry. That I don’t understand.
i guess he just means that you can build a model that looks like the SM at low energies without chiral fermions. and he is right – you can. you just need fine tuning, which is “ugly” but not
and of course you can add 2 carbon copy generations and by hand add CKM mixing. it’s not pretty – but who says that top-color assisted extending walking technicolor is? 🙂
49. I’ve had to delete repeated anonymous comments by someone who couldn’t be bothered to either look things up for himself or read Garrett’s previous response to the same question:
This entry was posted in Uncategorized. Bookmark the permalink. | {"url":"https://www.math.columbia.edu/~woit/wordpress/?p=3292&cpage=2","timestamp":"2024-11-05T22:40:09Z","content_type":"text/html","content_length":"155634","record_id":"<urn:uuid:ccd3c551-f5d5-4a49-8504-d2ad6d5f8151>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00143.warc.gz"} |
147 NTU Management Review Vol. 31 No. 2 Aug. 2021 approval of the Biopharmaceutical Act for both treated firms and control firms, and then comparing the difference between these two groups. Control
firms are found using the PSM procedure. In this study, we have two control groups, one for the intra-industry and the other for inter-industry analyses. Finally, we use the t statistic to examine
the significance of the DID estimator. The significance of the DID estimator can be used to explain that the innovation in approved biopharmaceutical firms is significantly different from the
innovation in control firms (i.e. unapproved biopharmaceutical firms or high-tech firms) after the Biopharmaceutical Act. 3.3.4 Difference-in-differences (DID) Regression The DID estimator may not be
sufficient to explain the influence of the Biopharmaceutical Act because it does not consider the heterogeneous dynamics from other variables (Buckley and Shang, 2002). In addition, most previous
studies conduct only DID regressions and do not use the DID estimator. Thus, we follow previous studies (Blundell and Costa-Dias, 2009; Buckley and Shang, 2002; Lechner, 2011) to simply incorporate
possible factors into the linear regression to estimate the influence of the Biopharmaceutical Act. The DID regression is: Yi,t = α0 + β∙Aftert + δ∙Treatmenti + γ∙Aftert × Treatmenti + π∙Control
variablesi,t + Year fixed effect + εi,t , (1) where Yi,t denotes the measure of innovation of firm i in year t; Aftert = 1 if the firm is in or after the approval year and 0 otherwise; Treatmenti = 1
if the firm is approved according to the Biopharmaceutical Act and 0 otherwise. The time period of this regression is from 2002 to 2017.24 We respectively use R&D intensity and patent adjusted
citations to measure the innovation activities in the regressions. We use firm size (natural logarithm of total assets; Huang, 2019), lagged R&D expenditure (pre-year R&D expenditure), ROA, and debt
ratio to explain the R&D investment (i.e. R&D intensity). When the patent adjusted citations are the innovation measure, we follow Lerner (1994) and Becker-Blease (2011) and use 24 All results for
DID regressions control for the year fixed effect. To save space, we do not show the results for the year fixed effect in the tables.
RkJQdWJsaXNoZXIy ODg3MDU= | {"url":"http://review.management.ntu.edu.tw/ebook/31.2/147/","timestamp":"2024-11-02T10:56:05Z","content_type":"text/html","content_length":"8157","record_id":"<urn:uuid:914c7f87-a671-40f2-8bcf-4bc10c209060>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00524.warc.gz"} |
code_saturne User's Forum
Dear developers,
I'am new user to Code_Saturne. I am running an incompressible RANS simulation of flow past a axisymmetric body, illustrated below. The computed pressure coefficient along the body is perfect.
However, the drag force is always ~20% higher than the measurement (which I believe is the integral of shear stress and pressure force over the surface and taking the axial direction.).
My near wall grid $y^+$ is less than 1. I've tried different numerical parameters and got nearly the same overestimate of the drag force. Giving the same mesh to ANSYS Fluent, I can obtain pretty
good drag force. I also attached the subroutine to collect the drag force in Code_Saturne.
In addition, I've tested my boundary force subroutine with a turbulent channel flow adding a body force in the flow direction, and got good prediction of the velocity profile. So I would expect the
subroutine at least work for a surface parallel to one direction of the axes.
Could you give me some advice please?
Thank you in advance,
Re: Drag force calculation
Which version do you use ?
Best Regards,
Re: Drag force calculation
Hi Martin,
I use the version 8.0
Best regards,
Re: Drag force calculation
In addition to my previous post, I extracted the shear stress along the body.
ANSYS Fluent gives 1% error compared with the total measured drag, so I suppose the shear stress predicted by Fluent is accurate. There is an overall shift of the Code_Sature results in the plot,
which roughly accounts for the 18% drag prediciton error. I guess there may be something wrong with the velocity gradient calculation.
Concerning the gradient calculation, I tried the default option, iterative option, and the least-square with extended neighbour option. None of them improves the total drag.
Best regards,
Re: Drag force calculation
Than you for the precisions.
Do you plot the norm of the shear stress or the x component (x being in the fluid direction)?
Pressure being defined up to a constant for an incompressible flow, we can shift it from a constant (but the integral of a constant over a closed solid is 0).
Another question: which turbulence model do you use?
Best regards
Re: Drag force calculation
Hi Martin,
Thank you for your help.
I plot the X-axis component of "Shear Stress" from the postprocessing output of Code_Satrune on the Boundary surface, which is the solver's standard output file. The result plotted is
circumferentially averaged. Yes, the x direction is the flow direction.
In answer to your second question, I use the k-omega SST turbulence model without wall function.
I have a little update since yesterday night. I gave a blending factor 0.8 to the spatial scheme for momentum equaiton (say velocity in the GUI) with "Automatic" scheme (I guess it is "Centred"). The
Drag force collected with the subroutine in my initial post gives an error with the measurement for about 10%, a bit better. This also mitigates the shear stress oscillation from my previous
simulation (which is almost the same issue as seen in viewtopic.php?p=11167).
I read another recent post (viewtopic.php?t=3187) saying the drag force prediction is around 4% with the version 8.1. So I am compiling this version now, and will give an update later.
Best regards,
Re: Drag force calculation
I haven't run version 8.1 yet as there is some issue with compilation on the cluster. But I did some parametric study with v8.0.
I ticked off the slope test option and gave a blending factor 0.95 both for the velocity scheme, both helped approach the measured drag force.
I run steady simulation with the temporal scheme selectable within the GUI (IDTVAR=2), rather than IDTVAR=-1 modifiable with vim.
Now the shear stress in the axial direction agrees well with ANSYS Fluent, and the total drag error is 5.6% compared to the measurement. There could be a further reduction as my inlet velocity has
1.3% different to the measured condition. Assuming the could contribute linearly to the drag within such small range, the total drag error is around 4%. This makes me comfortable.
Thank you! | {"url":"https://www.code-saturne.org/forum/viewtopic.php?p=17546&sid=7876752e5f5f810e82c9a4760e9d4066","timestamp":"2024-11-09T14:14:21Z","content_type":"text/html","content_length":"40597","record_id":"<urn:uuid:9bbd97e8-8e53-4334-9dad-eb901e48de5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00593.warc.gz"} |
5 great things about being a maths teacher
Posted by: Gary Ernest Davis on: May 1, 2012
This is a guest post written by Kimberley McCosh (@spyanki_apso on Twitter)
5 great things about being a maths teacher
Kimberley McCosh
I love maths. Â I have had a few jobs before becoming a maths teacher but the urge to teach was always there. Â I am a self confessed maths geek and I love nothing more than converting some of my
students to math lovers too! Â I teach 12 to 18 year olds in a secondary school in Scotland.
1. Â The interaction with pupils and knowing when you’ve really got through to them with maths. Â One particular highlight was when my class cut out triangles then stuck the angles from this down in
a line to prove that the angles in a triangle sum to 180 degrees. Â The next day one of the boys (aprox 13 years old) was so eager to tell me that after the class he went home and searched the
internet and found that all the angles did in fact always add up to 180 degrees. Â I know I had got through to him since he was choosing to look up maths in his own time.
2. Â Getting pupils interested in maths. Â I always try not to give “just a maths lesson” but also giving some background too. Â Ask any of my S3 class and they will be able to tell you more
interesting facts about Pythagoras and his life than they can about the latest boy band! Â I always try to make my lessons interesting, different but still always relevant. Â When the pupils are
interested, they are engaged and I have achieved my goal of sparking their interest in maths.
3. Helping pupils to think for themselves. Â Whether it be problem solving or applications of maths, whenever the pupils make the links for themselves it is always a real fantastic moment for me as a
teacher. Â They have learned the building blocks and are piecing them together and starting to see the big picture.
4. Â The feeling of achievement when the penny drops and the class “get it”. Â It’s all in that moment when the pupils say “Ahhh! Â So that means…”. Â Or even better, when the pupil who has been
struggling but working hard turns round and says “This is really easy!”. Â To know you have taught something which the pupils can now use in future years is what it’s all about.
5. Â Although not specific to maths, it is fabulous to make a difference in someone’s life. Â As a teacher you have daily interaction with pupils who may not always have the perfect home life but
when they come into your class they are praised, encouraged, challenged and motivated to be the best they can be. Â To see a whole class strive to be the very best they can is the biggest reward you
can ever receive.
I could go on – I just love my job! Â As a maths teacher you really make a difference. Â From teaching basic numeracy skills to complicated calculus, each lesson is important. Â I always try to
remember that we are preparing pupils for jobs that haven’t been invented yet so who knows what level of maths they will require in later life. Â As a teacher, you can get an amazing high from
something as simple as a pupil finally mastering percentages or cracking vector calculus. Â Each pupil, each class, and each lesson has highlights and I wouldn’t change my career for anything! | {"url":"http://www.blog.republicofmath.com/4882/","timestamp":"2024-11-03T13:19:41Z","content_type":"application/xhtml+xml","content_length":"57499","record_id":"<urn:uuid:a03c0c23-c72c-4e31-a312-244db13cb013>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00896.warc.gz"} |
• Author(s): Xuan Zuo, Zhi-Yuan Fan, Huai-Bing Zhu, and Jie LiExciton optomechanics, bridging cavity exciton polaritons and optomechanics, opens new opportunities for the study of light-matter
strong interactions and nonlinearities, due to the rich nonlinear couplings among excitons, phonons, and photon
• Author(s): Yan Liu, Zhentao Zhang, and Zhenshan YangWe study the generation of three-particle (one photon plus two phonons) states and phonon pairs in a cavity-waveguide system where the
optomechanical coupling is quadratic in the mechanical displacement.
• Author(s): Nils A. Krause and Ashton S. BradleyHomogeneous planar superfluids exhibit a range of low-energy excitations that also appear in highly excited states like superfluid turbulence.
• Author(s): E. Poli, D. Baillie, F. Ferlaino, and P. B. BlakieWe present a theoretical study of the excitations of the two-dimensional supersolid state of a Bose-Einstein condensate with either
dipole-dipole interactions or soft-core interactions.
• Author(s): Z. W. Wu and Y. Z. WangDielectronic recombination (DR) of highly charged ions with spin-polarized electrons is studied within the framework of density-matrix theory.
• Author(s): James L. Booth and Kirk W. MadisonAtoms constitute promising quantum sensors for a variety of scenarios including vacuum metrology.
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∼600 V/cm.
• Author(s): Jian-Dong Zhang, Mei-Ming Zhang, Chuang Li, and Shuai WangDetecting the presence of multiple incoherent sources is a fundamental and challenging task for quantum imaging, especially
within sub-Rayleigh region.
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of a pair of spins-1/2 with Ising coupling starting from the spin-down state, using bounded transverse field
• Author(s): Himanshu SahuWe study information scrambling (a spread of initially localized quantum information into the system's many degree of freedom) in discrete-time quantum walks.
• Author(s): Qi Hong, Wen-Xin Wu, Yuan-Peng Peng, Jie Qian, Can-Ming Hu, and Yi-Pu WangSynchronization and asynchronization are ubiquitous occurrences in a wide range of natural and artificial
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• 【主なニュース】▼アメリカ大統領選挙日本時間今夜から投票最後の訴えへ ▼政治資金萩生田氏の当時の秘書に不起訴不当の議決検察審査会 ▼漫画家の楳図かずおさん死去 88歳 など
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• Author(s): Timothy S. Chen, Congcheng Wang, Salem C. Wright, Kelsey Anne Cavallaro, Won Joon Jeong, Sazol Das, Diptarka Majumdar, Rajesh Gopalaswamy, and Matthew T.
• arXiv:2411.02369v1 Announce Type: crossAbstract: Assuming the polynomial hierarchy is infinite, we prove a sufficient condition for determining if uniform and polynomial size quantum circuits
over a non-universal gate set are not efficiently classically simulable in the weak multiplicative sense.
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• arXiv:2411.02087v1 Announce Type: crossAbstract: Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL
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• arXiv:2411.01992v1 Announce Type: crossAbstract: Since the success of GPT, large language models (LLMs) have been revolutionizing machine learning and have initiated the so-called LLM prompting | {"url":"https://rss.cafe/?utm_source=appinn.com","timestamp":"2024-11-05T06:06:44Z","content_type":"text/html","content_length":"9481","record_id":"<urn:uuid:4e032891-c76b-43e0-bee2-5aa388e4646e>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00747.warc.gz"} |
[Solved] Standard penetration tests were carried o | SolutionInn
Standard penetration tests were carried out in sands where the N 60 values at certain depths are
Standard penetration tests were carried out in sands where the N[60 ]values at certain depths are reported as follows. The unit weight of the sand is 18.5 kN/m^3. The water table is well below the
test depths.
Use Eq. (3.13) for C[N]. Determine the friction angle ϕ' using Eqs. (3.29), (3.30), and (3.31b).
Eq. (3.13)
Eq. (3.29)
Eq. (3.30)
Eq. (3.31b)
Transcribed Image Text:
Depth (m) 2.0 3.5 5.0 6.5 8.0 N60 17 23 26 28 29
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Step by Step Answer:
Answer rating: 71% (7 reviews)
Unit weight of sand 185kNm 3 P a Atmospheric pressure 100k...View the full answer
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Natalia Carbona Tobón, Georg-August Universität Göttingen, Germany
The contact process on dynamical random trees with degree dependence
In this talk we investigate the contact process in the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a connected locally
finite base graph we initially declare edges independently open with a probability that is allowed to depend on the degree of the adjacent vertices and closed otherwise. Edges are independently
updated with a rate depending on the degrees and then are again declared open and closed with the same probabilities. We are interested in the contact process, where infections are only allowed to
spread via open edges. Our aim is to analyse the impact of the update speed and the probability for edges to be open on the existence of a phase transition. For a general connected locally finite
graph, our first result gives sufficient conditions for the critical value for survival to be strictly positive. Furthermore, in the setting of Bienaymé-Galton-Watson trees, we show that the process
survives strongly with positive probability for any infection rate if the offspring distribution has a stretched exponential tail with an exponent depending on the percolation probability and the
update speed. In particular, if the offspring distribution follows a power law and the connection probability is given by a product kernel and the update speed exhibits polynomial behaviour, we
provide a complete characterisation of the phase transition.
This talk is based on join work with Marcel Ortgiese (University of Bath), Marco Seiler (University of Frankfurt) and Anja Sturm (University of Göttingen).
Henk Don, Radboud University Nijmegen, The Nederlands
The contact process on finite graphs
The contact process on a finite graph dies out with probability 1. Nevertheless, we will discuss how one can still identify a phase transition between quick extinction and long survival. In the long
survival phase, the process exhibits metastable behavior. In this talk we will review some results on the phase transition, the extinction time and the metastable distribution of the contact process
on finite graphs. In particular we will discuss the complete graph and the Erdös-Rényi graph.
Emmanuel Jacob, École Normale Supérieure de Lyon, France
Targeted immunisation thresholds for the contact process on power-law trees and scale-free networks.
Abstract: We consider the contact process on a Galton-Watson tree with power-law offspring distribution, which arises naturally as a local limit of some standard scale-free network models. These
trees have enough high-degree vertices to allow propagation and survival of the contact process even with an arbitrarily small infection rate. We then investigate the effect of immunisation of all
vertices with degree above a threshold, which is allowed to depend on the infection rate. Depending on the value of this threshold, we prove that the survival probability of the contact process after
immunisation is essentially unchanged, or severely reduced, or equal to zero. This is joint work with John Fernley.
Júlia Komjáthy, Delft University of Technology, The Nederlands
Degree dependent contact process on Galton Watson trees and the configuration model
In this talk we look at degree dependent contact processes. This is a variant of the usual contact process, where transmissions through edges depend on the degrees of the transmitter and the receiver
vertex. The dependence is such that in a unit time, any transmitter vertex infects in expectation a sublinear but increasing number of its neighbors. We unfold the phase transitions of this new
process with respect to the small lambda behavior on Galton Watson trees with degree distributions of unbounded support. More precisely, we show that there is a phase transition at the square root
function: when in expectation, a vertex infects more than square root of its neighbors, then the process behaves qualitatively similar to the classical process; while when it infects less on average,
then new phases occur in the dynamics.
The asymptotic shape theorem for some versions of the contact process
The first aim of the talk is the introduce our guest star: the contact process. Then I will focus on the case of the supercritical contact process on Zd,where, conditionaly on survival, the growth of
the contact process starting from a finite configuration is governed by a shape theorem. I will try to explain the main ingredients of the proof, possible extensions to some random environments and
still open questions. In the last part of the talk, Il will try to present current questions about the contact process and its variations, beyond the Zd case.
Bruno Schapira, Institut de Mathématiques de Marseille (I2M), Aix-Marseille Université, France
Contact process on a dynamic random regular graph
We consider the contact process on a dynamic random regular graph. We show that there exists a critical value for the infection parameter, below which the contact process dies out in a time which is
logarithmic in the size of the graph. This completes an earlier result of da Silva, Oliveira and Valesin, showing that above this critical value, the process survives a time exponential in the size
of the graph.
Joint work with Daniel Valesin.
Marco Seiler, Frankfurt Institute for Advanced Studies (FIAS) and Goethe University Frankfurt, Germany
Asymptotic behaviour of the contact process in an evolving random environment
We study a contact process in an evolving (edge) random environment on (infinite) connected and transitive graphs. We assume that the evolving random environment is described by an autonomous ergodic
spin systems with finite range, for example by dynamical percolation. This background process determines which edges are open or closed for infections.
In particular, we discuss the phase transition of survival and the dependence of the associated critical infection rate on the random environment and on the initial configuration of the system. For
the latter, we state sufficient conditions such that the initial configuration of the system has no influence on the phase transition between extinction and survival. We show that this phase
transition coincides with the phase transition between ergodicity and non-ergodicity and we discuss a complete convergences result of the process.
At the end of the talk we present some partial results regarding the expansion speed and asymptotic shape of the infection process conditioned on survival on a d-dimensional integer lattice.
This talk is based on joint work with Anja Sturm and on going work with Noemi Kurt and Michel Reitmeier.
Daniel Valesin, University of Warwick, United Kingdom
The interchange-and-contact process
We introduce a process called the interchange-and-contact process, which is defined on an arbitrary graph as follows. At any point in time, vertices are in one of three states: empty, occupied by a
healthy individual, or occupied by an infected individual. Infected individuals recover with rate 1 and infect healthy individuals in neighboring vertices with rate lambda. Additionally, each edge
has a clock with rate v, and when this clock rings, the states of the two vertices of the edge are exchanged. This means that particles perform an interchange process with rate v, moving around and,
when infected, carrying the infection with them. We study this process on Z^d, with an initial configuration where there is an infected particle at the origin, and every other vertex contains a
healthy particle with probability p and is empty with probability 1-p. We define lambda_c(v, p) as the infimum of the values of lambda for which the process survives with positive probability. We
prove results about the asymptotic behavior of \lambda_c when p is fixed and v is taken to zero and to infinity.
Joint work with Daniel Ungaretti, Marcelo Hilário and Maria Eulalia Vares.
Sonia Velasco, Université Paris-Cité, France
Extinction and survival in inherited sterility
We introduce an interacting particle system which models the inherited sterility method. Individuals evolve on Z^d according to a contact process with parameter lambda > 0. With probability p in
[0,1] an offspring is fertile and can give birth to other individuals at rate lambda. With probability 1-p, an offspring is sterile and blocks the site it sits on until it dies. The goal is to prove
that at fixed lambda, the system survives for large enough p and dies out for small enough p. The model is not attractive, since an increase of fertile individuals potentially causes that of sterile
ones. However, thanks to a comparison argument with attractive models, we are able to answer our question. | {"url":"https://contact-creteil.sciencesconf.org/resource/page/id/5","timestamp":"2024-11-15T02:21:10Z","content_type":"application/xhtml+xml","content_length":"21279","record_id":"<urn:uuid:d7ef22ff-236f-4ebf-82ae-288a65330476>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00509.warc.gz"} |
Husqvarna crankcase splitter
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Looking for usable dimensions to make a homemade crankcase splitter. I looked at one elsewhere and the measurements did not jive... Thanks!
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I made mine from a cardboard cutout that was gave to me. Send me your address and I'll send one your way.
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I took picture of oem splitter, If you wait until tomorrow i can post a picture, picture scale is A4 format
The Peanut Gallery
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I found a PDF over on AS. I think the user name was deprime?? Print it out a little big like 6 3/8" outside. This will give you the proper inside dimension when you cut out the final pieces which is
the important part. Cut out the paper trace it on some thin steel with a sharpie. Carefully cut it out with a plasma torch. Then clean it up with a die grinder. Use this as a template to cut your
actual jaws. They will come out nicer this way. Using 6" flat stock will make it easier to keep everything square. I used 3/16" and it's plenty strong.
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The Peanut Gallery
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I made these. Powdered
Pm if interested.
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Now that is being resourceful! I dig it. I now have an idea for saws the splitter will not fit. .
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Remember heat is your friend separating cases
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How do you like my Chicken"s Eye?
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This is the drawing from one Husky OEM cranksplitter (A4 papir):
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Remember heat is your friend separating cases
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Yep, heat is good. Even muh dog is in heat.
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the distance between the jaws must be 28mm for the big saws splitters example 372, 395
the distance between the jaws must be 24mm for the small saws splitters example 242, 346 ...
for the other dimensions no matter so much if you are out a little
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I made these. Powdered
Pm if interested.
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Nice work here
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Now that is being resourceful! I dig it. I now have an idea for saws the splitter will not fit. .
We split a 024 and a ms 880 with this one
Thats a pretty big range
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We split a 024 and a ms 880 with this one
Thats a pretty big range
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Yes we did!
Dr. Richard Cranium
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Yes, I like the colors as well. How much is the powdered finish?
Safety First !!!!!!
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Where did that original post by motor head go?
I don't see any pics?
Sent from my iPhone using Tapatalk | {"url":"https://opeforum.com/threads/husqvarna-crankcase-splitter.448/","timestamp":"2024-11-12T00:17:23Z","content_type":"text/html","content_length":"203938","record_id":"<urn:uuid:3e1aab75-7d86-4a9d-b064-6fc85e426098>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00645.warc.gz"} |
A Foundation for Computer Science
This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of
mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for
computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.
Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of
techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation
is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more
than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.
Major topics include:
• Sums
• Recurrences
• Integer functions
• Elementary number theory
• Binomial coefficients
• Generating functions
• Discrete probability
• Asymptotic methods
This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been
expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal
graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and
using them. | {"url":"https://doray.me/articles/series/concrete-mathematics","timestamp":"2024-11-03T10:01:21Z","content_type":"text/html","content_length":"30868","record_id":"<urn:uuid:8a8a0d8c-1912-407e-a297-86809fb07d45>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00592.warc.gz"} |
Element selection and discretization in Abaqus Unified FEA using the example of a linear load on a bending beam - PLM Blog & Knowledge Center
Abaqus offers a wide range of elements for the different analyses. Assistance for the correct selection of elements is provided by the manual, (Dassault Systèmes, 2023) and (Reichert, 2018).
A small example shows that it always depends on the use case and the goals of a calculation which element type and which discretization should be used. The small example of a bending beam with point
bending force shown here is a linear (geometry and material) case of structural mechanics. The example is intended to illustrate how the results change depending on the selected element and
discretization. So far, only hexahedral elements have been considered.
Problem definition and analytical solution #
The bending beam, 200 mm long, 6 mm wide and 12 mm high (see picture below), is held at one end in the x- and y-directions and in the center plane in the z-direction. The shape of the restraint is
intended to reduce the occurrence of stress peaks at the restraint. The material used is steel. The load is applied at the top two corners of the other end in point form (F = 2 x 50 N). Analytically,
the displacement in y-direction (u2) and the bending stress (corresponding to S11) at the clamping point are determined. The console (Kernel Command Line Interface) of Abaqus/CAE can also be used for
the calculation of the analytical solution.
4 parts of the same bending beam are used to view the results of different element types at the same time,
The element types used were (from top to bottom) C3D8R, C3D8R with Enhanced Hourglass Control, C3D8 and C3D8I.
• C3D8R: 8 nodes linear hexahedral element with reduced integration (1 integration point / element )
• C3D8R: as above, with Enhanded Hourglass Control
• C3D8 : 8 nodes fully integrated hexahedral element (8 integration points / element )
• C3D8I : 8 nodes modified fully integrated hexahedral element ( 8 integration points / element )
(see also [3])
All calculations were performed using Abaqus/Standard v2022 HF6. A Static General Step was used, leaving NLGEOM = OFF.
If one compares displacements and/or stresses between analytical and numerical solution, the problem always arises at which points to compare. Displacements are calculated at the nodes and can be
read directly there. However, stresses (and strains) are calculated at the integration points. However, these are not located on the surfaces of the elements and so a direct comparison is not
possible here. Here we use the values extrapolated to the nodes on the surface, which always results in some error.
For the different element types, the element size of the model was varied in each case (via the specification in Mesh/Seed/Part).
With quite sparse discretization with an edge length of 12 mm, an element over the height of the beam cannot represent the stress profile due to the bending.
In addition, the reduced integrated element with Default Hourglass Control behaves very softly. Only the C3D8I element provides a usable result here.
With an edge length of 8 mm and thus 2 elements over the height, the result is similar:
Only the C3D8I element shows a useful result.
With an edge length of 4 mm and thus 3 elements above the height, the results with regard to displacements become significantly better.
However, the suspense picture continues to be poorly presented.
Only with an edge length of 2 mm and thus 6 elements is there a further improvement.
However, the reduced integrated elements still do not represent the stress pattern correctly.
This small example illustrates that in case of “wrong” element selection even a supposedly simple problem concerning stress evaluation is not correctly reproduced. Here, the C3D8I element represents
the means of choice if one does not want to switch to square element approaches.
It should not go unmentioned that in Abaqus/CAE the C3D8R element is used as default. Certainly this element has its advantages for non-linear calculations and large deformations, but not for the
linear application and small deformations shown here.
Basically, we want to continue to address this issue.
A short test with C3D20R shows that good quality results can be achieved even with much coarser discretization.
Unfortunately, the element for all finite element calculations does not exist. Thus, depending on the problem to be calculated and the goals of a calculation, the user should determine the “right”
element and the correct discretization himself. The user is supported by the manual of the software and by individual contributions of different authors. With this paper we want to show how strongly
the results of a simple linear calculation of a bending beam can depend on the choice of the element type and the discretization. In this simple case, C3D8I or C3D20R are the means of choice. More
articles on this topic are to follow.
In part, this text and illustrations shown are based on research of evaluated literature. If this is the case, the sources are marked in the text by a number, e.g. [2], and the source is listed here.
In part, however, the sources listed here should simply be understood as recommendations for further reading.
[2] Reichert, A., 2018, “Element Selection in a Nutshell,” Dassault Systèmes User Conferences 2018 | {"url":"https://plm.systemworkx.de/en/docs/element-selection-and-discretization-using-the-example-of-a-linear-load-on-a-bending-beam/","timestamp":"2024-11-13T19:13:54Z","content_type":"text/html","content_length":"392740","record_id":"<urn:uuid:69da271f-09ec-4b8c-8239-80388994fbde>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00735.warc.gz"} |
Science:Math Exam Resources/Courses/MATH307/December 2008/Question 04
MATH307 December 2008
• Q1 • Q2 • Q3 • Q4 • Q5 • Q6 (a) • Q6 (b) • Q7 • Q8 • Q9 • Q10 •
Question 04
Find the QR decomposition of the matrix
${\displaystyle {\begin{bmatrix}0&1\\1&0\\0&-1\\-1&0\\0&1\\\end{bmatrix}}}$
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still
stuck, go for the next hint.
Hint 1
What do you notice when you take the dot product of the two columns of A? How does this simplify the computation?
Hint 2
Since the two columns of A are already orthogonal, we can simply use the normalized columns of A in Q.
Hint 3
As a last step, since A = QR and Q has orthonormal columns, we can find R by computing R = Q^TA.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
To begin with, we note that the columns of A are already orthogonal,
${\displaystyle {\begin{bmatrix}0\\1\\0\\-1\\0\end{bmatrix}}\cdot {\begin{bmatrix}1\\0\\-1\\0\\1\end{bmatrix}}=0.}$
So we can simply put the normalized columns of A into the matrix Q:
${\displaystyle Q={\begin{bmatrix}0&{\frac {1}{\sqrt {3}}}\\{\frac {1}{\sqrt {2}}}&0\\0&-{\frac {1}{\sqrt {3}}}\\-{\frac {1}{\sqrt {2}}}&0\\0&{\frac {1}{\sqrt {3}}}\end{bmatrix}}.}$
As a last step, we calculate R = Q^TA to find
${\displaystyle R={\begin{bmatrix}{\sqrt {2}}&0\\0&{\sqrt {3}}\end{bmatrix}}.}$
(Note that the question asks for the QR decomposition, not a QR decomposition. This is the unique QR decomposition of A such that Q has orthonormal columns and R is square upper triangular with
positive diagonal entries.)
Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Matrix decomposition, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag | {"url":"https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH307/December_2008/Question_04","timestamp":"2024-11-15T04:32:22Z","content_type":"text/html","content_length":"45344","record_id":"<urn:uuid:9a2c8b3e-c825-4b57-88fa-f105e4fc3ab7>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00070.warc.gz"} |
Pro Deep Learning With Tensorflow
Author : Santanu Pattanayak
Publisher : Apress
Total Pages : 412
Release : 2017-12-06
ISBN-10 : 9781484230961
ISBN-13 : 1484230965
Rating : 4/5 (61 Downloads)
Deploy deep learning solutions in production with ease using TensorFlow. You'll also develop the mathematical understanding and intuition required to invent new deep learning architectures and
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emphasized in this book. You will be able to use the prototypes demonstrated to build new deep learning applications. The code presented in the book is available in the form of iPython notebooks and
scripts which allow you to try out examples and extend them in interesting ways. You will be equipped with the mathematical foundation and scientific knowledge to pursue research in this field and
give back to the community. What You'll Learn Understand full stack deep learning using TensorFlow and gain a solid mathematical foundation for deep learning Deploy complex deep learning solutions in
production using TensorFlow Carry out research on deep learning and perform experiments using TensorFlow Who This Book Is For Data scientists and machine learning professionals, software developers,
graduate students, and open source enthusiasts | {"url":"https://inkedinchapters.net/version/pro-deep-learning-with-tensorflow/","timestamp":"2024-11-04T14:28:25Z","content_type":"text/html","content_length":"84362","record_id":"<urn:uuid:4be253da-b300-456c-85d6-62bd4939901b>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00026.warc.gz"} |
Re: [dev] Adding mathml in a document
Quick follow-up :
While this method produces documents whose MathML perfectly works in
lowriter, there is a minor difference between the way the xml is written
this way, and by the math component (i.e. by using libreoffice to
produce a document)
- in a libreoffice produced document (1), mathml element is using a
default namespace put on the parent math element
- in a document produced with the aforementioned method (2), each mathml
element uses a qualified name.
ie, in case (1) :
<math xmlns="http://www.w3.org/1998/Math/MathML" >
in case (2) :
<math:math xmlns="http://www.w3.org/1998/Math/MathML">
I understand that the meaning is the same in all alternative writings
but I would prefer producing results like (1), using a default namespace
for all math descendants and I cannot find a way to do that in
odftoolkit, having tried various createElement / createElementNS
It might be more of a java dom question than a odftoolkit question, but
any idea or pointer would be appreciated!
On 22/01/2024 11:12, vivien guillet wrote:
On 19/01/2024 21:09, Michael Stahl wrote:
not only the math node must be in the math namespace but all of the
descendants like "semantics" too - i think that's the most likely
explanation: create every element with the namespace.
That was it ! Thanks a lot.
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Discover Max element of each column in matrix
Hey guysđź‘‹, In this post we will be discuss the Program to find maximum element of each column in a matrix i.e., to find the maximum value in each column of the given matrix. Since it is considered
as an important problem to solve while practicing, hence thought to share🤝 with you all.
Problem Description
In a family, the people are arranged in rows and columns. Male persons in the families are arranged in a row and females are arranged in a column. Find the eldest women in each column. (Write a
program to find the maximum element in each column of the matrix.)
You can find the same set of problem in different way on the various coding platform.
Input Format:
The input consists of (m*n+2) integers.
The first integer corresponds to m, the number of rows in the matrix and the second integer corresponds to n, the number of columns in the matrix.
The remaining integers correspond to the elements in the matrix.
The elements are read in row-wise order, the first row first, then second row and so on.
Assume that the maximum value of m and n is 10.
Output Format:
Refer to the sample output for details.
Sample Input:
Sample Output:
Explaination :
In this we will be discussing the Program to find maximum element of each column in a matrix i.e., to find the maximum value in each column of the given matrix. This can be achieved by simple loop
and conditional statement. Initialize the max variable to first element of each column. If there is only one element present in each column of the matrix then the loop did not execute and max hold
the only present value in the matrix, thus that element becomes the maximum of each column. If matrix has more than one element, than loop executes and if any element found bigger than the previously
assigned value, then that element becomes the largest.
Logic to follow to come-up with the solution :
1. Declare the required sets of variables to use in the code.
2. Initialize the max variable to first element of each column.
3. If there is only one element present in each column of the matrix then the loop did not execute and max hold the only present value in the matrix, thus that element becomes the maximum of each
4. If matrix has more than one element, than loop executes and if any element found bigger than the previously assigned value, then that element becomes the largest.
5. At last maximum value of each column is displayed as the result output.
Coding Time 👨‍💻
#include <bits/stdc++.h>
using namespace std;
void largestInColumn(int mat[10][10], int rows, int cols)
for (int i = 0; i < cols; i++)
int maxm = mat[0][i];
for (int j = 1; j < rows; j++)
if (mat[j][i] > maxm)
maxm = mat[j][i];
cout << maxm << endl;
int main()
int n,m;
int mat[10][10];
for(int i=0;i<n;i++)
for(int j=0;j<m;j++)
largestInColumn(mat, n, m);
return 0;
Hence with the above set of logic and code you can easily understand and solve the problem to find maximum number in each column of a matrix.
Hope with this you learned and acquired some basic knowledge of C++ Programming.
Drop a Love❤ if you liked👍 this post, then share 🤝this with your friends and if anything is confusing or incorrect then let me know in the comment section.
Thanks from my side, this is Mayank, keep learning and exploring !!
If you liked the article then please consider Buying me a Coffee
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class Point(*args)¶
Real vector.
The number of components.
valuefloat, optional
The components value. Default creates a null vector.
Create a Point
>>> import openturns as ot
>>> x = ot.Point(3, 1.0)
>>> x
class=Point name=Unnamed dimension=3 values=[1,1,1]
Get or set terms
>>> print(x[0])
>>> x[0] = 0.0
>>> print(x[0])
>>> print(x[:2])
Create a Point from a flat (1d) array, list or tuple
>>> import numpy as np
>>> y = ot.Point((0.0, 1.0, 2.0))
>>> y = ot.Point(range(3))
>>> y = ot.Point(np.arange(3))
and back
Addition, subtraction (with compatible dimensions)
>>> print(x + y)
>>> print(x - y)
Multiplication, division with a scalar
>>> print(x * 3.0)
>>> print(x / 3.0)
add(*args) Append a component (in-place).
at(*args) Access to an element of the collection.
clear() Reset the collection to zero dimension.
dot(rhs) Compute the scalar product.
find(val) Find the index of a given value.
getClassName() Accessor to the object's name.
getDimension() Accessor to the vector's dimension.
getName() Accessor to the object's name.
getSize() Accessor to the vector's dimension (or size).
hasName() Test if the object is named.
isDecreasing() Check if the components are in decreasing order.
isEmpty() Tell if the collection is empty.
isIncreasing() Check if the components are in increasing order.
isMonotonic() Check if the components are in nonincreasing or nondecreasing order.
isNonDecreasing() Check if the components are in nondecreasing order.
isNonIncreasing() Check if the components are in nonincreasing order.
norm() Compute the Euclidean (
norm1() Compute the
normInf() Compute the
normSquare() Compute the squared Euclidean norm.
normalize() Compute the normalized vector with respect to its Euclidean norm.
normalizeSquare() Compute the normalized vector with respect to its squared Euclidean norm.
resize(newSize) Change the size of the collection.
select(marginalIndices) Selection from indices.
setName(name) Accessor to the object's name.
Append a component (in-place).
valuetype depends on the type of the collection.
The component to append.
>>> import openturns as ot
>>> x = ot.Point(2)
>>> x.add(1.)
>>> print(x)
Access to an element of the collection.
indexpositive int
Position of the element to access.
elementtype depends on the type of the collection
Element of the collection at the position index.
Reset the collection to zero dimension.
>>> import openturns as ot
>>> x = ot.Point(2)
>>> x.clear()
>>> x
class=Point name=Unnamed dimension=0 values=[]
Compute the scalar product.
pointsequence of float
Scalar product second argument
Scalar product
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> prod = x.dot([4, 5, 6])
Find the index of a given value.
valcollection value type
The value to find
The index of the first occurrence of the value, or the size of the container if not found. When several values match, only the first index is returned.
Accessor to the object’s name.
The object class name (object.__class__.__name__).
Accessor to the vector’s dimension.
The number of components in the vector.
Accessor to the object’s name.
The name of the object.
Accessor to the vector’s dimension (or size).
The number of components in the vector.
Test if the object is named.
True if the name is not empty.
Check if the components are in decreasing order.
>>> import openturns as ot
>>> x = ot.Point([3.0, 2.0, 1.0])
>>> x.isDecreasing()
>>> x = ot.Point([3.0, 3.0, 1.0])
>>> x.isDecreasing()
>>> x = ot.Point([1.0, 3.0, 2.0])
>>> x.isIncreasing()
Tell if the collection is empty.
True if there is no element in the collection.
>>> import openturns as ot
>>> x = ot.Point(2)
>>> x.isEmpty()
>>> x.clear()
>>> x.isEmpty()
Check if the components are in increasing order.
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> x.isIncreasing()
>>> x = ot.Point([1.0, 1.0, 3.0])
>>> x.isIncreasing()
>>> x = ot.Point([1.0, 3.0, 2.0])
>>> x.isIncreasing()
Check if the components are in nonincreasing or nondecreasing order.
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> x.isMonotonic()
>>> x = ot.Point([2.0, 2.0, 1.0])
>>> x.isMonotonic()
>>> x = ot.Point([1.0, 3.0, 2.0])
>>> x.isMonotonic()
Check if the components are in nondecreasing order.
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> x.isNonDecreasing()
>>> x = ot.Point([1.0, 1.0, 3.0])
>>> x.isNonDecreasing()
>>> x = ot.Point([1.0, 3.0, 2.0])
>>> x.isNonDecreasing()
Check if the components are in nonincreasing order.
>>> import openturns as ot
>>> x = ot.Point([3.0, 2.0, 1.0])
>>> x.isNonIncreasing()
>>> x = ot.Point([3.0, 3.0, 1.0])
>>> x.isNonIncreasing()
>>> x = ot.Point([1.0, 3.0, 2.0])
>>> x.isNonIncreasing()
Compute the Euclidean (
The Euclidean (
The vector’s Euclidean norm.
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> x.norm()
Compute the
The vector’s
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> x.norm1()
Compute the
The vector’s
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> x.normInf()
Compute the squared Euclidean norm.
The vector’s squared Euclidean norm.
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> x.normSquare()
Compute the normalized vector with respect to its Euclidean norm.
The normalized vector with respect to its Euclidean norm.
RuntimeErrorIf the Euclidean norm is zero.
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> print(x.normalize())
Compute the normalized vector with respect to its squared Euclidean norm.
The normalized vector with respect to its squared Euclidean norm.
RuntimeErrorIf the squared Euclidean norm is zero.
>>> import openturns as ot
>>> x = ot.Point([1.0, 2.0, 3.0])
>>> print(x.normalizeSquare())
Change the size of the collection.
newSizepositive int
New size of the collection.
If the new size is smaller than the older one, the last elements are thrown away, else the new elements are set to the default value of the element type.
>>> import openturns as ot
>>> x = ot.Point(2, 4)
>>> print(x)
>>> x.resize(1)
>>> print(x)
>>> x.resize(4)
>>> print(x)
Selection from indices.
indicessequence of int
Indices to select
Sub-collection of values at the selection indices.
Accessor to the object’s name.
The name of the object.
Examples using the class¶ | {"url":"https://openturns.github.io/openturns/latest/user_manual/_generated/openturns.Point.html","timestamp":"2024-11-11T14:23:15Z","content_type":"text/html","content_length":"207562","record_id":"<urn:uuid:82e7fabd-11a2-4ec4-a993-af99b355b39a>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00700.warc.gz"} |
Back in the summer we had this SO question:
by TemplateRex
From the article:
How to implement classic sorting algorithms in modern C++?
The std::sort algorithm (and its cousins std::partial_sort and std::nth_element) from the C++ Standard Library is in most implementations a complicated and hybrid amalgation of more elementary
sorting algorithms, such as selection sort, instertion sort, quick sort, merge sort, or heap sort.
There are many questions here and on sister sites such as http://codereview.stackexchange.com/ related to bugs, complexity and other aspects of implementations of these classic sorting
algorithms. Most of the offered implementations consist of raw loops, use index manipulation and concrete types, and are generally non-trivial to analyze in terms of correctness and efficiency.
Question: how can the above mentioned classic sorting algorithms be implemented using modern C++?
□ no raw loops, but combining the Standard Library’s algorithmic building blocks from <algorithm>
□ iterator interface and use of templates instead of index manipulation and concrete types
□ C++14 style, including the full Standard Library, as well as syntactic noise reducers such as auto, template aliases, transparant comparators and polymorphic lambdas
Here are four Live Examples (C++14, C++11, C++98 and Boost, C++98) testing all five algorithms on a variety of inputs (not meant to be exhaustive or rigorous). Just note the huge differences in
the LOC: C++11/C++14 need around 120 LOC, C++98 and Boost 180 (+50%) and C++98 more than +100% (note that heap sort could not even be done in terms of standard algorithms). | {"url":"https://isocpp.org/blog/2014","timestamp":"2024-11-04T11:11:03Z","content_type":"text/html","content_length":"43047","record_id":"<urn:uuid:a7b68e93-864d-4307-ac1a-b7b69d14e429>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00719.warc.gz"} |
Geometry in Daily Life - Meaning, Applications, Uses, and FAQs - Infinity Learn by Sri Chaitanya
MathsGeometry in Daily Life – Meaning, Applications, Uses, and FAQs
Geometry in Daily Life – Meaning, Applications, Uses, and FAQs
Application of Geometry in Day to Day Life
There are numerous ways that geometry is used in day to day life. A few examples include using geometry to measure distances, calculate volumes, and create shapes. Geometry is also used in
architecture and engineering to create plans and designs.
What is Geometry?
Geometry is the mathematics of shape and space. Geometric shapes are points, lines, planes, and solids. Points are the simplest geometric shape. A line is made up of points, and it extends in two
directions forever. A plane is a two-dimensional surface that is made up of an infinite number of lines. A solid is a three-dimensional object that is made up of an infinite number of planes.
Benefits of Geometry in Daily Life for Students:
– Geometry is used in many aspects of our lives, including architecture, engineering, and land surveying.
– Geometry can help students develop spatial skills, which are important in fields such as engineering and architecture.
– Geometry can also help students better understand mathematical concepts. | {"url":"https://infinitylearn.com/surge/maths/geometry-in-daily-life/","timestamp":"2024-11-02T21:08:47Z","content_type":"text/html","content_length":"158092","record_id":"<urn:uuid:4d32ed48-16c0-40a3-a9ee-ad3e20ae39a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00269.warc.gz"} |
Boosting Batch Arguments and RAM Delegation
We show how to generically improve the succinctness of non-interactive publicly verifiable batch argument (BARG) systems. In particular, we show (under a mild additional assumption) how to convert a
BARG that generates proofs of length poly (m)· k1-", where m is the length of a single instance and k is the number of instances being batched, into one that generates proofs of length poly (m,
logk), which is the gold standard for succinctness of BARGs. By prior work, such BARGs imply the existence of SNARGs for deterministic time T computation with succinctness poly(logT). Our result
reduces the long-standing challenge of building publicly-verifiable delegation schemes to a much easier problem: building a batch argument system that beats the trivial construction. It also
immediately implies new constructions of BARGs and SNARGs with polylogarithmic succinctness based on either bilinear maps or a combination of the DDH and QR assumptions. Along the way, we prove an
equivalence between BARGs and a new notion of SNARGs for (deterministic) RAM computations that we call "flexible RAM SNARGs with partial input soundness."This is the first demonstration that SNARGs
for deterministic computation (of any kind) imply BARGs. Our RAM SNARG notion is of independent interest and has already been used in a recent work on constructing rate-1 BARGs (Devadas et.al. FOCS
Original language English (US)
Title of host publication STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
Editors Barna Saha, Rocco A. Servedio
Publisher Association for Computing Machinery
Pages 1545-1552
Number of pages 8
ISBN (Electronic) 9781450399135
State Published - Jun 2 2023
Externally published Yes
Event 55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States
Duration: Jun 20 2023 → Jun 23 2023
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print) 0737-8017
Conference 55th Annual ACM Symposium on Theory of Computing, STOC 2023
Country/Territory United States
City Orlando
Period 6/20/23 → 6/23/23
All Science Journal Classification (ASJC) codes
• Batch Arguments
• Delegation of Computation
• RAM Delegation
Dive into the research topics of 'Boosting Batch Arguments and RAM Delegation'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/boosting-batch-arguments-and-ram-delegation","timestamp":"2024-11-15T00:28:08Z","content_type":"text/html","content_length":"52555","record_id":"<urn:uuid:bc5bf85d-f724-4281-85d3-e9173aa7a38b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00558.warc.gz"} |
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Knowledge-based Systems, Volume 289, 2024. 4, SCI, IF 8.8
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IEEE Signal Processing Letters, Volume: 31, pp. 491-495, 2024. 1
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IEEE Signal Processing Letters, Volume: 30, pp. 65 - 69, 2023. 1, IF 3.201
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IEEE Transactions on Fuzzy Systems, Volume 31, Issue 3, pp. 707-721, 2023 3, SCI, IF 12.029
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Electronics Letters, Wiley, Volume 58, Issue 16, pp,606-608 2022. 8, SCIE
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Locally Linear Reconstruction and Conditional Fuzzy C-Means Clustering"
International Journal of Approximate Reasoning, Volume 106, March 2019, pp. 228-243, SCI, IF 3.816.
D. Kyeong, K. Seo, "Two CPG Based Gait Generation Methods to Improve an Adaptation Ability on Slop Terrains for Humanoid Robots"
JEET(Journal of Electrical Engineering and Technology), March 2019, Volume 14, Issue 2, pp 941–946, Springer, SCIE, IF 1.069
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ECCV-2018 Workshop, Springer-Verlag, LNCS volumes 11134, pp. 1–10, 2019, SCOPUS
J. Yoon, D. Kyeong, K. Seo, "A hybrid method based on F-transform for robust estimators"
International Journal of Approximate Reasoning, Elsevier, Volume 104, January 2019, pp. 75-83, SCI, IF 3.816.
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Soft Computing Springer, March 2019, Volume 23, Issue 6,pp 1857–1872, SCIE, IF 3.643.
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Soft Computing, Springer, July 2019, Volume 23, Issue 13,pp 4969–4985, SCIE, IF 3.643
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IEEE Trans. on Evolutionary Computation, Vol. 19, No. 4, pp.465-473, Aug. 2015, SCI, IF 11.554.
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IEICE(Institute of Electronics, Information and Communication Engineers) ELECTRONICS EXPRESS, Vol. 8, No. 11, pp. 873-878. 2011.
K. Seo, Y. Kim, "Automated generation of rotation-robust corner detectors"
IEICE(Institute of Electronics, Information and Communication Engineers) ELECTRONICS EXPRESS, Vol. 7, No. 17, pp. 1226-1232. 2010. 10
K. Seo, S. Hyun E. D. Goodman, "Genetic Programming-Based Automatic Gait Generation in Joint Space for a Quadruped Robot"
Advanced Robotics, Vol. 24, No. 15, pp. 2199-2214. 2010.
K. Seo, Y. Kim, "Scale- and Rotation-Robust Genetic Programming-Based Corner Detectors"
Lecture Notes in Computer Science, Springer-Verlag, Vol. 6024/2010, pp. 381-391, 2010
S. Yoo, S. Oh, K. Seo, "Design of Face Recognition Algorithm Using Hybrid Data Preprocessing and Polynomial-Based RBF Neural Networks"
Lecture Notes in Computer Science, Springer-Verlag, Vol. LNCS 7368, Part 2, pp. 213-220, 2012.
S. Hyun, K. Seo, "Analysis of two evolutionary gait generation techniques for different coordinate approaches"
IEICE(Institute of Electronics, Information and Communication Engineers) ELECTRONICS EXPRESS, Vol. 8, No. 11, pp. 873-878. 2011.
K. Seo, Y. Kim, "Automated generation of rotation-robust corner detectors"
IEICE(Institute of Electronics, Information and Communication Engineers) ELECTRONICS EXPRESS, to be published in 2010.
K. Seo, S. Hyun, E. D. Goodman, "Genetic Programming-Based Automatic Gait Generation in Joint Space for a Quadruped Robot"
Advanced Robotics, to be published in 2010.
K. Seo, Y. Kim, "Scale- and Rotation-Robust Genetic Programming-Based Corner Detectors"
Lecture Notes in Computer Science, Springer-Verlag, Vol. 6024/2010, pp. 381-391, 2010
K. Seo, S. Hyun, "A Comparative Study between Genetic Algorithm and Genetic Programming Based Gait Generation Methods for Quadruped Robots?"
Lecture Notes in Computer Science, Springer-Verlag, Vol. 6024/2010, pp. 352-360, 2010
Y. Cho, K. Seo, H. Lee, "A Direct Adaptive Fuzzy Control of Nonlinear Systems with Application"
International Journal of Control, Automation, and Systems, ICASE, Vol. 5, No. 6, pp. 630-642. 2007.
J. Choi, S. Oh, K. Seo, "Simultaneous Optimization of ANFIS-Based Fuzzy Model Driven to Data Granulation and Parallel Genetic Algorithms"
Lecture Notes in Computer Science, Springer-Verlag, Vol. 4493/2007,pp. 225-230, 2007.
J. Hu, E. Goodman, K. Seo, Z. Fan, R. Rosenberg. "The Hierarchical Fair Competition (HFC) Framework for Continuing Evolutionary Algorithms"
Evolutionary Computation, The MIT Press, Vol. 13 Issue 2, pp. 241-277, 2005.
K. Seo, J. Hu, Z. Fan, E. Goodman, and R. Rosenberg, "Hierarchical Breeding Control for Efficient Topology/ Parameter Evolution"
Lecture Notes in Computer Science, Springer-Verlag, Vol. 3103/2004, pp. 722-723, 2004.
Z. Fan, K. Seo, J. Hu, R. Rosenberg, and E. Goodman, "System-Level Synthesis of MEMS via Genetic Programming and Bond Graphs"
Lecture Notes in Computer Science, Springer, Vol. 2723/2003, pp. 2058-2071, 2003.
K. Seo, Z. Fan, J. Hu, E. Goodman, and R. Rosenberg, "Dense and Switched Modular Primitives for Bond Graph Model Design"
Lecture Notes in Computer Science, Springer, Vol. 2724/2003, pp. 1764-1775, 2003.
J. Hu, K. Seo, Z. Fan, R. Rosenberg, and E. Goodman, "HEMO: A Sustainable Multi-Objective Evolutionary Optimization Framework"
Lecture Notes in Computer Science, Springer, Vol. 2723/2003, pp. 1029-1040, 2003.
Fan, K. Seo, J. Hu, E. Goodman, R. Rosenberg, "A Novel Evolutionary Engineering Design Approach for Mixed-Domain Systems"
Engineering Optimization, Taylor & Francis, Vol. 36, no. 2, pp. 127-147, 2003.
K. Seo, J. Hu, Z. Fan, E. D. Goodman, and R. C. Rosenberg, "Toward an Automated Design Method for Multi-Domain Dynamic Systems
Using Bond Graphs and Genetic Programming"
Mechatronics, Elsevier, Volume 13, Issues 8-9, 2003, pp. 851-885
K. Seo, J. Hu, Z. Fan, E. D. Goodman, and R. C. Rosenberg, "Automated Design Approaches for Multi-Domain Dynamic Systems
Using Bond Graphs and Genetic Programming"
The International Journal of Computers, Systems and Signals, vol.3, no.1, pp.55-70, 2002. | {"url":"http://intlab.skuniv.ac.kr/publications/International%20Journal.html","timestamp":"2024-11-04T20:11:15Z","content_type":"application/xhtml+xml","content_length":"17203","record_id":"<urn:uuid:8dd89a19-c128-40ab-9a04-ecb8d3dbcb82>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00074.warc.gz"} |
GRAPH PAPER - PRINTABLE GRAPH PAPER - GRAPH SHEET - Social News Daily
GRAPH PAPER – PRINTABLE GRAPH PAPER – GRAPH SHEET
What is graph paper and how to solve this problem? You will find all the answers about printable graph paper and its
The graph in mathematics and computer science is the basic material discussed in graph theory. Generally, a graph is a set of objects called dots, nodes, or vertices, objects that are linked to each
other through lines or edges or “edges”. An exact graph is defined by definition, and it is assumed to have the same object as the dotted line B from point A and point B from point A. On the other
hand, in a one-dimensional graph (a diagram or a directed graph), these two lines are assumed to be separate orientations (arcs or pointed edges).
Graph chart লেখার কৌশল
Graphs of real-life problems can be solved with graphs. For example, by imagining each city as a node and imagining the road between them, one can go the shortest route from one city to another. Thus
many problems can be solved by “modeling” different nodes and nodes in the graph.
Mathematician Leonard Euler is called the father of graph theory. In 7, he published a paper called “Seven Bridges of Königsberg.”
Online graph paper
Graph Theory in Mathematics and Computer Science is a topic that discusses graph-related topics. The “graph” is the sum of several vertexes or vertices, and some edges or line sums that connect the
various vertexes. The graph can be directional or non-directional, which means that there is no point in the connecting line of the two vertexes. The edges of the directional or side graphs have
specific points. See Graph (Mathematics) for a detailed definition.
The article written by Leonard Euler on the Seven Bridges problem in Konigsburg was published in 1 and has been taken as the first publication in the history of graph theory. Edge gives Euler’s
formula on the number of vertex and convex polyhedrons, and Kosi and L’Huillier study it and give a general description of the sources. This is how the topology is born.
Printable graph paper
Graph of statistics. I have shown the content of the statistics is accurate and easy to understand. A graph (ICTP, etc.), a band graph, a square graph, a bar graph (histogram, etc.), a pie chart
(sector graph), a line graph, and the like are used. (2) The function graph is represented by the point y = f (x) of several values of x and y about the function coordinate of the coordinate system
O-XY and x and y continuous changes (x, y) when by P. A curve drawn is called the function y = f (x) graph. (3) Graph theory graph. As a drawing of a stroke, a graph consisting of different points
and a line of their connection is called a graph In the theory of graphs, considering only the way of connecting lines, the shape is not a problem.
My article about Meat goat breeds.
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Home | Basics In Maths
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Begin Your Math Adventure: Maths Made Simple: BasicsInMaths”
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From beginner tutorials to advanced problem-solving techniques, embark on a journey of discovery and empowerment. Join us as we demystify mathematics and unleash your full potential!
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“www.basicsinmaths.com” స్వాగతం, ప్రాథమిక గణిత శాస్త్ర భావనలపై పట్టు సాధించడానికి, మీరు మీ పునాది నైపుణ్యాలను బలోపేతం చేయాలని చూస్తున్న విద్యార్థి అయినా లేదా వినూత్న బోధనా సామగ్రిని కోరుకునే విద్యావేత్త అయినా, మా వెబ్సైట్ మీ గణిత ప్రయాణానికి మద్దతుగా సమగ్ర ట్యుటోరియల్లు, ఇంటరాక్టివ్ వ్యాయామాలు మరియు
సహాయక వనరులను అందిస్తుంది.
అంకగణితం నుండి బీజగణితం వరకు, జ్యామితి నుండి కాలిక్యులస్ వరకు, మేము ఆన్ని టాపిక్స్ ను కవర్ చేసాము. గణితంలో బేసిక్స్తో గణిత శక్తిని అన్లాక్ చేయండి!
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సమాజ అభివృద్ధికి విద్య ఎంతగానో దోహధ పడుతుంది.
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ఇందులో ‘CBSE’ , ‘ ICSE’ కి సంబంధించిన Maths మెటీరియల్ , తెలుగు వ్యాకరణం Quantitative Aptitude మరియు English Grammar ఇవ్వబడింది.
CBSE (Central Board of Secondary Education) Mathematics curriculum covers a wide range of topics designed to develop students’ mathematical understanding, problem-solving skills, and analytical
Here’s a general overview of the CBSE Mathematics curriculum:
1. Number Systems: Understanding real numbers, rational and irrational numbers, laws of exponents, and expressing numbers in standard form.
2. Algebra: Topics include polynomials, pairs of linear equations in two variables, quadratic equations, arithmetic progressions, and more advanced concepts like linear programming and matrices.
3. Geometry: Covers Euclid’s geometry, lines and angles, triangles, quadrilaterals, circles, and constructions. Concepts like similarity, congruence, theorems related to circles, and coordinate
geometry are also included.
4. Trigonometry: Introduction to trigonometric ratios, trigonometric identities, heights and distances, and their applications in solving real-life problems.
5. Statistics and Probability: Basics of statistics including measures of central tendency, graphical representation of data, probability distribution, and its applications in various contexts.
6. Calculus: Introduction to calculus with the basics of differentiation and integration, including their applications.
Throughout the curriculum, there’s an emphasis on problem-solving techniques, reasoning, and application of mathematical concepts in different contexts.
Additionally, students are encouraged to develop critical thinking skills by solving real-life problems using mathematical principles.
CBSE also provides textbooks, sample papers, and other resources to help students effectively understand and practice the concepts.
It’s important for students to regularly practice problems, understand the underlying concepts, and seek help whenever needed to excel in CBSE Mathematics.
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Thin Lens Formula
1. Thin lens formula
Thin Lens Formula
`1/d_0 + 1/d_1 = 1/f`
Thin lens formula or Gaussian Lens formula.
Enter 'x' in the field to be calculated.
This tool is a thin lens formula calculator. It computes the image position of an object through a thin lens. All distances are measured from the center of the lens and obey a sign convention (see
To compute the position of the object by taking the lens focal points as origins, use this version Newton's version of thin lens equation.
d[0] : distance from object to lens in cm (directional distance AO in the diagram)
d[1] : distance from lens to image in cm (directional distance OA' in the diagram)
f : focal length in cm (OF' in the diagram)
All these distances obey the sign convention below.
Sign convention :
- All distances are measured in relation to the lens surface.
- The direction of light from object to lens is considered as the 'positive direction' (always from left to right).
- The focal length is positive (f > 0) for a converging lens and negative (f < 0) for a diverging lens.
The thin lens formula is then expressed as follows,
`1/d_0 + 1/d_1 = 1/f`
Calculation example : case of a converging lens
The purpose is to calculate the position of the image of an object which is 20 cm away from a converging lens. We suppose that the lens focal length is 5 cm. So we have,
- Object distance: d[0] = 20 cm (directional distance from object to lens).
- Focal length: f = 5 cm (positive for a converging lens)
We get (enter "x" in d[1] field),
d[1] = 6.67 cm
The image is a real image (ie can be viewed on a screen) located 6.67 cm to the right of the lens.
Case of a diverging lens
Under the same conditions as above (converging lens case) but with a diverging lens, we have,
- Object distance : d[0] = 20 cm.
- Focal length : f = -5 cm (negative sign for a diverging lens)
We get (enter "x" in d[1] field),
d[1] = -4 cm
The image is a virtual image (can not be seen on a screen) located 4 cm to the left of the lens.
See also
Newton conjugation relationship
Lens optical power
Optics Calculators | {"url":"https://www.123calculus.com/en/thin-lens-formula-page-8-40-200.html","timestamp":"2024-11-02T02:31:26Z","content_type":"text/html","content_length":"21150","record_id":"<urn:uuid:64c14125-eb88-459a-8d2d-8abdb4db31a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00486.warc.gz"} |
Learning random points from geometric graphs or orderings
Suppose that there is a family of n random points X_v for v ∈ V, independently and uniformly distributed in the square [-√(n)/2,√(n)/2]^2. We do not see these points, but learn about them in one of
the following two ways. Suppose first that we are given the corresponding random geometric graph G, where distinct vertices u and v are adjacent when the Euclidean distance d_E(X_u,X_v) is at most r.
Assume that the threshold distance r satisfies n^3/14≪ r ≪ n^1/2. We shall see that the following holds with high probability. Given the graph G (without any geometric information), in polynomial
time we can approximately reconstruct the hidden embedding, in the sense that, `up to symmetries', for each vertex v we find a point within distance about r of X_v; that is, we find an embedding with
`displacement' at most about r. Now suppose that, instead of being given the graph G, we are given, for each vertex v, the ordering of the other vertices by increasing Euclidean distance from v.
Then, with high probability, in polynomial time we can find an embedding with the much smaller displacement error O(√( n)). | {"url":"https://cdnjs.deepai.org/publication/learning-random-points-from-geometric-graphs-or-orderings","timestamp":"2024-11-06T05:29:56Z","content_type":"text/html","content_length":"154272","record_id":"<urn:uuid:e1d553d5-e6bb-498a-9b10-3ca30798f668>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00645.warc.gz"} |
Design a module named LFSR based on the given circuit using both structural and behavioral Verilog code. The output name is LFSR out . A clock signal clk is
Fully parametrizable combinatorial parallel LFSR/CRC module - alexforencich/verilog-lfsr
Implements an unrolled LFSR next state Testing. Running the included testbenches Thanks. In web I found that I need to tap 64,63,62,61 bit for 64 bit LFSR. Also, I am new to verilog and this stuff
and have very little knowledge. I want to drive a benchmark code using this lfsr. So what I want basically to design an lfsr that gives 64 bit output. I have modified the code – Emily Blake Oct 18
'16 at 6:56 Fully parametrizable combinatorial parallel LFSR/CRC module - alexforencich/verilog-lfsr Link: http://simplefpga.blogspot.co.uk/2013/02/random-number-generator-in-verilog-fpga.html This
tool generates Verilog or VHDL code for an LFSR Counter Read these posts: part1, part2, part3 for more information about the tool Download stand-alone application for faster generation of large
counters In computing, a linear-feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state.
Module Declaration module top_module( input clk, input reset, // Active-high synchronous reset to 32'h1 output [31:0] q ); If the taps positions are carefully chosen, the LFSR can be made to be
"maximum-length". A maximum-length LFSR of n bits cycles through 2 n-1 states before repeating (the all-zero state is never reached). The following diagram shows a 5-bit maximal-length Galois LFSR
with taps at bit positions 5 and 3. LFSR is a shift register whose random state at the output depends on the feedback polynomial. So it can count maximum 2n-1 states and produce pseudo-random number
of the output. By using FPGA, comparing 8 and 16 bit LFSR on the basis of memory, gates.
So it can count maximum 2n-1 states and produce pseudo-random number of the output. By using FPGA, comparing 8 and 16 bit LFSR on the basis of memory, gates.
After that, the LFSR changes value at each clock cycle but never reaches 0, so it goes through only 255 8bit values (out of 256 possible combinations of a 8bit output) before starting again. Now,
instead of outputting the full eight bits, we could output just one bit (i.e. choose any of the flip-flops to make the output, keep the other seven internal).
LFSRs are simple to synthesize, meaning that they take relatively Jun 23, 2020 The circuit implementation of 3, 4 and 5-bit LFSR circuit is built by Verilog HDL code and synthesis is carried out
using 90 nm CMOS technology ( VHDL, Verilog, SystemVerilog, SystemC, Xilinx, Intel(Altera), Tcl, ARM, The LFSR can be used to both generate the test sequence for the design that is to Verilog code
module … always@(posedge clk or posedge rst) begin if(rst). LFSR_reg=8'b0; else. LFSR_reg=Next_LFSR_reg; end Keywords: LFSR, FPGA, HDL, Cryptography and Verilog.
This page contains SystemVerilog tutorial, SystemVerilog Syntax, SystemVerilog Quick Reference, DPI, SystemVerilog Assertions, Writing Testbenches in SystemVerilog, Lot of SystemVerilog Examples and
SystemVerilog in One Day Tutorial.
LFSR_reg=8'b0; else. LFSR_reg=Next_LFSR_reg; end Keywords: LFSR, FPGA, HDL, Cryptography and Verilog. 1. INTRODUCTION.
By using FPGA, comparing 8 and 16 bit LFSR on the basis of memory, gates. Index Terms-LFSR, FPGA, Verilog HDL, pseudo-random number, TRNG, PRNG. I. INTRODUCTION Download Verilog Program from : http:/
/electrocircuit4u.blogspot.in/linear feedback shift register is implemented using Xillinx verilog HDL language. Design a 64-bit linear feedback shift register, using behavioral Verilog.
Stankskydd slapvagn
原理. 线性反馈移位寄存器（LFSR）的英文 Jan 12, 2017 Webinar Breakdown: • Introduc=on to pseudorandom number generator (LFSR) code.
For example, a 6th-degree polynomial with every term present is 2020年6月28日原理; Verilog实现; 仿真测试; 代码提示.
Kooperativa hyresrätter
에 있습니다. 웹에서 나는 64 비트 LFSR을 위해 64,63,62,61 비트를 사용할 필요가 있음을 발견했다. 또한, 나는 Verilog과이 물건을 처음 사용하고 지식이 거의 없다.
▫️ lfsr.v - Parametrizable 2020年6月1日写在前面相关博文博客首页注：学习交流使用！正文原理线性反馈移位寄存器（ LFSR）的英文全称为：Linear Feedback Shift Register。赛灵思 24 Mar 2018 (i.e, Verilog,
VHDL, or SystemC) and a synthesis tool. LFSR based Accumulator Generator proposed in [61] is a PRNG based on Digital Verilog Shift Register. Introduction¶. lfsr module.
Mellanchef starter kit
Pattern generators like LFSR (Linear Feed Back Shift Registers) can produce random patterns with low hardware requirements and is preferred choice for testing. It is categorized under pseudo-random
test pattern generators which can produce a random pattern for every clock cycle applied to it.
The sample Verilog code (lfsr_tb.v.) is written for an eight-cell autonomous LFSR with a synchronous (edge-sensitive) cyclic LFSR in an FPGA - VHDL & Verilog Code. How a Linear Feedback Shift
Register works inside of an FPGA. LFSR stands for Linear Feedback Shift Register and The Verilog description of this counter is shown. is really easy with Verilog. LFSR. There is a whole area of
mathematics devoted to this type of computation, 2020年6月1日线性反馈移位寄存器（LFSR）的英文全称为：Linear Feedback Shift Register。赛灵思公司的高速串口IP核示例程序经常以LFSR为例，例如Aurora 20
Jul 2020 Index Terms- LFSR, FPGA, Verilog HDL, pseudo-random number, TRNG, PRNG. I. INTRODUCTION.
Build a 32-bit Galois LFSR with taps at bit positions 32, 22, 2, and 1. Module Declaration module top_module( input clk, input reset, // Active-high synchronous reset to 32'h1 output [31:0] q );
large number of initial values are possible), then the generated numbers can be considered as random numbers for practical purposes.
HDL generation was correct and has not changed. Thanks for Rick Collins for bringing that to my attention. Updated 05/22/2000 - version 1.00 fixes the HDL code generation - looks good now. | {"url":"https://hurmanblirrikezylg.netlify.app/32092/84030","timestamp":"2024-11-07T10:53:07Z","content_type":"text/html","content_length":"10962","record_id":"<urn:uuid:db050685-c290-4e78-9c14-43cef2758e98>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00018.warc.gz"} |
Spawning missing zombie in clear quest.
Tried to google around and didn't find any reference to it. So I figured I'd share it here in case this would help anyone.
So, as the title suggest, I was in a POI that didn't spawn all zombie. Happened because of a switch/button that failed to reset properly. I tried to destroy the door it was controlling, but didn't
work. Destroying the switch itself on the other hand immediately spawned the zombies.
Might not work all the time, but a saved T6 is a saved T6!
21 minutes ago, Mastermind said:
Tried to google around and didn't find any reference to it. So I figured I'd share it here in case this would help anyone.
So, as the title suggest, I was in a POI that didn't spawn all zombie. Happened because of a switch/button that failed to reset properly. I tried to destroy the door it was controlling, but
didn't work. Destroying the switch itself on the other hand immediately spawned the zombies.
Might not work all the time, but a saved T6 is a saved T6!
That's clever. 👏
Way to keep at the problem and thinking outside the box.
That is thanks to the latest update. It didn't work before that. I'm glad it works now for those situations.
• 2 months later...
its not only the switches and buttons, the generators as well. in the shotgun messiah facility there are many of those generators to destroy.
7 hours ago, schwubdi said:
its not only the switches and buttons, the generators as well. in the shotgun messiah facility there are many of those generators to destroy.
You don't have to destroy them. You have the choice to destroy the generators (then the button will work if it is meant to work) or destroy the button and then the door will open if it is meant to
open. You don't need to do both. Note that is you are going to destroy generators, you only destroy the one that is closest to the button that isn't working. 😀
Edited by Riamus (see edit history) | {"url":"https://community.7daystodie.com/topic/32629-spawning-missing-zombie-in-clear-quest/","timestamp":"2024-11-13T05:25:46Z","content_type":"text/html","content_length":"128160","record_id":"<urn:uuid:d98cd32c-3861-44db-9a2e-01ddab11f12c>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00824.warc.gz"} |
Thermodynamics : Heat transfer question.
• Thread starter AlchemistK
• Start date
In summary, a solid body X with heat capacity C is initially at 400K in an atmosphere with temperature of 300K. It cools according to Newton's Law of cooling and at time t1, its temperature is 350K.
The body X is then connected to a large box Y at atmospheric temperature through a conducting rod with length L, cross sectional area A, and thermal conductivity K. The heat capacity of Y is much
larger than X, causing any vibration in its temperature rod to be negligible. To find the temperature of X at time t=3t1, we can use Newton's law of cooling and the equation for time it takes for the
body to reach a certain temperature. However, it is unclear if we
Homework Statement
A solid body X of heat capacity C is kept in an atmosphere whose temperature is 300K. At t=0 temperature of X is 400K. It cools according to Newton's Law of cooling.
At time t1 its temperature is found to be 350 K .
At this time,the body X is connected to a large box Y at atmospheric temperature through a conducting rod of length L, cross sectional area A and thermal conductivity K.
The heat capacity of Y is so large that any vibration in its temperature rod is small compared to the surface area of X.
Find temperature of X at time t=3t1
Homework Equations
Newton's law of cooling : dT/dt = -(4eσAT°^3)ΔT/ms
On integrating to calculate the time it takes for the body to reach a temperature T2 from T1:
t=(ms/4eσAT°^3) ln ((T1 - T°)/ (T2 - T°))
H = KA/L dT
The Attempt at a Solution
Now first, the box reaches the temperature of T=350K in t1 time
So, ti = (ms/4eσA°*300^3) ln ((400 - 300)/ (350 - 300))
= (ms/4eσA°*300^3) ln 2
Now after this, a rod is connected to it, at this point I don't know if I have to consider only heat transfer through the rod, or also by radiation into the atmosphere.
Also, how do I proceed in either case?
Newton's law of cooling : dT/dt = -(4eσAT°3)ΔT/ms
Please define your symbols.
FAQ: Thermodynamics : Heat transfer question.
1. What is heat transfer?
Heat transfer is the process of the transfer of thermal energy from one body or system to another. It occurs when there is a temperature difference between the two bodies or systems.
2. What are the three types of heat transfer?
The three types of heat transfer are conduction, convection, and radiation. Conduction is the transfer of heat through direct contact between two objects or substances. Convection is the transfer of
heat through the movement of fluids, such as air or water. Radiation is the transfer of heat through electromagnetic waves.
3. How is heat transfer related to thermodynamics?
Heat transfer is a fundamental concept in thermodynamics, which is the study of energy and its transformation. Thermodynamics helps explain how heat is transferred from one system to another and how
that affects the overall energy balance.
4. What is the difference between heat and temperature?
Heat and temperature are related but distinct concepts. Heat is a form of energy, while temperature is a measure of the average kinetic energy of particles in a substance. In other words, heat is the
energy being transferred, while temperature is a measure of how much heat is present.
5. How does heat transfer affect daily life?
Heat transfer is a crucial process in various aspects of daily life. For example, it is responsible for regulating the temperature in our homes through heating and cooling systems. It also plays a
role in cooking, transportation, and various industrial processes. Understanding heat transfer can help us make more efficient use of energy and improve our quality of life. | {"url":"https://www.physicsforums.com/threads/thermodynamics-heat-transfer-question.587748/","timestamp":"2024-11-10T21:05:30Z","content_type":"text/html","content_length":"75896","record_id":"<urn:uuid:3651c86c-555c-44fe-bca7-b978e1453046>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00683.warc.gz"} |
GUDe :: Browsing by browse.metadata.subject "Critical Phenomena"
Browsing by Subject "Critical Phenomena"
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Phonon renormalization and Pomeranchuk instability in the Holstein model
The Holstein model with dispersionless Einstein phonons is one of the simplest models describing electron-phonon interactions in condensed matter. A naive extrapolation of perturbation theory in
powers of the relevant dimensionless electron-phonon coupling λ0 suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform
compressibility at a critical value of λ0 of order unity. In this work, we re-examine this problem using modern functional renormalization group (RG) methods. For dimensions d>3 we find that the
RG flow of the Holstein model indeed exhibits a tricritical fixed point associated with a Pomeranchuk instability. This non-Gaussian fixed point is ultraviolet stable and is closely related to
the well-known ultraviolet stable fixed point of ϕ3-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling
λ0 and the adiabatic ratio ω0/εF have to be fine-tuned to assume critical values of order unity, where ω0 is the phonon frequency and εF is the Fermi energy. However, for dimensions d≤3 we find
that the RG flow of the Holstein model does not have any critical fixed points. This rules out a quantum critical point associated with a Pomeranchuk instability in d≤3. | {"url":"https://gude.uni-frankfurt.de/browse/subject?value=Critical%20Phenomena","timestamp":"2024-11-02T06:49:23Z","content_type":"text/html","content_length":"635270","record_id":"<urn:uuid:b69572e2-51a9-40fd-9d6c-49f105dcd791>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00102.warc.gz"} |
How To Add Zero In Front Of Number In ExcelHow To Add Zero In Front Of Number In Excel
How To Add Zero In Front Of Number In Excel
In this article, we will learn Different ways to add zeroes (0s) in front in Excel.
Adding Zero in front of the number in Excel. Default Excel doesn't take zeros in front of the number in the cell. Have you ever tried to enter some data like 000123 into Excel? You’ll probably
quickly notice Excel will automatically remove the leading zeros from the input number. This can be really annoying if you want those leading zeros in data and don’t know how to make Excel keep them.
Fortunately there are quite a few ways to pad your numbers with zeros at the start. In this article, I’ll explain different ways to add or keep those leading zeros to your numbers.
Different functions to add zero (0) in front
As we know TEXT function is the most used function to add 0 in front but here I’ll explain more ways to add zeros (0s) to numbers.
1. TEXT function
2. RIGHT function
3. BASE function
4. Add 0 in pivot table values
Format as Text
One of the solutions is converting the format of the number to text because if we input 00xyz, excel considers this value as text and keeps its 0 in front. There are different ways to convert a
number cell to text format.
• Using Custom format
• Using TEXT function
• Using Apostrophe ( ' ) in start of value while input
Custom Format
Let's understand these methods one by one. Using a custom format to change the format is the easiest way to change the already existing numbers in range. Just select the range and Use shortcut Ctrl +
1 or select More number format from the drop down list on the Home tab showing General.
Change the format from General to Text.
1. Select the range of cells you want to enter leading zeros in.
2. Go to the Home tab.
3. In the Numbers section click on the Format Dropdown selection.
4. Choose Text from the format options.
Now if you try to enter numbers with leading zeros, they won’t disappear because they are entered as text values instead of numbers. Excel users can add a custom formatting to format numbers with
leading zeros.
They will only appear to have leading zeros though. The underlying data won’t be changed into text with the added zeros.
Add a custom format to show leading zeros.
1. Select the range of cells you want to add leading zeros to and open up the Format Cells dialog box.
□ Right click and choose Format Cells.
□ Use the Ctrl + 1 keyboard shortcut.
2. Go to the Number tab.
3. Select Custom from the category options.
4. Add a new custom format in the Type input. If you want the total number of digits including any leading zeros to be 6 then add 000000 as the custom format.
5. Press the OK button.
TEXT Function
The TEXT function will let you apply a custom formatting to any number data already in your spreadsheet.
= TEXT ( Value, Format)
• Value – This is the value you want to convert to text and apply formatting to.
• Format – This is the formatting to apply.
= TEXT ( B3, "000000" )
If you wanted to add zeros to a number in cell B3 so that the total number of digits is 6, then you can use the above formula.
Using Apostrophe ( ' ) in start
You can force Excel to enter a number as text by using a leading apostrophe.
This means you’ll be able to keep those zeros in front as you’re entering your data.
This method is quick and easy while entering data. Just type a ' character before any numbers. This will tell Excel the data is meant to be text and not a number.
When you press Enter, the leading zeros will stay visible in the worksheet. The ' will not be visible in the worksheet, but is still there and can be seen in the formula bar when the active cell
cursor is on the cell.
RIGHT Function
Another way to get your zeros in front with a formula is using the RIGHT function. You can concatenate a string of zeros to the number and then slice off the extras using the RIGHT function.
The RIGHT function will extract the rightmost N characters from a text value.
= RIGHT ( Text, [Number])
• Text – This is the text you want to extract characters from.
• Number (Optional)- This is the number of characters to extract from the text. If this argument is not entered, then only the first character will be extracted.
= RIGHT ( "000000" & B3, 6 )
The above formula will concatenate several zeros to the start of a number in cell B3, then it will return the rightmost 6 characters resulting in some leading zeros.
BASE Function
You’ll notice this article describes 9 ways to add leading zeros, but my YouTube video only shows 8 ways.
That’s because I didn’t know you could use the BASE function to add leading zeros until someone mentioned it in the video comments.
The BASE function allows you to convert a number into a text representation with a given base.
The numbers you usually use are base 10, but you could use this function to convert a base 10 number into a base 2 (binary) representation.
= BASE ( Number, Base, [MinLength])
• Number – This is the number you want to convert to a text value in another base.
• Base – This is the base you want to convert the value to.
• MinLength (Optional) – This is the minimum length of the characters of the converted text value.
Use the formula
The above formula will convert a number in cell B3 into base 10 (it’s already a base 10 number but this will keep it as base 10) and convert it to a text value with at least 6 characters. If the
converted value is less than 6 characters, it will pad the value with zeros to get a minimum of 6 characters.
Definitely a cool new use for a function I otherwise never use.
Add zero (0) only in front of numbers not on text values
Here we will consider the scenario which covers both text and numbers values in the same list. Now excel considers each cell different in terms of format. Now how to let excel know that if the cell
has a number value then add 0 in front or else if the cell has text or any other value, then leave it as it is.
For this we will use a combination of two functions IF and ISNUMBER function. Here we have some values to try on.
The Generic formula goes on like
=IF(ISNUMBER(cell_ref),"0"&cell_ref, cell_ref)
cell_ref : cell reference of the corresponding cell
Use the formula in cell
=IF(ISNUMBER(F3),"0"&F3, F3)
Explanation: The ISNUMBER function checks the cell for number. IF function checks if the cell has a number then it adds zero in front of the number else it just returns the cell value.
As you can see in the above image the zero (0) is added in front of the number. Let's check if this function works on other values. Copy the formula in other cells, select the cells taking the first
cell where the formula is already applied, use shortcut key Ctrl+D
it will pad the value with zeros to get a minimum of one zero in front of characters. Definitely a cool new use for a function I otherwise never use.
Power Pivot Calculated Column
There is another option to add zeros into the pivot table. You can add them into a calculated column in Power Pivot. This way you can use the new column in the Filter, Rows or Column area of a pivot
= FORMAT ( Numbers[Number], "000000" )
In the Power Pivot add-in, you can add a new column and use the FORMAT function to create leading zeros in the column with a formula like above. A calculated column calculates a value for each row,
so there is no need to wrap the function inside a cell. Now you can use this new column inside a pivot table’s Rows area just like any other column of data.
It can be frustrating to see all your zeros disappear when we don’t know why it’s happening or how to prevent it. As you can see, there are lots of ways to make sure all those zeros stick around in
your data. Whatever your requirement, there is surely a good solution for you.
Here are all the observational notes using the formula in Excel
Notes :
1. The formula can be used for both texts and numbers.
2. The find and replace text should be sorted in a table. As we use indexes to find and replace, make sure that 'find' text has the same index as 'replace' text. It is better to have both lists side
by side as we have in the second example.
3. The generic formula has two SUBSTITUTE functions. You will need as many substitute functions as many you have replacements to do.
4. wildcards help with extracting value from the substrings.
5. The function supports logical operators like <, >, <>, = but these are used using double quote sign ( " ) with arguments. But if you are using cell reference for the criteria quotes can be
6. Criteria argument should use quote sign (") for texts ( like "Gillen" ) or numbers ( like ">50" ).
Hope this article about Different ways to add zeroes (0s) in front in Excel is explanatory. Find more articles on calculating values and related Excel formulas here. If you liked our blogs, share it
with your friends on Facebook. And also you can follow us on Twitter and Facebook. We would love to hear from you, do let us know how we can improve, complement or innovate our work and make it
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Related Articles :
Excel REPLACE vs SUBSTITUTE function: The REPLACE and SUBSTITUTE functions are the most misunderstood functions. To find and replace a given text we use the SUBSTITUTE function. Where REPLACE is used
to replace a number of characters in string.
Replace text from end of a string starting from variable position: To replace text from the end of the string, we use the REPLACE function. The REPLACE function use the position of text in the string
to replace.
How to Check if a string contains one of many texts in Excel: To check if a string contains any of multiple texts, we use this formula. We use the SUM function to sum up all the matches and then
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prepare your dashboard.
1. Saved me a lot of time. Great article.
2. Thank you so much. Bless!!
3. Exellent
How to add this eg 2,5,4a,6,7,3d
Ans should be 27 while addition we should neglect a and d
□ Hi,
You can split numbers from string and sum them up. You can use this article,
4. thanks so much! I've lost so much time trying to manipulate text to columns to get this done. amazing!
5. good job
6. Maybe this helps, I tried it seems to work out fine.
select the cells
Cell properties
custom : \0#
It will add a 0.
when saving to .csv and rename to file to .txt you will see the 0 added.
Name 4,055555
above ones with \0#
below is default
Name 4,55555
7. Thank you for helping me solve this issue.
8. Thanks!
9. What if i wanted to add the 0 in front of text?
□ use Concatenate Function to add 0 in front of a text
understand concatenate function here
10. how to convert later 254
19-20/ 254 formula pls helpme
□ You can use formula
=CONCATENATE ("19-20/ ",number)
"19-20/ "&number
11. THNX
12. this is not working in google sheet same result comes up
□ Try =text(cell number,"the zeros you need"). That should work. I just did this in google sheet
13. what if there are 2 types of digit numbers. 1 hv 10 digits, and other 11 digits. how to seperate it?
14. I want to use the zero in front of any number only in some cells, those are account numbers. Since the accounts belong to different banks so they all needn't zero in front of them.
This means What I type that should remain in that format only, like Word. But while I use TEXT format in Excel, it converts them in SCIENTIFIC format, Then I've to click each of them to correct
formatting of each such cell.
Isn't that possible to make a formula to convert the numbers (in the pasted sheets of 'Word' into 'Excel') starting with a special number must have zero & remains not?
15. THANKS FOR THE INFO 🙂
16. THANK YOU SO MUCH...
17. Right click on the tab—>Select Format Cells->select Custom->Select 00000->zero will be inserted before the number.
18. Right click on the tab--->Select Format Cells->select special->Select zip code->zero will be inserted before the number.
□ Hi I need help with home work and I cant do box methid sorry
19. This is great, thank you.
20. Very helpful technique !
21. non of these are working
22. What if the cell you want to add a 0 in front of includes numbers and text? For example, I want to add a 0 in front of:
909 - Design Project Engineer
910 - Design Project Engineer II
□ You can use concatenation, explained here in detail.
23. i love u so much thank you 🙂
24. thanx so much..you have made my work easier.
25. thanx for the quick solution 😉 you saved mah a lot of time | {"url":"https://www.exceltip.com/excel-functions/add-one-digit-zero-in-the-front-of-number.html","timestamp":"2024-11-12T22:21:09Z","content_type":"text/html","content_length":"146797","record_id":"<urn:uuid:679aa9a2-bc50-4e80-812a-940dbc521078>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00667.warc.gz"} |
(Answered) MATH399N Week 8 Assignment: Linear Regression Equations and Application
The following data represents the inches of rainfall y in month x of the year in a city in the U.S. (1 is January, 2 is February, etc.)
Using a calculator or statistical software, find the linear regression line for the data in the table.
Enter your answer in the form y=mx+b, with m and b both rounded to two decimal places.
x y
1 3.53
2 4.93
3 1.58
4 2.72
5 4.68
6 3.34
Using a calculator or statistical software, find the linear regression line for the data in the table below.
Enter your answer in the form y=mx+b, with m and b both rounded to two decimal places.
x y
0 2.12
1 2.19
2 1.92
3 2.79
4 3.81
5 4.72
Using the linear relationship graphed above, estimate the percent of return customers if 80% of customers wait more than 10 minutes in line.
A grocery store manager explored the relationship between the percent of customers that wait more than 10 minutes in line and the percent of return customers at the store. The manager collects
information from 6 checkout lines, shown in the table below.
Use the graph below to plot the points and develop a linear relationship between the percent of waiting customers and the percent of return customers.
Line % of Waiting Customers % of Return Customers
How much water should be consumed every two hours for a person to run 16 miles?
• Round your final answer to the nearest whole number.
A runner finds that the distance they run in miles, D, is dependent on the ounces of water consumed every two hours, x, and can be modeled by the function
Draw the graph of the distance function by plotting its D-intercept and another point.
Using the linear relationship graphed above, estimate the percent of snack purchases if 80% of customers wait in line for more than 15 minutes.
A ticket taker at a movie theater explored the relationship between the percent of customers that wait more than 15 minutes in line for tickets and the percent of customers that purchase snacks at
the theater. The ticket ticker collects information from 6 lines during a particular week, shown in the table below.
Use the graph below to plot the points and develop a linear relationship between the percent of waiting customers and the percent of snack purchases at a movie theater.
Line % of Waiting Customers % of Snack Purchases
How much time did a person spend at the library over a month for them to spend 40 hours reading a book?
• ;Round your final answer to the nearest whole number.
A librarian finds that the number of hours spent reading a book over a month, R, is dependent on the numbers of hours a person spends at the library over the course of a month, x, and can be modeled
by the function
The scatter plot below shows data relating total income and the number of children a family has. Which of the following patterns does the scatter plot show?
Using a calculator or statistical software, find the linear regression line for the data in the table below.
Enter your answer in the form y=mx+b, with m and b both rounded to two decimal places.
x y
0 3.16
1 4.72
2 4.5
3 6.6
4 7.72
5 7.39
6 11.11
7 10.57
8 11.06
Using a calculator or statistical software, find the linear regression line for the data in the table below.
Enter your answer in the form y=mx+b, with m and b both rounded to two decimal places.
x y
2 9.58
3 7.46
4 9.28
5 11.58
6 14.94
7 14.97
8 18.43
9 18.04
10 20.42
Rosetta owns a wedding photography business. For each wedding, she charges $100 plus $50 per hour of work. A linear equation that expresses the total amount of money Rosetta earns per wedding is y=
50x+100. What are the independent and dependent variables? What is the y-intercept and the slope?
Click link below to purchase full tutorial at $10 | {"url":"https://www.charteredtutorials.com/downloads/answered-math399n-week-8-assignment-linear-regression-equations-and-application/","timestamp":"2024-11-02T08:24:39Z","content_type":"text/html","content_length":"65297","record_id":"<urn:uuid:97edde38-1c4e-4317-9360-08bbd224da64>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00617.warc.gz"} |
Geometric Computing
Geometric Computingis one of the most coolest courses I've taken in EPFL. This course covers mathematical concepts and efficient numerical methods for geometric computing. It is a project-based
course, where we develop and implement algorithms to simulate and optimize 2D and 3D geometric models with an emphasis towards computational design for digital fabrication.
There are mainly 3 projects in this course: Make it stand, Elastic Material Fabriction, and Asymptotic Grid Shell Design.
1. Make it stand
The goal of this project is to optimize the shape of an object so that it stands in a desired pose under gravity by implementing a simplified 2D variant of the 2013 SIGGRAPH Paper "Make It Stand" by
Prévost et al.
We implemented BFGS algorithm to find the optimized the shape. The objective function is composed by the equilibrium energy, the shape energy, and the constrain of the face area. Therefore, the
deformed shape can best preserve the original shape, meaning that it looks like the original shape, but it can stand without falling down.
Aside from the example shape provided by the course for testing, we were encourged to try some other shape as well. Here is my final result (gray: original shape, yellow: optimized shape):
2. Inverse Design of Elastic Solids
If we want to manufacture the model using elastic material and fabricate the desired shape directly, gravity would deform it and result in an unwanted result, as shown on the left in the picture.
Therefore, to ensure that the final shape is what we desire despite the effects of gravity, we must determine the rest shape that will deform into the desired shape under the gravity forces.
The concept of this project is similar to the previous "make it stand", but with a more complex optimization target. In addition to BFGS, we also implemented the Newton-Conjugate Gradient to find the
optimized shape, which reduces the computation time dramatically and sometimes even finds a better result.
The final result is shown in the picture below. Originally, if we fabricate the desired shape (red) directly, we would get an unwanted shape(black). Now with inverse design, we can find the rest
shape of the elastic model for fabrication (blue / green), and the deformed shape (yellow) would be similar to the desired shape (red).
3. Asymptotic Grid Shell Design
The asymptotic curve comes with a special property that is beneficial for digital fabrication. The structure formed by a grid of asymptotic curves can be fabricated from straight and flat elements by
laser-cutter, making the manufacture process easier and minimizing the packing size.
The minimal surface, on the other hand, is an ideal structure to be form by an asymptotic grid. Since the minimal surface has zero mean curvature at all points, the asymptotic curve can be found at
every given point on the surface. Therefore, we can fabricate the minimal surface with an approximate uniformally distributed asymptotic grid.
The goal of this project is to find the minimal surface with a given border, and with the user input of the desired point on the surface, form the corresponding asymptotic grid. The result is shown
in the picture below.
At the end of the course, we also voted for some shape created by the students and laser cut the stripes and assemble them together. | {"url":"https://petingo.ch/projects/geometric-computing.html","timestamp":"2024-11-08T11:43:00Z","content_type":"text/html","content_length":"8584","record_id":"<urn:uuid:cd4ceac9-e9f8-42ec-873f-5bf90f086497>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00882.warc.gz"} |
Memory efficient simulation of frequency dependent Q
Kyle B. Withers
Kim B. Olsen
, &
Steven M. Day
Published July 1, 2015, SCEC Contribution #6002
Memory-variable methods have been widely applied to approximate frequency-independent Q in numerical simulation of wave propagation. The frequency-independent model is often appropriate for
frequencies up to about 1 Hz, but at higher frequencies is inconsistent with some regional studies of seismic attenuation. We apply the memory-variable approach to frequency-dependent Q models that
are constant below, and power-law above, a chosen transition frequency. We present numerical results for the corresponding memory-variable relaxation times and weights, obtained by non-negative least
squares fitting of the Q(f) function, for a range of exponent values; these times and weights can be scaled to arbitrary transition frequency and power-law prefactor, respectively. The resulting
memory-variable formulation can be used with numerical wave-propagation solvers based on methods such as finite differences or spectral elements, and may be implemented in either conventional or
coarse-grained form. In the coarse-grained approach, we fit ‘effective’ Q for low Q values (< 200) using a nonlinear inversion technique and use an interpolation formula to find the corresponding
weighting coefficients for arbitrary Q. A 3D staggered-grid finite difference implementation closely approximates the frequency-wavenumber solution to both a half-space and layered model with a
shallow dislocation source for Q as low as 20 over a bandwidth of two decades. We compare the effects of different power-law exponents using a finite-fault source model of the 2008 Mw 5.4 Chino Hills,
CA, earthquake and find that Q(f) models generally better fit the strong motion data than constant Q models for frequencies above 1 Hz.
Withers, K. B., Olsen, K. B., & Day, S. M. (2015). Memory efficient simulation of frequency dependent Q. Bulletin of the Seismological Society of America, 105, 3129-3142.
Related Projects & Working Groups
CME, Ground-Motion Prediction | {"url":"https://central.scec.org/publication/6002","timestamp":"2024-11-07T13:59:56Z","content_type":"application/xhtml+xml","content_length":"22683","record_id":"<urn:uuid:abd078a9-b622-43b2-b645-35239051256c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00505.warc.gz"} |
Primal Forms, a development over time
Primal forms have developed as a response to many years of continued exploration into natural form and the way nature builds.
The term primal form refers to the simple, fundamental nature of these forms. They are based on the sphere which has an important association with the three dimensional world. A circle is to two
dimensional as the sphere is to three dimensions. Three dimensional geometry is the geometry of the sphere. The curved surface of the sphere requires a different geometrical approach to that on a
flat plane.
In constructing the form, the first stage is the making of a smooth blank sphere, this then is ready to be marked out. The sphere is divided up so as to identify the solid to be represented. One
example is the cube which comprises six squares, as drawn on the sphere twelve interconnecting lines of equal length are marked over the sphere's surface. The other solids similarly have their
relevant number of equal lines plotted.
Further lines are drawn through the previously placed points. Once all the relevant points are positioned, spirals orbits and curves are drawn through and around them, as appropriate to the design.
This completes the two dimensional aspect. The integrity of the design becomes apparent in an intuitive process as the points are linked in this way. The aim is for the elements of the design, the
spirals, orbits and curves, to freely fit the pattern markers in a fluid unforced way.
In an almost uncanny way, maybe due to the play of sacred geometry and the integrity of the design, the result usually has an inevitability. The spirals, orbits and curves usually fit the design
pattern with an accuracy that is almost beyond expectation. The feeling being that the design is there just requiring to be uncovered, discovered rather than created or invented. The integrity and
logic of the linked points and the drawing that evolves, ensures that the result fulfils expectations. There is an intuitive feeling of completion, perhaps this is sacred geometry in action, ensuring
a harmonic outcome.
Celestial orbs, heavenly bodies, both refer to spherical forms and evoke comparisons to celestial and planetary phenomena. The interactive qualities of the orbits, spirals and curves in this
geometrical context produce intersections and alignments. These remind of the occurrences that result from planetary movement, particularly the eclipses that involve the interaction of the sun and
moon. Ancient monuments with their alignments are examples of mankind's historical understanding and interest in this.
After the drawing phase has established the fundamental design on the spherical surface, the drawn lines are cut. This begins the transformation journey into three dimensions.This gradual process
continues until all the flat areas have disappeared. The final honing process reveals subtle geometry and effectively "tunes" the form into full three dimensional completion where all the subtle
geometries are revealed. As this occurs tuning refers to the way all the harmonic elements combine, like a musical instrument in tune.
Geometric drawing on the sphere | {"url":"https://www.timothycresswell.co.uk/post/primal-forms-a-development-over-time-1","timestamp":"2024-11-13T02:53:05Z","content_type":"text/html","content_length":"1050488","record_id":"<urn:uuid:c21019b9-ef17-4138-a499-14fe99a41ab0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00361.warc.gz"} |
Get Organized When Doing Arithmetic - Ten Steps to Scoring Higher on the GRE - Crash Course for the New GRE
Crash Course for the New GRE, 4th Edition (2011)
Part II. Ten Steps to Scoring Higher on the GRE
Step 9. Get Organized When Doing Arithmetic
Questions involving ratios or averages can seem daunting at first. The math involved in these problems, however, generally involves little more than basic arithmetic. The trick to these problems is
understanding how to organize your information. Learning the triggers and set-ups for each of these problems can take a four-minute brain teaser and turn it into a 45-second cake walk.
Imagine you are asked to find the average of three numbers, 3, 7, and 8. This is not a difficult problem. Simply add the three together to get the total. Divide by three, the number of things, to get
the average. All average problems involve three basic pieces:
·Total: 18
·# of things: 3
·Average: 6
It is virtually assured that they will never give you a list of numbers and ask you for the average. That would be too easy. They will, however, always give you two out of these three pieces, and it
is your job to find the third. That’s where the average pie comes in. The minute you see the word “average” on a problem, draw your pie on your scratch paper. It looks like this:
Here’s how you would fill it in.
ETS won’t necessarily give you a list of numbers and ask you to find the average. That would be too easy. They might give you the average and the total and ask you to find the number of things, or
they might give you the number of things and the average and ask for the total. They will always give you two out of the three pieces of information. Just make your pie, fill in what you know, and it
becomes easy to find the missing piece. Here’s how it works:
The line in the middle means divide. If you have the total and the number of things, just divide and you get the average (18 ÷ 3 = 6). If you have the total and the average, just divide and you get
the number of things (18 ÷ 6 = 3). If you have the average and the number of things, simply multiply and you get the total (6 × 3 = 18). As you will see, the key to most average questions is finding
the total.
The benefit of the Average Pie is that you simply have to plug the information from the question into the Average Pie and then complete the Pie. Doing so will automatically give you all the
information you need to answer the question.
Let’s try this one:
Question 6 of 12
The average (arithmetic mean) of a set of 6 numbers is 28. If a certain number, y, is removed from the set, the average of the remaining numbers in the set is 24.
│Quantity A │Quantity B │
│y │48 │
The minute you see the word “average,” make your pie. If you see the word “average” a second time, make a second pie. Start with the first bite-sized piece, “The average of a set of 6 numbers is 28.”
Draw your pie and fill it in. With the average and the number of things you can calculate the total, like this:
Take your next piece of the problem, “If a certain number, y, is removed from the set, the average of the remaining numbers in the set is 24.” There’s the word “average” again, so make another pie.
Again, you have the number of things (5, because one number was removed from our set) and the average, 24, so you can calculate the total, like this:
The total for all six numbers is 168. When you take a number out, the total for the remaining five is 120. The number you removed, therefore, must be 168 − 120 = 48. y = 48. The answer is (C).
When working with fraction, decimals, and percentages, you are working with a part to a whole relationship. The fraction
Question 3 of 20
In a club with 35 members, the ratio of men to women is 2 to 3 among the members. How many men belong to the club?
The problem says, “the ratio of men to woman …” As soon as you see that, make your box. It should look like this:
In the top line of the box, list the items that make up your ratio, in this case, men and women. The last column is always for the total. In the second row of the box, fill in your ratio of 2 to 3
under Men and Women, respectively. The total is five. This doesn’t mean that there are actually two men and three women in the club. This just means that for every five members of this club, two of
them will be men and three of them will be women. The actual number of members, we’re told in the problem, is 35. This goes in the bottom right cell under Total. With this single number in the bottom
row we can figure out the rest. To get from 5 to 35, you need to multiply by 7. The multiplier remains constant across the ratio, so fill a 7 in all three cells of the third row, next to the word
“multiplier.” We now know that the actual number of men in the club is 14, just as the actual number of women is 21. Here’s what your completed ratio box looks like:
The fraction of the club that is male is
Median means the number in the middle, like the median strip on a highway. In the set of numbers 2, 2, 4, 5, 9, the median is “4” because it’s the one in the middle. If the set had an even number of
elements, let’s say: 2, 3, 4, 6, the median is the average of the two numbers in the middle or, in this case, 3.5. That’s it. There’s not much that’s interesting about the word “median.” There are
only two ways they can trick you with a median question. One is to give you a set with an even number of elements. We’ve mastered that one. The other is to give you a set of numbers which are out of
order. If you see the word “median,” therefore, find a bunch of numbers and put them in order.
“Mode” simply means the number that shows up the most. In the set 2, 2, 4, 5, 9, the mode is 2. That’s all there is to mode. If no number shows up more than another, then the set has no mode.
“Range” is even easier. It is the difference between the biggest number in a set and the smallest. In other words, find the smallest number and subtract it from the biggest number.
Let’s look at a problem:
Question 8 of 20
If in the set of numbers {20, 14, 19, 12, 17, 20, 24}, v equals the mean, w equals the median, x equals the mode, and y equals the range, which of the following is true?
v < w < x < y
v < x < w < y
y < v < w < x
y < v < x < w
w < y < v < x
In this question we’re asked to find the mean, the median, the mode, and the range of a set of numbers. The minute you see the word “median,” you know what to do. Put the numbers in order: 12, 14,
17, 19, 20, 20, 24. Do this on your scratch paper, not in your head, and while you’re at it, list A, B, C, D, and E so that you have something to eliminate. The minute we put the numbers in order,
three out of the four elements we are asked to find become clear. The range, 12, is equal to the smallest number, so y should be the element at the far left of our series. Cross off A, B, and E. The
average will be somewhere in the middle. Without doing some calculations, it’s not clear if it is larger than the median (19) or smaller, so skip to the mode. The mode is 20 and larger than the
median and certainly larger than the average. x should be the last element in our series. Cross off choice (D). The correct answer is (C). Always remember that the answer choices are part of the
question. Often it is far easier to find and eliminate wrong answers than it is to find the right ones.
Rates and Proportions
Rates are really just proportions. Just like ratios and averages, the basic math is straight forward, but the difficult part is organizing the information. Actually, organizing the information is the
whole trick. Set up all rates like a proportion and make sure you label the top and bottom of your proportion.
Let’s look at an actual problem:
Question 8 of 20
Stan drives at an average speed of 60 miles per hour from Town A to Town B, a distance of 150 miles. Ollie drives at an average speed of 50 miles per hour from Town C to Town B, a distance of 120
│Quantity A │Quantity B │
│Amount of time Stan spends driving│Amount of time Ollie spends driving │
In this problem we are comparing two separate rates and each rate consists of miles (distance) and hours (time). Start with Stan. Stan’s speed is 60 mph, which is to say that he drives 60 miles every
one hour. We’re asked to find how many hours it will take him to travel 150 miles. Just set it up as a proportion, like this:
Now we can compare miles to miles and hours to hours. There is an x in the second space for hours because we don’t yet know how many hours it’s going to take Stan. The nice thing about this set-up is
that you can always cross multiply to find the missing piece. If 60x = 150, then x = 2.5. This means that it took Stan 2.5 hours to drive 150 miles (at a rate of 60 miles per hour).
Now try Ollie. The set up is the same. Ollie drives 50 miles for every one hour. To find out how many hours he needs to drive 120 miles, just cross multiply. If 50x = 120, then x = 2.4. This means
that it took Ollie 2.4 hours to drive 120 miles (at a rate of 50 miles per hour). Quantity A is Stan, so the correct answer is (A).
Arithmetic Summary
│Trigger: When you see the word…│Response: │
│“Average” │Draw an Average Pie. │
│“Ratio” │Draw a ratio box. │
│“Median” │Find a bunch of numbers, and put them in order. │
│“Mode” │Find the number that appears the most often. │
│“Range” │Subtract the smallest from the biggest. │
│“Rate” │Set up a proportion; label top and bottom. │ | {"url":"https://schoolbag.info/test/gre_3/11.html","timestamp":"2024-11-02T12:12:39Z","content_type":"text/html","content_length":"24341","record_id":"<urn:uuid:48b4f786-b8c9-4d5b-8ca2-051243c204c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00614.warc.gz"} |
Small math problem. (rotations, with pictures!)
Here's a little visual for what I'm about to ask (I lied, there's only going to be a single picture):
1: First rotation at <0,0,0> (euler)
2: Second rotation
3: Third rotation
Black dot: Object center
White dot: Center of rotation
Grey circle: Projected location for object center given any angle around the X axis.
Given a prim (box/cylinder) with an abitrary size with one axis being longer and able to change its length while the script is running, how do I set an object's rotation while keeping one end
"anchored" in its place? (Rotating around an offset point.)
There's an example on the SL Wiki, but it results in relative rotation. For example, running the same code twice would cause two rotations with the same angle size. What I want to do is set the
rotation in global coordinates so that running the same code twice would cause one rotation and no change the second time.
In short, I want to know/understand the math required for this. I can script, I just can't do the math.
Additional notes for context:
- The rotation can be around multiple axes at once. (<45,90,30> euler)
- The center of rotation or "anchor point" is known.
- The prim is part of a link set and worn as an attachment.
- The length of the Z axis can be anything above 0.1 and max prim size.
Because rotations always happen around an objects center, what you want to do involves not only changing the rotation of your object, but its position as well. If your image accurately depicts what
you want to do, and the rotation and position of the object will always be one each of the three you illustrate, you can store them in a list, or use another value storage system, and change the
object's position/rotation with references to those variables.
The three rotations shown in the image are only examples. The rotation could be anything between 0-359 on each axis at once.
I also understand that the object's position has to change in addition to its rotation, but the math of it goes way over my head.
I can offer another solution based on what I can see of what you want to do. If you make profile cuts on your object, so that the object center and its center of rotation (its end or anchor
point) are the same, you would only have rotations to worry about. This is a strategy sometimes used to make doors. This would obviate the need to make any position adjustments. This option would
limit the object length to 32 meters.
I actually do that whenever I can, it's extremely useful for indirectly optimizing scripts.
However, sometimes path cuts are not an option, for example when trying to rotate flexi prims at the base. (Flexi cannot be sliced from beginning/end.) This is actually one of the cases I'm dealing
I'm not quite clear about what you are trying to do, but it looks like a basic door script for an uncut prim door. Something like this ...
rotation adjust;vector offset;default{ state_entry() { adjust = llEuler2Rot(<0.0,-90.0,0.0>*DEG_TO_RAD); vector Size = llList2Vector(llGetLinkPrimitiveParams(2,[PRIM_SIZE]),0); offset = <-(1.0-llCos(llRot2Angle(adjust)))*0.5*Size.x,0.0,-llSin(llRot2Angle(adjust))*0.5*Size.x>; } touch_start(integer total_number) { if (llDetectedLinkNumber(0) == 2) { list temp = llGetLinkPrimitiveParams(2,[PRIM_POS_LOCAL,PRIM_ROT_LOCAL]); vector Lpos = llList2Vector(temp,0); rotation Lrot = llList2Rot(temp,1); adjust = ZERO_ROTATION/adjust; llSetLinkPrimitiveParamsFast(2,[PRIM_POS_LOCAL, Lpos + (offset = -offset), PRIM_ROT_LOCAL,adjust*Lrot]); } }}
That really is a door script, so it's written to make a simple 90 degree rotation and then back again. It would be a fairly easy matter to put that stuff from the state_entry event into a user
defined function and then trigger it with a timer to make repeated small changes in the adjust variable and anything else that you want to modify.. Ubless I am missing your question.....
I guess you can think of it exactly like a door script for uncut prims like you said, the differences being that the "door" is more like a pole that can change its length and the hinge is at one end
of the pole.
But I'm having trouble following the script you posted, mainly in state_entry where you assign the offset for the first time.
Wulfie Reanimator wrote:
I guess you can think of it exactly like a door script for uncut prims like you said, the differences being that the "door" is more like a pole that can change its length and the hinge is at one end
of the pole.
[ .... ]
Exactly. That's why you have to offset the rotation point from the center of the "door" to one end before you rotate it. Now, if you want to change its length, you're going to have to do it in
increments as you make incremental rotations.
[ .... ] But I'm having trouble following the script you posted, mainly in state_entry where you assign the offset for the first time.
You're going to have to draw yourself a picture and do the trigonometry, I'm afraid. Fortunately, it's not hard. Your "door" is a rectangle, so it has right angles at its corners. When you rotate it
through an angle, though, it looks as if you have temporarily twisted the door into a parallelogram. The angle is a measure of that slight distortion, so you are calculating the "extra" bit of
length that results from it.
You are about to reply to a thread that has been inactive for 2828 days.
Please take a moment to consider if this thread is worth bumping. | {"url":"https://community.secondlife.com/forums/topic/398416-small-math-problem-rotations-with-pictures/","timestamp":"2024-11-03T10:07:02Z","content_type":"text/html","content_length":"154075","record_id":"<urn:uuid:141cb4b2-b391-4642-a0bf-3f20878bbc13>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00076.warc.gz"} |
Macroscopic entanglement witnesses
Macroscopic Entanglement Witnesses
It is commonly believed that for the understanding of the behaviour of large, macroscopic, objects at moderately high temperatures there is no need to invoke any genuine quantum entanglement. This is
because decoherence effects arising from many particles interaction with the environment would destroy all quantum correlations.
In a series of papers [1-4] we have shown that this belief is fundamentally mistaken and that entanglement is crucial to correctly describe macroscopic properties of solids. Moreover, we demonstrated
that macroscopic thermodynamical properties – such as internal energy, heat capacity or magnetic susceptibility – can reveal the existence of entanglement (i.e. are so called “entanglement witnesses)
within solids in the thermodynamical limit even at moderately high temperatures. We found the critical values of physical parameters (e.g. the high-temperature limit and the maximal strength of
magnetic field) below which entanglement exists in solids.
Figure: Detection of entanglement in magnetic solids, modelled by the xxx-Heisenberg spin-1/2 (a, top) and spin 1 (b, bottom) chains. The black solid curves are temperature dependences of the
zero-field magnetic susceptibilities per particle and the red curves are “entanglement witnesses”. All points below of the intersection points to the left (below the critical temperatures) indicate
the existence of entanglement in the solids.
[1] Č. Brukner and V. Vedral, Macroscopic Thermodynamical Witnesses of Quantum Entanglement, Preprint at
[2] Č. Brukner, V. Vedral and A. Zeilinger, Crucial Role of Quantum Entanglement in Bulk Properties of Solids,
Phys. Rev. A 73, 012110 (2006)
[3] M. Wiesniak. V. Vedral and Č. Brukner, Magnetic Susceptibility as Macroscopic Entanglement Witness,
New J. Phys. 7, 258 (2005)
[4] M. Wieśniak, V. Vedral, and Č. Brukner, Heat capacity as an indicator of entanglement,
Phys. Rev. B 78, 064108 (2008) | {"url":"https://www.quantumfoundations.org/macroscopic-entanglement-witnesses.html","timestamp":"2024-11-02T02:07:32Z","content_type":"text/html","content_length":"24814","record_id":"<urn:uuid:e2f0ef50-20d7-42b7-9645-898548f4d09f>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00206.warc.gz"} |
Informath >> Remarks on Keenan [Theor. Appl. Climatol., 2007] >> Response by authors
An early version of my critique of the paper of Chuine et al. [2004] was sent to the authors; following discusses the authors' response.
The relevant portion of the authors' response to my critique was as follows.
Keenan compares the simulated and observed anomalies of the 4 warmest years of his series from Dijon (2003, 1947, 1952, 1945) and concludes on the model failure. We find it a bit light to
conclude from 4 years taken in a series of more than 600 years that the "results of the paper are plainly highly unreliable".
First, we never claimed that our reconstructed yearly anomalies could be interpreted individually as observed anomalies. Each anomaly has to be interpreted in the light of the whole series. Even
if model simulation gives a higher anomaly than observed for 2003, it remains without contest the highest temperature of the whole measurement period with an anomaly nearly twice as large as the
second hottest year. Thus the conclusion of the paper remains absolutely correct.
Second, Keenan curiously only uses the three hottest years to conclude a model underestimation of all unusually warm years except 2003. If he had taken the following 3 hottest years in the
series, he would have seen on the contrary that 1976 and 1893 have a simulated anomaly higher than observed. Considering the seven warmest years of the series, 4 are actually underestimated and 3
are overestimated.
Likewise it seems also strange from a statistical point of view to consider that the years [Table 1 Keenan] with anomalies higher than one standard deviation are "nearly average". The four
warmest years in the observed 1880-2000 series [Table 1 Keenan] are actually all in the 17 warmest years in the simulated series. More generally among the 25 warmest years in the observed series,
more than 70% are found in the 25 hottest years in the simulated series and more than 80% in the 30 hottest years.
So in contradistinction to the conclusion of Keenan, the model does not fail at all, even for detecting the warmer years.
It is easy to get lost trying to follow the reasoning in the above—and then, upon finding the errors, rebut them in a way that is necessarily complicated. A complicated rebuttal would leave the
reader uncertain as to whether the rebuttal is valid. Perhaps that is what the authors hoped for.
Consider first the authors' assertion that 2003 “remains without contest the highest temperature of the whole measurement period … Thus the conclusion of the paper remains absolutely correct”. The
year 2003 has the highest modeled temperature. This is obvious and not in dispute. The authors then conclude that this implies their paper is correct. The conclusion is obviously not logical.
Consider next the authors' objection to my paper's use of the term “nearly average”. It is common in statistical practice to regard data within 1 std. deviation of the mean as being about average.
Moreover, it is common in statistics to require that the data lie more than 2 std. deviations away from the mean in order to be considered extreme (for a Gaussian distribution, this corresponds with
95%-confidence intervals, which are almost ubiquitous in science). Some studies require 2.5, or more, std. deviations away from the mean. Thus, saying that a year whose temperature was 1.05, 1.18, or
0.95 std. deviations above the mean is “nearly average” (as my paper did) is fine.
Third, consider the authors' assertion that “Keenan curiously only uses the three hottest years to conclude a model underestimation of all unusually warm years except 2003”. The authors seem to be
confused about what is required for a year to be extremely warm—yet it is extreme years in which we are interested. If, in order to be extreme, we require data to be outside the 95%-confidence
interval, then, given that there are 120 years of data, we would expect 0.05*120 = 6 years that are extreme. Of those 6, half would be expected to be extremely warm, and half extremely cool. My
paper's consideration of the 3 warmest years (prior to 2003) is thus consistent with that. The authors' use of 17 years would imply considering years that were >1.07 std. deviations above the mean as
extremely warm, which is obviously untenable. The authors' use of 25, or 30, years is worse.
A crucial point is that for the three hottest years during 1883–2002, the authors' model underestimates the temperatures so much that those years appear to be nearly average. Such a model should
obviously not be trusted to identify the hottest years prior to 1883.
It is possible to discuss the authors' objections further. It should be clear, though, that the authors have nothing substantive to say in rebuttal.
Acknowledgement: thanks to P. Claussen for reporting a mistake in an earlier version.
Chuine I., Yiou P., Viovy N., Seguin B., Daux V., Le Roy Ladurie E. (2004), “Grape ripening as a past climate indicator”, Nature, 432: 289–290. doi: 10.1038/432289a.
Keenan D.J. (2007), “Grape harvest dates are poor indicators of summer warmth”, Theoretical and Applied Climatology, 87: 255–256. doi: 10.1007/s00704-006-0197-9.
Instrumental temperature data is available from
Model-simulated temperature data is available from | {"url":"https://www.informath.org/apprise/a3200/b3.htm","timestamp":"2024-11-02T11:28:45Z","content_type":"text/html","content_length":"8216","record_id":"<urn:uuid:055eaad9-03d4-4c97-8bdf-de182ebd5495>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00888.warc.gz"} |
Codebook: Making and Breaking Promises
Ask here about the “Making and Breaking Promises” Codebook topic from the “Basic Quantum Algorithms” module.
I failed so far with Codercise A.4.1. - The multi-solution oracle.
I tried to create a matrix (oracle) that detects multiple solutions by creating
U_1, U_2, … U_n and then multiply them.
But this does not work - either because I made a programming error or my approach is wrong.
Could you please give me a hint what’s the correct way to create such an oracle? (some related theory).
Many thanks,
Hi @jomu ,
I don’t know what’s the exact cause for your program not to work. I can tell you however that there’s a way of creating the oracle matrix without needing to multiply individual matrices. Remember how
in A.2.1 you changed the entry corresponding to the solution from +1 to -1? You can do this for several solutions my changing more than one entry in the matrix.
I hope this helps!
Hi @jomu,
So in this codebook, the “indices” is a list of combination index, something like [2,6,7] where “2,6,7” represents the indexes of the secret combinations.
Now the goal of this exercise is to create an Oracle for each combination. You already did this before in “Magic 8 ball” but now you do the same thing (change the diagonal entry from “+1” to “-1”)
for the index of the secret combination and you do this for each combo. and by doing this you get your U1,U2 and so on.
Hope this helps
1 Like
Yes, that helps. I think i am quite close
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Intel Education: Digital Information - Intel
Digital Information
The Journey Inside℠, an Intel® Education program.
Lesson 1: What Is Binary Code?
People use all kinds of symbols, sounds, colors, and body motions to express themselves. These expressions are like codes or signals we use to communicate with one another.
Computers use a special language of their own to express the digital information they process. It's called the binary code, bi meaning two, because it consists of only two symbols—0s and 1s.
So why 0s and 1s? Because those are the only two numbers you need to express the flow of electricity through a transistor. A transistor is either on or off; on is 1, and off is 0. Everything you say
to a computer has to be put in terms of these two numbers.
Lesson 2: Binary Digit (Bit) and Machine Language
For a computer to execute or respond to a command, it must be translated into the only language a computer knows: the 0s and 1s of the binary number system. The 0s and 1s represent the on and off of
the transistors.
We call one of these 0s or 1s a bit. Just like how words are made of several letters, computers create numbers, colors, graphics, or sounds with 0s and 1s. They really are just a bit of something
Lesson 3: What Is a Pixel?
Imagine a computer that is made up of billions of electronic switches (transistors). They're either on or off.
Now imagine this. Your computer screen has hundreds of thousands, if not millions, of dots arranged in rows and columns. Each dot is a piece of a picture—otherwise known as a pixel—and the number of
pixels used is called the resolution. The higher the pixel count, the higher the resolution, and the better the picture quality. For example, a high-definition 1280 x 720p resolution screen means the
screen would have a width of 1,280 pixels and a length of 720 pixels. That's 921,600 total pixels!
Each of these pixels display some combination of red, green, and blue to create colors. Computers operate by mixing these three colors to create black, white, and millions of other color
For your next activity, think of the grid as a simplified view of a black-and-white computer screen. Each grid square represents a pixel. In these activities, the squares are much bigger than the
real thing, but doing this activity will show you how an image can be portrayed with just two instructions: on and off.
Try lesson 1: Work and play with pictures.
Try lesson 2: Pixel pictures.
Lesson 4: Binary Numbers
The binary number system, which computers use to store and process information, only uses two digits: 0 and 1. In fact, the bi in binary comes from the Latin prefix meaning two. Binary is a base 2
number system. The 2 represents the number of digits the system uses.
Compare this to the decimal number system you use. The decimal system includes 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Its name also tells you how many digits it includes, as dec comes from the
Latin prefix meaning ten. The decimal system is a base 10 number system.
So, if you saw the number 100, how would you know if it was in base 2 or base 10?
In math, a little subscript number is added to the right-most number in a set to tell you what base system the number set is in. When you see a number written as 100[2], the little 2 lets you know
the set of numbers is in base 2, or binary. If you see a number written as 100[10], the little 10 tells you that the set of numbers is in base 10, or decimal.
Reading Binary Numbers
Reading binary numbers is different than reading decimal numbers. In the decimal system, you read all the numbers together at once as a whole number¬—1, 10, 100. But in binary, you read the numbers
like a math equation. You have to solve a math problem for each individual number, or bit, and then add all the individual answers together to find out what the total whole decimal number equivalent
For example, in decimals, 10[10]=10, but in binary, 10[2]=2!
To read a binary number, first you need to look at the placement of each 1 and 0, and you always read binary numbers from right to left.
In decimal (base 10) numbers, you have a 1s place, a 10s place, a 100s place, and so on to represent value. Each place is 10 times greater than the place before it. The binary system (base 2) has
places, or columns, too. As binary only has two numbers, each place is worth double (two times) the one before it.
Binary places also have slightly different names. The right-most value in a binary number is your starting value, and it is in the "zero place." The next place to the left is considered the "first
place" because you've moved one spot from the start. The next place to the left after that is the "second place" because it is two spots over from the start.
Why does the place of the number matter?
Let's look at an example, the binary number 100[2].
• Starting at the right with the zero place, we see the number 0, which is also worth 0.
• Moving one place to the left (the first place), we see another 0. If we double 0, we still get a value of 0, as 0x2=0.
• Next, we move one more place to the left (the second place), and we see a 1. Since the 1 is two spots away from the start, to determine its decimal value, we have to double its value twice: 1+1=
2, then 2+2=4. So, in binary, the 1 in 100[2] is worth 4.
• Finally, we add the values of all three binary numbers together to find out the total whole decimal equivalent: 4+0+0=4. Therefore, 100[2] is the same as decimal number 4, or 4[10].
Tips for converting binary numbers to decimal numbers:
• Read binary numbers like a math equation.
• Treat each number like its own math problem. Once you've solved each individual math problem, add all the individual answers together to get the answer.
• In binary, a number doubles its value each time it moves a place to the left. The place of the bit, the 1 or the 0, tells you how many times you need to double the number.
• Always read binary numbers from right to left.
Try lesson 1: Finding decimal and binary number equivalents.
Try lesson 2: Converting decimal numbers to binary numbers.
Lesson 5: How to Add Binary Numbers
In lesson 4, you learned how to read a single binary number and convert it to its decimal equivalent, but what about adding binary numbers?
How would you solve this equation: 10[2]+11[2]=?
Let's break the problem down into smaller steps to find the decimal equivalents.
Right away, we see a subscript 2 on the end of each number set, which tells us these are binary numbers. Now we know we have some math to do on each individual number first.
Let's start with 10[2], which has two numbers: 1 and 0.
• Starting on the right, we see the number 0, which we know has a decimal value of 0.
• Next, we move to the 1. Since it is one place to the left of the starting spot, it is worth double its value, or 1+1, which equals 2.
• Finally, we add the two answers together: 0+2=2. Therefore, 10[2] is the same as decimal number 2.
Now, let's look at 11[2].
• Starting on the right, the first number is 1, which has a decimal value of 1.
• Next, we move one place to the left and see another 1. Since this number is one place to the left of the starting point, it is worth double its value, or 1+1, which equals 2.
• Now we add the two answers together: 1+2=3. Therefore, 11[2] is the same as decimal number 3.
Finally, we add the sum of both sets of equations, 2+3, to get the final whole decimal answer, 5. Written in binary, the answer is 10[2]+11[2]=101[2].
(Tip: If you want to double-check your work, look back at the lesson 4 activities.)
Converting Decimal Numbers to Binary Numbers
Now that you've learned how to convert a binary number to a decimal number, let's try working in reverse.
To convert a decimal number to a binary number, we again need to think about place.
│Place value │128 │64│32│16│8│4│2│1│
│Binary number │ │ │ │ │ │ │ │ │
In lesson 4, we learned that for every place we move to the left of the zero place, we have to double the value from the previous place.
When trying to convert a decimal to a binary number, first we need to find the place value that is as close to but not greater than the number.
Let's take the answer to the last example, decimal 5. If you were going to put a mark in the column with the place value closest to but not greater than 5, where would you put it? Under the 4, right!
│Place value │4│2│1│
│Binary number │1│ │ │
That leaves you with 1 remaining. Where does it go? Under the 1, right!
│Place value │4│2│1│
│Binary number │1│ │1│
We know that binary numbers must include either a 0 or a 1 and that there aren't any spaces. So what goes in the open space? The number 0, right!
│Place value │4│2│1│
│Binary number │1│0│1│
Now we've found that decimal number 5 is the same as binary number 101[2].
Try the lessons: Adding binary numbers.
Lesson 6: From Binary to ASCII
Bits—the 0s and 1s of binary code—can be used in many ways to represent information. Computers communicate with each other through a standard language: ASCII (American Standard Code for Information
ASCII is an 8-bit code, and 8 bits are called a byte. ASCII uses a byte to represent a letter, number, or punctuation mark. For instance, a lower case a is represented by 01100001[2]. The word cat
would be 01100011[2] 01100001[2] 01110100[2].
Try lesson 1: The name game.
Try lesson 2: Secret messages with ASCII.
Try lesson 3: The ASCII code chart.
Lesson 7: AND/OR Statements in Software Writing
You make decisions every day, like what movie to see or how to get home from school the fastest (by bus, bike, or by your own two feet). These are called OR situations where you can only select one
of the available options at a time.
Life is also filled with AND situations, such as trying to get both your homework and your chores done so you can go to the movies with friends. In this case, both must be done if you want the result
(being able to go to the movies).
When programmers write software, they frequently use AND and OR statements to determine a result. The word AND requires both conditions to be true (in other words, a yes to both parts) for the result
to happen.
The word OR requires either the first or the second statement to be true (a yes on one part and a no on the other) for the result to happen.
If you think of yes as a 1 and no as a 0, you can see how transistors in a computer that uses binary code can understand AND and OR statements.
Here is a quick summary of what we've covered:
• Binary is a code that consists of the numerals 0 and 1.
• Computers contain transistors that can be either on or off.
• If 1=yes and 0=no, then binary code can answer yes or no to simple questions.
Try the lesson: Can I go to the movies?
Explore the Curriculum
Discover more lessons with The Journey Inside™, from electricity to binary to the internet.
This unit provides a short history on the computer, introduces the four major components of a computer, and compares computer "brains" with the human brain.
This unit teaches students about electricity, electric circuits, and the difference between mechanical and nonmechanical (transistors) switches.
This unit explores the differences between the decimal and binary number systems and how the information is represented and processed using binary code.
This unit investigates how microprocessors process information, demonstrates the size and the complexity of their circuitry, and explains how they are manufactured.
This unit defines the internet, then explains the World Wide Web, hypertext, URLs, packets, bandwidth, connection choices, search engines, and the need to critically evaluate the quality of the
information found on the web.
This unit discusses the impact technological advances have on people's lives, with examples from the past and current day. Several readings provide insights on ways the digital age is already
affecting rate of change, and what we might expect to see in the near future. | {"url":"https://www.intel.com.tw/content/www/tw/zh/education/k12/the-journey-inside/explore-the-curriculum/digital-information.html","timestamp":"2024-11-03T19:18:10Z","content_type":"text/html","content_length":"179827","record_id":"<urn:uuid:f4491d2d-dee7-4986-a007-21412922a841>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00291.warc.gz"} |
replicate -package:Cabal -package:base -package:bytestring is:exact -package:memory -package:text -package:utility-ht package:rio
O(n) replicate n x
is a ByteString of length
the value of every element. The following holds:
replicate w c = unfoldr w (\u -> Just (u,u)) c
This implemenation uses
O(n) replicate n x
is a ByteString of length
the value of every element.
replicate n x
is a list of length
the value of every element. It is an instance of the more general
, in which
may be of any integral type.
replicate n x is a sequence consisting of n copies of x.
O(n*m) replicate n t
is a
consisting of the input
O(n*m) replicate n t
is a
consisting of the input
O(n) Vector of the given length with the same value in each position
O(n) Vector of the given length with the same value in each position
O(n) Vector of the given length with the same value in each position
O(n) Vector of the given length with the same value in each position | {"url":"https://hoogle.haskell.org/?hoogle=replicate%20-package%3ACabal%20-package%3Abase%20-package%3Abytestring%20is%3Aexact%20-package%3Amemory%20-package%3Atext%20-package%3Autility-ht%20package%3Ario","timestamp":"2024-11-04T03:08:31Z","content_type":"text/html","content_length":"209621","record_id":"<urn:uuid:880ae9c2-4e31-4479-92ed-01115ce1ca75>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00525.warc.gz"} |
The mp-wigner-eckart command transforms non-abelian symmetries into an abelian subgroup, for example {$SU(2) \supset U(1)$}.
mp-wigner-eckart [options] <symmetry-list> <input-psi> [output-psi]
show help message
-f, --force
overwrite the output file, if it exists
The mp-wigner-eckart command takes an input wavefunction with at least one {$SU(2)$} symmetry, and projects that symmetry down to {$U(1)$}. If the output file isn't specified, then the reflection is
performed in-place, overwriting the old input file.
If the output file already exists, then mp-wigner-eckart will refuse to overwrite it, unless you specify also the --force option.
The symmetry list parameter must match the symmetry list of the original wavefunction, except that every {$SU(2)$} symmetry must be replaced by a {$U(1)$} symmetry. For example, if the original
symmetry list is N:U(1),S:SU(2), then the new symmetry list could be N:U(1),Sz:U(1). The name of the new symmetry (Sz in this example) is arbitrary -- normally you would match it to the name of the
symmetry in a corresponding lattice.
To find out the symmetry list of an existing wavefunction, use mp-info.
1. Project an {$SU(2)$} symmetric spin chain to {$U(1)$}.
mp-wigner-eckart "Sz:U(1)" psi1 psi2
If psi2 already exists, then this will fail with an error, leaving the existing file psi2 untouched. To force overwriting psi2, add the -f option.
There are some limitations in the implementation of mp-wigner-eckart in the current version of the toolkit; these will be fixed in the future:
• If the wavefunction has more than one {$SU(2)$} symmetry, then unfortunately it isn't possible to project just one {$SU(2)$} symmetry, all of them need to be projected at the same time.
• The only projection that is currently implemented is {$SU(2) \supset U(1)$}. In the future it is hoped to generalize this to other projections, such as {$SU(2) \supset D_\infty \supset U(1)$},
and {$SU(2) \supset Z_3$}, and {$SU(2) \supset D_\infty \supset Z_2$}.
This command gets its name from the Wigner-Eckart theorem, which is the basic theorem that underlies the concept of non-abelian MPS. | {"url":"https://mptoolkit.qusim.net/Tools/MpWignerEckart","timestamp":"2024-11-14T21:02:28Z","content_type":"application/xhtml+xml","content_length":"13093","record_id":"<urn:uuid:df173c83-ecda-4ca7-aa55-8cb29a587cd3>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00379.warc.gz"} |
To describe something as "nonlinear" is to describe it by what it is not. But let's be more direct. First, let's define "linear".
Simply stated, something is linear if its output is proportional to its input. If, when you're reading late at night, you want twice as much illumination (output) to see the book, then you double the
number of light bulbs (input) by bringing over another similar lamp. If you want to buy twice as much buckwheat flour at the grocery store, you will pay twice as much.
Let's follow-up on this last example. Imagine that your store offers a bulk discount. Every additional pound of flour is 30% less that the previous pound. The incentive is to get you to buy more.
It's a nonlinear incentive: the more you buy, the bigger the discount becomes.
A more realistic example comes from an ecology of animals that compete for food, but in which there is only a fixed amount of food available each day. As long as the population is small, all the
animals get plenty of food. They grow and prosper, they reproduce and the population grows. But it can only grow so far. Once the population is beyond a balance with the available food, some animals
do not get enough. Eventually they cannot reproduce and the population size decreases. In this ecology then, the population growth is a nonlinear function of the available food. At low populations,
the growth is positive; at high populations, the growth is negative.
The concept of linearity is very closely related to that of reductionism. Reductionism is an approach to science that says that a system in nature can be understood solely in terms of how its parts
work. How can this be? If the system is a linear composition of its parts this works great, since the system as a whole is proportional to each of its parts separately. But for many phenomena this
doesn't work. For example, if you want to understand life, it is not possible to look only at the properties of the molecules in a living system. If the system is dismantled into all of the separate
molecules, it is no longer alive. Life is nonlinear; death is linear.
Both linearity and reductionism fail, at least as general principles, for complex systems. In complex systems there are often strong interactions between system parts and these interactions often
lead to the emergence of patterns and cooperation. That is, they lead to structures that are the properties of groups of parts, and not of the individual constituents. | {"url":"https://annex.exploratorium.edu/complexity/CompLexicon/nonlinear.html","timestamp":"2024-11-08T17:42:11Z","content_type":"text/html","content_length":"4835","record_id":"<urn:uuid:31958142-0769-4fff-bb49-251469be05e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00058.warc.gz"} |
Martingale in Forex Trading - Good strategy or hazard?
In this new article, we are going to have a look at the martingale system. What is martingale? Is it a viable money management strategy or pure hazard? Let's find out!
Martingale System Introduction
The martingale systems are widely used casino, sports betting, but the principles are also used by many traders in the financial markets.
And not only that. Martingale's principles are often part of the automated trading systems. How do you recognize a martingale trading system?
Usually by the fact that the system has an unbelievable balance curve.
The problem is that these systems are extremely risky. In this FX Experiment, we will examine the risk of these systems.
First, we need to clarify what Martingale is all about, the first part will be rather theoretical.
If you already know this system, you can jump straight to the second part where it is already being tested on historical data.
Let's imagine the classic casino roulette and color betting.
The result may be only that the ball ends in the black or red territory.
If the roulette works as it should be, the probability of a ball landing on a red number is the same as the probability the ball landing on the black number, 50%.
Suppose the player bets constantly on red $10.
If the ball actually falls on red in the first round, you get back your deposit and win an extra $10.
If the ball lands on a black number, then in the second round you need to deposit $20.
If you win this time, you will get back your bet of $20 and win another $20.
The winner will cover the first $10 bet and earn $10 extra. If the ball lands on the black number, the next round you will bet $40, and so on.
The principle is to double the deposit in the case of the bet is lost.
The player is expecting his or her color to fall sooner or later and make a profit of $10 regardless of the number of rounds.
If a player had unlimited capital and an unlimited number of rounds, then he would realize the endless risk-free profit.
The problem, however, is precisely the amount of capital that cannot be infinite. For illustration, the table below summarizes the Martingale principle:
The table assumes a capital of $10,000 and a $10 initial bet.
Even though we have a ten thousand times higher capital than the first bet size and therefore the expected winnings, we can lose it all fairly quickly.
As can be seen in the table, it is enough to have a streak of 9 bad colors and the player no longer has enough capital to make another bet.
There are reports that the same color fell even 30x or 40x in times in a row.
After the 30th round with an initial bet of $10, a player would have needed a capital of at least $10,737,418,230.
This is already the amount that few people have available. We encourage you to read this article written by a very famous Vegas trader on a similar subject.
The chart below shows a player who starts betting $10 on each round and has a capital of $1,000,000.
See how the bets are rapidly growing the same as the amount of capital declines.
After the unsuccessful 16th round, the player is already in minus.
The same principle is applied by traders to the financial markets.
Instead of bets on red and, they bet on the short or long side.
We will not deal with the reason why a trader enters long or short, the key is that when the market goes against the trader, the trader should open a new trade in the direction of the first trade,
but the volume is twice as big.
Traders using the Martingale systems are hoping that markets do not move in one direction without any retracement.
The obvious difference from the Martingale trading system from the casino roulette is the choice of payout ratio - the Takeprofit distance and the price range at which the new position opens if the
market goes against the trader
The Stoploss command in the Martingale system is usually completely missing, which is logical.
Instead, there are pre-prepared price levels for opening additional positions.
Just as in the case of roulette, where the underlying problem is a rising risk of bad bets, it is also true that what is seemingly impossible or unlikely will happen sooner or later and will have
fatal consequences for the trading account.
Until then, the system will be consistently profitable. However, as soon as an unfavorable scenario is reached, the result is a margin call.
In the next chapter, we will program an automatic trading system, which will try to show how this system performs in some markets.
Be very careful while considering using any form of martingale strategy on your funded forex accounts.
Results of martingale in forex trading
The automated trading system works as follows:
1. The first trade (long/short) is completely random.
2. The system immediately sets a fixed Profit target, the Stoploss order is not set.
3. If the position reaches a negative result, which equals the value of the profit target, the next position is open. Sequentially opened positions meet the following volume range: X, X, 2X, 4X, 8X,
4. Profit targets of all positions are always set according to the Profit target of the last position.
5. If Profit targets are filled, a new cycle starts from point 1.
Position sizes X, X, 2X, 4X, 8X are chosen so that the volume of the next position is equal to the sum of all previous positions. This ensures that each cycle ends with the same profit regardless of
how many positions the system opens.
Testing was performed on the most popular and volatile instruments, namely the EURUSD and the DAX stock index. We performed tests for the period 1.1.2016 - 31.12.2016 on 99.9% tick data sample. The
specific Take profit distance will be based on typical price movements and ranges. Since it is a random based system, the results will be different for the same parameters at each test, so we will do
3 tests for each instrument. We will present the test results for space reasons only in the form of the equity curve, from which you can see the profit and drawdown.
• Initial capital: $10,000
• Volume: 0,01 lot
• TP: 100 pips
This system achieved a 25% - 35% profit in all three test, which is a fairly decent result. However, three tests are not enough to fully see the level of the hidden risk. Let's do twenty cycles and
write down and see in how many cases the system survived the whole trading period (1.1.2016 – 31.12.2016). The month when the system lost its whole equity is also noted.
The table shows that the success in three consecutive tests is not such an exceptional situation. 7 tests out of 20 tests lost its entire capital. From the table, it can be assumed that the
probability of the system will bankrupt in 2016 is around 35%.
The average profit for the year is 34%.
Expected value (expectation) is: 0,65 * 3400 - 0,35 * 10,000 = -12%.
• Initial capital: $10,000
• Volume: 0,1 lot
• TP: 20 pips
The martingale system on Dax deleted the whole account in the first two tests in the first month, in the third test the Dax deleted the account in the second month.
It has been clearly shown that this system is capable of generating stable and relatively long-term gains, but they are redeemed at considerable risk.
This risk is hidden from historical results, so these systems are presented as holy grails, but they are also used by regular traders who are not aware of the risk associated with this technique and
believe that the market "can not rise/fall indefinitely without correction".
This is undoubtedly true, but just as in roulette and in markets, there are extraordinary situations that few people count on.
About FTMO
FTMO developed a 2-step Evaluation Process to find trading talents. Upon successful completion you can get an FTMO Account with a balance of up to $200,000. How does it work?. | {"url":"https://ftmo.com/en/martingale-forex-trading/","timestamp":"2024-11-02T01:24:11Z","content_type":"text/html","content_length":"99827","record_id":"<urn:uuid:9746af4d-50c7-4e0a-a5fd-8052fd346120>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00280.warc.gz"} |
Karger’s algorithm to find Minimum Cut
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In this article, we will explore how the Karger's algorithm works to find the minimum cut in a graph and its C++ implementation. This is a randomized algorithm based on Graphs.
Table of contents:
1. What is the minimum cut?
2. Karger's Algorithm
3. Success Probability and Time complexity
4. Code Implementation of Karger's algorithm
Pre-requisite: Randomized algorithm, Minimum Cut
Let us get started with Karger’s algorithm to find Minimum Cut.
What is the minimum cut?
The minimum cut in a graph refers to the number of edges in the graph that must be disconnected to split the graph into two disjoint components. Let's take a few pictorial examples to get a more
clearer idea.
In this example, the minimum cut is going to be the severing of the edges E and G. So, Minimum cut=2.
The minimum cut can also achieved by severing the edges B and D.
In this example, we see that the minimum cut is achieved by severing the edges A and B, which breaks the graph into two symmetric halves. So, Minimum cut=2.
We see that the size of the two disjoint components does not matter to us i.e. the two components can vary largely in size or be almost equivalent.
Karger's Algorithm
Karger's algorithm is a randomized algorithm to find the minimum cut in an unweighted and undirected graph in which we pick up a random edge and contract or coalesce the two ends of edge into a
single point. So, all points eventually combine into super-nodes and the size of the graph keeps decreasing. All the while, all self loops are removed. At the end of the algorithm, we are left with
two nodes connected by a number of edges. The number of edges connecting the two final super-nodes gives the required minimum cut.
So, we have the following algorithm:
While there are more than 2 vertices-
a) Pick a random edge (x, y) in the contracted graph.
b) Merge the points x and y into a single vertex (update the contracted graph).
c) Remove self-loops.
Return cut represented by two vertices.
Let us go back to our first example and see this algorithm in action visually.
• So, in the first step, we merge the vertices at the ends of edge B. So, we end up with a super node which now becomes one end of the edges A, C & D. The edge B will form a self-loop since both of
its ends have converged. So, we omit it.
• We contract the edge C (or D, same meaning). Both the edges C and D will form self loops and are thus omitted. We now have a super node which becomes the other end point of the edges A and F.
• We pick the edge A (or F) and converge the points at either end. So, both A and F form self loops and are removed from the graph. We end up with a graph with two vertices connected by the edges E
and G.
Since only two vertices remain, the algorithm terminates and 2(the number of edges remaining in the graph) is returned as the minimum.
Success Probability and Time complexity
Since Karger's algorithm is a randomised algorithm, it doesn't always arrive at the right answer in the first run. In fact, the probability of reaching the minimum cut is 2/n(n-1), which is low. So
to ensure the minimum cut, we must run the algorithm an adequate number of times. It has been observed that we need atleast n^2 log(n) runs to arrive at the optimal solution, where n is the number of
vertices in the graph.
The time complexity for merging of any pair of vertices as well as removal of self loops is O(n). Since both of these will be done on the graph until only 2 vertices remain, the time complexity for a
single run is O(n^2). But for optimality, we will be running the algorithm n^2 log(n) times, so our overall complexity shoots up to O(n^4 log(n)).
Code Implementation of Karger's algorithm
In the c++ implementation, we will be using an adjacency matrix to store the graph. This will be defined as a vector of vectors vector<vector<int>> edges. The number of rows and columns in the matrix
is equal to the number of vertices in the graph vertices. We shall make some set and get functions to set values in the graph and retrieve them as well. We also have get and set functions for matrix
Apart from these, we have three major functions essential for the algorithm- remove_self_loops(), merge_vertices() and kargers_algorithm(). Let us see these functions one by one.
• The remove_self_loops() iterates through the graph and removes all self loops i.e. edges that start and end at the same vertex. Since we are using an adjacency matrix, this function will set all
the diagonal elements as 0.
graph& remove_self_loops(){
for(int i=0;i<vertices;i++){
return *this;
We will see the set() function definition later when we examine the entire program.
• The merge_vertices() function takes two parameters u and v which are the vertices to be merged. The function iterates through the graph and for every vertex i, it adds all its edges (i,u) to the
vertex v and then, sets edges of vertex u to 0. This is also done for all (u,i) pairs as well since the graph is undirected.
graph& merge_vertices(int u, int v){
if(u< vertices && v< vertices){
for(int i=0;i<vertices;i++){
return *this;
• Now, we come to our kargers_algorithm() function. This function iterates while there are more than two vertices in the graph where it picks any two vertices at random, merges them and removes
self loops. So, we have the code:
void kargers_algorithm(graph& g)
while (g.count_vertices() > 2)
int u = 0, v = 0;
u = rand() % g.get_size();
v = rand() % g.get_size();
while (g.get(u, v) == 0);
// Merge both vertices
g.merge_vertices(u, v);
//Remove self-loops
The function defines only one run of the algorithm. So, for optimality, we will call it multiple times in our main() function.
Now, we can see the final code with a graph class and all of its member functions and see how it all comes together:
#include <bits/stdc++.h>
using namespace std;
class graph{
int vertices;
vector<vector<int>> edges;
void set(int r, int c, int d) {
edges[r][c] = d;
int get(int r, int c) {
return edges[r][c];
void set_size(int s) {
vertices = s;
edges.resize(vertices * vertices);
int get_size(){
return vertices;
int count_vertices() {
int v=0;
for(int i=0;i<vertices;i++){
for(int j=0;j<vertices;j++){
v++; break;
return v;
int count_edges(){
int e=0;
for(int i=0;i<vertices;i++){
for(int j=0;j<vertices;j++){
return e/2;
graph& remove_self_loops(){
for(int i=0;i<vertices;i++){
return *this;
graph& merge_vertices(int u, int v){
if(u< vertices && v<vertices){
for(int i=0;i<vertices;i++){
return *this;
void kargers_algorithm(graph& g)
while (g.count_vertices() > 2)
int u = 0, v = 0;
u = rand() % g.get_size();
v = rand() % g.get_size();
while (g.get(u, v) == 0);
// Merge both vertices
g.merge_vertices(u, v);
//Remove self-loops
int main()
graph g;
g.edges={{0 ,1 ,0, 1, 1, 0},
{1, 0, 1, 0, 1, 0},
{0, 1, 0, 0, 1, 1},
{1, 0, 0, 0, 1, 0},
{1, 1, 1, 1, 0, 1},
{0, 0, 1, 0, 1, 0}};
graph ming; ming.set_size(0);
cout << "Input vertex count: " << g.count_vertices() << endl;
cout << "Input edge count: " << (g.count_edges()) << endl;
int n = g.count_vertices();
float ln = log((float) n);
float runs = n * n * ln, mincut = INT_MAX;
for (int i = 0; i < runs; ++i)
graph copy = g;
int cut = copy.count_edges();
if (cut < mincut)
mincut = cut;
ming = copy;
cout << "Output vertex count: " << ming.count_vertices() << endl;
cout << "Output edge count: " << ming.count_edges()<< endl;
cout<< "Minimum cut for the graph is "<< mincut<<endl;
return 0;
In the main function, we have calculated the number of runs =n^2 log(n) using appropriate data types. We initialise a new graph ming to store the graph which has the minimum cut. We also create a
graph copy, which is initialised to our given graph g. This copy is modified by the algorithm to give the minimum cut. If the cut reached is lower than the current minimum cut mincut, then the mincut
value is updated and the graph ming stores the minimum cut graph.
Input vertex count: 6
Input edge count: 9
Output vertex count: 2
Output edge count: 2
Minimum cut for the graph is 2
Thus, in this article at OpenGenus, we have explored what the minimum cut for graphs is, Karger's algorithm for finding out the same as well as its C++ implementation. Keep learning! | {"url":"https://iq.opengenus.org/karger-algorithm-to-find-minimum-cut/","timestamp":"2024-11-04T12:20:54Z","content_type":"text/html","content_length":"71064","record_id":"<urn:uuid:19abaf80-6732-4a3f-a351-0ea3e826f310>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00026.warc.gz"} |
Modeling Pandemics (3) | R-bloggersModeling Pandemics (3)
[This article was first published on
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In Statistical Inference in a Stochastic Epidemic SEIR Model with Control Intervention, a more complex model than the one we’ve seen yesterday was considered (and is called the SEIR model). Consider
a population of size \(N\), and assume that \(S\) is the number of susceptible, \(E\) the number of exposed, \(I\) the number of infectious, and \(R\) for the number recovered (or immune)
individuals, \(\displaystyle{\begin{aligned}{\frac {dS}{dt}}&=-\beta {\frac {I}{N}}S\\[8pt]{\frac {dE}{dt}}&=\beta {\frac {I}{N}}S-aE\\[8pt]{\frac {dI}{dt}}&=aE-b I\\[8pt]{\frac {dR}{dt}}&=b I\end
{aligned}}\)Between \(S\) and \(I\), the transition rate is \(\beta I\), where \(\beta\) is the average number of contacts per person per time, multiplied by the probability of disease transmission
in a contact between a susceptible and an infectious subject. Between \(I\) and \(R\), the transition rate is \(b\) (simply the rate of recovered or dead, that is, number of recovered or dead during
a period of time divided by the total number of infected on that same period of time). And finally, the incubation period is a random variable with exponential distribution with parameter \(a\), so
that the average incubation period is \(a^{-1}\).
Probably more interesting, Understanding the dynamics of ebola epidemics suggested a more complex model, with susceptible people \(S\), exposed \(E\), Infectious, but either in community \(I\), or in
hospitals \(H\), some people who died \(F\) and finally those who either recover or are buried and therefore are no longer susceptible \(R\).
Thus, the following dynamic model is considered\(\displaystyle{\begin{aligned}{\frac {dS}{dt}}&=-(\beta_II+\beta_HH+\beta_FF)\frac{S}{N}\\[8pt]\frac {dE}{dt}&=(\beta_II+\beta_HH+\beta_FF)\frac{S}{N}-
\alpha E\\[8pt]\frac {dI}{dt}&=\alpha E+\theta\gamma_H I-(1-\theta)(1-\delta)\gamma_RI-(1-\theta)\delta\gamma_FI\\[8pt]\frac {dH}{dt}&=\theta\gamma_HI-\delta\lambda_FH-(1-\delta)\lambda_RH\\[8pt]\
frac {dF}{dt}&=(1-\theta)(1-\delta)\gamma_RI+\delta\lambda_FH-\nu F\\[8pt]\frac {dR}{dt}&=(1-\theta)(1-\delta)\gamma_RI+(1-\delta)\lambda_FH+\nu F\end{aligned}}\)In that model, parameters are \(\
alpha^{-1}\) is the (average) incubation period (7 days), \(\gamma_H^{-1}\) the onset to hospitalization (5 days), \(\gamma_F^{-1}\) the onset to death (9 days), \(\gamma_R^{-1}\) the onset to
“recovery” (10 days), \(\lambda_F^{-1}\) the hospitalisation to death (4 days) while \(\lambda_R^{-1}\) is the hospitalisation to recovery (5 days), \(\eta^{-1}\) is the death to burial (2 days).
Here, numbers are from Understanding the dynamics of ebola epidemics (in the context of ebola). The other parameters are \(\beta_I\) the transmission rate in community (0.588), \(\beta_H\) the
transmission rate in hospital (0.794) and \(\beta_F\) the transmission rate at funeral (7.653). Thus
epsilon = 0.001
Z = c(S = 1-epsilon, E = epsilon, I=0,H=0,F=0,R=0)
p=c(alpha=1/7*7, theta=0.81, delta=0.81, betai=0.588,
betah=0.794, blambdaf=7.653,N=1, gammah=1/5*7,
gammaf=1/9.6*7, gammar=1/10*7, lambdaf=1/4.6*7,
lambdar=1/5*7, nu=1/2*7)
If \(\boldsymbol{Z}=(S,E,I,H,F,R)\), if we write \(\frac{\partial \boldsymbol{Z}}{\partial t} = SEIHFR(\boldsymbol{Z})\)where \(SEIHFR\) is
SEIHFR = function(t,Z,p){
S=Z[1]; E=Z[2]; I=Z[3]; H=Z[4]; F=Z[5]; R=Z[6]
alpha=p["alpha"]; theta=p["theta"]; delta=p["delta"]
betai=p["betai"]; betah=p["betah"]; gammah=p["gammah"]
gammaf=p["gammaf"]; gammar=p["gammar"]; lambdaf=p["lambdaf"]
lambdar=p["lambdar"]; nu=p["nu"]; blambdaf=p["blambdaf"]
We can solve it, or at least study the dynamics from some starting values
times = seq(0, 50, by = .1)
resol = ode(y=Z, times=times, func=SEIHFR, parms=p)
For instance, the proportion of people infected is the following | {"url":"https://www.r-bloggers.com/2020/03/modeling-pandemics-3/","timestamp":"2024-11-05T00:23:33Z","content_type":"text/html","content_length":"95594","record_id":"<urn:uuid:4e7d9463-7333-4bde-b138-7b97d5c03a1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00521.warc.gz"} |
AutoEZ: Collected Short Examples Part 6
Examples of using AutoEZ have been created from time to time to address questions raised on various forums and reflectors. This is a collection of such examples, in some cases
slightly edited. Step by step instructions are omitted to maintain brevity.
40-30-20 Dipole Fed With Ladder Line
W5DXP has a nice page describing his No-Tuner All-HF-Band antenna, a 130 ft dipole used on 80m through 10m, fed in the center with varying lengths of ladder line. Here's how to see the effect of
different feedline lengths for a shortened 60 ft version on the 40/30/20m bands.
Define a simple dipole 60 ft long and 40 ft high (not shown), put the source at V1, and add a transmission line between the source (V1) and the feedpoint (Wire 1 / 50%) with length "=L" ft.
Use the Set Zo, VF, and Loss button to fill the remainder of the fields. Wireman 551 is used for this example.
Then create a series of test cases with L ranging from 60 ft to 90 ft, all at a constant frequency. Calculate.
Doing the same thing for two other frequencies and capturing the traces after each set of calculations yields this rectangular chart (reflection coefficient left scale, SWR right scale) on the
Custom tab.
Or this Smith chart (trace clockwise rotation = increasing line length).
Segment Lengths, AutoSeg, and Average Gain Test
K1STO discussed a spider beam antenna in the July 2005 QST. You can download the EZNEC model file from the QST Binaries page; look for the "0705ProdRev.zip" link.
Using that model merely as an example, and not because there is anything "wrong" with it, here's how to make use of the "For Information Only" columns and the "AutoSeg" dialog on the Wires tab to
quickly change segmentation levels. As a result the Average Gain Test results will improve slightly.
In model 'spiderbeam_3band_201510.ez' the segments on either side of the source are not the same length as the source segment. Besides that, at higher frequencies you'll get EZNEC segmentation
warnings for some of the other wires since the segmentation density is less than 20 per wavelength. This is what the original segmentation looks like at 29 MHz:
To improve the first issue set the segment length for wires 6 and 7 to (as close as possible) 0.1 m, the same as the short center segment for each of the driven elements. While you're at it, use
the same segment length for wires 4-5 and 8-9. That keeps the segment boundaries on those nearby wires close to alignment which is another way to improve AGT (left image below).
To eliminate the EZNEC segmentation warnings first set the "Freq" (cell C11 on the Variables tab) to 29 MHz, then set the segmentation density for all the other wires to at least 20 segs/WL at
the highest frequency of interest, assumed to be 29 MHz in this case (right image below).
With those changes the segmentation now looks like this:
Here's a comparison of the before and after AGT results (Ground Type = Free Space):
And here's a comparison of the before and after SWR values at three frequencies in each band:
Two-Element Yagi with Equal Length Elements
What? Doesn't a two-element Yagi have two different length elements, either a reflector and a driven element or a driven element and a director? Not necessarily.
The ARRL Antenna Book 10th edition (1964, cover price $2.00) has a sub-section on "Self-Resonant Parasitic Elements" for a two-element Yagi, found in Chapter 4 (Multielement Directive Arrays),
section "Parasitic Arrays", sub-section "The Two-Element Beam". KL7AJ mentioned this "Yagi with Self-Resonant Elements" in a forum posting and it gave me an excuse to dig out my earliest Antenna
Book from when I initially became interested in amateur radio.
First I created a dipole element in free space having length L, with XYZ coordinates specified in wavelengths rather than feet or meters, and used the Resonate button to set the length to
resonance at 28 MHz. Then I duplicated that element with a spacing between the two of S wavelengths. Hence both elements are the same length and both are self-resonant when stand-alone. Then I
ran a variable sweep with S ranging from 0.01λ to 0.4λ.
Here's the Triple sheet tab which shows several different parameters at a glance. The dip in the gain curve at spacing 0.11λ is where the parasitic (non-driven) element changes from being a
director to being a reflector; that is, where the main lobe azimuth angle changes from 0° to 180°. Remember, the length of the parasitic element has not changed, only the spacing.
And here's an animation of the free space azimuth patterns. The value of S for each frame of the animation is shown in the lower right corner. The outer ring is frozen at 7.33 dBi, the gain at a
spacing of 0.06λ, to allow comparison of relative strength as well as pattern shape.
As the author of this Antenna Book section said way back in 1964: "The special case of the self-resonant parasitic element is of interest, since it gives a good idea of the performance as a whole
of two-element systems, even though the results can be modified by detuning [changing the length of] the parasitic element. ... If front-to-back ratio is not an important consideration, a spacing
as great as 0.25 wavelength can be used without much reduction in gain, while the radiation resistance approaches that of a half-wave antenna alone."
Here's the model used for this exercise.
Dual-Band Vertical with "Autopilot" Matching Network
Here's a little modeling study which uses both the EZNEC "L Networks" feature and the AutoEZ optimizer in sort of a unique way.
In 1977 Wes Hayward, W7ZOI, described an interesting dual-band matching network. His Hints and Kinks QST submission is reproduced below.
Then in 2005 Dan Richardson, K6MHE, wrote a QST article titled A 20 and 40 Meter Vertical on 'Autopilot'. And in 2008 Ryan Christman, AB8XX, did a blog entry with a corresponding YouTube video
for a similar 80m/40m antenna. For those who might be interested in doing something similar, it's possible to model the entire system, antenna plus "autopilot" matching network, with EZNEC or
with the AutoEZ/EZNEC combination. (Please look over at least the K6MHE article before continuing.)
A conventional low pass L network has an inductor in the series branch and a capacitor in the shunt branch. However, EZNEC L networks can have compound components in each branch. Hence the
matching network can be pictured like this (from the K6MHE article Fig 2).
Here is the AutoEZ L Networks table with variables K-L-M being used for the component C1-L1-C2 values. The source is on V1, the network input port. The network output port is the antenna
The following extract from the AutoEZ Variables sheet tab shows the initial values for the K6MHE 40m/20m setup. Instructions for setting the initial network component values (variables K-L-M) are
found in additional comments on the sheet, not shown. (The scratchpad area of the sheet has the necessary Excel formulas for computing the initial K-L values for an impedance match at the upper
frequency and the initial M value for a net zero reactance at the lower frequency.)
Note that ground loss due to a less than perfect radial field is simulated using variable D in conjunction with MININEC-type ground; actual radials are not included in the model (although you may
add them if you wish and change to High Accuracy ground).
With the C1-L1-C2 component values as set above, the L network looks like this when the model is passed to EZNEC.
Once the initial network values are set, the AutoEZ optimizer can be used to adjust components C1-L1-C2 (variables K-L-M) to produce the lowest possible SWR at the two midband frequencies.
After optimization, here are the final SWR values for seven 40m frequencies and eight 20m frequencies. This is similar to Fig 5 in the K6MHE article except that all frequencies are shown on a
single chart. (Blue markup added for clarity.)
Doing the same kind of analysis for 160m/80m (with radiator length B = 132 ft, although not many folks have a full-size vertical for 160) gives these SWR values, this time shown with a different
scale for SWR (on the right). The midband SWR values are fine (remember, right scale) but because of the relatively larger widths of 160 and 80 the band edge SWR values are higher.
The examples above were for 40/20 and 160/80 but the two bands need not be harmonically related. Here is an example for 40/30 with three scenarios: First was a 35 ft vertical at 7.15 and 10.125
MHz with no matching network. Second was adding a "dual-band autopilot" L network with optimized component values. And third was allowing the optimizer to vary the wire length as well as the
network values; that is, optimizing on 4 variables instead of 3 variables. In all cases the model had 10 ohms of assumed ground loss.
Here is the model used for this study.
Download that file, use the Open Model File button, then tab to the Variables sheet and read the comments to get started. As downloaded, the model is ready to be optimized (Optimize tab, Start
button) and then calculated over multiple frequencies (Calculate tab, Calculate All button).
Of course, results are only as good as the model. If you have an analyzer and can measure the actual unmatched feedpoint impedance (R±jX) at your chosen target frequency in each band, you can
then adjust variables B and D (with E = False, no network) to make the calculated R and X match the measured R and X. Then model the matching network.
Ground Mounted Vertical, Resonant Height vs Element Diameter
If a "shortening factor" is defined as the ratio between the height of a resonant ground mounted vertical and a free space quarter wavelength, the following curves show the factors for element
diameters ranging from 0.125" (~ #8 AWG) to 4" at both 3.75 MHz and 28.5 MHz.
For 3.75 MHz the factors vary from 0.973 to 0.953; for 28.5 MHz, from 0.964 to 0.917. Note that the 28.5 MHz factors do not correspond to Table 1 in the Antenna Compendium Volume 2 article by
Doty et al. The Doty ACV2 measurements were done with the vertical element mounted over an elevated counterpoise, not ground mounted.
Calculations for the above chart are easy using the Resonate button of AutoEZ. The procedure is shown here.
If you'd like to duplicate these results, or run calculations for other frequencies, the following model file is suitable for use with the free demo version of AutoEZ in conjunction with any
version of EZNEC v. 5.0.60 or newer including the EZNEC Pro+ v. 7.0 program which is now free.
Download that file, use the AutoEZ Open Model File button, tab to the Calculate sheet, select all the H values (cells D11-D27), and click the Resonate button. Or change all the frequencies to
your new choice and set all the H values to approximately λ/4 at that frequency (initial value not critical), then select all the H values and click the Resonate button.
To produce a plot of the results, tab to the Custom sheet and choose "Variable 3" (Shortening Factor) for the Y axis and "Variable 1" (Diameter) for the X axis.
Modeling 8-Circle Arrays
Concerning 8-circle receiving arrays, I was curious about the relationship between array size, element phasing, and number of active elements so I created models where the array diameter and
phasing can be controlled via variables. For example, here's the AutoEZ Variables sheet tab for a "W8JI type" (aka BroadSide/End-Fire or BSEF, 4 elements in use) 8-circle array.
Since everything is controlled by variables you can run "variable sweeps" changing one or more parameters. Here's the W8JI array with the spacing (B) held constant while the phase delay (P) is
swept from 80 to 140 degrees. For each test case AutoEZ will automatically calculate the RDF (last column).
When the calculations finish you can step through the 2D patterns. Here are the elevation and azimuth (at 20° TOA) patterns for 0.604 wl broadside spacing and 125° phasing, as shown by B and P in
the lower right corner.
You can run similar sweeps changing the array size while holding the phase delay constant, or hold both size and phase constant and do a frequency sweep, or use the "Generate Test Cases" button
to create any combination. For example, the setup below would vary broadside spacing B from 0.50 to 0.70 wavelengths; for each B the phase delay P would be varied from 115 to 135 degrees; all at
a constant frequency of 1.85 MHz.
The model for a "Hi-Z type" (8 elements in use) 8-circle array is similar except that array size is specified in feet (or meters) rather than wavelengths. Here's the 3D pattern for a Hi-Z array
with diameter 200 ft and ±106° phasing, along with the 2D elevation pattern.
And here's how the RDF for a 200 ft Hi-Z array varies as the phase is swept from ±100° to ±112°.
The models also have a variable to control the azimuth direction of the main lobe. That lets you change the direction just by changing (or doing a sweep on) a single variable. So you can easily:
1) see how the azimuth patterns overlap as the array is steered in 45° steps, and 2) compare patterns against other models which may be fixed in a particular direction.
For case 1, here's how the Hi-Z type array patterns would overlap as you turn the knob on the control box. This is for a 200 ft diameter array with ±106° phasing, 20° TOA for the patterns.
Compass rose angles are shown in parenthesis on the polar chart.
For case 2, this is how the W8JI type array with 0.604 wl broadside spacing and 125° phasing compares against a 4-square receiving array with 0.125 wl element spacing and Crossfire feeding. Both
arrays use the same basic element, the top hat model RXvrhat.ez from the W8JI Small Vertical Arrays page.
At a 20° TOA the 4-square has a gain of -22.43 dBi compared to -10.95 dBi for the 8-circle. (But the RDF is only about 1 dB lower for a diameter of 94 ft compared to 348 ft for the 8-circle.) In
order to get an accurate comparison of the pattern shapes on the same polar chart, the gain of the 4-square has been normalized to that of the 8-circle by reducing the value of the swamping
resistors. Original gain below on left, normalized gain below on right.
The 4-square can be fed either as Crossfire or BroadSide/End-Fire using exactly the same phasing lines, a neat idea picked up from IV3PRK. In the patterns below, the swamping resistors are back
at the normal values (to give 75+j0 at the feedpoint for a standalone element). With Crossfire feeding the gain at 20° TOA is down 7.45 dB compared to BSEF feeding but the RDF is 2.1 dB better,
below left. Note that a design criteria for IV3PRK was good rejection at backside 45° elevation, below right.
For both of the array types, I created one model using W8JI-style top hat loaded vertical elements and a second model using simple aluminum tube elements per the four-section, 23.25 ft, Hi-Z
AL-24 antenna. Here are the models.
In all the models, a single variable (X) controls the segmentation. You can reduce that to speed up the calculations. You can also run a sweep on X to do a convergence test for model accuracy.
Here's the 4-squareRX model with a variable that switches between Crossfire feed and BSEF feed.
For comparison with the above arrays, here's the W8WWV "Benchmark Beverage" model. With this one you can "sweep" the length and/or other parameters.
ON4UN's Low-Band DXing by John Devoldere, Chapters 7 and 11:
For "W8JI type" arrays:
For "Hi-Z type" arrays:
For the IV3PRK dual-feed 4-square (follow the links on the left side of the page):
For the W8WWV Beverage:
2014 Contest Univesity presentation by W3LPL on Receiving Antennas: | {"url":"https://ac6la.com/aecollection6.html","timestamp":"2024-11-04T04:53:02Z","content_type":"text/html","content_length":"24291","record_id":"<urn:uuid:d296d5d0-fd26-49ec-8f7f-66db007e56d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00200.warc.gz"} |
next → ← prev
Micro, Macro Weighted Averages of F1 Score
Two techniques combine the F1 scores of several classes in a classification task: macro and micro-weighted averages. A model's accuracy is determined by calculating its F1 score, which accounts for
precision and recall. It ranges from 0 to 1, 1 representing the best possible score. It is the harmonic mean of precision and recall.
If the classification problem is multi-class, the F1 score can be computed separately for every class. The average of the F1 scores for each class, without considering the percentage of samples in
each class 1, is known as the macro-average F1 score. Adding up all of the true positives, false negatives, and false positives for class 1 yields the micro-average F1 score. The average of the F1
scores for each class, weighted by the quantity of samples in each class 1, is the weighted average F1 score.
Micro F1 Score:
The total number of true positives, false negatives, and false positives across all classes is considered when calculating the micro F1 score. By adding up all of the true positives, false negatives,
and false positives, it calculates the F1 score globally. A micro F1 score is appropriate when you wish to assign each data point the same amount of weight regardless of class.
TPi: True Positive for class i
FPi: False Positive for class i
FNi: False Negative for class i
Let's understand Micro F1 with an example. Suppose we have the following prediction results for multiple class problems:
Class TP (True Positive) FP (False Positive) FN (False Negative) F1 Score
0 10 2 3 0.8
1 20 10 12 0.6
2 5 1 1 0.8
Sum 35 13 16
As you can see, each class's normal F1 score has been determined. We only need to determine the mean of the three class F1 scores to return the macro F1 score, which is as follows:
Micro F1 Score is: 35 / (35 + 0.5 * (13 + 16)) = 0.71
Macro F1 Score:
The average of the F1 scores across all classes determines the macro F1 score.
The macro F1 score is obtained by taking the average of the F1 scores independently computed for each class. When evaluating the model's performance in each class equally, regardless of class
imbalance, the macro F1 score is appropriate.
n: Total number of classes
F1i: F1 Score or class i
Let's look at an example to solve the Macro F1 problem. Use the following prediction results for a multi-class problem.
Class TP FP FN F1 Score
0 10 2 3 0.8
1 20 10 12 0.6
2 5 1 1 0.8
Sum 35 13 16
As you can see, each class's normal F1 score has been determined. We only need to determine the mean of the three class F1 scores to return the macro F1 score, which is as follows:
Macro F1 Score is: (0.8+0.6+0.8)/3 = 0.73
Weighted F1 Score?
The weighted F1 score is a metric used in machine learning to evaluate the performance of a model, especially in scenarios where class imbalance exists. Let's break it down:
F1 Score:
The F1 score consists of precision and recall into a single value. It is computed as the harmonic mean of precision and recall. Precision represents the accuracy of positive predictions, while recall
measures how well the model identifies actual positive cases. The F1 score ranges from 0 to 1, with 1 being the best.
Weighted-Averaged F1 Score:
The weighted-averaged F1 score considers each class's support (i.e., the number of classes in the dataset). It is calculated by taking the mean of all per-class F1 scores, weighted by their support.
For example, if there's only one observation with an actual label of Boat, its support value would be 12.
Sample-Weighted F1 Score:
It is Ideal for class-imbalanced data distributions. It's a weighted average of class-wise F1 scores, where the number of samples in each class determines weights. Remember that the F1 score ranges
between 0 and 1 only, and it's a valuable metric for assessing a model's overall performance.
How to Calculate Weighted F1 Score?
Calculate the weighted average by allocating a weight to each class's F1 scores based on the number of instances in that class.
N is the total number of classes.
Support[i] is the number of instances in the class i.
Calculation Through Python Code
This is an example of Python code that calculates the precision and recall scores on a micro- and macro-average basis for a model trained on the SkLearn IRIS dataset, which comprises three distinct
classes: setosa, versicolor, and virginica. The model is trained using a single feature to produce a confusion matrix with numbers in every cell. Observe the training data X that iris has been
assigned.data[:, [1]].
With a single feature, the model trained on the IRIS data set would have this confusion matrix. The Python code used to calculate the precision scores for the micro- and macro-averages looks like
True positive prediction (diagonally) for all the classes
precisionScore_manual_microavg, precisionScore_manual_macroavg
The Sklearn recall_score, f1-score, and precision_score methods can also be used to calculate the same thing. To find the micro-average, macro-average, and weighted average scores, the parameter
"average" must be passed through three levels.
The F1 score, also called the F-measure, is a frequently used indicator to see how a classification model performs. We use averaging techniques to compute the F1 score when dealing with multi-class
classification, producing different average scores like macro, micro, and weight scores., in the classification report. The following section will describe the average scores, how to calculate the
average scores using Python code, and why and how to choose the best one.
Which is Better for Imbalanced Datasets?
Both micro and macro F1 scores have advantages and considerations for imbalanced datasets:
Micro F1 Points Macro F1 Score
The total number of TP, FP, and FP across all classes is used to calculate the micro F1 The macro F1 score calculates the F1 score for each class individually before averaging the results for all
score, which determines F1. classes.
It assigns the same weight to every data point, neglecting class inequality. Nonetheless, it assigns each class with the same weight.
When emphasizing the performance of the major class in the dataset, the Micro F1 score is When you want to ensure that the model performs well across all classes, regardless of their imbalance, the
more applicable. macro F1 score comes in handy.
Why Does the Scikit-Learn Classification Report Not Have a Micro Average?
Yes, a micro average for precision, recall, and F1-score is given in the scikit-learn classification_report. Nevertheless, the output of the classification report does not specifically identify it as
Micro. Rather, it is the weighted average of all classes' precision, recall, and F1 score, with each class contributing in accordance with the number of instances in which it occurs.
How Does it Work?
1. Precision Micro Average: To calculate the micro average precision, add all of the true positives in all classes, then divide that total by the total of all the false positives and true positives
in all classes.
2. Micro Average Recall: To calculate the micro average recall, add all of the true positives in every class, then divide that total by the total of all the false negatives and true positives in
every class.
3. Micro Average F1-Score: The harmonic mean of micro average recall and precision is the micro average F1-score.
These values, weighted by the total number of instances in each class, represent the aggregated performance metrics across all classes and are presented in the classification_report without being
explicitly labelled as micro averages.
Differentiates of Micro & Macro F1
Micro F1 Macro F1
The micro F1-score first determines the F1-score for each class individually before averaging the results The macro F1-score computes the unweighted average over all classes after determining the
across all classes and weighting them according to the number of instances in each class. F1-score for each class..
Larger classes are given more weight, and all classes are treated equally. All classes are treated equally, irrespective of their size..
When evaluating the model's overall performance, considering the total number of true positives, false When assessing the model's performance on each class separately and then averaging the
positives, and false negatives across all classes, the Micro F1-score is a useful tool. results across all classes, the macro F1-score comes in handy..
Because it guarantees that the evaluation is not skewed toward the majority class, it is especially It can be a more accurate gauge of the model's overall performance across all classes, even
helpful in class imbalances. imbalanced classes, because it is more responsive to small-class performance.
Which One Should I Go With, Micro F1 or Macro F1?
Selecting between the Micro and Macro F1 scores will rely on the particulars of your classification task as well as your goals:
When Use Micro F1?
• When the model's overall performance across all classes is your main concern-especially when there is a class imbalance-use the Micro F1-score.
• When the dataset is unbalanced, the Micro F1-score can be a more accurate measure of the model's overall performance by giving larger classes more weight.
• It works well when the classes are noticeably different in size or you wish to prioritise the majority class's performance.
When Use Macro F1?
• When you wish to assess the model's performance equally across all classes, regardless of size, use the macro F1-score.
• The macro F1-score gives information about the model's performance on an individual class basis before averaging the results across all classes.
• It is helpful when you don't want the evaluation skewed in favour of the majority class and want to ensure the model performs well across all classes.
The weighted averages, macro, and micro of the F1 score are selected based on the goals and class distribution of the classification task. While the macro F1-score treats all classes equally, the
micro F1-score prioritizes overall performance and favours larger classes. Weighted averages balance class sizes. Knowing these metrics makes choosing the best evaluation technique easier and
guarantees a thorough model assessment.
← prev next → | {"url":"https://www.javatpoint.com/micro-macro-weighted-averages-of-f1-score","timestamp":"2024-11-10T08:07:18Z","content_type":"text/html","content_length":"91738","record_id":"<urn:uuid:9203bc9d-9881-45d1-846d-8b3f98f5ee7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00324.warc.gz"} |
1.2. Linear and Quadratic Discriminant Analysis
1.2. Linear and Quadratic Discriminant Analysis¶
Linear Discriminant Analysis (LinearDiscriminantAnalysis) and Quadratic Discriminant Analysis (QuadraticDiscriminantAnalysis) are two classic classifiers, with, as their names suggest, a linear and a
quadratic decision surface, respectively.
These classifiers are attractive because they have closed-form solutions that can be easily computed, are inherently multiclass, have proven to work well in practice, and have no hyperparameters to
The plot shows decision boundaries for Linear Discriminant Analysis and Quadratic Discriminant Analysis. The bottom row demonstrates that Linear Discriminant Analysis can only learn linear
boundaries, while Quadratic Discriminant Analysis can learn quadratic boundaries and is therefore more flexible.
1.2.1. Dimensionality reduction using Linear Discriminant Analysis¶
LinearDiscriminantAnalysis can be used to perform supervised dimensionality reduction, by projecting the input data to a linear subspace consisting of the directions which maximize the separation
between classes (in a precise sense discussed in the mathematics section below). The dimension of the output is necessarily less than the number of classes, so this is in general a rather strong
dimensionality reduction, and only makes sense in a multiclass setting.
This is implemented in the transform method. The desired dimensionality can be set using the n_components parameter. This parameter has no influence on the fit and predict methods.
1.2.2. Mathematical formulation of the LDA and QDA classifiers¶
Both LDA and QDA can be derived from simple probabilistic models which model the class conditional distribution of the data \(P(X|y=k)\) for each class \(k\). Predictions can then be obtained by
using Bayes’ rule, for each training sample \(x \in \mathcal{R}^d\):
\[P(y=k | x) = \frac{P(x | y=k) P(y=k)}{P(x)} = \frac{P(x | y=k) P(y = k)}{ \sum_{l} P(x | y=l) \cdot P(y=l)}\]
and we select the class \(k\) which maximizes this posterior probability.
More specifically, for linear and quadratic discriminant analysis, \(P(x|y)\) is modeled as a multivariate Gaussian distribution with density:
\[P(x | y=k) = \frac{1}{(2\pi)^{d/2} |\Sigma_k|^{1/2}}\exp\left(-\frac{1}{2} (x-\mu_k)^t \Sigma_k^{-1} (x-\mu_k)\right)\]
where \(d\) is the number of features.
1.2.2.1. QDA¶
According to the model above, the log of the posterior is:
\[\begin{split}\log P(y=k | x) &= \log P(x | y=k) + \log P(y = k) + Cst \\ &= -\frac{1}{2} \log |\Sigma_k| -\frac{1}{2} (x-\mu_k)^t \Sigma_k^{-1} (x-\mu_k) + \log P(y = k) + Cst,\end{split}\]
where the constant term \(Cst\) corresponds to the denominator \(P(x)\), in addition to other constant terms from the Gaussian. The predicted class is the one that maximises this log-posterior.
Relation with Gaussian Naive Bayes
If in the QDA model one assumes that the covariance matrices are diagonal, then the inputs are assumed to be conditionally independent in each class, and the resulting classifier is equivalent to the
Gaussian Naive Bayes classifier naive_bayes.GaussianNB.
1.2.2.2. LDA¶
LDA is a special case of QDA, where the Gaussians for each class are assumed to share the same covariance matrix: \(\Sigma_k = \Sigma\) for all \(k\). This reduces the log posterior to:
\[\log P(y=k | x) = -\frac{1}{2} (x-\mu_k)^t \Sigma^{-1} (x-\mu_k) + \log P(y = k) + Cst.\]
The term \((x-\mu_k)^t \Sigma^{-1} (x-\mu_k)\) corresponds to the Mahalanobis Distance between the sample \(x\) and the mean \(\mu_k\). The Mahalanobis distance tells how close \(x\) is from \(\mu_k
\), while also accounting for the variance of each feature. We can thus interpret LDA as assigning \(x\) to the class whose mean is the closest in terms of Mahalanobis distance, while also accounting
for the class prior probabilities.
The log-posterior of LDA can also be written [3] as:
\[\log P(y=k | x) = \omega_k^t x + \omega_{k0} + Cst.\]
where \(\omega_k = \Sigma^{-1} \mu_k\) and \(\omega_{k0} = -\frac{1}{2} \mu_k^t\Sigma^{-1}\mu_k + \log P (y = k)\). These quantities correspond to the coef_ and intercept_ attributes, respectively.
From the above formula, it is clear that LDA has a linear decision surface. In the case of QDA, there are no assumptions on the covariance matrices \(\Sigma_k\) of the Gaussians, leading to quadratic
decision surfaces. See [1] for more details.
1.2.3. Mathematical formulation of LDA dimensionality reduction¶
First note that the K means \(\mu_k\) are vectors in \(\mathcal{R}^d\), and they lie in an affine subspace \(H\) of dimension at most \(K - 1\) (2 points lie on a line, 3 points lie on a plane, etc).
As mentioned above, we can interpret LDA as assigning \(x\) to the class whose mean \(\mu_k\) is the closest in terms of Mahalanobis distance, while also accounting for the class prior probabilities.
Alternatively, LDA is equivalent to first sphering the data so that the covariance matrix is the identity, and then assigning \(x\) to the closest mean in terms of Euclidean distance (still
accounting for the class priors).
Computing Euclidean distances in this d-dimensional space is equivalent to first projecting the data points into \(H\), and computing the distances there (since the other dimensions will contribute
equally to each class in terms of distance). In other words, if \(x\) is closest to \(\mu_k\) in the original space, it will also be the case in \(H\). This shows that, implicit in the LDA
classifier, there is a dimensionality reduction by linear projection onto a \(K-1\) dimensional space.
We can reduce the dimension even more, to a chosen \(L\), by projecting onto the linear subspace \(H_L\) which maximizes the variance of the \(\mu^*_k\) after projection (in effect, we are doing a
form of PCA for the transformed class means \(\mu^*_k\)). This \(L\) corresponds to the n_components parameter used in the transform method. See [1] for more details.
1.2.4. Shrinkage and Covariance Estimator¶
Shrinkage is a form of regularization used to improve the estimation of covariance matrices in situations where the number of training samples is small compared to the number of features. In this
scenario, the empirical sample covariance is a poor estimator, and shrinkage helps improving the generalization performance of the classifier. Shrinkage LDA can be used by setting the shrinkage
parameter of the LinearDiscriminantAnalysis class to ‘auto’. This automatically determines the optimal shrinkage parameter in an analytic way following the lemma introduced by Ledoit and Wolf [2].
Note that currently shrinkage only works when setting the solver parameter to ‘lsqr’ or ‘eigen’.
The shrinkage parameter can also be manually set between 0 and 1. In particular, a value of 0 corresponds to no shrinkage (which means the empirical covariance matrix will be used) and a value of 1
corresponds to complete shrinkage (which means that the diagonal matrix of variances will be used as an estimate for the covariance matrix). Setting this parameter to a value between these two
extrema will estimate a shrunk version of the covariance matrix.
The shrunk Ledoit and Wolf estimator of covariance may not always be the best choice. For example if the distribution of the data is normally distributed, the Oracle Shrinkage Approximating estimator
sklearn.covariance.OAS yields a smaller Mean Squared Error than the one given by Ledoit and Wolf’s formula used with shrinkage=”auto”. In LDA, the data are assumed to be gaussian conditionally to the
class. If these assumptions hold, using LDA with the OAS estimator of covariance will yield a better classification accuracy than if Ledoit and Wolf or the empirical covariance estimator is used.
The covariance estimator can be chosen using with the covariance_estimator parameter of the discriminant_analysis.LinearDiscriminantAnalysis class. A covariance estimator should have a fit method and
a covariance_ attribute like all covariance estimators in the sklearn.covariance module.
1.2.5. Estimation algorithms¶
Using LDA and QDA requires computing the log-posterior which depends on the class priors \(P(y=k)\), the class means \(\mu_k\), and the covariance matrices.
The ‘svd’ solver is the default solver used for LinearDiscriminantAnalysis, and it is the only available solver for QuadraticDiscriminantAnalysis. It can perform both classification and transform
(for LDA). As it does not rely on the calculation of the covariance matrix, the ‘svd’ solver may be preferable in situations where the number of features is large. The ‘svd’ solver cannot be used
with shrinkage. For QDA, the use of the SVD solver relies on the fact that the covariance matrix \(\Sigma_k\) is, by definition, equal to \(\frac{1}{n - 1} X_k^tX_k = \frac{1}{n - 1} V S^2 V^t\)
where \(V\) comes from the SVD of the (centered) matrix: \(X_k = U S V^t\). It turns out that we can compute the log-posterior above without having to explicitly compute \(\Sigma\): computing \(S\)
and \(V\) via the SVD of \(X\) is enough. For LDA, two SVDs are computed: the SVD of the centered input matrix \(X\) and the SVD of the class-wise mean vectors.
The ‘lsqr’ solver is an efficient algorithm that only works for classification. It needs to explicitly compute the covariance matrix \(\Sigma\), and supports shrinkage and custom covariance
estimators. This solver computes the coefficients \(\omega_k = \Sigma^{-1}\mu_k\) by solving for \(\Sigma \omega = \mu_k\), thus avoiding the explicit computation of the inverse \(\Sigma^{-1}\).
The ‘eigen’ solver is based on the optimization of the between class scatter to within class scatter ratio. It can be used for both classification and transform, and it supports shrinkage. However,
the ‘eigen’ solver needs to compute the covariance matrix, so it might not be suitable for situations with a high number of features. | {"url":"https://scikit-learn.org/1.2/modules/lda_qda.html","timestamp":"2024-11-02T21:24:33Z","content_type":"text/html","content_length":"37609","record_id":"<urn:uuid:608cbc5c-9a67-4e6d-9f49-e02d27d0189c>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00456.warc.gz"} |
Some Fun Facts About Eleven - Doc's Knife Works
Eleven is a two-digit natural number. It is also a prime number and is one more than ten. Let’s learn more about this number. There are a lot of interesting facts about eleven. Listed below are some
of these facts. The number 11 is also one of the prime numbers. Hence, it is a lucky number. It is also one more than nine, and is one more than ten. Here, we’ll take a look at some of its
characteristics and what makes it so special.
Eleven is a two-digit number
In mathematics, eleven is a two-digit natural number. It follows 10 and precedes 12. In other words, it is the first repdigit. It is also the smallest two-digit positive integer requiring three
syllables. In addition, it is the only two-digit number that does not have a T. Eleven also has its own name in most Germanic and Latin-based languages. Eleven is the first compound number in many
other languages.
The atomic number of sodium is 11. Group 11 of the Periodic Table of Elements includes copper, gold, and silver. Recently synthesized element roentgenium is also found in this group. Eleven is also
the first manned spacecraft to land on the moon. The approximate period of the sun’s sunspots is eleven years. This number also has many other interesting facts. Eleven is often used in sports,
including football.
In mathematics, a two-digit number has two places for each ‘digit.’ The one’s place is on the right side of the number while the ten’s place is on the left. In addition, the value of the ten’s place
is ten times its original value. These are two different kinds of two-digit numbers. When you look at them, you will notice that they are similar to each other and are written differently.
It is a prime number
11 is a prime number because it has exactly two factors, 1 and itself. This makes it an extremely rare number to find, so this fact is especially important. Prime numbers are the only ones that can
be divided evenly, making them a unique type of number. The prime factor of a number is its divisor. Likewise, a composite number is a number that can be divided by more than two factors. Therefore,
11 is a prime number.
There are several reasons why 11 is a prime number. One of them is because it isn’t an even number. There are no divisors, like two and three, that can divide it evenly. A prime number, on the other
hand, has two factors. As a result, 11 cannot be divided into two parts. For instance, an even number is made up of two divisors, such as three, four, and six.
The informal notion of a prime number excludes the number 1. The reason for this is that the number can only be broken apart into two parts if it is the product of two positive integers. Therefore, 1
cannot be a prime number, but it is an exception to this rule. Moreover, 11 is not a perfect square. Hence, the prime number criteria does not apply to it. However, 11 is an exceptional case and is
considered a prime number.
There are 25 prime numbers between one thousand. However, they can go beyond one thousand. In addition to eleven being a prime number, 17 is not a prime number and is not divisible by three and five.
Hence, 17 and 11 are sexy primes. If you consider their factors, both are prime numbers. If you consider all of them, 11 is the prime number of 29. One other reason why 11 is a prime number is that
it is close to two other prime numbers.
Prime numbers are numbers that have no other positive integers but themselves. This means that 11 cannot be divided by two negative numbers without resulting in a remainder. In addition to this, 11
is the smallest integer that requires three syllables. And finally, 11 is the only two-digit number that does not contain a T. A prime number is often a very important number in a mathematical
It is 1 more than 10
What is 1+1 more than 20? This depends on the number being more than or less than 10. If 10 is more than 11, then it is 11 and vice versa. However, if you want to add a 10 to a higher number, it
would be 20. And so on. If there is one more number than ten, then the next number is 11.
It is a natural number
The natural number 11 is a two-digit number that comes after the digits 10 and preceding the digits 12. It is the only two-digit positive integer that does not require a T. 11 has its own name in
Germanic languages and Latin-based languages. It is also the first compound number in many languages. Here are some fun facts about 11.
There are two types of numbers: natural and whole. Natural numbers fall on the right side of the number line, excluding the zero. Natural numbers are a subset of whole numbers and are arranged in a
pattern. This arrangement makes them easier to memorize. The number 11 is the second most frequently used number in English, and is the eighth most common number in the world. It is also known as a
prime number.
Another type of natural number is n. A natural number is any number not divisible by a base ten. The set of natural numbers is one, two, and three. The only natural number not in this category is -1.
The natural number N is a composite number, with all other natural numbers being integers. In addition, N is closed, associative, and commutative. It is also divisible by two.
As natural numbers, eleven is a positive integer. In fact, it is the only positive integer that begins at zero. There are no negative numbers, and they continue to infinity. The sum and product of
two natural numbers is 8000. Natural numbers are not decimal, which makes 11 an ideal natural number for mathematical calculations. The sum of two natural numbers is 2004, while the product of two
natural numbers is 8000. All natural numbers are whole numbers, except for zero.
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How to Keep Up With Health News | {"url":"https://www.docsknifeworks.com/some-fun-facts-about-eleven/","timestamp":"2024-11-13T11:48:10Z","content_type":"text/html","content_length":"277890","record_id":"<urn:uuid:d9c8c13e-fd32-4c53-b3c1-a7c6c92bdc9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00234.warc.gz"} |
Eat all apples in the maze
Hello All,
I’ve just started to learn how to solve competitive problems. I met such a problem, but not sure what would be a right way to solve it. Would really appreciate if you can help with that
My thoughts:
I already solved one problem using BFS for finding the shortest path from start of maze to the end, but not sure if that algorithm would be useful for solving this problem as well.
I would keep coordinates of apples in the binary search tree (e.g. node 2 would contain all coordinates of apples with size 2). Then I would apply in-order traversal through this BST finding shortest
paths to the next apple using BFS. If there are many coordinates contained in one node (e.g. 2), compare and choose the shortest path to the next apple. Does it sound correct? I just believe that
there is an easier solution.
• Given sizes of a matrix: N and M.
• Given matrix itself.
Matrix contains the following values:
1. 1s - cells where snake can go
2. 0s - where snake can’t go
3. bigger than 1 - cells with apple sizes
Snake should eat all the apples in ascending order (2s, 3s, 4s, etc.) Multiple apples could have the same size.
Find the shortest path to do that. Return -1 if snake can’t finish all the apples.
this link might help you to solve your problem,
Thanks, that is what I was thinking about. Do you think that it is an efficient way to keep track of apples in ascending order and just apply this algorithm to each “supposed next” apple? one by one
yes this is one of the best way to solve this kind of problem | {"url":"https://discusstest.codechef.com/t/eat-all-apples-in-the-maze/16675","timestamp":"2024-11-11T00:37:21Z","content_type":"text/html","content_length":"26127","record_id":"<urn:uuid:4fe40af5-1a19-4d37-85dd-220f90fc98c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00664.warc.gz"} |
Efficiency calculation: test conditions and final considerations
Hello everyone!My question is related to this thread.I didn’t quite understand whether the last comment (by Kris) actually confirms the assumptions made about calculating efficiency with a thermal
Is it correct to include Psw at the denominator as the user suggests?
I noticed this problem also in my simulations and you don’t notice it until you calculate low loads efficiency but in general, it should slightly change the efficiency also at higher load in some
For example, I attached a modified version of the DAB demo model. I decreased by a factor of 100 all the MOSFET and diodes resistances (and Co_esr) to not consider them in the efficiency calculation
(Pohm=0) and to get the same power at the input and output… Let’s say I only want to consider switching and conduction losses from the thermal model to calculate the efficiency in two ways:
• eff=1-Psw/Pin- eff_real=1-Psw/(Pin+Pws)
If the output resistance is set to 1kW at a nominal voltage (Vout=380V), the two efficiencies are almost the same also for lower power.Let’s say now that the DAB is in a voltage mismatch condition
between primary and secondary DAB sides (Vout=200V) so we have an increase of the reactive power and the RMS current.If the resistor is set to 1kW, efficiency does not change too much (more or less
0.2%) and in this condition, we have more or less Psw=40W (only MOSFET losses without diode losses but If I included diode losses, Psw would get worse.).
Now, if I keep the voltage mismatch condition (Vout=200V) but I reduce the output resistor power to 20W, here we have eff<0 and eff_r=32%. I know that 20W is ridiculous compared to the nominal power
of the converter but this test is only to stress out this behavior and try to understand the “real” efficiency formula. In general, the efficiency difference increase with the voltage mismatch
Are my considerations correct? Am I being too fussy? Am I overlooking something?Actually, since the Psw are not supplied by Pin because it’s the electrical model, they should always be included at
the denominator, right?
Thank you for your help!
dab_mod_eff_calc.zip (42.3 KB)
The assumption that the semiconductor device losses are much smaller than the processed power is fundamental to the lookup table based approximation PLECS uses for thermal modeling. That is, Psemi <<
Pin. This is the goal with most power conversion systems and so the simplified formula of 1-Psemi/Pin in the documentation is a very good approximation of the efficiency in most cases. The formula is
simple and easy to understand.
Thermal models should only be applied under the conditions outlined above. Variations on the efficiency formula can provide additional insight, but may not be accurate as the different formulations
impact the estimated efficiency only when the above conditions are not met.
Thank you very much for your reply!
This is exactly what I expected… | {"url":"https://forum.plexim.com/t/efficiency-calculation-test-conditions-and-final-considerations/1071","timestamp":"2024-11-07T12:33:55Z","content_type":"text/html","content_length":"28782","record_id":"<urn:uuid:b913b7da-0e3c-48fc-88f6-5d5fbad3ceba>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00113.warc.gz"} |
What is the gain of instrumentation amplifier?
What is the gain of instrumentation amplifier?
The overall gain of the amplifier is given by the term (R3/R2){(2R1+Rgain)/Rgain}. The overall voltage gain of an instrumentation amplifier can be controlled by adjusting the value of resistor Rgain.
The common mode signal attenuation for the instrumentation amplifier is provided by the difference amplifier.
What is gain in instrumentation?
The most commonly used instrumentation amplifier circuit is shown in the figure. The gain of the circuit is. The rightmost amplifier, along with the resistors labelled and is just the standard
differential-amplifier circuit, with gain = and differential input resistance = 2ยท .
What is the output of instrumentation amplifier?
The output impedance of the instrumentation amplifier is the output impedance of the difference amplifier, which is very low. The CMRR of the op-amp 3 is very high and almost all of the common mode
signal will be rejected.
How do you find common-mode gain of instrumentation amplifier?
Each half of the amplifier can be seen as a simple noninverting amplifier (with Gain=Rf/Rin+1). Note that the gain set resistor is also split in half, so the gain of each half is Gain=2Rf/Rg+1. Also
note that the common-mode voltage (Vcm) is transferred to the output of both halves of the amplifier.
What is common-mode gain?
1.3. Common-mode voltage gain refers to the amplification given to signals that appear on both inputs relative to the common (typically ground). You will recall from a previous discussion that a
differential amplifier is designed to amplify the difference between the two voltages applied to its inputs.
What is the common-mode gain formula?
To measure common mode gain, connect both inputs of the instrumentation amplifier to a sine wave generator and measure Vin and Vout vs frequency. Gc = Vout/Vin. To measure differential gain, ground
one input and connect the other to a sine wave generator and measure Vin and Vout vs frequency.
How do you find common mode gain of instrumentation amplifier?
What is the difference between differential and common mode gain?
Common mode voltage gain results from the same signal being given to both the inputs of an op-amp. If both signals flow in the same direction, it creates common mode interference, or noise.
Differential mode is the opposite of common mode, in that the direction of the signals are different.
What does gain 1 mean?
A gain of factor 1 (equivalent to 0 dB) where both input and output are at the same voltage level and impedance is also known as unity gain.
What is gain in dB?
It is usually defined as the mean ratio of the signal amplitude or power at the output port to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units
(“dB gain”).
How is dB gain calculated?
So if a circuit or system has a gain of say 5 (7dB), and it is increased by 26%, then the new power ratio of the circuit will be: 5*1.26 = 6.3, so 10log10(6.3) = 8dB….Decibel Table of Gains.
dB Value Power Ratio 10log(A) Voltage/Current Ratio 20log(A)
6dB 4 2
10dB 10 โ 10 = 3.162
20dB 100 10
30dB 1000 31.62
What is amplifier gain in dB?
In an amplifier, gain is simply the ratio of the output divided by the input. Gain has no units as it is a ratio. However, amplifier gain is often expressed in decibel units, abbreviated dB. This is
the base-10 logarithm of the output/input ratio multiplied by a factor of 10. Gain = 10log10[Output/Input] dB.
What is the gain of differential amplifier?
The gain of a difference amplifier is the ratio of the output signal and the difference of the input signals applied. From the previous calculations, we have the output voltage VOUT as. VOUT = R2 /
R1 (V1 โ V2) So, Differential Amplifier Gain AD is given by. AD = VOUT / (V1 โ V2) = R2 / R1. | {"url":"https://www.davidgessner.com/life/what-is-the-gain-of-instrumentation-amplifier/","timestamp":"2024-11-03T19:52:49Z","content_type":"text/html","content_length":"42779","record_id":"<urn:uuid:9d53df0d-c404-40e8-ac42-dbf51bde5518>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00630.warc.gz"} |
Online Betting Id | The Mathematics Behind Satta Matka: Probability and Predictions
The Mathematics Behind Satta Matka: Probability and Predictions
By merging chance with strategy, Satta Matka is deeply embedded in mathematical ideas. One can understand the mathematics behind Satta Matka and particularly probability as well as predictions that
will go a long way in boosting your bets. In this article, we are going to look into numerical aspects of Satta Matka such as how probability theory as well as mathematical analysis can help you make
informed decisions thus increasing your chances of success.
Basic Probability
Probability is at the heart of Satta Matka and other forms of gambling. It measures the likelihood of an event happening and it is expressed by numbers ranging between 0 and 1 where 0 means
impossible while 1 means surety. Here are some fundamental concepts about probability that pertain to SattaMatka:
Single Event Probability: To get the single event probability divide favorable outcome number by total outcomes possible.
Multiple Events Probability: The probabilities for several events may be combined using addition or multiplication rules depending on whether these events are independent or mutually exclusive.
For instance, when there’s a simple drawing for a satta matka game from which you select one number out of ten possible numbers, how probably would you win? When dealing with multiple events,
probabilities can be combined either through addition or multiplication depending on whether the events are independent or mutually exclusive which would resultantly lead to situations such as
illustrated above e.g. in a simple drawing for a satta matka game if you pick one number out of ten possible numbers then what are your chances if winning?
Applying Probability to Satta Matka
Knowing about probability in Satta Matka helps one make better bets strategically. Below are some ways that you can apply probability concepts to your betting strategies:
Calculating Odds: Employing probability enables us to compute odds for various outcomes hence enabling us to judge risks and rewards tied to each bet.
Diversifying Bets: Wager on different numbers as well as combinations in order to heighten the overall winning likelihood. Besides reducing the weight of single loss, this approach increases the odds
of striking a winning combination.
Historical Data Analysis: Analyzing historical data helps to track significant patterns and trends. Past results do not necessarily predict future outcomes but they give some insights into
probabilities of certain numbers or combinations.
Predictive Analytics in Satta Matka
Predictive analytics uses statistical techniques and historical data to make forecasts about what will happen next. In relation to Satta Matka, predictive analytics comes in handy in the following
Trend Analysis: Delve deep into past outcomes looking at long-term trends and patterns that are common among them. This will help us find out more frequently appearing numbers or combinations.
Regression Analysis: Regression models use different variables to analyze relationships between them and forecast future results. For example, you might explore how the occurrence of certain numbers
correlates with specific draws.
Machine Learning: Advanced players and platforms employ machine learning algorithms which can analyze large volumes of data to generate predictions; these complex algorithms recognize difficult
patterns; they become more accurate over time too.
Statistical Tools for Satta Matka
Several statistical tools can aid in your analysis and prediction efforts:
Probability Distributions: Learn about various probability distributions like uniform distribution and normal distribution among others used for analyzing satta matka results.
Variance and Standard Deviation: Variability across outcomes tells you how consistent different numbers or combinations are by measuring variance while lower variance means that there is higher
Bayesian Inference: Use Bayesian methods where you update what the chances are for an event happening based on new information or evidence received. As more data is gathered this approach makes you
refine your predictions accordingly.”
Tips you can use in Satta Matka Using Mathematics
Below are practical tips on how to integrate mathematical elements into your Satta matka strategy.
The simplest beginning: Start with the basic probability computations, and then gradually include more complicated techniques as you get comfortable.
Keep Tracks: Keep a record of detailed information about your wagers, results, and analysis. This information is crucial for refining your strategies and enhancing your predictions.
Stay Updated: Keep adjusting your analysis with recent data. The accuracy of your forecasts depends on how current the statistics are.
Embrace Technology: Explore applications that have statistical analysis and predictive tools. You can take advantage of advanced functions available through platforms such as Online Betting Id which
will enhance analytical power.
The math behind satta matka , specifically probabilities and predictions, has a big role to play in improving one’s betting strategy. With knowledge about these calculations torn into action, one is
capable of making informed choices, managing risks effectively and improving chances for success. Trust Online Betting Id to provide the tools and resources you need to navigate the numerical aspects
of Satta Matka. Use mathematics to enjoy strategic betting experience that rewards highly!
Leave a Comment | {"url":"https://online-bettingid.in/blog/the-mathematics-behind-satta-matka-probability-and-predictions/","timestamp":"2024-11-09T23:28:39Z","content_type":"text/html","content_length":"106348","record_id":"<urn:uuid:7bf2b6dc-a04b-49f3-b3bd-ae53d161caf6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00209.warc.gz"} |
Derivative of $\sec x$
The derivative of $\sec x$ is $\boxed{\sec x\tan x}$. We will use our knowledge of the derivatives of $\sin x$ and $\cos x$ to prove this result. Recall that
\begin{align*} \dfrac d{dx} \sin x = \cos x\end{align*}
\begin{align*} \dfrac d{dx} \cos x = -\sin x\end{align*}
For a detailed discussion and proof of the derivative of $\cos x$, click here. For a discussion and proof proof of the derivative of $\sin x$, click here. Since
\begin{align*} \sec x = \frac{1}{\cos x}\end{align*}
it makes sense to use the quotient rule, which states that for two functions $f(x)$ and $g(x)$
\begin{align*} \frac d{dx}\bigg(\frac{f(x)}{g(x)}\bigg) = \frac{f'(x) g(x) – g'(x) f(x)}{g^2(x)}\end{align*}
In this case, will let $f(x) = 1$ and $g(x) = \cos x$. Then $f'(x) = 0$ and $g'(x) = -\sin x$. We will now substitute these values into the quotient rule to get
\begin{align*} \frac d{dx}(\sec x) = \frac d{dx}\bigg(\frac{1}{\cos x}\bigg) = \frac{0\cdot\cos x – (-\sin x)1}{\cos^2 x} = \frac{\sin x}{\cos^2 x}\end{align*}
At this point, we recall that
\begin{align*} \frac{\sin x}{\cos x} = \tan x\end{align*}
\begin{align*} \frac{1}{\cos x} = \sec x\end{align*}
In conclusion, we combine these two facts to get
\begin{align*} \frac d{dx}(\sec x) = \frac{\sin x}{\cos^2 x} = \frac{1}{\cos x}\cdot\frac{\sin x}{\cos x} = \sec x\tan x\end{align*}
Similarly, we may use this technique to find the derivative of other trigonometric functions, like $\tan x$, $\csc x$, and $\cot x$. You can find the detailed steps for finding the derivative of $\
tan x$ here. Similarly, the steps for finding the derivative of $\cot x$ are here and for $\csc x$ are here. | {"url":"https://iacedcalculus.com/derivative-of-sec-x/","timestamp":"2024-11-07T22:02:36Z","content_type":"text/html","content_length":"607829","record_id":"<urn:uuid:45872089-6d1d-4d09-b054-f173f3c910bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00119.warc.gz"} |
Option Pricing | SpiderRock Documentation
Version: 8.4.10.2
The SpiderRock Connect trading platform includes a family of proprietary pricing models that are used to compute prices, implied volatilities, common option greeks (delta, gamma, theta, vega, rho,
and phi), and various scenario risk slides for equity and futures options. These pricing models are used in many contexts throughout the platform, including the live servers, GUI tools, and SRSE
proxy tables. This document provides an overview of these pricing models and related topics.
Generalized Black-Scholes
SpiderRock Connect's pricing models are solutions (sometimes numeric) to the standard generalized Black-Scholes equation (Haug, 2007):
$\frac{δV}{δt} + \frac{1}{2}σ^2S^2\frac{δ^2V}{δS^2} + (r-q)S\frac{δV}{δS}0-rV=0$
where 𝑉(𝑆,𝑡) is the price of a derivative as a function of stock price S and time t, the variables r and q are the risk-free rate and dividend rate, respectively, and σ is volatility. The equation
was originally introduced to price stock options but can be extended to price options on other underlying instruments as well.
The volatility input is not directly measurable, so any pricing solution to this equation is implemented alongside a corresponding inverse function for volatility, known as the implied volatility
Within SpiderRock Connect, each solution to the differential equation is implemented in the following functional forms:
$OptionPrice = PriceFunction(ex, cp, strike, uPrc, years, vol, rate, sdiv, dividends)$
$Volatility = VolatilityFunction(ex, cp, strike, uPrc, years, opx, rate, sdiv, dividends)$
• ex = Exercise type [American, European]
• cp = Option type [Call, Put]
• strike = Option strike price
• years = Years to expiration (see section on Time to Expiration)
• uPrc = Underlying price
• vol = Volatility
• opx = Option price (premium)
• rate = Average discount rate to expiration
• sdiv = Average continuous stock dividend rate to expiration
• dividends = List of discrete dividend payment dates and amounts.
The general form of this equation has several regions of interest. Some regions can be priced using fast closed form solutions (e.g. European calls with no dividends and sdiv = 0), and other regions
allow pricing with other analytical methods that are very accurate (e.g. American puts with no dividends and fewer than 2 weeks to expiration). There are also cases that require more complex and
significantly slower numerical methods to accurately compute solutions (e.g. American puts with non-trivial discrete dividends).
To address these different pricing situations in practice, the SpiderRock general pricing model is a patchwork of distinct sub-models for a variety of regions of interest. Whenever a closed form
solution exists for a region, it will be used. Otherwise, an appropriate numerical method will be chosen. Note that most numerical methods offer only approximate solutions and usually feature a
trade-off between accuracy (number of tree steps, grid size, etc.) and computational time (number of calculations per second). SpiderRock Connect typically targets numerical model accuracy that works
out to around 1/10th of the minimum price variation in the market in question. However, it is impossible to guarantee this level of theoretical accuracy for all options all the time.
As an example, consider the convergence plot below for a tree calculation of an OTM put option on a stock with an sdiv of 1.25%. For price comparison purposes, the benchmark price of the option is
calculated with a 10,000 step CRR binomial tree (Cox, Ross, & Rubenstein, 1979), and the SpiderRock pricing model is used to calculate the option price across a range of tree steps. The plot shows
the decrease in pricing error (relative to the benchmark) of a tree calculation as the number of time steps is increased. The same calculation is also shown for a standard CRR tree across the same
range of time steps.
For this specific set of market inputs, the SpiderRock pricing model will use a tree with 301 time steps. Note that the pricing error is roughly half of $0.001, which is well below the stated
accuracy goal of 1/10th of the minimum price variation (assuming a minimum tick increment of 0.01). Moreover, the convergence is significantly better than that of the basic CRR tree, which would
require more than twice the number of steps to achieve similar results.
The SpiderRock pricing models have been tested and calibrated to known accurate (but slow) models as well as to the published academic literature and are suitable for precise pricing and trading of
all listed options. Our core pricing models have been in use for many years and have held up well in a variety of markets.
In practice, accurate pricing and trading of options requires both high-quality pricing models and precisely defined inputs. Differences between values computed from one trading system to the next
can often reflect differences in the inputs rather than differences in the choice and implementation of pricing models. Some inputs are universally the same (e.g. cp, strike, vol, opx). Others are
precise, but there can be differences depending on implementation (e.g. uPrc can use mid-market, opposite side, weighted average, latency, etc.). The interest rate inputs, rate and sdiv, can require
interpolation or estimation from other markets, and they also depend on how a system represents the passing of time. Moreover, both rates and volatilities are expressed in time-dependent units.
Discrete dividends, which can require projection from historical payment patterns, also generate differences when the dividend times and/or amounts are not the same between systems.
The SpiderRock Connect trading platform, by default, attempts to make appropriate choices for all inputs and can be used as-is without concern in most reasonable actively traded markets. Note that it
is possible, in many but not all contexts, to override the default inputs supplied by the platform.
Underlying Price
SpiderRock Connect defaults to using the current (live) underlying price. The underlying price is usually taken to be mid-market for markets less than 5 ticks wide and the last print (or bid/ask if
print is outside of current bid/ask) for wider markets.
When working orders, it is also possible to choose the opposite side of the underlying market (the bid or offer). This is done by picking one of the ‘X’ limit variants (eg. DeX, VolX). The platform
will determine which direction you need to trade in the underlying market to hedge the option you are buying or selling, and it will price options under an assumption that you will need to cross the
underlying market to hedge.
It is also possible in the SymbolViewer to override the current live market price with any price by specifying it as an override.
Platform Defaults
uPrc: Mid-market (tight markets) or last print (wide markets).
Dividend, Interest, and Carry Rates
The generalized Black-Scholes model involves 3 interest rates: risk-free rate (rate), dividend rate (sdiv), and carry rate (carry). They are related via the formula:
$carry = rate - sdiv.$
This relationship between rates means that the generalized Black-Scholes pricing model depends on only two interest rates, since any of the three rates can be expressed in terms of the other two.
Typically for equity pricing, as is the case with SpiderRock Connect, the inputs are entered in terms of rate and sdiv.
The risk-free rate determines how to value cash flows at any given point in time. For option evaluation, it can be viewed as the rate used to discount the future expected payout. The carry rate
relates to the expected value of the underlying asset in the future. Under the simplest of assumptions (no dividends and not hard-to-borrow), the carry rate is equal to the risk-free rate for equity
options. This can change, for example, when there are discrete dividends to be paid in the future, and the market prices don’t line up with the estimated dividend times and/or amounts in place.
For futures option pricing, the carry rate is set to zero (as with the Black-76 model). In SpiderRock Connect, this is accomplished by setting sdiv equal to rate.
Platform Defaults
• rate = Term rate interpolated from risk-free rates
• sdiv = Option-market implied rate (see section on Dividend Rate Calibration)
• rate = Term rate interpolated from risk-free rates
• sdiv = rate
For more information, please read our technical note on Interest Rate Term Structure.
Discrete Dividends
The SpiderRock pricing models use a numerical method (modified binary trees w/splicing) to price options with discrete dividends for American calls and puts (Vellekoop & Nieuwenhuis, 2006). This
method is accurate for both small and large dividend values, and while it is slow in absolute terms, it is relatively fast compared to other reasonably accurate alternatives.
SpiderRock Connect includes estimates of future dividend dates for all equities that are known to be dividend-paying. If a company has announced a future dividend date or amount, the announced values
will typically be used. In addition, for companies that regularly pay dividends, estimated payment dates and amounts will be used to supplement any announced dividends.
The default discrete dividend streams currently in use for pricing options in the platform can be accessed via the SRSE/SRAnalytics table msgSRDiscreteDividend.
It is also possible to override discrete dividends in the SymbolViewer by double-clicking on the dividend panel. However, such overrides only apply to calculations performed in the SymbolViewer, as
well in any volatility limit orders generated from the SymbolViewer. These local (per-user) overrides will not affect risk or other calculations performed by background servers.
In addition, discrete dividend overrides can be supplied for individual parent orders when uploading orders via SRSE.
Dividend Rate Calibrations
SpiderRock Connect continuously calibrates call and put implied volatilities for expirations by adjusting the sdiv pricing parameter to minimize the mismatch between call and put surfaces for each
individual expiration. This results in an implied sdiv value being computed for each option expiration with reasonable public markets.
This implied parameter can be interpreted as either a market-based correction to a discrete dividend estimate or as an implied estimate of the hard-to-borrow (HTB) rate of the underlying security or
Most equity securities that are considered general collateral can be borrowed or lent at the overnight risk-free rate (± borrow fee). If a security goes hard-to-borrow then the borrow fee typically
increases from the small general collateral value to a (potentially) much larger value. This has the same effect on cash flows as a nightly dividend (percentage of the underlying price) paid or
received when holding or lending the underlying security. As a result, we treat this effective cash flow as a continuous stock dividend (sdiv) in our option pricing models.
Note that most clearing firms quote an overnight HTB rate for securities that go hard-to-borrow. This rate is related to the sdiv rate via the formula:
sdiv = Overnight Rate - HTB Rate.
As the HTB rate gets smaller and becomes more negative, the sdiv rate gets larger and becomes more positive. Also, the sdiv for a general collateral with a borrowing rate of 0% would also be expected
to be 0%. In effect, the sdiv rate can be thought of as a measure of how much harder to borrow any given security has become versus the general collateral rate.
The HTB rate quoted by most clearing firms is the overnight HTB rate and is usually good for one day only. This rate, of course, is subject to change over time and may or may not reflect the average
expected HTB rate to option expiration. Implied sdiv values, on the other hand, are computed for each individual expiration and (in the case of no or accurate discrete dividends) reflect the average
or expected borrow rate to that expiration date.
Note that high short-term HTB rates often revert to more moderate rates over time and that this effect is typically observable in implied market sdiv rates across option expirations.
Also, indexes such as the SPX that have many components that themselves pay dividends can be treated as paying a continuous dividend rather than a large series of discrete dividends. In these cases,
the estimated implied sdiv values represent the expected cumulative average dividend payment stream to expiration rather than a HTB rate or an actual continuous dividend rate, and the net effect on
the underlying is a decrease in the expected forward price of the index.
By looking at differences between the implied call and put volatilities, it is possible to solve for an implied forward underlying price. SpiderRock Connect's pricing engine will automatically adapt
to the market consensus of the forward price by calibrating the sdiv input so that the near ATM implied volatilities match between calls and puts.
As an example, consider the above snapshot of the June 2017 volatility surface, taken in October 2016.
By adjusting the sdiv input in the upper right-hand corner of the tool, it is possible to demonstrate the impact of a change in the input sdiv rate to the implied volatility markets.
At the time of the market snapshot, the sdiv was calibrated to 23 basis points. The following graphic demonstrates the effect of a 50-bp adjustment up and down from that level:
Note that the call and put markets are misaligned on either side of the calibrated 23-bp level.
Over the course of the trading session, the sdiv input is continuously adjusted to keep ATM implied volatilities in line. Generally, the rate level is stable, but it can and will change as the
markets move. When the sdiv is small, as in this example, it can be considered as a slight adjustment to account for uncertainties such as the ones mentioned above. In this particular example, the
adjustment is small, as evidenced by the dollar value implied by applying the sdiv interest rate to the underlying price and option time to expiration.
$Forward \space price \space adjustment = S \times (1-e^{sdivT})$ $= 28.85 \times (1- e^{0.0023 \times \frac{160.7}{252}})$ $= 0.0423$
Note the resulting 4.2-cent adjustment is relatively small, especially considering that the dividends paid to expiration are estimated at 3*23=69 cents.
Time to Expiration
When an interest rate is established, it includes an assumption of day count, or rather, the convention used to measure time. This same concept is also integral to volatility. If an option is priced
using a 365-day convention, the volatility implied from the market will be different than when the pricing model uses a business day count convention. The time convention in place also affects the
amount of theta decay that occurs both during and in-between market hours.
When a trading system is configured to use a business day calendar convention, the time-to-expiration only decays during market trading hours, whereas in between each trading day, during holidays,
and over weekends, no time can elapse. In prior versions of SpiderRock Connect's trading platform (v6 and earlier), a business day count convention was used. The rule-of-thumb for calculating time
was to start with the number of days left to expiration (including the current day) and subtract the fraction of the trading day that had elapsed since the market opened. This fraction was easily
calculated by taking the number of trading hours that had occurred since 8:30 AM CT and dividing by the 6.5 hours in a typical US equity market trading day.
This approach oversimplifies the relationship between the underlying price movement during the trading session, and after the market is closed. It assumes that no volatility occurs unless the market
is open, and it ignores the possibility that a contract might have trading activity outside of these hours.
Alternately, a continuous, constant-rate time could be employed, where time ticks down minute by minute, at a constant rate, regardless of whether the market is open or closed.
However, neither approach fits the market conditions well. There is observable price volatility that occurs in between market hours, so applying a naïve business day count convention is not
consistent when comparing end-of-day volatility with opening volatility during the following trade window. Further, the measurable volatility in between trading sessions is observed to be less than
the volatility during market hours, so applying a constant time clock that treats open market hours and closed market hours as equal, will similarly be inconsistent.
The latest version of SpiderRock Connect (v8 and higher) applies a hybrid time convention that takes both observations into account. During market hours, time is weighted more heavily. In other
words, time decays more rapidly during trading hours than it does during overnight and weekend hours. Every minute of the day is treated differently, depending on whether the market session is open
or closed, and the resulting time is used in conjunction with volatility.
One minute during trading hours does not amount to the same fraction of a year as a minute during non-trading hours. Time is accounted for, minute by minute so that it is measured in units of years,
but the difference in time-to-expiration between two separate moments depends specifically on how many hours are trading hours versus how many are non-trading hours.
Consider an option expiring exactly 48 hours from now, after two consecutive days of trading. For the sake of simplicity, if a trading session is 8 hours long, then there will be twice as much time
spent when the market is closed as during market hours (i.e. 16 hours off, then 8 hours on). If the forecast volatility used to price the option is 20%, but 70% of the price variance is expected to
occur during the two 8-hour trading sessions, then the time decay can be shifted to make the total variance accrue at a constant rate:
Note that the two plots represent the same relationship between time and volatility. The difference comes from a change in perspective, between the actual hours that pass and the time inferred
from an assumption of constant variance. In order to make volatility decay at a constant rate, the “volatility time” needs to reflect the fraction of variance which occurs over each period of
time. Simply put, if the non-trading hours generate 30% of the variance, then the volatility time which should elapse over each non-trading period is 0.3 days.
Extending this concept to a business-day count convention for equities, suppose there are 252 trading days, each with 6.5 hours of trading time, representing a total of [2526.5 =1638] hours. Under a
252-trading-day year assumption, the total number of hours in a year is [36524 = 8760] hours, and the remaining non-trading hours in the year amounts to [8760 - 1638 = 7122] hours. If expected
volatility is the same between market and non-market hours, each hour should count as 1/8760 of a year.
In this constant volatility framework, the entire year can be broken up into 1638 trading and 7122 non-trading time segments, each representing an hour, all of which add up to 1 year:
$1638 \times \frac{1}{8,760} + 7,122 \times \frac{1}{8,760} = 1$
However, in a business day count convention, where the 7122 non-trading hours account for none of the variance, the time unit used for a single trading hour needs to be adjusted so that the time
measured during each trading session accounts for all of the time accumulated in the year. This is accomplished by counting all non-trading hours as equal to zero and counting each trading hour as 1/
1638 of a year:
$1638 \times \frac{1}{1,638} + 7,122 \times \frac{0.3}{7,122} = 1$
Any weight can be used to determine the volatility time for a trading hour and for a non-trading hour, by taking a weighted average where α is the percent of volatility attributed to trading time:
$1638 \times \frac{α}{1,638} + 7,122 \times \frac{1-α}{7,122} = 1$
With this approach, we account for time in terms of fractions of hours during trading and fractions of hours outside of trading, each multiplied by their respective volatility “times” for a trading
market hour and a non-trading market hour.
$Time \space to \space Expiration = (Trading \space Hours \space Remaining) \times \frac{α}{1,638} + (NonTrading \space Hours \space Remaining) \times \frac{1-α}{7,122}$
To give an idea of how a 70% trading-time volatility convention (α=0.7) compares against the original 252-day convention in previous versions of SpiderRock Connect (α=0.0), the following plot shows
both approaches applied to the last two weeks leading up to the 20 January 2017 expiration. For ease of comparison, the time has been converted into volatility “days” by multiplying time by 252.
Option Greeks
Delta (Δ) measures the rate of change of the option price with respect to a 1 point change in the underlying price. For analytic and approximation models, the greek is calculated analytically, and
for numerical models, the calculation is performed alongside the price calculation, for example, within the same binomial tree used to calculate the option price.
Gamma (Γ) measures the rate of change of the option delta with respect to a 1 point change in the underlying price. It is a second-order derivative of the option price with respect to the underlying.
For analytic and approximation models, the greek is calculated analytically, and for numerical models, the calculation is performed alongside the price calculation, for example, within the same
binomial tree used to calculate the option price.
Vega measures the rate of change of the option price with respect to a 1% change in volatility. For analytic and approximation models, the greek is calculated analytically, and for numerical models,
the calculation is performed by recalculating the option price at 1% increments both up and down and evaluating the slope via a centered difference.
Theta (θ) measures the rate of change of the option price with respect to a 1-day change in time to expiration. It is always measured numerically, by calculating the difference between the current
option price and the option price calculated with volatility time decreased by 1/252 years. For analytic and approximation models, the calculation is performed through a direct call to the model, and
for tree methods, the calculation is performed within the tree. When time to expiration is less than 1 volatility day, the next-day option price is taken to be the payout value of the option at
The value of theta is reported in terms of decay, that is, its value is reported as a positive value of how much option premium is expected to decay in 1 days’ time. There are special cases when the
reported value can be negative, in which case, the option premium will be expected to increase in the future. For example, negative call theta values can occur for large values of sdiv, and they can
also be observed when an Ex-dividend date is expected within the next day.
Rho (ρ) measures the rate of change of the option price with respect to a 1% change in the risk-free rate. For analytic and approximation models, the greek is calculated analytically, and for
numerical models, the calculation is performed by recalculating the option price at a 1% higher rate and evaluating the slope via a right-handed difference quotient.
Phi (φ) measures the rate of change of the option price with respect to a 1% change in the dividend rate. For analytic and approximation models, the greek is calculated analytically, and for
numerical models, the calculation is performed by recalculating the option price at a 1% higher rate and evaluating the slope via a right-handed difference quotient.
Pricing Model Performance
The core platform pricing model, while accurate, can be too slow in some regions for common tasks. As a result, we have constructed a distributed pricing solution that has two separate components.
First, we continuously compute and broadcast, throughout the platform, a collection of “calibration records” which are computed using our high precision models. These calibration records then allow
us to quickly and accurately compute prices, volatilities, and greeks using much faster analytic techniques, while still maintaining high precision and consistent pricing throughout the system. The
computation and storage of calibration records occur within SpiderRock Connect's data centers and are typically not visible to users.
This distributed pricing solution is highly accurate for most situations. However, in cases where the pricing inputs are significantly different from the current market environment, SpiderRock
Connect will use the high precision models. The typical situation where this occurs is in calculating some risk metrics, such as to market shocks and other scenario calculations.
Note that the Symbol Viewer tool uses the distributed pricing solution as the default method for calculating all prices. There are a few specific situations where it may be desirable to
temporarily access the high-precision pricing calculations. For example, if input overrides are entered, such as volatility, adjustments to dividends, or price shocks, using the faster
analytics-based approach will no longer guarantee sufficient accuracy of the option prices and greeks. The high-precision models can be accessed in the Symbol Viewer by selecting “Use exact
It is important to note that when exact pricing is enabled in the Symbol Viewer, “Auto-refresh” will be disabled. As soon as any input overrides have been cleared, it is recommended to return to the
default setting with auto-refresh turned back on.
Root Definitions
All pricing configuration for options is handled by the platform and set at the option root level. These settings can be viewed in the SymbolViewer and via the msgRootDefinition table in SRSE. For
example, the following SRSE query lists the types of pricing methods in use:
SELECT root_tk, root_ts, pricingModel FROM srLive.msgRootDefinition;
Currently, there are three pricing methods employed by SpiderRock Connect: ‘Equity’, ‘Future’, and ‘Normal’.
Equity option pricing uses the generalized Black-Scholes approach described above.
Future option pricing simplifies the pricing problem by always using a combination of the Ju-Zhong model for American options (Ju & Zhong, 1999) and the Black-Scholes closed-form equation for
European options. The Ju-Zhong pricing model is an improvement to the approach used by the more well-known Whaley approximation model (Barone-Adesi & Whaley, 1987).
For pricing options on Eurodollars and other short-term interest rates (STIR), SpiderRock Connect uses an analytic pricing model based on the Ornstein-Uhlenbeck SDE and similar to the equation
published by Iwasawa (Iwasawa, 2001, December 2).
Other Notes
We automatically switch American options to European on expiration day (there is no early exercise option on expiration day).
We also automatically revert to rate = sdiv = carry = 0 on expiration day, as intraday interest is not typically assessed for most market participants.
It is possible to configure a binary cutoff time (on an account by account basis), after which all hedge delta calculations become binary and take one of the values: [-1, -0.5, 0, +0.5, +1]. If the
underlying market is straddling the strike price, a hedge delta of +0.5 or -0.5 is used. Otherwise, for ITM and OTM options, a hedge delta of +1, -1, or 0 is used. The default (which can be changed
on request) is that the binary cutoff occurs at 0.5 trading days to expiration.
Barone-Adesi, G., & Whaley, R. E. (1987). Efficient analytic approximation of American option values. The Journal of Finance, 42(2), 301-320.
Cox, J. C., Ross, S. A., & Rubenstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7, 229-263.
Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas (2nd ed.). McGraw-Hill.
Iwasawa, K. (2001, December 2). Retrieved from http://www.math.nyu.edu/~iwasawa/normal.pdf
Ju, N., & Zhong, R. (1999). An approximate formula for pricing American options. The Journal of Derivatives, 7(2), 31-40.
Vellekoop, M., & Nieuwenhuis, J. (2006). Efficient pricing of derivatives on assets with discrete dividends. Applied Mathematical Finance, 13(3), 265-284. | {"url":"https://docs.spiderrockconnect.com/docs/8.4.10.2/Documentation/PlatformFeatures/Analytics/OptionPricing/","timestamp":"2024-11-05T07:16:07Z","content_type":"text/html","content_length":"108979","record_id":"<urn:uuid:9289e1eb-d823-4439-ab4f-ccc3a112bdf4>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00394.warc.gz"} |
Evolution of Leibniz’s Thought in the Matter of Fictions and Infinitesimals
In this chapter, we offer a reconstruction of the evolution of Leibniz’s thought concerning the problem of the infinite divisibility of bodies, the tension between actuality, unassignability, and
syncategorematicity, and the closely related question of the possibility of infinitesimal quantities, both in physics and in mathematics. Some scholars have argued that syncategorematicity is a
mature acquisition, to which Leibniz resorts to solve the question of his infinitesimals – namely the idea that infinitesimals are just signs for Archimedean exhaustions, and their unassignability is
a nominalist maneuver. On the contrary, we show that syncategorematicity, as a traditional idea of classical scholasticism, is a feature of young Leibniz’s thinking, from which he moves away in order
to solve the same problem, as he gains mathematical knowledge. We have divided Leibniz’s path toward his mature view of infinitesimals into five phases, which are especially significant for
reconstructing the entire evolution. In our reconstruction, an important role is played by Leibniz’s text De Quadratura Arithmetica. Based on this and other texts, we dispute the thesis that
fictionality coincides with syncategorematicity, and that unassignability can be bypassed. (In this chapter, we employ “syncategorematic” as a shorthand for “eventually identifiable with a procedure
of exhaustion” and, as a consequence, “involving only assignable quantities.” We also identify syncategorematicity with potentiality, as suggested by some part of the scholastics, and in Sect. 2.2,
we show that the two characterizations are equivalent, provided that “potentiality” is intended in the correct way: namely as in the unending iterative procedures of Greek mathematics.) On the
contrary, we maintain that unassignability, as incompatible with the principle of harmony, is the ultimate reason for the fictionality of infinitesimals.
Original language English
Title of host publication Handbook of the History and Philosophy of Mathematical Practice
Subtitle of host publication Volume 1-4
Publisher Springer International Publishing
Pages 341-384
Number of pages 44
Volume 1
ISBN (Electronic) 9783031408465
ISBN (Print) 9783031408458
State Published - 1 Jan 2024
Bibliographical note
Publisher Copyright:
© Springer Nature Switzerland AG 2024.
• Fictionality
• Infinita terminate
• Infinite divisibility
• Infinitesimals
• Leibnizian metaphysics
• Potential infinity
• Syncategoremata
• Unassignability
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Surface forces: Surface roughness in theory and experiment
A method of incorporating surface roughness into theoretical calculations of surface forces is presented. The model contains two chief elements. First, surface roughness is represented as a
probability distribution of surface heights around an average surface height. A roughness-averaged force is determined by taking an average of the classic flat-surface force, weighing all possible
separation distances against the probability distributions of surface heights. Second the model adds a repulsive contact force due to the elastic contact of asperities. We derive a simple analytic
expression for the contact force. The general impact of roughness is to amplify the long range behaviour of noncontact (DLVO) forces. The impact of the elastic contact force is to provide a repulsive
wall which is felt at a separation between surfaces that scales with the root-mean-square (RMS) roughness of the surfaces. The model therefore provides a means of distinguishing between "true zero,"
where the separation between the average centres of each surface is zero, and "apparent zero," defined by the onset of the repulsive contact wall. A normal distribution may be assumed for the surface
probability distribution, characterised by the RMS roughness measured by atomic force microscopy (AFM). Alternatively the probability distribution may be defined by the histogram of heights measured
by AFM. Both methods of treating surface roughness are compared against the classic smooth surface calculation and experimental AFM measurement.
Dive into the research topics of 'Surface forces: Surface roughness in theory and experiment'. Together they form a unique fingerprint. | {"url":"https://researchportalplus.anu.edu.au/en/publications/surface-forces-surface-roughness-in-theory-and-experiment","timestamp":"2024-11-11T06:41:33Z","content_type":"text/html","content_length":"48762","record_id":"<urn:uuid:dd33213b-29cd-487c-9907-b8fdb792b7cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00151.warc.gz"} |
Open Textbook Initiative
Book of Proof
Richard Hammack
Digital versions PDF
Latex source No
Exercises Yes
Solutions Short answers to odd numbered questions
License Creative Commons Attribution-NonCommercial-No Derivative Works
• Sophomore level text for an introduction to proofs course
• Third edition (copyright 2018) in print and PDF. The third edition is a slightly expanded version of the second edition, but the two editions are otherwise compatible (exercises have not been
renumbered, etc.).
• Widely used (more than 20,000 printed copies sold) with an extensive course adoption list available from the book’s home page
• Softcover version (380 pages) from Amazon or Barnes and Noble for about $21
• Comprehensive review in the MAA Digital Library linked from the book’s home page
From the author’s preface:
This text is an expansion and refinement of lecture notes I developed while teaching proofs courses over the past ten years. It is written for an audience of mathematics majors at Virginia
Commonwealth University, a large state university….However, I am mindful of a larger audience. I believe this book is suitable for almost any undergraduate mathematics program.
Designed for the typical bridge course that follows calculus and introduces the students to the language and style of more theoretical mathematics, Book of Proof has 13 chapters grouped into four
sections: (I) Fundamentals, (II) How to Prove Conditional Statements, (III) More on Proof, (IV) Relations, Functions, and Cardinality. One math professor who has used the book writes:
Hammack’s book is great. I’ve used the book twice now, will use it again, and have recommended it to other instructors. I have used it in a discrete math course which serves as a “transition”
course for our majors. | {"url":"https://textbooks.aimath.org/textbooks/approved-textbooks/hammack/","timestamp":"2024-11-03T06:15:19Z","content_type":"text/html","content_length":"29518","record_id":"<urn:uuid:f753aca9-c989-4163-8f51-57077886af48>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00617.warc.gz"} |
Taking a Closer Look at Hitting with Runners in Scoring Position
In baseball, part of what is commonly debated is how important it is to hit with runners in scoring position. Viewers of their teams will often have their sad sigh when their team leaves runners
stranded in scoring position and will look up how their team does in those situations and say, “this is why we don’t score runs” or “this is why we don’t win games.” They will also look at other
teams and see how good of an offense the other team might have and immediately make the assumption that they are going to be better at hitting with runners in scoring position than most other teams
if their offense is better. But just how much of a team’s success is based on hitting with runners in scoring position and how much of hitting with runners in scoring position is based on team
I. Impact of Hitting with Runners in Scoring Position
One of the old clichés in baseball is, “you can’t win without hitting with runners in scoring position.” Many people link that to why the Cardinals had done so well in the past and why they haven’t
really been able to get going this year. In years past, they have consistently been not only one of the best teams in baseball, but also the best at hitting with runners in scoring position.
Many people in the game consider it also to be one of the most important stats when it comes to judging a player’s hitting ability. In a press conference at the beginning of the season, Matt Williams
had sabermetricians finally thinking that someone with their ideology was becoming the manager of the Washington Nationals when he said, “If you don’t get with the times, bro, you better step aside.”
When I heard that, I immediately thought that he would be talking about more advanced hitting metrics than batting average and home runs and RBI’s. He followed that comment up with, “My favorite stat
right now and always has been the stat of hitting with runners in scoring position. Because batting average and on-base percentage and all of those things are great, but who is doing damage and how
can they hit with guys in scoring position.” When I heard that, I immediately slunked back in my chair and placed him in the category of old-school.
And listening to one of the Reds games (as I always do), listening to Marty Brennaman (who I think is a good broadcaster for his catchy phrases and also because he’s from where I’m from), I heard him
talk about Votto and he said, “Votto will take a 3-0 pitch an inch off the outside corner, when he could do with it what he did Wednesday. I believe in expanding your strike zone when you’ve got guys
on base.” For those who don’t know, what he did on Wednesday (a while ago), was drive a 3-0 pitch from Matt Harvey (that shows how long ago it was) for a home run to left field in New York.
Unfortunately, for a while now Marty Brennaman has been seemingly leading a war of the old-school against his own team’s star first baseman Joey Votto over hitting. Namely hitting with runners in
scoring position or men on base. Again, while listening, I slide back in my chair, disappointed in Marty for being so illusioned and confused and broadcasting his wrong opinion to many of the people
who listen to him on the radio.
Williams and Brennaman aren’t the only people that have this mindset though. The thing that they and many other people think is that if you can’t hit with runners in scoring position, you can’t win
games and you can’t score runs. For these people, it is for the most part a blind hypothesis, just assuming it is true because it seems that it should be true.
For examining this data, I am going to look at the coefficient of determination, or R2 (I have below this the formula for R, correlation coefficient, that when squared equals the coefficient of
determination). For those who don’t know, when looking at the data and calculating a formula of best fit, R2 shows a percentage value of how many of the samples of the x-value fit the line of best
fit (the line that in perfect situations can calculate the y-values). I am going to call the dependent variable, or y-value, wins and runs and the independent variable, or x-value, the various
offensive statistics that I will use to test my hypothesis (hitting with runners in scoring position does not have much to do with determining how many wins a team gets in a season or how many runs a
team scores). Basically it is how dependent team wins and runs are on hitting with runners in scoring position. Before I look at hitting with runners in scoring position, it is important to establish
which three offensive statistics are the best at determining wins and runs.
In terms of influencing the scoring of runs from 2002 to 2013, the three best offensive statistics are:
1. OPS with an R2 of .9132 (91% of the OPS x-values fit the formula: y = 2059.2x – 791.27)
2. ISO with an R2 of .5801 (58% of the ISO x-values fit the formula: y = 3279.75x + 238.02)
3. wOBA with an R2 of .3999 (40% of the wOBA x-values fit the formula: y = 3482.9x – 389.93).
When it comes to which statistics determine wins the most, the three best statistics are:
1. WAR with an R2 of .5329 (53% of the WAR x-values fit the formula: y = 1.1243x + 59.614)
2. wRC+ with an R2 of .4302 (43% of the wRC+ x-values fit the formula: y = 0.8977x – 5.4636)
3. wRAA with an R2 of .3632 (36% of the wRAA x-values fit the formula: y = 0.1033x + 81.239)
There are a couple things to notice when looking at this data. One of those things is that most offensive statistics have a much weaker coefficient of determination when looking at wins, largely in
part to the fact that pitching is kept completely out of the equation. Another thing to know is that if there was a bigger sample size, the R2 values would be different but using this sample size
(which I will use for RISP), these are the R2 values that show up.
The purpose behind collecting those statistics in terms of offense in general as opposed to just RISP is because this way there will be statistics to use when looking at how much RISP influences
offense. Looking at determining runs scored in an overall season with RISP numbers:
1. OPS has an R2 of .3099 (31% of the OPS x-values fit the formula: y = 948.7x + 19.173)
2. ISO has an R2 of .2395 (24% of the ISO x-values fit the formula: y = 1812.2x + 470.92)
3. wOBA has an R2 of .2898 (29% of the wOBA x-values fit the formula: y = 2391.5x – 35.754)
It is quite a dramatic change, especially when looking at OPS that clearly had a big hand in determining runs scored in a season. While some of them still have some modest effect in determining runs
scored, it is still not quite at the same level as those that covered a full season and not just a given scenario. Now looking at how those other statistics determine wins with runners in scoring
1. WAR has an R2 of .29 (29% of the WAR x-values fit the formula: y = 2.5609x + 68.94)
2. wRC+ has an R2 of .2739 (27% of the wRC+ x-values fit the formula: y = 0.5518x + 27.727)
3. wRAA has an R2 of .2366 (24% of the wRAA x-values fit the formula: y = 0.2366x + 80.996)
As I had mentioned before, it should be expected that these numbers ought to be low because there is much more that goes into a win than just offensive ability. There has to be great pitching too
that is not put into account. With that said, these numbers are quite far from being great in determining wins as is evidenced by their still being far away from even the 50% mark that they should be
close to.
For Matt Williams’ sake, I also looked at how much batting average with runners in scoring position determines wins and runs:
1. For scoring runs, AVG has R2 value of .181 (18% of AVG x-values fit the formula: y = 2005.8x + 213.05)
2. For wins, AVG has R2 of .1427 (14% of AVG x-values fit the formula: y = 257.76x + 13.255)
So Matt, not to rain on your parade, but batting average with runners in scoring position has very little to do with determining runs or wins. And Marty, it’s just limiting Votto’s overall production
to a small sample size that doesn’t have a whole lot to do with winning games. No one will argue that hitting with runners in scoring position can help to win games because it does often result in
scoring a run but it should not be looked at as one of the key stats in a player’s production.
II. Is it dependent on overall strength of offense?
Now back to those St. Louis Cardinals. Last year, with runners in scoring position, they put up not only unreal numbers, they put up numbers that are really just plain stupid. I mean, they batted
.330 with runners in scoring position, had a .370 wOBA, and a 138 wRC+, and won 97 games, 32 games over .500. Like I have previously established, those numbers are intrinsically worthless considering
that it is such a small sample size but those are still just gaudy numbers. This year, for lack of a better word, they’re awful with runners in scoring position. A .244 batting average, .293 wOBA,
and 86 wRC+ all those with runners on second or third and have won 39 games, only 4 over .500.
Many people look at that and think that clearly, their inability to hit with runners in scoring position this year has caused the drop off in production. Of course, the low .303 wOBA, 92 wRC+, OPS of
.681, and AVG of .250 are a bit of a drop off from the .322 wOBA, 106 wRC+, .733 OPS, and .269 AVG of last year might have something to do with that drop off in offense too. The Cardinals offense is
also scoring about a run less this year than they did last year (4.83 Runs/9 innings in 2013 and 3.67 Runs/9 innings in 2014) meanwhile their pitching has practically been identical to last year with
a FIP of 3.31, xFIP of 3.66, and SIERA of 3.60 this season compared to last year’s 3.39 FIP, 3.63 xFIP, and SIERA of 3.57. But is hitting with runners in scoring position dependent on how the offense
overall is? I’m sure you can already see what coefficient we’re going back to.
The process was similar to last time, with the dependent variable, or y-value, being hitting with runners in scoring position, and the independent variable, or x-value, being the same statistic only
looking at the value over the course of a full season. I found that wRC in a year has by far the strongest effect in determining how a team hits with RISP with an R2 of .7527 with 75% of the x-values
fitting into the equation of y = 0.3364x – 51.232. OPS is after that with an R2 of .6487 and 65% of the x-values fitting the equation of y = 1.0184x + 0.0025. And then there is wOBA that has an R2 of
.6258 and 63% of the x-values fitting the equation of y = 0.9807x + 0.0062. Some other values are:
• wRAA that has an R2 of .5811 (58% of the x-values fit into the equation: y = 0.2586 + 0.5721)
• wRC+ that has an R2 of .5558 (56% of the x-values fit into the equation: y = 0.9678x + 3.3038)
• WAR that has an R2 of .3831 (38% of the x-values fit into the equation: y = 0.2005x + 0.8901)
So a case could be made that the strength of a team’s offense overall does dictate how that same team hits with runners in scoring position. While by no means is it an overwhelmingly strong
coefficient of determination in any of the cases, in most cases the strength of an offense determines at least 50% of hitting with runners in scoring position which is good enough to at the very
least say that better offensive teams are more likely to hit better with runners in scoring position than weak offensive teams.
Fantasy writer covering prospects for Rotoballer.com, about as big of a Reds fan as you will ever find.
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10 years ago
Wait a minute. You debunk Matt Williams and then at the end of the article you say this, “….which is good enough to at the very least say that better offensive teams are more likely to hit better
with runners in scoring position than weak offensive teams.”
Isn’t that the point Williams was trying to make?
Average with RISP makes a difference. It is not the end-all in determining a team’s offensive strength.
Now, if you could write something on Productive Outs that would be great.
10 years ago
RISP does matter for individual players because it’s a collection of those several individuals that contribute to the team’s success. For example, the Detroit Tigers hit 282 as a team with RISP in
2013. Why was this the case? It’s very simple, they made the most of the RISP opportunities they had regardless of the sample size. Here are the general numbers with RISP for some of the hitters in
Detroit’s lineup in 2013:
Cabrera Avg 397 OBP 529
Fielder Avg 282 OBP 371
Infante Avg 275 OBP 304
Martinez Avg 264 OBP 340
Peralta Avg 344 OBP 414
Hunter Avg 281 OBP 298
Detroit had 4 guys hitting 280 or better with RISP and yes some of this is brought up greatly by Cabrera and Peralta. However when you line up Cabrera, Fielder, Martinez, Hunter, and Peralta in a
row, you can see why the 2013 Tigers didn’t have RISP slumps that lasted for an longer period of time compared to the average team. If Detroit had struggled with RISP immensely for the first 2-3
months of the season, they would have not hit 282 with RISP.
Detroit may not have had as many opportunities with RISP as they could have, but they didn’t disappoint fans because they came through to an amount that was satisfying enough because they maximized
the given opportunities presented. This is why fans get on players for stinking with RISP, because you benefit from multiple guys stacked in a row in a lineup that are hitting well for the season
with RISP.
**** Think about it though. It only makes sense regardless of a hitter’s overall numbers and sample size with RISP for a pitcher to pitch around a guy hitting over 300 with RISP than it is to pitch
to a pull hitting guy who is at 240 with RISP. I mean between having to face Edgar Martinez and Mark Texeira with RISP, I’ll take Texeira all day because he’s the type of hitter that if you locate,
he won’t get a hit whereas Martinez can flick a perfectly spotted fastball on the outside corner to right field for a hit to drive in a run. Why would I want to deal with that headache? | {"url":"https://community.fangraphs.com/taking-a-closer-look-at-hitting-with-runners-in-scoring-position/","timestamp":"2024-11-09T13:49:47Z","content_type":"text/html","content_length":"160597","record_id":"<urn:uuid:a32278ce-5029-4831-9187-757546d43e94>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00474.warc.gz"} |
On Hölder continuity of solutions of the Beltrami equations on the boundary - R Discovery
In the present paper, it is found conditions on the complex coefficient of the Beltrami equations with the degeneration of the uniform ellipticity in the unit disk under which their generalized
homeomorphic solutions are continuous by Hölder on the boundary. These results can be applied to the investigations of various boundary value problems for the Beltrami equations. In a series of
recent papers, under the study of the boundary value problems of Dirichlet, Hilbert, Neumann, Poincare and Riemann with arbitrary measurable boundary data for the Beltrami equations as well as for
the generalizations of the Laplace equation in anisotropic and inhomogeneous media, it was applied the logarithmic capacity, see e.g. Gutlyanskii V., Ryazanov V., Yefimushkin A. On the boundary value
problems for quasiconformal functions in the plane // Ukr. Mat. Visn. - 2015. - 12, no. 3. - P. 363-389; transl. in J. Math. Sci. (N.Y.) - 2016. - 214, no. 2. - P. 200-219; Gutlyanskii V., Ryazanov
V., Yefimushkin A. On a new approach to the study of plane boundary-value problems // Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. - 2017. - No. 4. - P. 12-18; Yefimushkin A. On Neumann
and Poincare Problems in A-harmonic Analysis // Advances in Analysis. - 2016. - 1, no. 2. - P. 114-120; Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations in
quasidisks // Ukr. Mat. Visn. - 2015. - 12, no. 2. - P. 190–209; transl. in J. Math. Sci. (N.Y.) - 2015. - 211, no. 5. - P. 646–659; Yefimushkin A., Ryazanov V. On the Riemann–Hilbert Problem for the
Beltrami Equations // Contemp. Math. - 2016. - 667. - P. 299-316; Gutlyanskii V., Ryazanov V., Yakubov E., Yefimushkin A. On Hilbert problem for Beltrami equation in quasihyperbolic domains //
ArXiv.org: 1807.09578v3 [math.CV] 1 Nov 2018, 28 pp. As well known, the logarithmic capacity of a set coincides with the so-called transfinite diameter of the set. This geometric characteristic
implies that sets of logarithmic capacity zero and, as a consequence, measurable functions with respect to logarithmic capacity are invariant under mappings that are continuous by Hölder. That
circumstance is a motivation of our research. Let \(D\) be a domain in the complex plane \(\mathbb C\) and let \(\mu: D\to\mathbb C\) be a measurable function with \( |\mu(z)| \lt 1 \) a.e. The
equation of the form \(f_{\bar{z}}\ =\ \mu(z) f_z \) where \( f_{\bar z}={\bar\partial}f=(f_x+if_y)/2 \), \(f_{z}=\partial f=(f_x-if_y)/2\), \(z=x+iy\), \( f_x \) and \( f_y \) are partial
derivatives of the function \(f\) in \(x\) and \(y\), respectively, is said to be a Beltrami equation. The function \(\mu\) is called its complex coefficient, and \( K_{\mu}(z)=\frac{1+|\mu(z)|}{1-|\
mu(z)|}\) is called its dilatation quotient. The Beltrami equation is said to be degenerate if \({\rm ess}\,{\rm sup}\,K_{\mu}(z)=\infty\). The existence of homeomorphic solutions in the Sobolev
class \(W^{1,1}_{\rm loc}\) has been recently established for many degenerate Beltrami equations under the corresponding conditions on the dilatation quotient \(K_{\mu}\), see e.g. the monograph
Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami equation. A geometric approach. Developments in Mathematics, 26. Springer, New York, 2012 and the further references therein. The main
theorem of the paper, Theorem 1, states that a homeomorphic solution \( f:\mathbb D\to\mathbb D \) in the Sobolev class \( W^{1,1}_{\rm loc} \) of the Beltrami equation in the unit disk \(\mathbb D\)
has a homeomorphic extension to the boundary that is Hölder continuous if \(K_{\mu}\in L^1(\Bbb D)\) and, for some \(\varepsilon_0\in(0,1)\) and \(C\in[1,\infty)\), $$ \sup\limits_{\varepsilon\in(0,\
varepsilon_0)} \int_{\mathbb D\cap D(\zeta,\varepsilon)}K_{\mu}(z) dm(z) \lt C \qquad \forall \zeta \in \partial \mathbb{D} $$ where \(D(\zeta,\varepsilon)=\left\{z\in{\Bbb C}: |z-\zeta| \lt \
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k-map, the weird cousin of k-anonymity - Ted is writing things
Suppose that you're a doctor who studies human sexual behavior. You want to run a study with all the patients that you can find, but you don't find a lot of volunteers. You only end up with about 40
After you've ran your study and collected data, you want to share this data with other researchers. You look at the attributes, and deduce that ZIP code and age are likely to be used in
reidentification attacks. To share it in a safe way, you're thinking of \(k\)-anonymity.
When trying to find a strategy to obtain \(k\)-anonymity, you find out that you would have to lose a lot of information. For \(k=10\), a rather small value, you end up with buckets like \(20\le age\
lt 50\). That makes sense: you have only few people in your database, so you have to bundle together very different age values.
But when you think about it, you start questioning whether you really need \(k\)-anonymity. Who are the attackers, in your scenario? The researchers with whom you share the data, and possibly unknown
parties if the data ever leaks. None of these people have background information about who is in the dataset. Thus, the attacker doesn't just have to distinguish between different records, but to
actually find the real identity of a record based on its information. This attacker has significantly weaker capabilities than for \(k\)-anonymity!
Let's look at two different rows in this database.
ZIP code age
At first glance, the amount of information for this two individuals seems to be the same. But let's take a look at the values…
• 85535 corresponds to a place in Arizona named Eden. Approximately 20 people live in this ZIP code. How many people do you think are exactly 79 years old in this particular ZIP code? Probably only
• 60629 corresponds to a part of the Chicago metropolitan area. More than 100,000 people live there. How many of them are 42 years old? A thousand, at least, and probably more!
It seems that it would be very easy to reidentify the first row, but that we don't have enough information to reidentify the second row. But according to \(k\)-anonymity, both rows might be
completely unique in the dataset.
Obviously, \(k\)-anonymity doesn't fit this use case. We need a different definition: that's where \(k\)-map comes in.
Just like \(k\)-anonymity, \(k\)-map requires you to determine which columns of your database are quasi-identifiers. This answers the question: what can your attacker use to reidentify their target?
But this information alone is not enough to compute \(k\)-map. In the example above, we assumed that the attacker doesn't know whether their target is in the dataset. So what are they comparing a
given row with? With all other individuals sharing the same values in a larger, sometimes implicit, dataset. For the previous example, this could be "everybody living in the US", if you assume the
attacker has no idea who could have this genetic disease. Let's call this larger table the reidentification dataset.
Once you picked the quasi-identifiers and the reidentification dataset, the definition is straightforward. Your data satisfies \(k\)-map if every combination of values for the quasi-identifiers
appears at least \(k\) times in the reidentification dataset.
In our example, this corresponds to counting the number of people in the US who share the quasi-identifier values of each row in your dataset. Consider our tiny dataset above:
ZIP code age
We said earlier than the values of the first row matched only one person in the US. Thus, this dataset does not satisfy \(k\)-map for any value of \(k\ge 2\).
How do we get a larger \(k\)? We could generalize the first value like this:
ZIP code age
85*** 79
ZIP codes between 85000 and 85999 include the entire city of Phoenix. There are 36,000+ people between 75 and 84 years old in Phoenix, according to some old stats. It's probably safe to assume that
there are more than 1,000 people who match the quasi-identifiers values of the first row. We saw earlier that the second row also matched 1,000+ people. So this generalized dataset satisfies
Attack model considerations
Wait a second, why does this feel like cheating? What happened there, to give us such a generous number so easily? This comes from the generous assumptions we made in our attack model. We assumed
that the attacker had zero information on their target, except that they live in the US (which is implied by the presence of ZIP codes). And with only the information (ZIP code, age), you don't need
a lot of generalization to make each row of your dataset blend in a large crowd.
To make this attack model stronger, you could assume that the attacker will use a smaller reidentification database. For example, suppose that your genetic disease you're studying requires regular
hospital check-ups. The attacker could restrict their search only to people who have visited a hospital in the last year. The number of possible "suspects" for each value tuple gets smaller, so the \
(k\) of \(k\)-map decreases too^1.
\(k\)-map is inherently a weak model. So when choosing the quasi-identifiers and reidentification dataset, you have to think hard at what an attacker could do. If your attacker doesn't have lots of
resources, it can be reasonable to assume that they won't get more data than, say, the voter files from your state. But if they can figure out more about your users, and you don't really know which
reidentification dataset they could use, maybe \(k\)-anonymity is a safer bet^2.
And now, some practice
OK, enough theory. Let's learn how to compute \(k\)-map in practice, and anonymize your datasets to make them verify the definition!
… There's one slight problem, though.
It's usually impossible.
Choosing the reidentification dataset is already a difficult exercise. Maybe you can afford to make generous assumptions, and assume the attacker doesn't know much. At best, you think, they'll buy
voter files, or a commercial database, which contains everyone in your state, or in the US. But… then what?
To compute the maximum \(k\) such as your dataset verifies \(k\)-map, you would first need to get the reidentification dataset yourself. But commercial databases are expensive. Voter files might not
be legal for you to obtain (even though an evil attacker could break the law to get them).
So, most of the time, you can't actually check whether your data satisfies \(k\)-map. If it's impossible to check, it's also impossible to know exactly which strategy to adopt to make your dataset
verify the definition.
Exception 1: secret sample
Suppose you're not releasing all your data, but only a subset (or sample) of a bigger dataset that you own. Then, you can compute the \(k\)-map value of the sample with regard to the original, bigger
dataset. In this case, choosing \(k\)-map over \(k\)-anonymity is relatively safe.
Indeed, your original dataset is certainly smaller than the reidentification dataset used by the attacker. Using the same argument as above, this means that you will obtain a lower bound on the value
of \(k\). Essentially, you're being pessimistic, which means that you're on the safe side.
Even if the attacker has access to the original dataset, they won't know which records are in the sample. So if the original dataset is secret, or if you've chosen the sample in a secret way, \(k\)
-map is a reasonable definition to use, and you can compute a pessimistic approximation of it.
Exception 2: representative distribution
This case is slightly different. Suppose that you can make the assumption that your data is a representative (or unbiaised) sample of a larger dataset. This might be a good approximation if you
selected people (uniformly) at random to build your dataset, or if it was gathered by a polling organization.
In this case, you can compute an estimate of the \(k\)-map value for your data, even without the reidentification dataset. The statistical properties which enable this, and the methods you can use,
are pretty complicated: I won't explain them in detail here. They are mentioned and compared in this paper, which has references to the original versions of each of them.
Exception 3: using humans
For the case of our doctor earlier, if the dataset is small enough, a motivated data owner could actually do the job of an attacker "by hand". Go through each record, and try to map it to a real
person, or estimate the chances of it being possible. We pretty much did that in this article!
This is very approximative, and obviously not scalable. But for our imaginary doctor, it might be a reasonable solution!
ARX implements the methods from exceptions 1 and 2. Documentation for the first one can be found here. Instructions to estimate the number of unique values assuming uniformity can be found here.
Originally, μ-ARGUS was the first software with this feature, but I couldn't run it on my machine, so I can't say much about it.
You might wonder why I wrote an entire article on a definition that is hardly used because of how impractical it is. In addition to the unique problems that we talked about in this article, the
limitations of \(k\)-anonymity also apply. It's difficult to choose \(k\), non-trivial to pick the quasi-identifiers, and even trickier to model the reidentification database.
The definition also didn't get a lot of attention from academics. Historically, \(k\)-anonymity came first^4. Then, people showed that \(k\)-anonymity was sometimes not sufficient to protect
sensitive data, and tried to find stronger definitions to fix it. Weaker definitions were, of course, less interesting.
Nonetheless, I find that it's an interesting relaxation of \(k\)-anonymity. It shows one of its implicit assumptions: the attacker knows that their target belongs to the dataset. This assumption is
sometimes too pessimistic: it might be worth considering alternate definitions.
Choosing a privacy model is all about modeling the attacker correctly. Learning to question implicit assumptions can only help!
1. There is a generic version of this argument. Let's call your database \(D\), and suppose \(R\) and \(R^\prime\) are two possible reidentification databases. Suppose that \(R^\prime\) is "larger"
than \(R\) (each element of \(R\) appears in \(R^\prime\)). Then if \(D\) satisfies \(k\)-map with regard to \(R\), it also satisfies \(k\)-map with regard to \(R^\prime\). The reverse is not
true. ↩
2. One simple consequence of the previous footnote is that if a dataset \(D\) verifies \(k\)-anonymity, then it automatically verifies \(k\)-map for any reidentification dataset^3. ↩
3. I didn't say this explicitly, but the reidentification dataset is always assumed to contain all rows from your dataset. It's usually not the case in practice because data is messy, but it's a
safe assumption. Hoping that your attacker will just ignore some records in your data would be a bit overly optimistic. ↩
4. Latanya Sweeney first mentioned the idea behind \(k\)-map in this 2002 paper^ (pdf), several years after the introduction of \(k\)-anonymity. ↩ | {"url":"https://desfontain.es/blog/k-map.html","timestamp":"2024-11-11T09:45:11Z","content_type":"text/html","content_length":"26967","record_id":"<urn:uuid:ee3d12e7-075c-459f-9158-9f4317d04e73>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00312.warc.gz"} |
Apparently, Now The Mountains Are Eating The Warming - Pirate's Cove
Here’s yet another excuse as to why the Earth is refusing to warm in a statistically significant manner, causing the vast majority of climate models from the Believers to fail
New warmist excuse: Mountains eating the global warming. Mountain rainfall removes CO2 causing lower temps. http://t.co/K1UQj6KmzG
— Steve Milloy (@JunkScience) January 26, 2014
(Climate Central) U.S. scientists have measured the rate at which mountains make the raw material for molehills – and found that if the climate is rainy enough, soil gets made at an astonishing
speed. And in the course of this natural conversion of rock to fertile farmland and forest loam, carbon is naturally removed from the atmosphere.
Isaac Larsen of the University of Washington in Seattle and colleagues from California and New Zealand took a closer look at rates of weathering on the western slopes of the Southern Alps in New
Zealand. They report in Science that, according to their measurements, rock is being transformed into soil more than twice as fast as previously believed. (snip)
The research matters because – once again – it throws new light on one of the dark regions of the climate machine: how carbon dioxide is removed from the atmosphere, at what rate, and where
it goes and where it all ends up.
So,let’s see
Climate Depot Analysis: ‘There have been at least seven separate explanations for the standstill in global warming’ – 1) Low Solar Activity; 2) Oceans Ate Warming; 3) Chinese Coal Use; 4)
Montreal Protocol; 5) Readjusted past temps to claim ‘pause’ never existed 6) Volcanoes 7) Decline in Water Vapor
Now we add “rain falling on mountains”.
If you liked my post, feel free to subscribe to my rss feeds.
Can someone tell me where the CO2 is? Jeff has not been able to find it in the oceans. The CO2 in the lower atmosphere is the same, the only CO2 that is elevated is in the upper atmosphere, yet we
don’t have a model to put it there, according to Jeff.
You’ve been duped by the hacks at Climate Depot and JunkScience. Please find in the article where the authors make the claims that the Climate Depot and JunkScience fossil fuel lobbyists and
propagandists say they make. A change in atmospheric CO2 significant enough to impact climate change is significant enough to be measured. This is the dumbest thing you’ve regurgitated from these
lying whores yet.
Here’s the abstract from the Science article:
“Evaluating conflicting theories about the influence of mountains on carbon dioxide cycling and climate requires understanding weathering fluxes from tectonically uplifting landscapes. The lack of
soil production and weathering rate measurements in Earth’s most rapidly uplifting mountains has made it difficult to determine whether weathering rates increase or decline in response to rapid
erosion. 10Be concentrations in soils from the western Southern Alps, New Zealand, demonstrate that soil is produced from bedrock more rapidly than previously recognized, at rates up to 2.5 mm per
year. Weathering intensity data further indicate that soil chemical denudation rates increase proportionally with erosion rates. These high weathering rates support the view that mountains play a key
role in global-scale chemical weathering and thus have potentially important implications for the global carbon cycle.”
david, You are wrong. There is CO2 dissolved in the oceans and CO2 in the atmosphere. If you evidence to the contrary please present it and stop playing your childish games.
J- So which of the excuses are the climateers going with (this week)?
Mountain rainfall removes CO2
My question, is how does CO2 only know to fall out with the rainfall in the mountains? Is the CO2 racist against lowlands? Is CO2 a-hatin’ on the beaches? Does CO2 not reside around boreal forests?
What is it about mountains that CO2 only rains down upon them, and not other places?
Of course, I jest as this is complete and utter hogwash. Even a 3-year old can see through this BS.
This is like J saying only man-made CO2 raises the temperature of the air. This is like J saying that only CO2 is responsible for our greenhouse effect.
As I pointed out, the Pirate was duped by the lying whores Morano and Milloy. There was nothing in the article that says anything remotely close to what the Pirate, Morano and Milloy claim. The list
of “excuses” they claim climate realists make are a collection of falsehoods.
There is no need for excuses. The Earth is warming because of increased atmospheric CO2 from burning fossil fuels.
The reason you cultists rely on non-scientists paid by fossil fuel interests for your information is pretty obvious.
Sadly, most of what you think you know is false.
You should try to read and understand the article.
You are confused about CO2. All CO2 contributes to the greenhouse effect, but human-generated CO2 is upsetting the carbon cycle and is responsible for the increase in atmospheric CO2 from 280 ppm to
400 ppm. Not all the CO2 generated from fossil fuels stays in the atmosphere, as about half dissolves in the oceans. CO2 is the predominant greenhouse gas but there are others.
There was nothing in the article that says anything remotely close to what the Pirate, Morano and Milloy claim.
Actually Jeffery, if you read the abstract you cite, the articles use the same conclusions presented in the abstract.
I fear that you have been drinking too much tonight and have forgotten how to read and comprehend what people say.
Those liberal blinders really affect your ability to understand anything above 2+2=4.
So, all the CO2 that comprised the difference ‘tween 400 and 280ppm is all man produced? Really?
CO2 is the predominant greenhouse gas? Really?
as about half dissolves in the oceans.
at any given moment.. maybe. over time all of it is absorbed in to the ecosystem. recycled. re-emitted. re-absorbed. re-recycled.
it is carbon.. it never dies. It is always here. always will be. | {"url":"https://www.thepiratescove.us/2014/01/30/apparently-now-the-mountains-are-eating-the-warming/","timestamp":"2024-11-04T01:31:55Z","content_type":"application/xhtml+xml","content_length":"104452","record_id":"<urn:uuid:f452c129-ce7e-44db-a1f8-be2090977bf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00439.warc.gz"} |
1d Heat Equation Python Code - Tessshebaylo
1d Heat Equation Python Code
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The One Dimensional Diffusion Equation
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Solving The Heat Diffusion Equation 1d Pde In Python You
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Trending Posts
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.tessshebaylo.com/1d-heat-equation-python-code/","timestamp":"2024-11-06T14:33:09Z","content_type":"text/html","content_length":"59920","record_id":"<urn:uuid:36215bd6-1f2e-4344-b37b-72b44c682d6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00140.warc.gz"} |
If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n? - Crackverbal
Get detailed explanations to advanced GMAT questions.
If the sum of the consecutive integers from –42 to n inclusive is 372, what is the value of n?
Difficulty Level
Option D is the correct answer.
Option Analysis
Number of terms =42+1+n=(n+43)
To find the sum = Formula = (n/2)(a+l), where n is the number of terms and ‘a’ and ‘l’ is the first term and last term.
744=(n+43)∗(n−42) n=50
42 terms after zero and 42 terms below zero will total 0. So, our new question will be consecutive integers with first term 43 have sum 372, what is the last term: ((43+n)/2)∗(n−43+1)=372 (n+43)∗
(n−42)=744 n=50 | {"url":"https://www.crackverbal.com/solutions/sum-consecutive-integers-372/","timestamp":"2024-11-09T19:16:32Z","content_type":"text/html","content_length":"98553","record_id":"<urn:uuid:63d58579-a930-408c-be4d-fbb99bc59c15>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00548.warc.gz"} |
Limit Cycle Bifurcations of a Piecewise Linear Dynamical System
In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number of dropping sections and approximating some continuous nonlinear function.
Studying all possible local and global bifurcations of its limit cycles, we give a sketch of the proof of the theorem stating that such a piecewise linear dynamical system with k dropping sections
and 2k+1 singular points can have at most k+2 limit cycles, k+1 of which surround the foci one by one and the last, (k+2)-th, limit cycle surrounds all of the singular points of this system. | {"url":"http://lib.physcon.ru/doc?id=7731f3c6ec53","timestamp":"2024-11-02T02:02:05Z","content_type":"text/html","content_length":"4718","record_id":"<urn:uuid:b959bcda-8c8c-440d-88b3-2fb8f31138f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00493.warc.gz"} |
Ti-83 log scale
ti-83 log scale
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Complex Fractions
Fractions, Ratios, Money, Decimals and Author Message
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Author Message
BantJamir Posted: Thursday 28th of Dec 15:13
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From: On the Interweb
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Posted: Thursday 28th of Dec 15:13
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Posted: Friday 29th of Dec 08:09
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Re: Improvements to formulas
Jun 06, 2019 10:36 AM
One of our biggest feature requests is that we would love to see gigantic improvements to formulas — in particular, way more flexibility with typing in formulas, and better improvements to error
More specifically, these are our top 3 requests for formulas:
1. Please allow us to have a much larger typing area for formulas, more like a real programming language would allow. Currently, we can’t press the “return” key within our formulas — nor can we
space out the formulas to be more readable.
2. On that very same note, the formulas should be waaaaay more understanding & waaaaaay more forgiving when it comes to spaces & returns. All other programming languages simply ignore extra spaces
or extra carriage returns, but any of this causes Airtable to give us an error message. For example, the first formula below works just fine. But the 2nd & 3rd formulas below don’t work at all.
The ONLY difference is one extra space in the 2nd & 3rd formulas. All other programming languages would understand that these are the exact same formula, but not Airtable.
Formula #1 — this one works:
IF(AND(City=BLANK(), {Event Name}=BLANK()),Country,IF({Event Name}=BLANK(),City,{Event Name}))
Formula #2 - this one doesn’t work:
IF (AND(City=BLANK(), {Event Name}=BLANK()),Country,IF({Event Name}=BLANK(),City,{Event Name}))
Formula #3 - this one doesn’t work:
IF(AND (City=BLANK(), {Event Name}=BLANK()),Country,IF({Event Name}=BLANK(),City,{Event Name}))
1. Because Airtable is so rigid & strict & unforgiving when it comes to formulas, we need better formula errors. Airtable should highlight EXACTLY where the error is, so we don’t spend 2 hours (like
I did the other night) troubleshooting a formula, when the only problem was simply an extra space.
Jun 06, 2019 11:35 AM
Jun 06, 2019 11:35 AM
I completely identify with this request, and agree they would all be really nice quality-of-life improvements to have – but there are a lot of other things I’d rather see first, mostly because this
issue can be worked around by using a code editor to write and store copies of all my formulas.
I just use a code editor (Visual Studio Code) to create .py documents (just a personal choice for the sake of syntax highlighting) to write out my Airtable formulas in, and then copy paste them into
the field editor. If it’s a big, important formula that I anticipate I may need to update in the future, I save the document with a name that refers me back to where that formula lives in my base.
Since this can be worked around by using an external editing environment, I wouldn’t expect it to be high on their list of enhancements to make.
Jun 06, 2019 07:51 PM
Jun 06, 2019 07:51 PM
Thanks, @Jeremy_Oglesby!
I can see how an external code editor could help us write & visualize our Airtable formulas.
Question: Is your Visual Studio Code program configured to recognize the syntax of Airtable’s formulas? In other words, can Visual Studio Code alert you to errors in your Airtable formulas?
Jun 06, 2019 08:16 PM
Jun 06, 2019 08:16 PM
Re: #2, my gut says that spaces in certain places aren’t allowed because a good chunk of Airtable’s formulas revolve around functions. Take IF, for example. In JavaScript, the format for an IF
statement looks like this:
if (x > 15) doSomething();
In that line, “if” is a reserved keyword. While spacing around such keywords is encouraged for clarity, it still works to leave the space out.
In Airtable’s formula system, however, IF is a function. Functions often have more strict syntax rules, but it’s up to the language designer to choose whether doSomething() is just as valid as
doSomething (). In JavaScript, both versions are valid, though it’s encouraged to leave spaces out when calling functions. Why? My gut says that spaces are discouraged with function calls for the
same reason that they’re encouraged around keywords: clarity. You can tell at a glance that “if” is a keyword and “doSomething” is function name by how whitespace is used (or not) around each.
With Airtable, the developers have decided that spaces are not allowed with function calls, so IF(Field, "True", "False") is valid, but IF (Field, "True", "False") is not. In a sense this choice
feels similar to one that the developers of Python made when creating rules about defining code blocks. With many languages, indentation is optional when defining a block of code for function
definitions, if statements, etc. With Python, though, they force you to indent, and the indentation itself is what defines what code is in a block.
While it may seem like a simple thing from our perspective to allow spaces between a function name and the opening parenthesis, and while I somewhat agree that it would help increase formula clarity
in some cases, it might actually be a huge headache from a development standpoint based on how the Airtable devs implemented the tool’s formula system. I also feel that the lack of space is a helpful
reminder that IF, AND, OR, etc. are functions in Airtable, not keywords, and should be treated as such.
Jun 06, 2019 08:37 PM
Jun 06, 2019 08:37 PM
Thanks, @Justin_Barrett! I get what you’re saying, but I’m coming from the FileMaker Pro world — which is completely designed for maximum developer-friendliness — and FileMaker Pro allows for any
number of spaces or carriage returns for optimal readability (and typeability) within all functions & formulas.
It makes Airtable feel extremely restrictive & unforgiving, especially when Airtable doesn’t tell us WHERE the error in our formulas lie.
I spent 2 hours trying to troubleshoot a perfectly valid formula the other night, but because there was an extra space in it, Airtable wouldn’t accept it.
And even worse, Airtable wouldn’t tell me WHERE the problem was in my formula.
Of course, now that I know this information, I will be way more careful in the future, but it just seems like formulas still have a long way to go.
Another unrelated example of how far formulas can still go: instead of nested “IF” functions, we should have a “CASE” function (like FileMaker Pro has). This eliminates the need for nested IF
statements. I think they tried to get close to this with the “SWITCH” function, but I don’t believe that the “SWITCH” function allows for multiple expressions to be evaluated throughout the formula.
FileMaker’s “CASE” function allows this. I should probably just make this one another feature request. :stuck_out_tongue_winking_eye:
Jun 07, 2019 10:16 AM
Jun 07, 2019 10:16 AM
No, I’m not aware of any existing tools in Visual Studio Code that perform syntax highlighting based on Airtable formula standards. It might be possible to build an extension for this, as Visual
Studio Code is very extensible… but I don’t have time for that right now.
I use the built in python syntax highlighting just to get the color distinctions on operators and what not – the python highlighter does pretty well at making my Airtable formulas easy to read.
The main advantage, though, is the ease with which I can structure out a formula with new lines and still have it be valid in Airtable’s formula editor when copy-pasted in. In the video below you can
see how parentheses are automatically closed for me as I type, and formulas automatically indent their parameters for me as I type. And all of it is valid when copy-pasted as-is into Airtable, so it
makes it much easier to prevent errors like an extra space, even if it’s not necessarily easier to spot them (although it may be due to the syntax highlighting).
And, also as you can see, there’s an advantage of being able to collapse formulas, or parts within formulas when they are getting really long, and if you are saving a long list of formulas, you can
add comments to them to remind you what the formula was for.
The only thing you miss in this is the auto-completing of Formula names and field names that you get in Airtable’s editor.
Jun 07, 2019 10:42 AM
Jun 07, 2019 10:42 AM
I can echo the benefits that @Jeremy_Oglesby lists for using an external editor. For simple formulas that aren’t nested too deeply, I’ll build them directly in Airtable (though I do agree that a
larger editor space would be better). Anything more than that and I’ll move over to BBEdit, then copy my results back to Airtable. I haven’t tried applying syntax highlighting yet, but might give
that a shot eventually.
@Jeremy_Oglesby Not sure why, but the embedded video isn’t playing. Does it play when you view the post?
Jun 07, 2019 10:44 AM
Jun 07, 2019 10:44 AM
No, it doesn’t play for me either – probably too big a file. I linked it from dropbox, but I’m sure Discourse still has limits on how big a file it will stream… :man_shrugging: | {"url":"https://community.airtable.com/t5/product-ideas/improvements-to-formulas/idc-p/62046/highlight/true","timestamp":"2024-11-06T05:28:13Z","content_type":"text/html","content_length":"629384","record_id":"<urn:uuid:28368e46-ba58-4eec-a860-8698ca717748>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00290.warc.gz"} |
Topic summary
Rounding simplifies a number. Imagine walking into a shop an 5 items are 99p. Your brain will round each item to a pound and realise the total cost is about £5!
To the nearest...
If we round 236 to the nearest hundred, we will either get 200 or 300. If the number is 250 or more, it will round up to 300, if less then it will round down to 200. 236 is closer to 200, so will
round to 200.
Decimal places
Decimals can be very long (or can go on forever!) To round a decimal we can use decimal places. We count the number of digits past the decimal place we want.
Rounding 0.1351 to 2 decimal places will either give us 0.13 or 0.14. We look at the 'next digit' to work out what will happen. If the 'next digit' is 5 or more, it will round up, if less then it
will round down. 5 is 5 or more, so will round up to 0.14.
Significant figures
Some numbers, like distances in space, can get very big. Significant figures will work with any number, even if its not a decimal. The rules are the same for decimal places, but we do not start
counting at the decimal point any more. Significant figures count the number of digits after the first non-zero digit.
Rounding 36781 to 2 significant figures will either give us 36000 or 37000. We look at the 'next digit' to work out what will happen. If the 'next digit' is 5 or more, it will round up, if less then
it will round down. 7 is 5 or more, so will round up to 37000. | {"url":"https://www.onmaths.com/resource/rounding/?cid=132700","timestamp":"2024-11-08T20:15:44Z","content_type":"text/html","content_length":"84249","record_id":"<urn:uuid:0bbc8cb6-e26e-4b50-9fd1-2ff99f3ba233>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00204.warc.gz"} |
Comprehensive Guide to Range Operations in Excel | MBT
In this post, we would be examining range operations in excel. To begin, let us define what a range is in excel.
What is a Range
A range is a collection of 2 or more cells. Range in excel is used mainly when writing functions, it can be used to create dynamic charts, for pivot table configuration and to bookmark data.
Types of Range
There are two types of range in excel namely contiguous and non contiguous range. Now let’s do a deep dive into each of these categories:
Contiguous range
A contiguous range is a collection of cells that are next to each other, either horizontally or vertically. The easiest example is when you highlight cells manually from top to bottom, left to right
or right to left. For example, the range A1:A10 above is a contiguous range when selected alone. So also is the range C1:C10 a contiguous range when selected alone.
Non-Contiguous range
A non-contiguous range on the other hand consists of two or more separate blocks of cell. These blocks can be separated by rows of columns. For example, in the image above, when range A1:A10 and
range C1:C10 are taken together, that constitutes a non-contiguous range because they are separated by column B
Selecting Ranges with Mouse / Touchpad
In excel, both contiguous and non-contiguous ranges can be selected using the mouse or trackpad and there are some instant benefits to this. In subsequent paragraphs, we would examine how to
highlight each type of range and the benefits of doing so.
Selecting a Contiguous Range
It is very easy to select a contiguous range, all you need to do is pick a starting cell and then right click and drag either from top to bottom or left to right.
The image on the left (first image for mobile users) is a contiguous range of B2:B7 whilst the image on the right (second image for mobile users) is a contiguous range of B2:G2
Note: selecting a Contiguous range is not only limited to a single column or row, we can highlight multiple rows or columns as a contiguous range so far they are not separated by any cell, row or
column. Examples are the images below.
The image on the left (first image for mobile users) has a range of B2:F4 whilst the image on the right (second image for mobile users) has a range of B2:C7
Selecting a Non-Contiguous Range
Remember we have established that non-contiguous ranges are separated by a cell, row, column or combination of row and column. How then do we select them since selecting contiguous range required us
dragging our selection continuously. Well, it’s easy, just follow these steps:
Step 1: Identify the first cell for the first range and select it.
Step 2: Now that you have identified the first cell, drag either from top to bottom or left to right. In our example, we would drag it down to row 10
Step 3: We have selected the first range, supposing we want to add range D2:D10, all we need to do is to hold down CTRL and highlight D2 to D10
Instant Benefits of Highlighting a Range in Excel
Selecting ranges in excel allows you a quick glance of some key metrics if dealing with numbers. These metrics are three in number and can be seen at the bottom of the document window when you
highlight a range with numbers in it. The metrics are
As stated earlier, highlighting a range with numbers give you a quick view of key parameters namely the average, count and sum of the set of numbers
Filling a Range (How to Auto Fill in Excel)
Excel gives you the ability to fill a range by dragging and releasing, and in the examples below, you would see how.
Fill Subsequent Cells with First
This is for when you want to fill subsequent cells in a range with the value in the first cell
For example, assume we have the number 50 in cell B2, how do we fill the cell B3 to B10 with this same number? Easy, all you need to do is to execute the following steps:
Step 1: Enter the value 50 into cell B2
Step 2: Click on the lower right corner of cell B2 and drag down till row 10 (the last row in our target ending cell B10)
Step 3: You’re done, you should see 50 from Cell B2 to B10
Auto Fill as a Sequence
This works for a sequence of numbers or dates. Once you fill the first two cells, then you can highlight those two cells and drag down as detailed above for excel to populate with the same spacing.
For example, if the first 2 numbers are 1 and 3 (difference of two), excel fills the third cell with number 5, the fourth with number 7 and so on.
Number Sequence
Below are the steps for filling a number sequence as discussed above:
Step 1: Input numbers in the first and second cell of the planned congruous range. In our example, we are choosing 1 as the first number and 3 as the second number. Our last row would be row 10
Step 2: Highlight the cells B2 and B3, click on the right corner of cell B3 and drag down till B10 and you’re done
Date Sequence
The steps in filling a date sequence is similar to that of filling a number sequence. All you need to do is to enter the first two dates and drag down as discussed in the previous sequence.
In our example, assume the starting cell as a date of 1st of May 2022, and subsequent cells in the sequence should be one week after the previous date.
Step 1: Input the dates (1st of May 2022 and 8th of May 2022) into the first and second cells
Step 2: Highlight the cells B2 and B3, click on the right corner of cell B3 and drag down till B10 and you’re done
Named Range
A named range is a collection of cells that form a range and have been given a name. They are usually used when writing formulas in excel. The modeler can define ranges to make it easier for others
to understand.
Take for example the snapshot of our Instant benefit of highlighting a range section, we have a list of enquiries to a call center and the count in a given year. We can calculate KPIs for the call
center like the highest number of enquiries, lowest number of enquiries, average number of enquiries and sum of enquiries and we don’t need to select the range every instance once we define the range
from the get go. What do I mean? See the guide below
Step 1: Define the range by highlighting the relevant cells and giving it a name you would remember in the name box. In the example below, I am defining cell B4:B24 as enquiries. To do this, type
your chosen name (enquiries in this case) into the name box at the top left corner and press enter
Step 2: Type in the formula using the name you defined instead of the actual range
Remember “enquiries” is the named range B4:B24. In that case, the formulas used for cells G7:G10 would be as follows:
=MAX(enquiries) // To calculate the highest number of enquiries in cell G7
=MIN(enquiries) // To calculate the lowest number of enquiries in cell G8
=AVERAGE(enquiries) // To calculate the average number of enquiries in cell G9
=SUM(enquiries) // To calculate the sum of all enquiries in cell G10
Copying a Range
Copying a range is as simple as copying any cell and pasting it somewhere else. You however need to ensure that in case of an analysis that picks a cell outside the range as input, you anchor ($ sign
in front and at the back of the Column letter) said cell because excel would pick the equivalent number of rows down / up and equivalent number of columns to the right / left of the cell outside the
copied range. Confusing? Dont worry, would discuss in detail after I show you how to copy a range.
Step 1: Highlight the range you intend to copy. From the example above, assume we want to copy the range B4:B24 to column G
Step 2: Right click and select copy or hit Ctrl + C
Step 3: Navigate to cell G4 or whichever cell you desire to paste, right click and click past under paste options. You can also hit Ctrl + V to paste
Step 4: That’s it, you are done.
Copying a range with dependencies outside the range
Ok, back to what we were discussing about anchoring cells outside the range we intend to copy. Let’s imagine that in our previous example, we forecasted a 5% increase in all enquiries for 20X3 on
column C and later on we want to copy all enquiries for 20X3 to column G. What do we do? Read on to find out.
In the image above, all enquiries under 20X3 are dependent on the growth rate OF 5% we have specified in cell F8. We are projecting an increase in growth rate of 5% from 20X2 and the formula used for
Package clarification Enquiries under 20X3 for example is:
Now let’s copy C4:C24 to column G
You can see from the result that something is not right. That is because we didn’t anchor / fix cells outside the range we were copying. For starters, the formula in column C is dependent on the
previous year enquiries in column B. Copying to column G will then see excel move references initially picking column B to now pick column F because it is one column to the left of where we are
pasting it.
The same applies to the cell containing the growth rate. Excel would count the equivalent number of columns to the right of column G and pick that as the growth rate cell. See below >>>
As you can see from the above, the growth rate cell in column F is three columns to the right of column C (the original range we copied), and because of that, excel picks three columns after
destination G as the growth rate.
Excel’s default when handling references is relative cell reference and in the example above we need to maintain absolute reference for the growth rate cell and column for 20X2 enquiries. To correct
this, we need to anchor column B for each row in column C and then anchor cell F8 that contains the growth rate %
Anchoring Columns or Rows
In our example above, we only need to anchor the column B for every formula in our range C4:C24. This means that as I copy the formula from C4 to C5, it should pick column row 5 of column B. In
essence, the row reference is relative whilst column reference is absolute.
Anchoring rows only is the flip side, column reference would be relative and row reference would be absolute. The third anchoring is absolute anchoring where both column and row reference are
Column Anchor
To anchor a column only, take for example column B, let a $ (dollar) sign appear before the column letter as seen below.
$B4 // Press F4 on your keyboard three times continuously to achieve this
Row Anchor
To anchor a row only, take for example row 4, let a $ (dollar) sign appear before the row number as seen below.
B$4 // Press F4 on your keyboard two times continuously to achieve this
Absolute Anchor
Absolute anchor is anchoring both the row and column meaning that no matter where you copy the formula to, it would still pick the same cell reference. To achieve this, let a dollar sign appear
before the column letter and before the row number as seen below.
$B$4 // Press F4 on your once to achieve this
Back to our Example
As stated above, we need to anchor column B in our formula, and to do this, enter into the formula in C4, click on B4 in the formula bar and press F4 three times
Good, but we are not done yet. We still need an absolute anchor for cell F8. To do this, click on F8 in the formula bar and press F4 once
Now press enter and copy and paste the formula in C4 to the range C5:C24
Almost done, just need to check that everything is working well. Highlight range C4:C24 and paste it in column G
You should see the exact same numbers as those on column C.
Moving a range
Moving a range can be done in two ways. One is with your mouse / touchpad and the other with your keyboard.
Moving a range with your mouse / touchpad
Step 1: To move a range with your mouse / touchpad, highlight the range and click on the border of the range
Step 2: Drag the range to its new location. For example let’s move the range in column G to column J.
That’s it, you are done.
Moving a range with your keyboard
Step 1: Highlight the range and press Ctrl + X to cut. You should see some cell selector dots (marching ants) at the borders of the range.
Step 2: Navigate to the new location and paste with Ctrl + V
Voila, and you are done.
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Connecting Math and Science through Decoding Models
Thank you for taking the time to watch this video! Our team would be interested in your feedback, especially in regard to the following: 1. Are you using or are aware of a similar approach to using
"decoding" or interpreting the disciplinary ideas and processes embodied in code for math/science learning? 2. Have you used or are aware of assessment metrics or indicators that students are
connecting a. science & code; b. math & code; and, c. math & science through code? Feel free to comment on any other aspects of the project that you have questions about.
Hi Aditi, this is an interesting idea. If I understood it correctly, students are given both a physical phenomenon (i.e.: water filtration) and a computer program describing that phenomenon (btw, are
you using scratch?), and they have to arrive to an understanding of how the code does model the phenomenon. Is this correct? So they are not directly engaged with actual coding, it's more like
debugging the give program to understand how it works?
Hello Andres - great question. You are right that one of our goals is for students to reason about or "decode" how the code in a model represents a mathematical or scientific idea. However, our aim
is also to have students modify the code in some way to represent a particular math/science idea. We couldn't focus on this in this past year because of the challenges of collaborative coding online.
One of our RQs is around investigating relationships between decoding in science class and modifying code in math class as a way to help students bridge math and science learning. So we have been
reflecting on how to incorporate more coding in our PD and curriculum for next year. The modeling environment we're using is StarLogo Nova.
Hi Aditi, thanks for your response to my question. I find the idea of decoding very interesting, as a programmer my self I do it all the time, but haven't thought of its pedagogical potential!
Thank you for sharing your work and I appreciate how the idea of decoding blossomed from the challenges associated with collaborative programming during the last year. This reminds me of how the WISE
program (web-based inquiry science environment) integrates of computational thinking (not necessarily coding) and science/math concepts (e.g., photosynthesis). One thing I appreciate about the WISE
curriculum is that it integrates interactive graphs and figures to enhance student’s understanding (or a type of “decoding”?) of what is happening computationally. Do you think that might serve as an
approach towards moving towards the most sophisticated kinds of decoding?
Hi Michael! That's a great question. We're characterizing "decoding" specifically as interpreting the code to be able to describe the mathematical and scientific abstractions embedded in it. So in
the WISE example, we would not describe students interpreting graphs and figures as a type of decoding. But I agree with you that additional representations (including graphs and figures) in the
computational model can play a powerful role in enabling students to interpret and read the mathematical/scientific abstractions embedded in the code. They all serve as additional forms of feedback
essentially to understand what the code is doing. In our filtration unit, we use "single step runs" to enable students to step through the code one tick at a time to understand how a loop works. In a
unit on heat absorption/reflection, we have students interpret a graph of trapped/reflected heat. If you have ideas/leads for how to use multiple representations to support interpretation of code,
would love to hear!
Thanks for sharing your work. This looks like a great way to connect coding to real world problems. If you're looking for an extension problem set after your students go through the water filtration,
it looks like modeling a dialysis machine would fit well. Similarly, you need holes in a membrane sized appropriately to allow smaller waste molecules to pass through but prevent larger red blood
cells from passing through.
Hi Joseph - those are great ideas! We're always looking to add new examples to our teacher guide for teachers to have handy when teaching our units. We'll include these ones too. Thanks for stopping
Thanks for sharing your work. This looks like a great way to connect coding to real world problems. If you're looking for an extension problem set after your students go through the water filtration,
it looks like modeling a dialysis machine would fit well. Similarly, you need holes in a membrane sized appropriately to allow smaller waste molecules to pass through but prevent larger red blood
cells from passing through.
Hi Aditi, thanks for your video and great to learn about your "decoding" approach. I especially appreciated the researcher-practitioner partnership sensibility of your work, and that you managed to
get some data with students even during a pandemic! I am thinking about your question -- (a) who else is doing decoding? Asked this specifically, nothing comes to mind. But more generally, I think
you are asking "is anyone exploring how connecting multiple representations helps students to learn mathematics" -- and within that frame, there's a lot of research that could situate and inform your
work. I'm a fan of Hiebert's suggestion that what it means to "make sense" of mathematics is to "make connections" especially across representations. In the SimCalc work I did (ask Eric), I came to
think of the valuable space for making connections as having two dimensions - visual (simulations) vs. linguistic representations (code) and familiar vs. formal representations. Visual-formal=a graph
or a bar representation of ratio | Visual-familiar=a phenomena | Linguistic formal = code | Linguistic familiar = a story or explanation. Really supporting students to make all four types of
connections takes time, but I believe it does result in deeper learning. Is this helpful? Is it generative for anything you might do or try?
Hi Jeremy, good to see you here, and thanks for watching our video! I like the idea of thinking about sense making in math as connecting across representations along the two dimensions you mention.
Our RQs are situated around investigating how students come to "see" the mathematical relationships/processes embedded in the linguistic-formal, and I can see how the tools we're using to support
them in that work could be described as visual-familiar, linguistic-familiar and even visual-formal.
I do think though that in this project, we're giving special status to the linguistic-formal (code) as a representational form that can support bridging math and science learning. My question to the
audience was to ask if others were doing a kind of connecting math+science learning.
I am familiar with the SimCalc work - a lot of the work coming from the Kaput Center really grounded my theoretical and methodological understanding of how to think about and study the role of
technologies in learning :)
Cool to hear that SimCalc was helpful to you. Good luck with this work going forward! | {"url":"http://stemforall2021.videohall.com/presentations/2050.html","timestamp":"2024-11-07T05:55:39Z","content_type":"application/xhtml+xml","content_length":"60187","record_id":"<urn:uuid:d68c48e1-4691-4a06-88ed-9388cfcc501f>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00619.warc.gz"} |
How do you solve abs(2/3x+2)=10? | HIX Tutor
How do you solve #abs(2/3x+2)=10#?
Answer 1
$\textcolor{g r e e n}{x = 12}$ or $\textcolor{g r e e n}{x = - 18}$
If #abs(2/3x+2)=10# then #{: ("either ", 2/3x+2 = 10," or ",2/3x+2=-10), (,rarr 2/3x=8,,rarr 2/3x = -12), (,rarr x=12,,rarr x=-18) :}#
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Answer 2
To solve the equation abs(2/3x + 2) = 10:
1. Start by isolating the absolute value expression: 2/3x + 2 = 10 or 2/3x + 2 = -10
2. Solve each equation separately: For 2/3x + 2 = 10: Subtract 2 from both sides: 2/3x = 8 Multiply both sides by 3/2: x = 12 For 2/3x + 2 = -10: Subtract 2 from both sides: 2/3x = -12 Multiply both
sides by 3/2: x = -18
So the solutions are x = 12 and x = -18.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Julian Calendar 2024: Today's Julian Date Converter
What is Julian Date?
The International Astronomical Union unites many astronomical societies from around the world under its roof. This union, which is a member of the International Science Council, is the official
institution of the astronomical community.
The Julian calendar, originally called the “Radical Calendar“, is a calendar that was used by several ancient cultures, from the middle of the second millennium BC until the first century BC. However
technically, a today’s julian date can mean different things.
Additionally, we offer an array of 2024 calendar templates that users can personalize and print out for organizing and scheduling. Our 2024 calendars have various formats – such as annual, monthly,
and weekly layouts – and enable you to incorporate your own images, holidays, or remarks. Perfect for home, academic, or business purposes. There are also 2024 calendars tailored specifically for
educators, learners, mothers, fathers, and more. Peruse our selection of free printable and customizable 2024 calendars.
Julian Calendar 2024
Julian Calendar 2024
The Julian calendar for 2024, like its predecessor introduced by Julius Caesar in 46 BC, treats 2024 as a leap year, adding an extra day to February. This calendar follows a simpler leap year system
than the Gregorian calendar, designating every fourth year as a leap year without the more complex exceptions found in the Gregorian system, such as skipping centurial years not divisible by 400.
In 2024, the Julian year begins on a Saturday and ends on a Sunday, making it a year with 52 full weeks and one extra day. This additional day brings February’s total to 29 days. The remaining months
follow the standard pattern of alternating between 30 and 31 days, with July and August being the only consecutive months with 31 days each.
The difference between the Julian and Gregorian calendars grows over time due to their varying leap year rules. By 2024, this discrepancy will be 13 days. For example, when it’s January 1, 2024, in
the Gregorian calendar, it will correspond to December 19, 2023, in the Julian calendar.
Some Orthodox Christian churches continue to follow the Julian calendar, which results in different dates for religious observances compared to those using the Gregorian calendar. For instance,
Christmas is celebrated on January 7 in the Julian calendar, aligning with December 25 in the Gregorian calendar.
The Julian calendar does not accommodate daylight saving time adjustments. Therefore, regions adhering to this calendar do not change their clocks in 2024.
Despite its simplicity and regular month and leap year pattern, the Julian calendar’s less accurate accounting of the solar year means it gradually drifts out of alignment with the astronomical
seasons. Nonetheless, it remains significant for historical studies, religious observances, and in astronomy, particularly through its influence on the Julian Day Number system, which provides a
continuous count of days for astronomical calculations.
Today’s Julian Date
Today's date is
. Today's Julian Date is
Julian Date Converter
Julian Day, a time measurement system, is included in the Julian date system. The Julian Day proposed by the International Astronomical Union for use in astronomical studies, BC. Presents the time
interval of a day in fractions of days starting from January 1, 4713.
How is Julian Date Calculated?
Julian Day, BC. It presents the time interval of a day starting from Noon with Universal Time (UT) on Monday, January 1, 4713, in fractions of the day and the day. In other words, noon was chosen as
the starting time of the day. The first day is considered to be the 0. Julian Day. In this way, the multiples of 7 always correspond to Monday.
Negative values โ โ can also be used, but these values โ โ invalidate any history saved. While each day is expressed as an integer, any hour of the day is added to this integer as a fraction of
the day. The date January 1, 4713 is the beginning. The days that come after this date have been counted consecutively and added to this number. These numbers are indicated in the astronomy annals as
charts by years, months, and days. The Julian Date, abbreviated as JT, is the date determined by these numbers. The Julian Date of any date can be found in 2 ways:
The charts in the astronomy annals are checked.
The Julian Day calculation formula is used.
The formula for calculating the Julian Day is as follows:
JT = 2415020 + 365 x (year – 1900) + N + L – 0.5
This formula; It gives the Julian Date that coincides with any date since 1990. Universal time is EZ = 0 in this calculation. N is the number of days after the new year, and L is the number of leap
years between 1901 and the date to be calculated. In the formula; 2415020 refers to the Julian Day corresponding to January 1, 1900, and the amount 0.5 denotes the decimal equivalent of the half day
resulting from the start of the Julian Day in the middle of the day.
The formula for finding the number N is as follows:
N = <275M / 9> -2 <(M + 9) / 12> + I-30
In this formula, M is the number of months, and I is the day of the month. Also <> indicates that the full part of the number in the brackets will be taken. For a leap year, the factor of 2 in the
2nd term is removed.
Correspondence of the decimal parts of Julian Date in hours, minutes and seconds:
0.1 = 2.4 hours or 144 minutes or 8640 seconds
0.01 = 0.24 hours or 14.4 minutes or 864 seconds
0.001 = 0.024 hours or 1.44 minutes or 86.4 seconds
0.0001 = 0.0024 hours or 0.144 minutes or 8.64 seconds
0.00001 = 0.00024 hours or 0.0144 minutes or 0.864 seconds.
Julian Calendar
The Julian Calendar is the most famous of the solar calendars. B.C. It was prepared by the Roman Emperor Jules Caesar (Jules Ceasar) in 46 BC and was used in the western world until the 16th century.
It is also known as the Julius Calendar. It is considered to be the first version of the Gregorian calendar.
Aiming to solve the confusion and problems in the calendars used in the past years, Caesar received help from the Alexandrian astronomer Sosigenes. With the suggestion of Sosigenes, this calendar was
prepared based on the movements of the sun, not the moon. Aiming to correct seasonal shifts, Sosigenes calculated a year as 356.25 days.
In this way, while the years that cannot be divided into 4 are 365 days, the increasing quarter days from these years were added to the 4th year and the leap year was increased to 366 days. In order
for 1 year to be 12 months, leap years are arranged to be 6 months 30 days, and the other 6 months 31 days. In non-residual years, 1 day is removed from the last month of the year.
At that time, New Year’s Eve was in March. Therefore, February, which is the last month of the year, has been reduced to 30 days in leap years and 29 days in other years. Caesar wanted to immortalize
his name as the organizer of the calendar and changed the name of July to July.
The arrangements made in the calendar after the death of Caesar could not be implemented properly. The fact that Pontifeksler, who made arrangements in the calendar, made a leap year application in 3
years, caused confusion again.
During the 40 years of application in this way, there has been 3 days of slippage. Roman Emperor Augustus, BC. In the 8th year, it corrected this shift by stopping the leap year application for 12
years. He also changed the name of August to his own name, Augustus. In the regulations made; It was taken 1 day from February, and added to August, after July 31 and August took 30 days. In this
way, February took 29 days in leap years and 28 days in other years. Julian Calendar, BC. It was used from the 46th to the 16th century.
Leap year practice was applied in Julian Calendar for the first time in history. As a result of a small difference in this calculation, a 1-day shift occurred approximately every 128 years. Due to
the confusion created by this shift, the Julian Calendar was abandoned in the 16th century and the Gregorian Calendar was adopted.
Understanding the Differences between the Julian and Gregorian Calendars
The calendar is a crucial tool in our daily lives, helping us to organize time and plan our schedules. However, did you know that there are actually two different calendars in use today? The Julian
calendar and the Gregorian calendar are the two most widely used calendar systems in the world, and while they may seem similar at first glance, there are some key differences between the two.
The Julian Calendar
The Julian calendar was first introduced by Julius Caesar in 45 BC as a way to align the Roman calendar with the solar year. The calendar was based on a year of 365 days, with an extra day added
every four years to account for the extra time it takes the Earth to orbit the Sun. This extra day was added to February, creating what is known as a “leap year.”
One of the main features of the Julian calendar is its simplicity. With only one rule to remember (a leap year every four years), it was easy for people to use and understand. However, the Julian
calendar’s approximation of 365.25 days for the tropical year (the time from one spring equinox to the next) was slightly too long. As time passed, the excess (11 minutes and some 14 seconds
according to modern measurements) slowly added up, causing the calendar date of the equinox to move ever earlier (by one day every 128 years or so).
The Gregorian Calendar
To correct the problem with the Julian calendar, Pope Gregory XIII introduced the Gregorian calendar in 1582. The new calendar was based on the same principles as the Julian calendar, but with a few
key changes. The most significant change was the introduction of a new rule for leap years. In the Gregorian calendar, a leap year is still added every four years, but years that are divisible by 100
are not leap years unless they are also divisible by 400. This means that the year 2000 was a leap year, but the year 1700 was not.
The introduction of this new rule has helped to keep the calendar more closely aligned with the solar year. In fact, the Gregorian calendar is accurate to within just 26 seconds per year. This means
that the calendar will not need to be corrected for over 3,000 years.
One of the most significant differences between the Julian and Gregorian calendars is that the latter is in use today by most of the world. The Catholic Church and many Catholic countries adopted the
Gregorian calendar immediately, while other countries followed suit over the next several centuries. However, some countries, such as Greece and Russia, did not adopt the Gregorian calendar until
much later.
Which calendar is more accurate?
The Gregorian calendar is more accurate than the Julian calendar.
When was the Julian calendar introduced?
The Julian calendar was introduced by Julius Caesar in 45 BC.
When was the Gregorian calendar introduced?
The Gregorian calendar was introduced by Pope Gregory XIII in 1582.
Which countries still use the Julian calendar?
Some countries, such as Greece and Russia, did not adopt the Gregorian calendar until the early 20th century.
How often is a leap year in the Julian calendar?
In the Julian calendar, a leap year is added every four years.
How often is a leap year in the Gregorian calendar?
In the Gregorian calendar, a leap year is added every four years, but years that are divisible by 100 are not leap years unless they are also divisible by 400. | {"url":"https://www.typecalendar.com/julian-date","timestamp":"2024-11-12T10:25:09Z","content_type":"text/html","content_length":"301106","record_id":"<urn:uuid:11ec6951-ca5d-4bb6-8d91-1b3359335fa3>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00053.warc.gz"} |
Arithmetic and Geometric Averages in Trading and Investing: Position Sizing and the Kelly Criterion - QuantifiedStrategies.com
Arithmetic and Geometric Averages in Trading and Investing: Position Sizing and the Kelly Criterion
The arithmetic vs geometric averages can be difficult to grasp. Albert Einstein is famous for saying that compounding is the eighth wonder. But what if he is wrong? Perhaps multiplicative compounding
is the most destructive force in the universe? The sequence of returns and the different alternative paths ultimately determine your geometrical average which might be far away from the arithmetic
average. Don’t be fooled by the arithmetic average.
The arithmetic and geometric averages/means and returns differ in trading and investing because the arithmetic average is mainly a theoretical average, while the geometric average takes into account
the sequence of returns (or paths) of an investment.
The arithmetic average might be positive, but you can still end up with losses – even ruin. The reason is the volatility tax. The multiplicative effects of compounding might leave you with losses you
never recover from. Your sequence of returns is dependent on your position sizing. Thus, we end the article by explaining the Kelly Criterion which is all about finding the optimal position/betting
What is the arithmetic average?
If your strategy has a positive expected average gain per trade, the end result still might be catastrophic. The reason is due to path/sequence and compound average growth rate (CAGR). CAGR
(geometric return), mean, or average, is the correct measurement and is different from the arithmetic return.
Let’s assume you have a strategy that returns the following sequence of ten trades measured in percentage: 11, 33, 6, -5, 7, 21, -19, -9, 29, and -24. If you add all the numbers and divide by the
number of observations (10), you get 5%. This means that the average gain per trade was 5%. This is the arithmetic average.
But the problem with the arithmetic average is that it doesn’t indicate your compounded return on your trades and your end result.
The difference between your starting capital and how much you end up with is not the arithmetic return, but the CAGR or geometric average/geometric mean:
What is the geometric average?
Now, if you start with 100 000 and add or deduct the above ten trades, you end up with 139 092. This equals a geometric return (CAGR) of 3.35%, much lower than the arithmetic average of 5%.
The compound annual growth rate (CAGR) is the return on an investment over a certain period of time. This is why it differs from the arithmetic average. It takes into account the compounding from the
start to the finish. This means that the arithmetic and geometric average are two completely different things.
Arithmetic averages vs geometric averages: Why do they differ?
Arithmetic averages and geometric averages differ because the arithmetic average is mainly a theoretical value while the geometrical average is what you get in real life based on the sequence of
returns. Mark Spitznagel writes in his Safe Haven book: you get what you get, not what you expect.
If you roll the dice you might not get the theoretical average because of the sequence and order of the rolling of the dice. You might get 1, 4, and 3 when you roll, while the next sequence might be
2, 6, and 1. Depending on the expected gain on each number of the dice, the end result fluctuates wildly. We show this by an example further down.
In the example above we got a lower CAGR and geometric average than the arithmetic average. Why do we get worse results when using the geometric return? It’s because of the volatility tax:
What is the volatility tax?
Over my 25yrs in the game of quantitative investing, and drenched in equations and models, I’ve honed in on the only math that really matters: Minimize drawdowns during unfavorable times. The
multiplicative nature of positive returns during good times does all the rest.
Wayne Himelsein on Twitter
The name volatility tax is taken from Mark Spitznagel’s book called Safe Haven – Investing For Financial Storms. Spitznagel defines the volatility tax like this:
It’s a tax extracted by the multiplicative dynamics of compounding, what I have dubbed the volatility tax.
The volatility tax is a tax well hidden in the arithmetic average. The geometric average is lower because you suffer drawdowns that are hard to recover from. If you have a 33% loss in one year, you
need a 49% return to get back to even. If you lose 50%, you need to get 100% to recoup.
This is why Spitznagel calls this a volatility tax. Just ONE devastating loss might not lower the arithmetic average significantly, but it can lower the geometric average significantly – even lead to
ruin! This is why you need to control your losses and control your position sizing.
An example of arithmetic average vs geometric average
Mark Spitznagel has a very illustrative example in his book about Safe Haven Investing. He made up a gamble of rolling dice with the following three different outcomes:
• If 1 comes up, you lose 50% of your equity.
• If 2, 3, 4, and 5 come up, you gain 5% of your equity.
• If 6 comes up, you gain 50% of your equity.
The arithmetic return of the game is 3.3%: (-50, 5, 5, 5, 5, 50)/6.
This means your expected gain per roll is 3.3%. If you start with 1 dollar and roll the dice 300 times, you end up with 18 713 (1.033 x 1.033 x 1.033….. 300 times). Compounding the 3.3% gain per roll
leaves you with almost 19 000 times your starting wealth, on average. Not bad!
However, before you buy champagne and order escorts you really need to think things through. The problem is that when you roll the dice you will not get the arithmetic average. You only get to walk
one path. You better aim for survival!
In fact, you will never get the arithmetic average because it’s just a theoretical number. You either lose 50%, gain 5%, or gain 50%. That is the only possible outcome of each roll of the dice. An
arithmetic average is just a theoretical number that exists on paper. This is the difference between being street smart vs. book smart.
The huge difference between each roll of the dice implies you face the risk of an unlucky roll – something called sampling error. On average one in six rolls will cut your equity in half! You might
take comfort in that this is a fair game and if you keep rolling the dice your expected gain will regress to the mean of 3.3%.
But wait a minute. Let’s test if the latter reasoning is correct:
You start rolling the dice and the first six rolls return 3, 6, 1, 5, 4, and 2. What is your equity after those six rolls?
1 (starting capital) x 1.05 x 1.5 x 0.5 x 1.05 x 1.05 x 1.05 = 0.912
Each side of the dice turned up once and your equity is still less than what you started with. Shouldn’t you get the average 3.3% gain when all sides of the dice turned up once?
The problem in the real world is that you only get one path or sequence – not all of the 300 rolls. As Spitznagel writes in his book:
You get what you get, not what you expect!
What happens if you play out different paths ten thousand separate times (each with 300 rolls of the dice)?
The illustration below from Mark Spitznagels Safe Haven Investing shows the frequency distribution (read more about Monte Carlo simulation in trading and investing):
In real life, you can only traverse only one of these ten thousand paths. You better pick the right one!
Even though you can expect each side to turn up equally on all sides about 50 times in a 300 roll (one path), the arithmetic average doesn’t translate into the same real-life geometric average:
If each side of the dice comes up an equal 50 times (300/6) on each side, your entire equity is almost wiped out:
0.5^50 x 1.05^50 x 1.05^50 x 1.05^50 x 1.05^50 x 1.5^50 = 0.01
You risk losing 99% of your life savings! Mark Spitznagel calls multiplicative compounding the most destructive force in the universe. Perhaps Einstein was wrong in labeling compounding as the eighth
What is the geometric average of sampling ten thousand paths of 300 rolls?
Unfortunately, the geometric average is a negative 1.5%: 0.01^(1/300). Your 3.3% positive arithmetic average will most likely end up with a loss in the real world.
Most retail investors perform much worse than the benchmarks, and one reason for that is big losses that multiplicates. If you “choose” the wrong path, you are unlikely to ever recover. You are lost.
This is why you should always trade with a smaller size than you’d like.
What is the optimal betting size? Let’s look at the Kelly Criterion:
What is the Kelly Criterion?
Based on the difference between the arithmetic and geometric averages a mathematician named John Kelly made a formula in 1956 for the optimal betting size when the expected returns are known. The
criterion is based on the expected geometric return and not the arithmetic average.
The Kelly Criterion is based on the logarithmic scale of the geometric expected return. It maximizes the expected value by considering the risk of ruin and losses.
How to calculate the optimal betting size by using the Kelly Criterion
The Kelly Criterion is straightforward to calculate: you only need two inputs to determine the optimal betting size:
1. The win/loss ratio – the win percentage of your trading strategy (R) (dividing the total gains of the winning trades by the total loss of the losing trades)
2. The win ratio of the trading strategy (W) (the number of trades that showed a profit divided by the number of trades that made a loss)
These two variables you need to put into this formula:
Kelly % = W – [(1 – W) / R]
Let’s assume you have a strategy you want to trade after a successful out-of-sample backtest. The backtest had these numbers:
• Total gains of the winning trades: 9 229 137
• Total loss of the losing trades: 3 206 730
• (The win/loss ratio is thus 2.88)
• The win ratio is 74% (thus 26% of the trades showed a loss)
Putting this into the formula above yields this result:
Kelly % = 0.74 – [(1 – 0.74) / 2.88] = 0.6497
Thus, the optimal betting size according to the Kelly formula is 0.65 of your equity.
Of course, we don’t know the future outcome of a trading strategy so the Kelly Criterion should be used carefully. Also, keep in the back of your mind that the biggest drawdown is yet to come. As we
always emphasize, always trade a smaller size than you’d like.
Why betting size is extremely important
Let’s return to Spitznagel’s example of rolling the dice.
Instead of betting 100% of your equity on each roll, you bet only 40% of your equity while the remaining 60% sits idle and yields nothing. This means your arithmetic average drops to 1.32%.
But a funny thing happens to your expected end result: the geometric average improves from a negative 1.5% to a positive 0.6%!
Mark Spitznagel has the following diagram which shows the expected geometric average return for the different betting sizes:
There is even a more famous and illustrative betting game that illustrates the importance of the betting size:
The optimal betting size in a coin toss
In a behavioral study done by scientists, a group of 61 people was given 25 USD and asked to place even money on a game that would land heads 60% of the time. There was a time limit of 30 minutes and
the coin was tossed 10 times a minute, hence the participants could place a maximum of 300 bets. The prizes were capped at 250 USD.
What is the optimal betting size? According to the Kelly Criterion, the optimal betting size is about 20% which equals about 2% gain per toss. This equals a maximum potential gain of 10 500 USD if it
wasn’t for the capped 250 USD winning size.
How did the participants perform in this game with a highly positive expectancy? It turns out 28% went bust and the average payout was just 91 USD. The max limit was reached by only 21% of the
Furthermore, 18% of the 61 participants bet everything on one toss, and even 67% of the group at one point bet on tails at least once. If everyone had followed the betting size of the Kelly
Criterion, about 95% of the players would have reached the limit of max profits.
A practical trading example of position sizing in trading:
Below is the backtest report of a trading strategy in the S&P 500. The backtest is from SPY’s inception in 1993 until October 2021. The backtest had the following statistics:
• 533 trades
• CAGR 15.4% if 100% equity allocation (buy and hold was 10.4%)
• The average gain per trade was 0.8%
• The win ratio was 74%
• The average winner was 1.68%
• The average loser was a negative 1.72%
• The profit factor was a solid 2.88
• The Sharpe Ratio was 2
If we allocate 100% of our equity on each trade for this strategy, Amibroker gave the following results for a simulated Monte Carlo analysis:
Even with such good backtest statistics, the strategy has a pretty high chance of going bust (10%).
If we lower the allocation to 70% of the equity, the optional allocation is between 65 and 70%, we get the following numbers:
The annual return is lower for all simulations because we allocate less capital, but we have a zero chance of going bust. As Nassim Nicholas Taleb often reminds us, first we must survive. All else is
More about averages and moving averages:
If you are interested in reading more about averages we also recommend our main article about moving averages: moving averages strategies That page also links to about 20 different backtests we have
done on the different moving averages that exist.
Conclusion – arithmetic averages vs geometric averages:
Don’t be fooled by the arithmetic average: The arithmetic and geometric averages differ. The sequence of returns might lead to completely different geometrical returns and averages.
In real life, you can only traverse one path, and you better make sure you have a margin of safety in case that path turns out to be wrong. You get only one path to survival, you don’t get the
average of paths. Multiplicative compounding is fantastic if you manage to get on the right track, but detrimental if you get it wrong.
Even with a positive expectancy, you can highly likely suffer losses – even ruin – if you don’t have the optimal position size. We end the article by yet again emphasizing our main trading rule:
Always trade smaller than you’d like.
How does the arithmetic average differ from the geometric average in measuring returns?
The arithmetic average calculates the average gain per trade without accounting for the compounding effect. On the other hand, the geometric average (CAGR) considers the compounding from start to
finish, providing a more accurate measure of the actual return.
Can positive arithmetic averages lead to losses or ruin in trading?
Yes, even with a positive arithmetic average, losses or ruin are possible due to the volatility tax and the multiplicative effects of compounding. Geometric averages, considering drawdowns and
compounding, offer a more realistic view of potential outcomes.
How can one calculate the optimal betting size using the Kelly Criterion?
The Kelly Criterion requires two inputs: the win/loss ratio (R) and the win ratio of the trading strategy (W). The formula for Kelly % is: Kelly % = W – [(1 – W) / R]. This provides the optimal
betting size for maximizing expected returns while considering risk. | {"url":"https://www.quantifiedstrategies.com/arithmetic-and-geometric-averages-in-trading-and-investing-position-sizing-and-the-kelly-criterion/","timestamp":"2024-11-08T20:53:50Z","content_type":"text/html","content_length":"238373","record_id":"<urn:uuid:e0c1ee26-9b60-43dc-bf89-03d4a506c2a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00641.warc.gz"} |
Doppler Effect for Sound
The Doppler effect is the change in the observed frequency of a source due to the motion of either the source or receiver or both. Only the component of motion along the line connecting the source
and receiver contributes to the Doppler effect. Any arbitrary motion can be replaced by motion along the source-receiver axis with velocities consisting of the projections of the velocities along
that axis. Therefore, without loss of generality, assume that the source and receiver move along the x-axis and that the receiver is positioned further out along the x-axis. The source emits a
continuous tone of frequency, f[0], equally in all directions. First examine two important cases. The first case is where the source is stationary and the receiver is moving toward or away from the
source. A receiver moving away from the source will have positive velocity. A receiver moving toward the source will have negative velocity. If the receiver moves towards the source, it will
encounter wave crests more frequently and the received frequency will increase according to
${f}^{\prime }={f}_{0}\left(\frac{c-{v}_{r}}{c}\right)$
Frequency will increase because v[r] is negative. If the receiver is moving away from the source, the v[r] is positive and the frequency decreases. A similar situation occurs when the source is
moving and the receiver is stationary. Then the frequency at the receiver is
${f}^{\prime }={f}_{0}\left(\frac{c}{c-{v}_{s}}\right)$
The frequency increases when v[s] is positive as the source moves toward the receiver. When v[s] is negative, the frequency decreases. Both effects can be combined into
${f}^{\prime }={f}_{0}\left(\frac{c-{v}_{r}}{c}\right)\left(\frac{c}{c-{v}_{s}}\right)={f}_{0}\left(\frac{c-{v}_{r}}{c-{v}_{s}}\right)={f}_{0}\left(\frac{1-{v}_{r}}{c}}{1-{v}_{s}}{c}}\right).$
There is a difference in the Doppler formulas for sound versus electromagnetic waves. For sound, the Doppler shift depends on both the source and receiver velocities. For electromagnetic waves, the
Doppler shift depends on the difference between the source and receiver velocities.
[1] Halliday, David, R. Resnick, and J. Walker, Fundamentals of Physics, 10th ed. Wiley, New York, 2013. | {"url":"https://uk.mathworks.com/help/phased/ug/doppler-effect-for-sound.html","timestamp":"2024-11-10T11:39:46Z","content_type":"text/html","content_length":"69818","record_id":"<urn:uuid:21aba3e7-8ad7-44e1-9053-43c4114c3433>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00192.warc.gz"} |
Types of Fractions - Math Steps, Examples & Questions
In order to access this I need to be confident with:
Here you will learn about the types of fractions, including how to model and write numbers as proper and improper fractions and mixed numbers.
Students will first learn about different types of fractions as part of number and operations – fractions in 4th grade.
Types of fractions are different ways to show numbers that include parts of a whole. Types of fractions are typically grouped by values that are less than 1 and greater than 1.
For numbers that are less than one, you use proper fractions. In this type of fraction, the numerator (top number) is smaller than the denominator (bottom number).
When a proper fraction has a numerator of 1, it is called a unit fraction.
When a fraction has the same numerator and denominator, it is equal to \bf{1}.
For numbers larger than one, you use improper fractions and mixed numbers.
An improper fraction is a fraction where the numerator (top number) is equal to or larger than the denominator (bottom number).
A mixed number has a whole number part and a fractional part. Sometimes mixed numbers are called mixed fractions.
Any number greater than 1 can be shown as an improper fraction AND a mixed number.
Proper fractions
Improper fractions
A fraction where numerator (top number)
is smaller than the denominator A fraction where the numerator (top number)
(bottom number): is equal to or larger than the denominator
(bottom number):
Fractions equal to \bf{1}
Mixed numbers
A fraction where the numerator and
A number with a whole number and denominator are the same:
fractional part.
Proper fractions A fraction where numerator (top number) is smaller than the denominator (bottom number):
Improper fractions A fraction where the numerator (top number) is equal to or larger than the denominator (bottom number):
Mixed numbers A number with a whole number and fractional part.
Fractions equal to \bf{1} A fraction where the numerator and denominator are the same:
Use this worksheet to check your grade 4 studentsβ understanding of types of fractions. 15 questions with answers to identify areas of strength and support!
Write the value shown by the shaded part of the model.
The model is broken up into 6 equal parts, so 6 is the denominator.
2Decide if the fraction is larger than \bf{1} whole. If it is less, skip to step \bf{4} .
The number is less than 1 whole, skip to step 4.
Numbers less than 1 whole, can be written as proper fractions. The numerator is 4, because 4 parts are shaded in.
The model shows the proper fraction \, \cfrac{4}{6} \, shaded in.
The last whole in the model is broken up into 9 equal parts, so 9 is the denominator.
Decide if the fraction is larger than \bf{1} whole. If it is less, skip to step \bf{4} .
The value is larger than 1 whole, so move to the next step.
Decide whether to write the fraction as an improper fraction or a mixed number.
The model shows 2 wholes with parts left over, which can be used to write a mixed number.
The numerator is 5, because 5 parts are shaded in the last whole.
The model shows the mixed number \, 2 \cfrac{5}{9} \, shaded in.
Each whole in the model is broken up into 10 equal parts, so 10 is the denominator.
The model is shown in parts, which can be used to write an improper fraction.
The numerator is 18, because 18 parts are shaded in.
The model shows the improper fraction \, \cfrac{18}{10} \, shaded in.
The denominator is 8, so the model should be shown with 8 equal parts.
The numerator is less than the denominator, so \, \cfrac{3}{8} \, is a proper fraction and is less than 1 whole. Skip to step 4.
The numerator is 3, so there should be 3 parts shaded in.
The model shows the proper fraction \, \cfrac{3}{8} \, shaded in.
The denominator is 4, so the fractional part of the model should be shown with 4 equal parts.
There are 4 wholes with parts left over, so \, 4 \cfrac{1}{4} \, is a mixed number.
Show 4 wholes shaded, and 1 out of 4 parts shaded for the fractional part.
The model shows the mixed number \, 4 \cfrac{1}{4} \, shaded in.
The denominator is 12, so the model should show twelfths.
There are just parts, so \, \cfrac{13}{12} \, is an improper fraction.
The numerator is 13, so 13 parts should be shown.
The model shows the improper fraction \, \cfrac{13}{12} \, on a number line.
1. What is the number shown by the shaded part of the model?
Each whole in the model is broken up into 4 equal parts, so 4 is the denominator.
Numbers greater than 1 whole can be written as improper fractions.
The numerator is 14, because 14 parts are shaded in.
The model shows the improper fraction \, \cfrac{14}{4} \, shaded in.
2. What is the number shown by the shaded part of the model?
The model is broken up into 8 equal parts, so 8 is the denominator.
Numbers less than 1 whole can be written as proper fractions.
The numerator is 2, because 2 parts are shaded in.
The model shows the proper fraction \, \cfrac{2}{8} \, shaded in.
3. What is the number shown by the shaded part of the model?
Numbers greater than 1 whole can be written as mixed numbers.
The last whole in the model is broken up into 3 equal parts, so 3 is the denominator.
The numerator is 1, because 1 part is shaded in.
The model shows the mixed number \, 3 \cfrac{1}{3} \, shaded in.
The denominator is 6, so the model should show 6 equal parts.
The numerator is 11, so there should be 11 parts shaded in.
5. Which model shows \, 2 \cfrac{3}{10} \, shaded in?
The denominator is 10, so the last whole should show 10 equal parts.
The numerator is 3, so there should be 3 parts shaded in the last whole.
The denominator is 12, so the model should show 12 equal parts.
The numerator is 7, so there should be 7 parts shaded in.
The numerator shows the number of parts and the denominator shows the size of the parts.
Students work with all of the fraction types overviewed on this page, which all involve only natural numbers as the numerators and denominators. In middle and high school, they use more advanced
topics (like negative numbers and variables) to work with more complex fractions.
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Posts - Investment Consultancy
In this post I a want to give a short derivation of the replication portfolio and the risk neutral probabilities in the binomial model from Cox-Ross-Rubinstein. Let
Next we replicate the option value in twin security
In efficient markets there exist no profitable arbitarge opportunities. Therefore the outcome of the option value
law of one price
tells us that the value of assets that lead to the same cash flows must be the same. That means that the value of the option at time
We create a new variable
Hence we obtain:
risk-neutral probabilities
. Note that the value of the option does not explicitly involve the actual probabilities
Discounting at the risk-free rate is the main difference between decision tree analysis (DTA) and contingent claim analysis (CCA) or real options analysis (ROA). DTA does not take into account that
the risk of the cash flow streams changes when you consider options and opportunities. ROA implements this issue correctly.
WACC, Return Rates & Betas with Debt
Joachim Kuczynski, 02 April 2021
In this post I want to summarize some interesting results concerning equity return rates , betas and WACC of a levered company. Regarding the market value balance sheet of a firm we can state that
the value of the unlevered firm VU plus the present value of the tax shield VTS must be the same as the sum of levered equity E and debt D:
Further the rates of return on each side of the balance sheet are the weighted average of the component rates of return:
Substituting VU in the rate of return expression we get a general form of the equity return rate:
Consequently the general form of CAPM beta is given by:
The WACC is defined as the weighted average of equity and debt return rates including tax shield at corporate income tax rate WACC with Tax Shield), the WACC is given by:
Substituting the equity return rate we get a general form of the WACC:
Modigliani and Miller: Constant debt value
If the firm keeps its dept value D constant, there are no specific market risks concerning the tax shield. Therefore we can set the tax shield discount rate
Hence we get simplified expressions for equity return, equity beta and WACC:
Assuming that debt interest rate does not depend on the market return rate (CAPM) we can set Hamada equation for levered beta:
It is important to realize that Hamada’s equation is only valid if the value of debt is kept constant over time.
Harris and Pringle: Constant leverage ratio
Constant leverage ratio means that debt value is proportional to the value of the unlevered firm. According to Harris and Pringle that results in
But we have to take care. Miles and Ezzell, Arzac and Glosten have shown that you have a tax shield discount rate of
Miles and Ezzell
With a perpetuity growing rate g of debt and discounting in the first period with
Harris and Pringle
Taking the formula of Miles and Ezzell and setting
General debt ratio
If the amount of leverage is flexible and not constant or growing with a constant growth rate over time, the previous formulas do not work. In this case you have to use the APV method, in which you
calculate the tax shield in each time period seperately.
Baldwin Rate of Return | MIRR
Joachim Kuczynski, 05 March 2021
Baldwin rate definition
The modified internal rate of return (MIRR), or Baldwin rate of return respectively, is an advancement of the internal rate of return (IRR). But also the MIRR can be misleading and can generate false
investment decisions. The MIRR is defined as:
FV means the final value at the last considered period, PV stands for the present value.
Pitfalls of the Badwin rate
These points have to be considered carefully when applying the MIRR:
• Cash flows in different countries, with different currencies, equity betas, tax rates, capital structure, etc. should be evaluated with specific risk-adjusted discount rates. In MIRR the project
is profitable, if the rate of return is higher than the required WACC. But which WACC do we mean in projects with various differing cash flows ? No diversification of cash flows can be taken into
account in MIRR. But that is crucial in evaluating international projects.
• You need premises about the reinvestment rate of the contribution cash flows that affect the profitability of the project. These premises are not required in the (e)NPV concept. Hence you add an
additional element of uncertainty in your calculation when using the MIRR, without any need.
• Reinvesting contribution cash flows (numerator of the root) with the risk adjusted WACC means that the return of the project increases when the project risks and WACC increases. That cannot be
true. In pinciple you should not use key figures that require assumptions about reinvestment return rates. You are evaluating a certain project and not of other unknown investment sources. In
general you can discount all cash flows with its appropriate discount rate and capitalize it to the last period.
• Does capitalizing (or rediscounting) a cash flow with a risk-adjusted discount rate to a future period make sense in general? I do not think so. To rediscount cash flows with a risk adjusted
discount rate including a risk premium means that you are increasing the risk of the project. The project does not remain the same, because its risk increases. Only taking a riskless discount
rate for reinvestments would not increase the risk of the project. The MIRR comes from a classical perspective with no risk adjustment of the discount rates. If you are using risk-adjusted
discount rates, you are mixing two concepts that do not fit.
• An additional positive cash flow must improve the profitability of the project. If you add an additional, small cash flow in an additional period
• If you have e.g. an after sales market with small positive cash flows, the Baldwin rate decreases by considering these cash flows in your calculation. This is because the number of periods
increases and the
• You have to define clearly, which cash flow is in the numerator and which is in the denominator of the root. There is no clear and logic distinction. Thus you can find different definitions in
literature. Anyway avoid to take balance sheet definitions of “investment”. Note that besides investments also fixed costs and leasing payments have to be discounted with a default free discount
rate in general.
• You cannot compare mutual exclusive investment projects, if the investments or the project periods are different.
• You can also not evaluate investment projects with negative value contribution to the firm. But anyway such projects exist and have to be decided.
• All cash flows should be considered as expected value of a probability distributions. The expected value of the Baldwin rate is not the Baldwin rate of the expected values of the cash flows.
The (e)NPV concept is much better than the MIRR or Baldwin Rate of Return. The (e)NPV does not have all the pitfalls mentioned above. Further you can also evaluate and compare value-loosing
investment alternatives and do not need any premises about reinvestment rates. There are only disadvantage of the MIRR / Baldwin rate compared to the (e)NPV, try to avoid the application of MIRR /
Baldwin rate.
Operating Leverage
Operating leverage is the sensitivity of an asset’s value on the market development caused by the operational cost structure, fixed and variable costs. The asset can be a company, a project or
another economic unit. A production facility with high fixed costs is said to have high operating leverage. High operating leverage means a high asset beta caused by high fixed costs. The cash flows
of an asset mainly consists of revenues, fixed and variable expenses:
cash flow = revenues – fixed expenses – variable expenses
Costs are variable if they depend on the output rate. Fixed costs do not depend on the output rate. The (present) value of the asset,
Rearranging leads us to:
Those who receive the fixed expenses are like debtholders in the project. They get fixed payments. Those who receive the net cash flows of the asset are like shareholders. They get whatever is left
after payment of the fixed expenses. Now we analyze how the beta of the asset is related to the betas of revenues and expenses. The beta of the revenues is a weighted average of the betas of its
component parts:
The fixed expense beta is close to zero, because the fixed expenses do not depend on the market development. The receivers of the fixed expenses get a fixed stream of cash flows however the market
develops. That means
This is the relationship of asset beta to the beta of turnover. The asset beta increases with increasing fixed costs. As an accounting measure we define the degree of operating leverage (DOL) as:
The degree of operating leverage measures the change in profits when revenues change.
Valuing the equity beta is a standard issue in DCF analysis. In many cases you take an industry segment beta and adjust it to your company or project. The adjustment of the industry beta also
includes the adjustment of operating leverage. We assume that
For detailed information see: Brealey/Myers/Allen: Principles of Corporate Finance, 13th edition, p. 238, McGraw Hill Education, 2020)
After-Tax Discount Rate
In this post I want do derive the after-tax discount rate from the before-tax discount rate. “Before tax” means that the tax shield is not considered in the discount rate. It does not mean that the
tax expenses (without tax shield) are not considered in the free cash flow. The tax expenses (without tax shield) are a part of the free cash flow in the before-tax and in the after-tax discount
rate. For further information have a look at my other post WACC with Tax Shield. Abbreviations:
We assume that the values of before-tax discount rate is:
Rearranging the above to solve for
The after-tax discount rate at a constant leverage rate is:
This is the famous equation most financial analysts might know. The factor “-t” comes from the tax shield and decreases the discount rate. Hence the discount rate after taxes is lower than the return
rate before taxes. But you have to take care. This after-tax formula is only valid if the leverage rate this post. By substituting
This formula can be useful, because you do not have to know the equity return rate to calculate the after-tax return rate. But have in mind that this is only valid, if the leverage ratio is constant
and the total tax shield amount can really be deducted from the tax expenses.
Pitfalls of Discounted Cash Flow Analysis
Correctly appraising capital projects with DCF analysis methods requires knowledge, practice and acute awareness of potentially serious pitfalls. I want to point out some important errors in project
appraisal and suggest ways to avoid them. For many people DCF analysis seems to be quite easy, but it can be very difficult for complex projects. Here are some crucial issues from my point of view:
• Decision focus: The calculation is focused on making the right decision concerning a project or an investment. That can be different from a calculation including all expenditures of the project
or investment, e.g. sunk costs. For further comments concering this topic see Incremental Free Cash Flows.
• Point of view: It has to be defined clearly from which perspective you are doing the decision and calculation. For example, the calculation can be different from the view of a business area and
from the view of the overall company. The right perspective to the decision problem determines the relevant incremental cash flows.
• Investment: Define clearly what you mean when talking about “investment”. Avoid the balance sheet view, look at investment as initial expenditures required for later contribution cash flows. In
my point of view the term “investment” is best defined as commitments of resources made in the hope of realizing benefits that are expected to occur over a reasonably long period of time in the
• Cash flows: A clear view of cash flow is important, avoid views from accounting and cost accounting, e.g. depreciation. And take into account tax effects.
• Incremental cash flows: The correct definition of incremental cash flow is crucial. It is the difference between the relevant expected after-tax cash flows associated with two mutually exclusive
scenarios: (1) the project goes ahead, and (2) the project does not go ahead (zero scenario). Sunk costs must not considered. For further comments see Incremental Free Cash Flows.
• Comparing scenarios: Alway be aware of having a relative sight between the cash flow scenarios. Sometimes it is not so easy to define what would happen in the future without the project (zero
• Risk-adjusted discount rates: Risk adjustment of discount rates has to be done for all (!) cash flows of the investment project that have significant risk differences: Fixed costs, investment
expenses, one time expenses and payments, expenses for working capital, leasing, tax shields and contribution cash flows (turnover and variable costs) in various markets. For more infos
concerning risk adjusted discount rates see Component Cash Flow Procedure.
• Key figures: The only key figure that is valid for all types of projects and investment decision is the famous NPV. All other well-known figures like IRR, Baldwin rate, … are leading to false
decisions in some cases. NPV also allows to build the bridge to financial calculation approaches like option valuation. Payback and liquidity requirements have to be considered carefully
additionally to NPV.
• Expected versus most likely cash flows: Quite often analysts take most likely cash flows. The right way is to consider the expected value (mathematical definition) of the cash flows.
• Limited capacity: Do not forget internal capacity limitation when regarding market figures. Limited capacity has also to be considered when constructing the event tree in real options analysis.
Besides that the temporal project value development with contribution cash flow’s discount rate has to be ensured in the binomial tree.
• Hurlde rates: Avoid hurlde rates for project decisions, because the can also lead to false decisions. Especially when you take one hurdle rate for different projects.
• Cash flow forecasting: Forecasts are often untruthful. Try to verify and countermeasure cash flows from different sources.
• Inflation: Be careful considering inflation. In multinational project it might influence the foreign currency location’s required return. You can also consider a relationship between inflation
rate and expected future exchange rates according to the purchasing power parity (PPP).
• Real and nominal discount rates and cash flows: The procedure should be consistent for cash flows and discount rates. Usually we take nominal values for the calculation.
• Real options: A DCF analysis should always be linked to a real options analysis. The more flexibility is in the project the more important is a real options analysis. Risk adds value to real
• Precise cash flow timing: The influence of timing intervals can be significant. You can choose smaller time intervals in crucial time periods to increase accuracy.
Option to Wait
This is a simple example of an option to wait. We consider a 15 year project which requires an investment of 105 M€, that can be done anytime. Arbitrage Pricing Theory provides a yearly risk-adjusted
capital discount rate (WACC) of 15%. Investment and internal risk cash flows are discounted by the risk-free rate. We assume for all years equal free net cash flow present values of 100/15 M€.
Classical incremental cash flow analysis provides a present value of the market-related net cash flows of 100 M€. That means that the classic NPV of the project is -5 M€. Because of the negative NPV
management should reject the project.
But management has an option to wait. It can wait with the decision and invest only if the market development is profitable. For sure the company loses revenues because of the delayed investment, but
on the other hand management gets more information about market development. The question is: What is the value of this option to wait and how long should management wait with that investment
decision? Can the project become profitable?
Monte-Carlo-Simulation of the project provides a project volatility of 30%. The risk-free rate is 5%. Next we are performing a real option analysis (ROA) of the waiting option with the binomial
approach regarding 15 time steps, one for each year.
Real option analysis provides a project value of 21 M€. That means that the value added by the waiting option is 26 M€. Because the project value with waiting option is positive we should not reject
the project any more. Management should go on with the project. Including the option in the project valuation leads to the opposite management decision. And besides the waiting option there might be
additional options like the option to abandon or the option to expand/contract. They would bring additional value to the project.
Real option analysis also provides the information that there should be no investment done before the second year. Dependent from the market development management can decide when to invest according
to the time value of the expected free cash flows.
In this example we assumed yearly cash flows that results in a decrease of the expected future cash flows. This corresponds to paying dividends at financial securities. Considering options in the
lifetime of a project requires binomial valuation with leakage. If you assume relative leakage you get a recombining tree, with absolute leakage values you get a non-recombining binomial tree.
Sequential Compound Options
This is a simple example of a sequential compound option, which is typical in projects where the investment can be done in sequential steps. The option is valued by the binomial approach.
The project is divided into three sequential phases: (1) Land acquisition and permitting, (2) design and engineering and (3) construction. Each phase must be completed before the next phase can
start. The company wants to bring the product to market in no more than seven years.
The construction will take two years to complete, and hence the company has a maximum of five years to decide whether to invest in the construction. The design and engineering phase will take two
years to complete. Design and engineering has to be finished sucessfully before starting construction. Hence the company has a maximum of three years to decide whether to invest in the design and
engineering phase. The land acquisition and permitting process will take two years to complete, and since it must be completed before the design phase can begin, the company has a maximum of one year
from today to decide on the first phase.
Investments: Permitting is expected to cost 30 million Euro, design 90 million Euro, and construction another 210 million Euro.
Discounted cash flow analysis using an appropriate risk-adjusted discount rate values the plant, if it existed today, at 250 million Euro. The annual volatility of the logarithmic returns for the
future cash flows for the plant is evaluated by a Monte-Carlo-Simulation to be 30%. The continuous annual risk-free interest rate over the next five years is 6%.
Static NPV approach: The private risk discount rate for investment is 9%. With that we get a NPV without any flexibility and option analysis of minus 2 million Euro. Because of its negative NPV we
would reject the project neglecting any option flexibility.
ROA: Considering the real options mentioned above we calculate a positive project value of 41 million Euro. That means that the compound options give an additional real option value (ROV) of 43
million Euro. Thus we should implement the project.
Binomial valuation tree of a sequential compound option
The real option analysis additionally provides the information when and under which market development to invest in each phase. The investment for the first phase should be done in year 1, for the
second phase in year 3 and for the third phase in year 5. The option valuation tree tells management what to do in which market development.
The valuation can be done in smaller time steps to increase accuracy. But the purpose of this example is to illustrate the principle of a sequential compound option valuation.
Details are specified in Kodukula (2006), p. 146 – 156.
CRR Binomial Method
Das Binomialmodell von Cox-Ross-Rubinstein aus dem Jahr 1979 ist der Grundstein für die klassische Optionsbewertung von Finanztiteln. Aber auch für die Bewertung von Realoptionen ist es das zentrale
Modell, mit welchem flexibel unterschiedliche Optionsarten simultan analysiert werden können.
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[Solved] The gra a quadratic function with vertex | SolutionInn
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The gra a quadratic function with vertex (0, 1) is shown in the Find the unge and the domain. 10 8 -2 24 6
The gra a quadratic function with vertex (0, 1) is shown in the Find the unge and the domain. 10 8 -2 24 6 8.10 Write the range and domain using interval notation. (a) range: (0,0) [0,0] (0,0)
There are 3 Steps involved in it
Step: 1
The domain of a quadratic function or any polynomial function is all real numbers because we can ...
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Transactions Online
Ock-Kyung YOON, Dong-Min KWAK, Bum-Soo KIM, Dong-Whee KIM, Kil-Houm PARK, "Automated Segmentation of MR Brain Images Using 3-Dimensional Clustering" in IEICE TRANSACTIONS on Information, vol. E85-D,
no. 4, pp. 773-781, April 2002, doi: .
Abstract: This paper proposed an automated segmentation algorithm for MR brain images through the complementary use of T1-weighted, T2-weighted, and PD images. The proposed segmentation algorithm is
composed of 3 steps. The first step involves the extraction of cerebrum images by placing a cerebrum mask over the three input images. In the second step, outstanding clusters that represent the
inner tissues of the cerebrum are chosen from among the 3-dimensional (3D) clusters. The 3D clusters are determined by intersecting densely distributed parts of a 2D histogram in 3D space formed
using three optimal scale images. The optimal scale image results from applying scale-space filtering to each 2D histogram and a searching graph structure. As a result, the optimal scale image can
accurately describe the shape of the densely distributed pixel parts in the 2D histogram. In the final step, the cerebrum images are segmented by the FCM (Fuzzy c-means) algorithm using the
outstanding cluster center value as the initial center value. The ability of the proposed segmentation algorithm to calculate the cluster center value accurately then compensates for the current
limitation of the FCM algorithm, which is unduly restricted by the initial center value used. In addition, the proposed algorithm, which includes a multi spectral analysis, can achieve better
segmentation results than a single spectral analysis.
URL: https://global.ieice.org/en_transactions/information/10.1587/e85-d_4_773/_p
author={Ock-Kyung YOON, Dong-Min KWAK, Bum-Soo KIM, Dong-Whee KIM, Kil-Houm PARK, },
journal={IEICE TRANSACTIONS on Information},
title={Automated Segmentation of MR Brain Images Using 3-Dimensional Clustering},
abstract={This paper proposed an automated segmentation algorithm for MR brain images through the complementary use of T1-weighted, T2-weighted, and PD images. The proposed segmentation algorithm is
composed of 3 steps. The first step involves the extraction of cerebrum images by placing a cerebrum mask over the three input images. In the second step, outstanding clusters that represent the
inner tissues of the cerebrum are chosen from among the 3-dimensional (3D) clusters. The 3D clusters are determined by intersecting densely distributed parts of a 2D histogram in 3D space formed
using three optimal scale images. The optimal scale image results from applying scale-space filtering to each 2D histogram and a searching graph structure. As a result, the optimal scale image can
accurately describe the shape of the densely distributed pixel parts in the 2D histogram. In the final step, the cerebrum images are segmented by the FCM (Fuzzy c-means) algorithm using the
outstanding cluster center value as the initial center value. The ability of the proposed segmentation algorithm to calculate the cluster center value accurately then compensates for the current
limitation of the FCM algorithm, which is unduly restricted by the initial center value used. In addition, the proposed algorithm, which includes a multi spectral analysis, can achieve better
segmentation results than a single spectral analysis.},
TY - JOUR
TI - Automated Segmentation of MR Brain Images Using 3-Dimensional Clustering
T2 - IEICE TRANSACTIONS on Information
SP - 773
EP - 781
AU - Ock-Kyung YOON
AU - Dong-Min KWAK
AU - Bum-Soo KIM
AU - Dong-Whee KIM
AU - Kil-Houm PARK
PY - 2002
DO -
JO - IEICE TRANSACTIONS on Information
SN -
VL - E85-D
IS - 4
JA - IEICE TRANSACTIONS on Information
Y1 - April 2002
AB - This paper proposed an automated segmentation algorithm for MR brain images through the complementary use of T1-weighted, T2-weighted, and PD images. The proposed segmentation algorithm is
composed of 3 steps. The first step involves the extraction of cerebrum images by placing a cerebrum mask over the three input images. In the second step, outstanding clusters that represent the
inner tissues of the cerebrum are chosen from among the 3-dimensional (3D) clusters. The 3D clusters are determined by intersecting densely distributed parts of a 2D histogram in 3D space formed
using three optimal scale images. The optimal scale image results from applying scale-space filtering to each 2D histogram and a searching graph structure. As a result, the optimal scale image can
accurately describe the shape of the densely distributed pixel parts in the 2D histogram. In the final step, the cerebrum images are segmented by the FCM (Fuzzy c-means) algorithm using the
outstanding cluster center value as the initial center value. The ability of the proposed segmentation algorithm to calculate the cluster center value accurately then compensates for the current
limitation of the FCM algorithm, which is unduly restricted by the initial center value used. In addition, the proposed algorithm, which includes a multi spectral analysis, can achieve better
segmentation results than a single spectral analysis.
ER - | {"url":"https://global.ieice.org/en_transactions/information/10.1587/e85-d_4_773/_p","timestamp":"2024-11-06T16:58:12Z","content_type":"text/html","content_length":"64677","record_id":"<urn:uuid:53063ea5-1aa5-4bc3-a392-4391e24d2016>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00575.warc.gz"} |
Mho to Megasiemens Converter (ʊ to MS) | Kody Tools
1 Mho = 0.000001 Megasiemens
One Mho is Equal to How Many Megasiemens?
The answer is one Mho is equal to 0.000001 Megasiemens and that means we can also write it as 1 Mho = 0.000001 Megasiemens. Feel free to use our online unit conversion calculator to convert the unit
from Mho to Megasiemens. Just simply enter value 1 in Mho and see the result in Megasiemens.
Manually converting Mho to Megasiemens can be time-consuming,especially when you don’t have enough knowledge about Electrical Conductance units conversion. Since there is a lot of complexity and some
sort of learning curve is involved, most of the users end up using an online Mho to Megasiemens converter tool to get the job done as soon as possible.
We have so many online tools available to convert Mho to Megasiemens, but not every online tool gives an accurate result and that is why we have created this online Mho to Megasiemens converter tool.
It is a very simple and easy-to-use tool. Most important thing is that it is beginner-friendly.
How to Convert Mho to Megasiemens (ʊ to MS)
By using our Mho to Megasiemens conversion tool, you know that one Mho is equivalent to 0.000001 Megasiemens. Hence, to convert Mho to Megasiemens, we just need to multiply the number by 0.000001. We
are going to use very simple Mho to Megasiemens conversion formula for that. Pleas see the calculation example given below.
\(\text{1 Mho} = 1 \times 0.000001 = \text{0.000001 Megasiemens}\)
What Unit of Measure is Mho?
Mho is a unit of measurement for electric conductance. Mho is derived from spelling ohm backwards. Also, its symbol is upside-down capital Greek letter Omega ℧.
What is the Symbol of Mho?
The symbol of Mho is ʊ. This means you can also write one Mho as 1 ʊ.
What Unit of Measure is Megasiemens?
Megasiemens is a unit of measurement for electric conductance. Megasiemens is multiple of electric conductance unit siemens. One megasiemens is equal to 1000000 siemens.
What is the Symbol of Megasiemens?
The symbol of Megasiemens is MS. This means you can also write one Megasiemens as 1 MS.
How to Use Mho to Megasiemens Converter Tool
• As you can see, we have 2 input fields and 2 dropdowns.
• From the first dropdown, select Mho and in the first input field, enter a value.
• From the second dropdown, select Megasiemens.
• Instantly, the tool will convert the value from Mho to Megasiemens and display the result in the second input field.
Example of Mho to Megasiemens Converter Tool
Mho to Megasiemens Conversion Table
Mho [ʊ] Megasiemens [MS] Description
1 Mho 0.000001 Megasiemens 1 Mho = 0.000001 Megasiemens
2 Mho 0.000002 Megasiemens 2 Mho = 0.000002 Megasiemens
3 Mho 0.000003 Megasiemens 3 Mho = 0.000003 Megasiemens
4 Mho 0.000004 Megasiemens 4 Mho = 0.000004 Megasiemens
5 Mho 0.000005 Megasiemens 5 Mho = 0.000005 Megasiemens
6 Mho 0.000006 Megasiemens 6 Mho = 0.000006 Megasiemens
7 Mho 0.000007 Megasiemens 7 Mho = 0.000007 Megasiemens
8 Mho 0.000008 Megasiemens 8 Mho = 0.000008 Megasiemens
9 Mho 0.000009 Megasiemens 9 Mho = 0.000009 Megasiemens
10 Mho 0.00001 Megasiemens 10 Mho = 0.00001 Megasiemens
100 Mho 0.0001 Megasiemens 100 Mho = 0.0001 Megasiemens
1000 Mho 0.001 Megasiemens 1000 Mho = 0.001 Megasiemens
Mho to Other Units Conversion Table
Conversion Description
1 Mho = 1 Siemens 1 Mho in Siemens is equal to 1
1 Mho = 0.000001 Megasiemens 1 Mho in Megasiemens is equal to 0.000001
1 Mho = 0.001 Kilosiemens 1 Mho in Kilosiemens is equal to 0.001
1 Mho = 1000 Millisiemens 1 Mho in Millisiemens is equal to 1000
1 Mho = 1000000 Microsiemens 1 Mho in Microsiemens is equal to 1000000
1 Mho = 1 Ampere/Volt 1 Mho in Ampere/Volt is equal to 1
1 Mho = 1000000 Gemmho 1 Mho in Gemmho is equal to 1000000
1 Mho = 1000000 Micromho 1 Mho in Micromho is equal to 1000000
1 Mho = 1e-9 Abmho 1 Mho in Abmho is equal to 1e-9
1 Mho = 899000042253 Statmho 1 Mho in Statmho is equal to 899000042253 | {"url":"https://www.kodytools.com/units/conductance/from/mho/to/megasiemens","timestamp":"2024-11-02T12:47:12Z","content_type":"text/html","content_length":"76876","record_id":"<urn:uuid:4182a77d-2129-407c-91f2-0c681665f180>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00849.warc.gz"} |
Direct Air Capture, Part 1: the Entropy Penalty
Direct air capture, or DAC, is what climate industry professionals call any process that takes atmospheric air and removes (some) carbon dioxide. DAC gets people, myself included, very excited.
Wouldn’t it be great if we could undo CO$_2$ emissions?
The idea is not, strictly speaking, that new. Various working technologies already exist: submarines have had “CO$_2$ scrubbers” that heat and cool monoehtanolamine to absorb CO$_2$ from the air and
release it outside since the mid-20th century. Despite this, several large, well-funded DAC projects since 2010 have failed to deliver. IEA.
DAC today has two problems: 1), nobody will pay you to take CO$_2$ out of the air, and even if someone did, 2), it would cost too much in electricity per CO$_2$ removed.
You can solve part of the second problem with better technology. But there is also a fundamental physical limit on the energy required for any process that pulls one gas out of a mix of gases.
Soherman, Jones, and Dauenhauer amusingly call this the ‘entropy penalty’ of DAC, because it is due to the entropy of mixing.
I’d like to do a series of posts about DAC. I’ll start with what I’m most qualified to discuss, which also happens to be the least solvable problem.
Here is a simple sketch of the entropy penalty using physics found in most thermodynamics textbooks.
Ideal mixing and separation; maximum efficiency
Physically, the difference between DAC and capture at, e.g., a powerplant flue is the composition of the initial mixture, and in particular the initial concentration of CO$_2$. I will treat both.
Under conditions of atmospheric temperature and pressure, air can be treated as an ideal gas with molar mass 28.97 g$\cdot$mol$^{−1}$, and with 44.01 g$\cdot$mol$^{−1}$.
Call air with all CO$_2$ removed A, and pure CO$_2$ B. Because the initial concentration of B is low compared to the major constituents of air (O$_2$ at 21% and at N$_2$ at 78%), I can neglect the
effect of removal of B on the change in ideal gas properties of atmospheric air versus A. In addition, when B is removed from air, there is a negligible change in internal energy. Although air with B
removed is not a single gas, summing over the partial pressures of the constituents of air, and neglecting the change when CO$_2$ is removed, the air mixture can be treated as a binary mixture.
The entropy of a binary mixture of two ideal gases is higher than the sum of the separate entropies. Assuming constant U and V, with $x = \frac{N_B}{N_A + N_B}$:
$\Delta S_{\text{mixing}} = -R \left[ x \ln x + \left( 1 - x \right) \ln \left(1 - x \right) \right]$
$\text{and given } G = -TS + U + PV$,
$\Delta G _{\text{mixing}} = -T\Delta S _{\text{mixing}}$ (Schroeder 187).
At constant pressure, each additional molecule of $B$ replaces a molecule of A, so we have $x = \frac{N_A}{N_{init} }$, where $N_{init}$ is constant and given by the ideal gas law. Therefore $\Delta
G_{\text{mixing}}$ is a function of $N_B$ only.
At the emitter, CO$2$ mixes into the atmosphere, and it is completely mixed by the time atmosphere touches the separator. The change in Gibbs free energy due to the entropy of mixing is therefore not
Assuming that other (e.g. mechanical) processes in the separator are reversible, $\Delta G_{\text{mixing}}$ is then the minimum energy that an ideal separator must supply to unmix the gases at
constant pressure and capture $N_B$ molecules of CO$_2$. A real separator will use some higher amount of energy E for the same result, and will have an efficiency $r = \frac{\Delta G_{mixing}}{E} <
Under this model, the minumum energy required to separate from atmospheric air is $\Delta G_{\text{mixing}}$=$RT(x\ln x +(1− x)\ln (1− x))$
The goal is to capture some fixed $N_B$ of CO$_2$ . Estimate the energy required to capture 1 ton, or $N_B = 2.3 \cdot 10^4 mol$, depending on the intial concentration, and compare values for capture
at the plant and directly from air.
As a function of input concentration. Figure 1 gives the energy to extract 1 ton as a function of input concentration. When concentration approaches 0, as expected, the work approaches infinity
(Schroeder). However, in the range of interest (atmospheric to tailpipe emissions) the energy varies by less than 1 order of magnitude.
Direct air capture. Atmospheric air consists of 415 (molar) ppm CO$_2$, or 0.0415% (NOAA). Using this value,$x = \frac{N_B}{ N_{init}} = 0.0415$. Under approximate atmospheric temperature and
pressure (300 K, 100 kPa), the energy required for direct air capture is 0.498 MJ$\cdot$t$^{−1}$.
It is interesting to note that carbon capture is subject to a negative feedback loop: as the average atmospheric temperature increases, (i.e. as global warming occurs) the energy required to extract
also increases, but since average temperature increases are only of a few degrees the effect is small, on the order of $\frac{1}{300}$.
Power plant exhaust. Typical concentrations at a power plant exhaust, in contrast, are approximately 15%. The corresponding value for $\Delta G$ is 0.160 MJ $\cdot$t$^{−1}$, or approximately 3 times
lower than direct air capture.
Underground storage. As an aside, Figure 2 gives the work required to compress by atmospheric volume into a theoretical 1 million cubic meter depeted oil field, over a volume range from 1 to
approximately 100 tons of air. The energy for storage alone is close to the separation step.
Thus, the theoretical energy requirement of unmixing depends on the initial concentration but does not vary by orders of magnitude accross the range. In the extremes: for CCS, the energy of
separation due to $\Delta S_{\text{mixing}}$ is lower than for DAC by a factor of 3. Underground storage of high volumes of CO$_2$, however, comes at an even higher energy cost.
Current technology
The state of the art for separation technology is in chemical methods. The most efficient of those, however, only approach 2.5 GJ t^−1, which is $r = \frac{0.2}{1000} = \frac{1}{5000}$ .
The energy requirements for a carbon capture system are not limited to the change in Gibbs: they also include pumping air accross the separator. Here, there is a tradeoff between running the
separator at saturation and the energy cost of the airflow.
Finally, with respect to storage, the formation of stable chemical compounds is a more encouraging avenue than pumped storage .
I estimated the energy requirements of a perfectly efficient carbon capture and storage system. Even in the ideal case, the entropy of mixing at the point of emissions results in unrecoverable
energy. This is the so-called entropy penalty. It places an unavoidable thermodynamic limit on the efficiency of a DAC process, which current technologies do not approach.
Next, if I can learn enough chemistry, I’d like to take a look at recent progress in the solvable part of the efficiency problem (planned references: Nohra, et al., Zhu, et al.). Later, I’d like to
model the economic incentives that would be required for a sustainable DAC project assuming perfect technology.
Questions? Comments? Email me at blog@silasbailey.com. | {"url":"https://silasbailey.com/blog/","timestamp":"2024-11-02T10:47:13Z","content_type":"text/html","content_length":"66735","record_id":"<urn:uuid:9513bbd2-79de-47a5-900b-c3affe3a6a7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00843.warc.gz"} |
Fixed-term Joustosähkö - Keravan Energia
How is the consumption effect calculated?
We calculate the average price for the month by multiplying the exchange price of each hout with your consumption. Then we divide the total of that with your total consumption for the month. The
difference of this weighted average and the general average of the month is your consumption effect that is either added to your price or discounted from it.
We calculate the weighted average of the consumption effect that is compared to the month’s average the following way: ∑ (X*Y) / Z = c/kWh
∑ = sum of all days of the month
X = consumption of an hour
Y = the exchange price of an hour
Z = the total electricity consumption of the month
Here are a couple of examples based on real cases:
A detached house with electric heating system
Total hourly consumption for the month is 1 800 kWh (Z). Your combined hourly consumption total is 17 388 cents (X*Y). 17 388 c/1 800 kWh = 9,66 c/kWh, which is your weighted average price. The
monthly average of the exchange price has been 11 c/kWh.
This makes your consumption effect to be 9,66–11 = –1,34 c/kWh.
When the consumption effect is negative, that lowers the final price you pay for the month. For example, if your fixed basic price is 9,90 c/kWh, your final price for the month would be 8,56 c/kWh in
this case.
Apartment duplex with district heat
Total hourly consumption for the month is 160 kWh (Z). Your combined hourly consumption total is 1974,4 cents (X*Y). 1974,4 c/160 kWh = 12,34 c/kWh, which is your weighted average price. The monthly
average of the exchange price has been 11 c/kWh.
So this makes your consumption effect to be 12,34–11 = 1,34 c/kWh.
When the consumption effect is positive, that raises the final price you pay. For example, if your fixed basic price is 10 c/kWh, your final price for the month would be 11,34 c/kWh. | {"url":"https://www.keravanenergia.fi/en/fixed-term-electricity-contract/fixed-term-joustosahko/","timestamp":"2024-11-05T23:51:01Z","content_type":"text/html","content_length":"148149","record_id":"<urn:uuid:093ebcf8-70e4-4b2c-b0cf-93cfef1e983c>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00640.warc.gz"} |
Compactification (physics)
In physics, compactification means changing a theory with respect to one of its space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that
this dimension has a finite length, and may also be periodic.
Compactification plays an important part in thermal field theory where one compactifies time, in string theory where one compactifies the extra dimensions of the theory, and in two- or
one-dimensional solid state physics, where one considers a system which is limited in one of the three usual spatial dimensions.
At the limit where the size of the compact dimension goes to zero, no fields depend on this extra dimension, and the theory is dimensionally reduced.
The space \( M \times C \)is compactified over the compact C and after Kaluza–Klein decomposition, we have an effective field theory over M.
Compactification in string theory
In string theory, compactification is a generalization of Kaluza–Klein theory.[1] It tries to reconcile the gap between the conception of our universe based on its four observable dimensions with the
ten, eleven, or twenty-six dimensions which theoretical equations lead us to suppose the universe is made with.
For this purpose it is assumed the extra dimensions are "wrapped" up on themselves, or "curled" up on Calabi–Yau spaces, or on orbifolds. Models in which the compact directions support fluxes are
known as flux compactifications. The coupling constant of string theory, which determines the probability of strings splitting and reconnecting, can be described by a field called a dilaton. This in
turn can be described as the size of an extra (eleventh) dimension which is compact. In this way, the ten-dimensional type IIA string theory can be described as the compactification of M-theory in
eleven dimensions. Furthermore, different versions of string theory are related by different compactifications in a procedure known as T-duality.
The formulation of more precise versions of the meaning of compactification in this context has been promoted by discoveries such as the mysterious duality.
Flux compactification
A flux compactification is a particular way to deal with additional dimensions required by string theory.
It assumes that the shape of the internal manifold is a Calabi–Yau manifold or generalized Calabi–Yau manifold which is equipped with non-zero values of fluxes, i.e. differential forms, that
generalize the concept of an electromagnetic field (see p-form electrodynamics).
The hypothetical concept of the anthropic landscape in string theory follows from a large number of possibilities in which the integers that characterize the fluxes can be chosen without violating
rules of string theory. The flux compactifications can be described as F-theory vacua or type IIB string theory vacua with or without D-branes.
See also
Dimensional reduction
Kaluza–Klein theory
Dean Rickles (2014). A Brief History of String Theory: From Dual Models to M-Theory. Springer, p. 89 n. 44.
Chapter 16 of Michael Green, John H. Schwarz and Edward Witten (1987). Superstring theory. Cambridge University Press. Vol. 2: Loop amplitudes, anomalies and phenomenology. ISBN 0-521-35753-5.
Brian R. Greene, "String Theory on Calabi–Yau Manifolds". arXiv:hep-th/9702155.
Mariana Graña, "Flux compactifications in string theory: A comprehensive review", Physics Reports 423, 91–158 (2006). arXiv:hep-th/0509003.
Michael R. Douglas and Shamit Kachru "Flux compactification", Reviews of Modern Physics 79, 733 (2007). arXiv:hep-th/0610102.
Ralph Blumenhagen, Boris Körs, Dieter Lüst, Stephan Stieberger, "Four-dimensional string compactifications with D-branes, orientifolds and fluxes", Physics Reports 445, 1–193 (2007). arXiv:hep-th/
String theory
Strings History of string theory
First superstring revolution Second superstring revolution String theory landscape
Nambu–Goto action Polyakov action Bosonic string theory Superstring theory
Type I string Type II string
Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism
GS formalism
String duality
T-duality S-duality U-duality Montonen–Olive duality
Particles and fields
Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon
D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition
Conformal field theory
Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model
Gauge theory
Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G2, F4, E6, E7, E8) ADE classification Dirac string p-form electrodynamics
Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold
Calabi–Yau manifold Hyperkähler manifold
K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory (physics) Twisted K-theory
Supergravity Superspace Lie superalgebra Lie supergroup
Holographic principle AdS/CFT correspondence
Matrix theory Introduction to M-theory
String theorists
Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Cleaver Curtright Dijkgraaf Distler Douglas Duff Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson
Harvey Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov
Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher
Steinhardt Strominger Sundrum Susskind 't Hooft Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License | {"url":"https://www.hellenicaworld.com/Science/Physics/en/Compactificationphysics.html","timestamp":"2024-11-11T20:26:01Z","content_type":"application/xhtml+xml","content_length":"13654","record_id":"<urn:uuid:cfc0a82b-bc4e-452a-860d-aaedd80c3366>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00492.warc.gz"} |
Paul Irofti | DDNET
[1] P. Irofti, F. Stoican, and V. Puig, “Fault Handling in Large Water Networks with Online Dictionary Learning,” Journal of Process Control, vol. 94, pp. 46--57, 2020. [ bib | DOI | http ]
[2] A. Pătrașcu and P. Irofti, “Stochastic proximal splitting algorithm for composite minimization,” Optimization Letters, pp. 1--19, 2021. [ bib | DOI | .pdf ]
[3] C. Rusu and P. Irofti, “Efficient and Parallel Separable Dictionary Learning,” in Proceedings of the IEEE 2021 27th International Conference on Parallel and Distributed Systems (ICPADS). 2021,
pp. 1--6, IEEE Computer Society. [ bib | http ]
[4] A. Pătrașcu and P. Irofti, “Computational complexity of Inexact Proximal Point Algorithm for Convex Optimization under Holderian Growth,” pp. 1--42, 2021. [ bib | arXiv ]
[5] P. Irofti, L. Romero-Ben, F. Stoican, and V. Puig, “Data-driven Leak Localization in Water Distribution Networks via Dictionary Learning and Graph-based Interpolation,” 2021, pp. 1--6. [ bib |
arXiv ]
[6] P. Irofti, C. Rusu, and A. Pătrașcu, “Dictionary Learning with Uniform Sparse Representations for Anomaly Detection,” 2021, pp. 1--6. [ bib | arXiv ]
[7] P. Irofti, A. Pătrașcu, and A.I. Hîji, “Unsupervised Abnormal Traffic Detection through Topological Flow Analysis,” in 2022 14th International Conference on Communications (COMM). 2022, pp. 1--6,
IEEE. [ bib | DOI | http ]
[8] A. Pătrașcu and P. Irofti, “On finite termination of an inexact Proximal Point algorithm,” Applied Mathematics Letters, vol. 134, pp. 108348, 2022. [ bib | DOI | http ]
The main goal of this project, called DDNET, is to adapt and propose new dictionary learning methods for solving untractable fault detection and isolation problems found in distribution networks.
Given a large dataset of sensor measurements from the distribution network, the dictionary learning algorithms should be able to produce the subset of network nodes where faults exist. Since a
model-based approach is impractical, we consider here the data-driven alternative where the data is provided by the network sensors and processed as signals laying on a graph. Graph signal processing
is a new and very active field where data-driven methods such as sparse representations with dictionary learning have shown promising results. Sparse representations are linear combinations of a few
vectors (named atoms) from an overcomplete basis (called dictionary). Formulating sensor placement as a graph sparse representation problem and modeling large-scale utility networks via a dictionary
that is trained from sensor data, was attempted only very recently and as far as we are aware only by our team members. The project objectives are to provide sparse modeling for sensor placement and
to perform and improve data-driven fault detection and isolation with dictionary learning through multi-parametric reformulations of the standard algorithms while exploiting the underlying
geometrical and topological structure of the data. The research team covers the positions of principal investigator Paul Irofti with extensive expertise in dictionary learning and his mentor Florin
Stoican, fault tolerant control expert. The team had a fruitful collaboration in the past during two national research projects. | {"url":"https://cs.unibuc.ro/~pirofti/ddnet.html","timestamp":"2024-11-11T10:05:07Z","content_type":"text/html","content_length":"12368","record_id":"<urn:uuid:41da90da-c1c5-4142-a6ff-cf50c85617e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00006.warc.gz"} |
Profile Evaluation [UK International Student]
Hey all, I'm going to be applying to PhD programs in a few months and wanted to get a second pair of eyes since I'm very naive wrt North American unis.
Undergrad: T10 UK university for maths
Masters: Part III maths at Cambridge
Major(s): Mathematics for both (technically "Pure Mathematics" at Cambridge)
Minor(s): N/A
GPA: High(?) First Class in my undergrad
Type of Student: International White Male
GRE Revised General Test: Not taking
GRE Subject Test in Mathematics: Ditto (would have to fly over to France or Bulgaria to take)
Program Applying: PhD Pure Math, looking to specialise in functional analysis/operator theory. Very ambivalent towards PDEs, I'm only really looking at FA/operators for now.
Research Experience: Not really research per se but did a project on Leray-Hopf solutions to the Navier-Stokes equations as part of my undergrad (not original, more of a survey). I will be doing a
Part III essay somewhere in functional analysis, tending towards geometry of Banach spaces considering the specialisms of my desired supervisor.
Awards/Honors/Recognitions: None since high school (one of the downsides of small fish big pond)
Pertinent Activities or Jobs: I wrote much of ProofWiki's coverage of functional analysis and measure theory. Competed in a small inter-university integration competition this year and plan on
helping coordinate next year's.
Any Miscellaneous Points that Might Help: My coursework is probably the strongest thing about my application. (if it even matters much) Grad-level courses would be three courses in functional
analysis going up to C* algebras & Borel functional calculus and another course "Unbounded Operators and Semigroups", manifolds, analytic theory of PDEs, non-linear analysis, modular forms,
measure-theoretic probability. Otherwise decent mix of undergrad courses, courses in groups/rings, linear algebra, real/complex/multivariable analysis, point-set topology, basic non-Euclidean
geometry, number theory, measure theory, set theory, logic and a first course in undergrad statistics. My last year was slightly disappointing (had issues with insomnia around exams) with a few 2:1s
but still did fairly well. (my overall grade is mid 80s)
Currently my list of North American schools is:
Dartmouth (throwing this in because it doesn't have an application fee mainly)
Kent State
Maybe throwing in Oregon too
This'd be in addition to 4 or 5 in the UK to get a total of like 11-13 schools. My first reference letter will be from my undergraduate tutor who has a pretty high opinion of me, and who I met with
quite a lot during my second year. Very recognisable name internationally but in an area that's not mine. Second letter writer was the supervisor of my third year project and was very impressed with
my work. While he's in PDEs and harmonic analysis more so than functional, he has a low collaborative distance with quite a few people at the universities I'm applying to, so he might be a known name
to some. (though I'm not sure if any will know him well) Third letter writer is a bit murkier, I will basically have the option between my first year undergrad tutor (who I met a few times and wrote
one of my letters for Cambridge) and my tutor in Cambridge who I'll have only met a few times by time of applying. Neither are in relevant areas but the latter is a bit more well known than the
(1) Are there any big names that I'm missing? Texas A&M is the no brainer but unfortunately they're requiring the subject GRE. I asked if they'd consider an application without it (nothing to lose
just asking) but haven't got a response.
(2) I'm worried my letters will seem a bit weak, especially going out with a whimper when they see the third. Is that justified here?
(3) Am I shooting a bit too high? Should I look into more safeties? I think if I fail to get a (funded) PhD offer anywhere this year, I will probably just go off into industry, I don't really want to
be reapplying year after year, so I really want to take my best shot this year.
Thanks a bunch!
Re: Profile Evaluation [UK International Student]
I honestly think you are aiming too low. You should definitely apply to top 10 and top 20 programs
Re: Profile Evaluation [UK International Student]
matejxx1 wrote: ↑
Mon Aug 29, 2022 8:29 am
I honestly think you are aiming too low. You should definitely apply to top 10 and top 20 programs
Are there any in particular I should be looking at? I did have a brief look at Berkeley, (which has a few people in the area of operator algebras) and I could revisit it. For other top choices my
problem was that a lot of them concentrated more on harmonic analysis & PDEs rather than functional analysis/operator algebras. Not sure if this means I should bend my will a bit and apply a bit more
broadly (just shooting for "analysis" more broadly) or stay on my current trajectory and have a less flashy university attached to my PhD.
Re: Profile Evaluation [UK International Student]
How did you in part III? I think regardless you should aim higher (and apply to European schools too). If you get into a T5 UK school then is it is probably way better than any of the North American
schools you are considering.
Re: Profile Evaluation [UK International Student]
LetMeIn2401 wrote: ↑
Wed Aug 31, 2022 6:05 pm
How did you in part III? I think regardless you should aim higher (and apply to European schools too). If you get into a T5 UK school then is it is probably way better than any of the North
American schools you are considering.
Haven't started it yet. I'm aiming for a Merit.
I'm thinking of expanding out into harmonic analysis (maybe analytic number theory) to bulk the list out with more reaches. (just been having a glance at Brown, Maryland, Caltech etc. as well as
Berkeley. Would have applied UCLA if not for GRE. Haven't checked where my supervisor has coauthors yet) Will probably go back to my undergrad supervisor and see what he has to say. | {"url":"https://mathematicsgre.com/viewtopic.php?f=1&t=5882","timestamp":"2024-11-12T03:50:20Z","content_type":"text/html","content_length":"31225","record_id":"<urn:uuid:684ef341-1578-4c10-bd94-b9c7250771a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00505.warc.gz"} |
Numerical methods for nonlinear optimal control problems, 2nd edition
Title data
Grüne, Lars:
Numerical methods for nonlinear optimal control problems, 2nd edition.
In: Samad, Tariq ; Baillieul, John (ed.): Encyclopedia of Systems and Control. - London : Springer , 2020
ISBN 978-1-447-15057-2
DOI: https://doi.org/10.1007/978-1-4471-5102-9_208-3
Abstract in another language
In this article we describe the three most common approaches for numerically solving nonlinear optimal control problems governed by ordinary differential equations. For computing approximations to
optimal value functions and optimal feedback laws, we present the Hamilton-Jacobi-Bellman approach. For computing approximately optimal open-loop control functions and trajectories for a single
initial value, we outline the indirect approach based on Pontryagin’s maximum principle and the approach via direct discretization.
Further data | {"url":"https://eref.uni-bayreuth.de/id/eprint/53045/","timestamp":"2024-11-09T02:58:24Z","content_type":"application/xhtml+xml","content_length":"21537","record_id":"<urn:uuid:46a26707-3d1c-46fa-891c-35e3046c40ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00104.warc.gz"} |
Class 9 Maths Chapter 4 Exercise 4.2 Q.3 - e Guru
NCERT Solutions Class 9 Maths Chapter 4 Linear Equations In Two Variable Exercise 4.2 Introduction: In this exercise/article we will learn about Linear Equations In Two Variable. You have seen that
every linear equation in one variable has a unique solution. What can you say about the solution of a linear equation involving two […]
NCERT Solutions Class 9 Maths Chapter 4 Linear Equations In Two Variable Exercise 4.2 Read More » | {"url":"https://eguru8.com/tag/class-9-maths-chapter-4-exercise-4-2-q-3/","timestamp":"2024-11-02T01:48:53Z","content_type":"text/html","content_length":"185237","record_id":"<urn:uuid:d0b42904-07cd-417b-8c8d-6c69f2fe9f34>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00306.warc.gz"} |
Marginal Cost: Definition, Formula, and Examples
Now that you’ve been introduced to the basics, there are a few nuances you should be aware of to maximize your marginal cost experience. Marginal costing values closing inventory at a lower cost per
unit than absorption costing and this means that the cost of goods sold figure is higher using the marginal method. Understanding marginal and absorption costing How to Void Check for Direct Deposit
should be relatively straightforward, as it’s covered, in one form or another, at all levels of the AAT qualification. It is important as it helps understand the profit-maximizing level of output.
Variable costs, on the other hand, are those that rise or fall along with production, such as inventory, fuel, or wages that are directly tied to production.
• Each instance of a student not having achieved a maths or English GCSE at grade 4 or above is counted.
• If an alternative aim that meets these criteria cannot be identified, the withdrawn aim must remain as the core aim.
• When students are on a 2 year programme and they complete the first year, they will be counted as retained in that academic year.
• Some students will have programmes planned in blocks that extend over multiple funding years – that is, they do not have start and end dates within the usual August to July pattern.
• Marginal cost includes all of the costs that vary with that level of production.
• The first step is to calculate the total cost of production by calculating the sum of the total fixed costs and the total variable costs.
To calculate marginal cost, divide the change in production costs by the change in quantity. The purpose of analyzing marginal cost is to determine at what point an organization can achieve economies
of scale to optimize production and overall operations. If the marginal cost of producing one additional unit is lower than the per-unit price, the producer has the potential to gain a profit. The
total cost per hat would then drop to $1.75 ($1 fixed cost per unit + $0.75 variable costs).
Block 2: GCSE maths and English
You may find a marginal cost calculator under different names, such as an incremental cost calculator or a differential cost calculator, but they are all related to the same topic. However, marginal
cost is not the same as margin cost described in our margin calculator! In this article, you can find more details on how to calculate the marginal cost and the marginal cost formula behind it.
Inputting the total cost for different quantities into an Excel spreadsheet and applying the formula can yield marginal costs for different production levels — providing valuable insights for
business decision-making. It’s essential to understand that the marginal cost can change depending on the level of production. Initially, due to economies of scale, the marginal cost might decrease
as the number of units produced increases.
• In this case, the cost of the new machine would need to be considered in the marginal cost of production calculation as well.
• Imagine a company decides to increase its production from 10 units to 12 units, and the total cost increases from $20 to $26.
• When performing financial analysis, it is important for management to evaluate the price of each good or service being offered to consumers, and marginal cost analysis is one factor to consider.
• When represented on a graph, the Marginal Cost curve often takes a U-shape.
• Marginal cost is different from average cost, which is the total cost divided by the number of units produced.
In a perfectly competitive market, a company arrives at the volume of output to be produced based on marginal costs and selling price. Now, let us consider the following two scenarios to understand
the relevance of the marginal cost formula. Marginal cost is the change in the total cost which is the sum of fixed costs and the variable costs. Fixed costs do not contribute to the change in the
production level of the company and they are constant, so marginal cost depicts a change in the variable cost only. So, by subtracting fixed cost from the total cost, we can find the variable cost of
Divide the revenue by the quantity
Therefore we will use the percentage of 14 to 16 year old direct-funded students who are eligible for block 1 funding to derive a factor for block 2 funding, which we will pay at the rate of £1,118
per instance. Block 1 funding recognises that there are additional costs incurred in engaging, recruiting, and retaining young people from economically disadvantaged backgrounds. We determine whether
a student is eligible for block 1 funding by their home postcode and the level of deprivation recorded for that area in the Index of Multiple Deprivation (IMD) 2019. For vocational programmes, we
determine the weighting by the core aim’s sector subject area (SSA) tier 2 classification. When a student stops studying for and does not complete their core aim, providers must only record a
replacement core aim when it is a substantial and core component of the study programme. If an alternative aim that meets these criteria cannot be identified, the withdrawn aim must remain as the
core aim.
When a company knows both its marginal cost and marginal revenue for various product lines, it can concentrate resources towards items where the difference is the greatest. Instead of https://
intuit-payroll.org/personal-income-tax/ investing in minimally successful goods, it can focus on making individual units that maximum returns. Marginal cost includes all of the costs that vary with
that level of production.
Marginal decisions in economics
For example, management may be incurring $1,000,000 in its current process. Should management increase production and costs increase to $1,050,000, the change Outsourced Accounting Nonprofit Services
in total expenses is $50,000 ($1,050,000 – $1,000,000). Marginal cost is calculated as the total expenses required to manufacture one additional good.
Use the EAR calculator to compute the effective annual rate of an investment or a loan. Our net operating working capital calculator can help you to measure a company’s liquidity. Our wallet maker
usually retails their product for £30 each at a market stall. However, they decide to supply the surplus wallet at a wholesale rate of £20, to a stall holder on the other side of town. | {"url":"http://chixaroluz.com.br/marginal-cost-definition-formula-and-examples/","timestamp":"2024-11-11T22:37:54Z","content_type":"text/html","content_length":"156520","record_id":"<urn:uuid:575d5d93-0f88-4585-b58a-bf412a601777>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00863.warc.gz"} |
Parameters: Static Solver
Model ElementParam_Static defines the solution control parameters for Static and Quasi-static analysis, where the parameters control the accuracy the solution and the method to be used for solution.
supports two distinct methods for static analyses:
• A Maximum Kinetic Energy Attrition Method.
• A Force Imbalance Method.
The Maximum Kinetic Energy Attrition Method:
1. The model is formulated as a dynamics problem from which all damping is removed. The result is a conservative system whose total energy, defined as the sum of kinetic and potential energies,
should remain invariant with time.
2. Numerical integration is started. During integration, the kinetic energy of the system is monitored. When a peak (maximum) is detected, integration stops and backtracks as necessary to locate the
peak in time, within some precision. Since the system is conservative, this instant also corresponds to a valley (minimum) for potential energy.
3. At this point, all velocities and accelerations in the model are set to zero, leading to zero kinetic energy. Then, the integration is restarted.
4. Steps 1-3 constitute one iteration. If the peak is located perfectly, then the model already is at the equilibrium configuration. However, due to the discrete nature of the integrator, this is
usually not the case. Thus, it is necessary to repeat steps 1-3 until the process converges as defined by three convergence parameters:
□ Maximum Kinetic Energy Tolerance (max_ke_tol): This is the maximum residual kinetic of the system at the static equilibrium point. This should be a small number.
□ Maximum state tolerance (max_dq_tol): This specifies the upper limit for the change in system states at the static equilibrium point.
□ Maximum number of iterations (max_num_iter): This is the maximum number of iterations that are allowed before simulation stops.
The Kinetic Energy Minimization method has following advantages:
□ It only finds the stable equilibria.
□ It is suitable for problems where the equilibrium configuration is far from the model configuration. Such situations are problematic for the Force Imbalance Method.
□ It works well for contact dominated models.
The Maximum Kinetic Energy Attrition Method has following disadvantages:
□ The method can be slow.
□ It does not work with quasi-statics in the current version.
The Force Imbalance Method:
1. This method sets all the velocity and acceleration terms in the equations of motion to zero to obtain a system of nonlinear algebraic, force balance equations [ F(q, l) = 0 ].
2. The generalized coordinates (q) and constraint forces (λ) are the unknowns.
3. The nonlinear equations are solved using the Newton-Raphson method to find the equilibrium configuration.
4. The iterations are stopped when:
• The imbalance in the equations of motion is reduced below a user-specified tolerance value specified.
• The equations F(q, λ) = 0 is satisfied to a user specified tolerance.
• The maximum number of iterations is reached.
The force imbalance method has the following advantages:
• It is a fairly reliable method and works well for a large class of problems.
• It finds the solution relatively quickly.
• It works for both static and quasi-static methods.
The force imbalance method has the following disadvantages:
• It does not distinguish between stable, unstable, or neutral equilibrium configurations. It is equally capable of finding any of these solutions. It converges to the static solution closest to
the initial configuration.
• It has difficulty in cases where the equilibrium configuration is far away from the model configuration.
• It has difficulty for models dominated by contact.
The two methods, Maximum Kinetic Energy Attrition Method and Force Imbalance Method, are complementary.
[ method = "MKEAM" ]
[ max_ke_tol = "real" ]
[ max_dq_tol = "real" ]
[ max_num_iter = "integer" ] >
|[ method = { "FIM_S" | "FIM_D" } ]
[ max_imbalance = "real" ]
[ max_error = "real" ]
[ stability = "real" ]
[ max_num_iter = "integer" ]
[ compliance_delta = "real" ]
Specifies the choice of the algorithm to be used for Static or Quasi-static simulation. For static equilibrium, choose one of the following:
MKEAM is a method based on minimization of maximum kinetic energy.
FIM_S is a modified implementation which supports all MotionSolve elements except Force_Contact.
For Quasi-static simulation, the MKEAM method is not applicable. However, there is an additional choice called FIM_D. FIM_D is a time integration based approach to quasi-static solution. It
is not applicable for a pure static solution. In addition to the parameters specified in the Param_Static element, FIM_D also uses the parameters specified in the Param_Transient element to
control the DAE integration (in particular, dae_constr_tol to control the error tolerance). FIM_D defaults to FIM_S when a pure static solution is required.
In summary, both for static and quasi-static solutions, there are three choices of solvers. See Comments for their strengths and weaknesses
The default is FIM_D.
Applicable if MKEAM was chosen. Specifies the maximum allowable residual kinetic energy of the system at the static equilibrium point. This should be a small number. The default value for
max_ke_tol is 10-5 energy units.
Applicable only if MKEAM was chosen.
This specifies the upper limit for the change in system states at the static equilibrium point. The iterations are deemed to have converged when the maximum relative change in the states is
smaller than this value. The default value for max_dq_tol is 10-3.
Specifies the maximum number of iterations that are allowed before simulation stops. If max_ke_tol and max_dq_tol are not satisfied at this point, the equilibrium iterations should be considered
as having failed. The default value for max_num_iter is 100.
Applicable if force_imbalance was chosen. Specifies the maximum force imbalance in the equations of motion that is allowed at the solution point. This should be a small number. The default value
for max_imbalance is 10-4 force units.
Applicable if force_imbalance was chosen. This specifies the upper limit for the change in residual of the system equations at the static equilibrium point. The iterations are deemed to have
converged when the maximum residual in the equations of motion is smaller than this value. The default value for max_error is 10-4.
Specifies the fraction of the mass matrix that is to be added to the Jacobian (see discussion on Newton-Raphson method in the Comments section) to ensure that it is not singular. The Jacobian
matrix can become singular when the system has a neutral equilibrium solution and the initial guess is close to it. To avoid this, a fraction of the mass matrix (known to be non-singular) is
added to the Jacobian in order to make it non-singular. The value of stability does not affect the accuracy of the solution, but it may slow the rate of convergence of the Newton-Raphson
iterations. stability should be a small number. The default value for stability is 1e-10.
Note: The square root of the value specified by stability is multiplied to the mass matrix and then added to the Jacobian to make it non-singular.
Delta used during compliance matrix calculation (default = 0.001).
This example shows the default settings for the Param_Static element that uses the MKEAM method.
method = "MKEAM"
max_ke_tol = "1.000E-05"
max_dq_tol = "0.001"
max_num_iter = "100"
This example shows the default settings for the Param_Static element that uses the FIM_S solution method.
method = "FIM_S"
max_residual = "1.000E-04"
max_imbalance = "1.000E-04"
max_num_iter = "50"
1. For Quasi-static simulation, MKEAM is not supported.
2. While the MKEAM, FIM_D and FIM_S methods support all the following elements:
□ Body_Point
□ CVCV
□ PTSF
□ CVSF
□ SFSF
□ Constraint_Gear
□ Constraint_Usrconstr
□ Force_Contact
□ Force_Field
□ Control_Diff
For more details, refer to the following tables in the MotionSolve User's Guide:
Map ADAMS and MotionSolve Modeling Elements.
Map ADAMS and MotionSolve Command Elements.
Map ADAMS and MotionSolve Functions.
Map ADAMS and MotionSolve User Subroutines.
MotionSolve Quasi-statics refers to the Force Imbalance method.
3. The Newton-Raphson algorithm is an iterative method used to solve nonlinear algebraic equations.
Assume, a set of equations F(Q) = 0 is to be solved. Assume also that an initial guess Q* is available.
Let Q be partitioned into two sets of coordinates Q[t], Q[r], where Q[t] are the translational coordinates, and Q[r] the rotational coordinates:
QT = [Q[t]T, Q[r]T ]
Let NORM(x) be a function that returns the infinity norm of any array x. In other words, the maximum of the absolute value of the components of x.
If NORM(F(Q*)) < max_imbalance, Q = Q* is the solution. However, this does not always happen. It is much more common to find that NORM(F(Q*)) >> max_imbalance.
The Newton-Raphson method is an iterative process for refining the initial guess Q* so that the equations F(Q) = 0 is satisfied. The algorithm proceeds as follows:
a. Set iteration counter j = 0. Set Q[j] = Q*.
b. Evaluate F(Q[j]).
c. If NORM (F(Q[j])) < max_imbalance, skip to step 10.
d. Evaluate Jacobian, J = [ $\partial$F[i]/ $\partial$Q[j]].
e. Calculate Δ Q[j] by solving the linear equation J*ΔQ[j] = -F(Q[j]).
f. Set Q[j]+1 = Q[j] + ΔQ[j].
g. Set j = j + 1.
h. If j ≤ max_num_iter, go to step 2.
i. Else: iterations did not converge to a solution; exit.
j. Iterations converged; exit.
4. Finding the static equilibrium configuration for nonlinear, non-smooth systems is difficult. You may find that the default parameters do not work very well for all models. Here are some tips for
obtaining successful static solutions using "The Force Imbalance Method":
□ Make sure that your system starts in a configuration close to a static equilibrium position.
□ Look at the animation of the static equilibrium iterations to understand what the algorithm is trying to do. Visual inspection is crucial for gaining an insight into the behavior of the
□ Sometimes, it is useful to have several static iterations to obtain the final solution. Here is how the outer iterations could work:
Figure 1.
5. Avoid non smooth forces if possible. Newton-Raphson assumes that the equations have smooth partial derivatives. If the forces acting on the system are not smooth, Newton-Raphson will have
6. Many systems have neutral equilibrium solutions. Examples of neutral equilibrium are (a) a spherical ball on a table, and, (b) a car standing still on a flat road. Use larger values of stability
(example stability = 0.01) to deal with such problems. Remember, stability does not change the static equilibrium solution.
7. When used for quasi-static, FIM_S involves a sequence of static simulations. In contrast, FIM_D method uses the DAE integrator DASPK to perform quasi-static simulation. Hence, FIM_D uses the
DSTIFF parameters specified in the Param_Transient element to control the integration process (in particular, dae_constr_tol to set the error tolerance), in addition to the parameters specified
in the Param_Static element. FIM_D always uses I3 DAE formulation.
8. FIM_D uses FIM_S at the start and the end of the quasi-static solution. FIM_S is also used when the integrator encounters difficulties. Thus, the error tolerance setting for FIM_D is
Param_transient::dae_constr_tol except for the first and last steps, which use the error tolerances in Param_Static for FIM_S.
9. FIM_D is usually much faster than FIM_S for quasi-static simulation. | {"url":"https://2021.help.altair.com/2021/hwsolvers/ms/topics/solvers/ms/xml-format_78.htm","timestamp":"2024-11-04T01:48:05Z","content_type":"application/xhtml+xml","content_length":"125237","record_id":"<urn:uuid:afa7f1a0-f68c-451a-a4e5-9b32003b08eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00803.warc.gz"} |
Strength Training
One Rep Max Calculator
Weight Reps
Your One-Rep Max (one-rm): ?
95%: ?
90%: ?
85%: ?
80%: ?
75%: ?
70%: ?
65%: ?
60%: ?
55%: ?
50%: ?
1RM Calculator: How to Calculate Your One-Rep Max for Optimal Strength Training
In strength training, understanding your limits and pushing them appropriately can be the key to unlocking significant progress. One of the most effective ways to gauge strength levels is through the
concept of the one-rep max (1RM). A 1RM calculator enables athletes, gym-goers, and fitness enthusiasts to estimate the maximum amount of weight they can lift in a single repetition for a given
exercise. Knowing your 1RM helps tailor workout programs, track progress, and achieve specific fitness goals safely and effectively. But what exactly is a 1RM, and why is it so impactful in shaping
our workouts?
What is a One-Rep Max (1RM)?
A one-rep max, commonly abbreviated as 1RM, refers to the maximum amount of weight you can lift for a single repetition in a specific exercise. This measurement serves as a baseline for your absolute
strength, indicating the maximum exertion your muscles can sustain. Whether you’re lifting for personal satisfaction or as part of a competitive sport, your 1RM reveals much about your physical
capacity. Because of its intensity, the 1RM is rarely tested directly, especially for beginners; instead, estimates based on repetitions at lower weights provide a safe and effective way to gauge
Importance of Knowing Your 1RM
Knowing your 1RM serves multiple purposes, from planning workouts effectively to minimizing the risk of injury. Here’s why it matters:
• Tailored Training Programs: When you know your 1RM, you can adjust the intensity of your workouts to match your goals, whether that's building muscle, improving endurance, or increasing pure
• Tracking Progress: Monitoring changes in your 1RM over time can provide a clear indication of strength gains, allowing you to celebrate milestones and stay motivated.
• Injury Prevention: Exercising beyond your capacity can lead to injuries. Knowing your 1RM helps you lift safely within your limit and ensures that any progression is sustainable.
In essence, a 1RM acts like a GPS, helping you navigate your fitness journey with precision and avoiding detours caused by under- or overestimating your strength levels.
How a 1RM Calculator Works
A 1RM calculator uses established formulas to estimate your one-rep max. Calculators often take input values like the weight lifted and the number of reps completed to provide an estimated 1RM.
Although there are various formulas, the Epley and Brzycki formulas are among the most popular:
• Epley Formula: 1RM = Weight × (1 + Reps / 30)
• Brzycki Formula: 1RM = Weight × (36 / (37 - Reps))
Common 1RM Calculation Formulas
Several 1RM formulas are commonly used to provide the most accurate result based on the information available. Some of the popular ones include:
• Epley Formula: Suitable for mid-range repetitions (around 6-12).
• Brzycki Formula: Ideal for lower reps (1-10) and is widely accepted for its accuracy.
• Lombardi Formula: Adjusts more dynamically for higher repetitions.
• O’Conner Formula: Used for higher rep ranges and endurance-focused workouts.
How to Use a 1RM Calculator Step-by-Step
Using a 1RM calculator can be straightforward, but following the steps precisely helps ensure accuracy. Here’s how:
1. Warm-Up: Begin with a general warm-up, followed by a specific warm-up with lighter weights to prepare your muscles.
2. Choose Weight and Reps: Select a weight that you can comfortably lift for 3-10 reps without reaching absolute muscle failure.
3. Input Data: Enter the weight lifted and number of reps performed into the calculator.
4. View Results: The calculator will display an estimated 1RM based on the provided formula.
Benefits of Using a 1RM Calculator
The 1RM calculator brings several key advantages to lifters of all experience levels:
• Accuracy: Calculators provide a close estimate, sparing you from having to attempt a potentially risky one-rep lift.
• Convenience: It offers a quick and effective method to gauge strength, perfect for busy schedules.
• Preventing Overtraining: Knowing your limits helps you avoid overexerting, which reduces the risk of overtraining and injury.
Manual Calculation vs. Online 1RM Calculators
While manual calculations provide valuable insight, online calculators simplify the process:
• Manual Calculation: Allows for more personal control and understanding but requires familiarity with formulas.
• Online Calculators: Save time and often include more detailed metrics, making them ideal for beginners or those with limited math experience.
Factors Affecting Your 1RM
Several physiological and situational factors can influence your 1RM:
• Fatigue: Tired muscles perform less efficiently, so it's best to test your 1RM when well-rested.
• Muscle Fiber Composition: Individuals with a higher ratio of fast-twitch fibers may exhibit greater strength in single lifts.
• Experience Level: Novices may see rapid changes in their 1RM due to neuromuscular adaptations.
Tips for Safely Testing Your 1RM
Testing your 1RM involves exerting maximum effort, so safety is crucial:
• Warm-Up Thoroughly: Proper warm-up increases blood flow to muscles, reducing injury risk.
• Use a Spotter: Especially important for exercises like bench press and squat.
• Progress Gradually: Don’t jump directly to your heaviest weight; incrementally build up.
Using Your 1RM to Structure Your Workouts
Knowing your 1RM opens doors to effective training by setting intensity benchmarks:
• Strength Goals: Train at around 80-90% of your 1RM for 3-6 reps.
• Hypertrophy (Muscle Growth): Aim for 60-75% of your 1RM, focusing on 8-12 reps.
• Endurance: Utilize 40-60% of your 1RM for higher reps, usually 15+.
1RM by Exercise Type
The 1RM calculation can vary greatly depending on the type of exercise you're performing. Compound exercises, like the bench press, squat, and deadlift, often yield higher 1RM values due to the
involvement of multiple muscle groups. Isolation exercises, such as bicep curls or tricep extensions, generally have lower 1RMs since they target specific muscles and limit support from larger muscle
1RM for Different Fitness Goals
The 1RM is a versatile tool that can be applied to various fitness objectives. Whether your goal is to increase strength, muscle mass, or endurance, knowing your 1RM helps structure your training
plan accordingly.
Updating Your 1RM Over Time
As your strength progresses, it’s essential to recalculate your 1RM periodically to ensure your training remains challenging and effective.
Understanding and utilizing your 1RM is an invaluable approach for anyone serious about strength training. A 1RM calculator provides a reliable way to estimate your max strength, helping you
customize workout programs, track progress, and set specific, attainable goals. With the correct application, you can optimize training intensity, improve results, and reduce injury risk. As you
progress in your fitness journey, periodically revisiting your 1RM ensures you stay aligned with your goals, pushing your limits safely and effectively.
What if I don't lift heavy regularly?
If you don’t regularly engage in heavy lifting, a 1RM calculator can still estimate your strength. Use lighter weights and higher reps to get an approximate 1RM without risking injury.
How accurate are 1RM calculators?
1RM calculators are generally accurate, though they’re based on formulas and assumptions. The results are close estimates, ideal for gauging strength without direct one-rep testing.
Can beginners use a 1RM calculator?
Yes, beginners can use a 1RM calculator to estimate strength levels and guide workout planning. Starting with lighter weights and progressing gradually will improve accuracy over time.
Is it necessary to test my 1RM for all exercises?
Testing 1RM is most beneficial for compound exercises like squats and deadlifts, which involve multiple muscle groups. Isolation exercises don’t typically require 1RM testing but can still
benefit from percentage-based training.
How can I increase my 1RM over time?
To increase your 1RM, progressively overload your muscles by increasing weight, reps, or sets over time. Consistency and proper technique are key to safely improving your maximum strength. | {"url":"https://calculatorsee.com/1rm-calculator/","timestamp":"2024-11-13T02:41:36Z","content_type":"text/html","content_length":"45415","record_id":"<urn:uuid:1baa72be-16c6-4f32-a439-b402528fce3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00429.warc.gz"} |
SPSR - Silvia Sellán - Talking Papers Podcast
Talking Papers Podcast
🎙️ Welcome to the Talking Papers Podcast: Where Research Meets Conversation 🌟
Are you ready to explore the fascinating world of cutting-edge research in computer vision, machine learning, artificial intelligence, graphics, and beyond? Join us on this podcast by researchers,
for researchers, as we venture into the heart of groundbreaking academic papers.
At Talking Papers, we've reimagined the way research is shared. In each episode, we engage in insightful discussions with the main authors of academic papers, offering you a unique opportunity to
dive deep into the minds behind the innovation.
📚 Structure That Resembles a Paper 📝
Just like a well-structured research paper, each episode takes you on a journey through the academic landscape. We provide a concise TL;DR (abstract) to set the stage, followed by a thorough
exploration of related work, approach, results, conclusions, and a peek into future work.
🔍 Peer Review Unveiled: "What Did Reviewer 2 Say?" 📢
But that's not all! We bring you an exclusive bonus section where authors candidly share their experiences in the peer review process. Discover the insights, challenges, and triumphs behind the
scenes of academic publishing.
🚀 Join the Conversation 💬
Whether you're a seasoned researcher or an enthusiast eager to explore the frontiers of knowledge, Talking Papers Podcast is your gateway to in-depth, engaging discussions with the experts shaping
the future of technology and science.
🎧 Tune In and Stay Informed 🌐
Don't miss out on the latest in research and innovation.
Subscribe and stay tuned for our enlightening episodes. Welcome to the future of research dissemination – welcome to Talking Papers Podcast!
Enjoy the journey! 🌠
#TalkingPapersPodcast #ResearchDissemination #AcademicInsights
Talking Papers Podcast
SPSR - Silvia Sellán
• Yizhak Ben-Shabat • Season 1 • Episode 17
In this episode of the Talking Papers Podcast, I hosted Silvia Sellán. We had a great chat about her paper "Stochastic Poisson Surface Reconstruction”, published in SIGGRAPH Asia 2022.
In this paper, they take on the task of surface reconstruction with a probabilistic twist. They take the well-known Poisson Surface reconstruction algorithm and generalize it to give it a full
statistical formalism. Essentially their method quantifies the uncertainty of surface reconstruction from a point cloud. Instead of outputting an implicit function, they represent the shape as a
modified Gaussian process. This unique perspective and interpretation enables conducting statistical queries, for example, given a point, is it on the surface? is it inside the shape?
Silvia is currently a PhD student at the University of Toronto. Her research focus is on computer graphics and geometric processing. She is a Vanier Doctoral Scholar, an Adobe Research Fellow and the
winner of the 2021 UoFT FAS Deans Doctoral excellence scholarship. I have been following Silvia's work for a while and since I have some work on surface reconstruction when SPSR came out, I knew I
wanted to host her on the podcast (and gladly she agreed). Silvia is currently looking for postdoc and faculty positions to start in the fall of 2024. I am really looking forward to seeing which
institute snatches her.
In our conversation, I particularly liked her explanation of Gaussian Processes with the example "How long does it take my supervisor to answer an email as a function of the time of day the email was
sent", You can't read that in any book. But also, we took an unexpected pause from the usual episode structure to discuss the question of "papers" as a medium for disseminating research. Don't miss
Silvia Sellán, Alec Jacobson
shapes from 3D point clouds. Instead of outputting an implicit function, we represent the reconstructed shape as a modified Gaussian Process, which allows us to conduct statistical queries (e.g., the
likelihood of a point in space being on the surface or inside a solid). We show that this perspective: improves PSR's integration into the online scanning process, broadens its application realm, and
opens the door to other lines of research such as applying task-specific priors.
📚Poisson Surface Reconstruction
📚Geometric Priors for Gaussian Process Implicit Surfaces
📚Gaussian processes for machine learning
📚 Paper
💻Project page
To stay up to date with Silvia's latest research, follow him on:
👨🏻🎓Google Scholar
🎧Subscribe on your favourite podcast app: https://talking.papers.podcast.itzikbs.com
📧Subscribe to our mailing list: http://eepurl.com/hRznqb
🐦Follow us on Twitter: https://twitter.com/talking_papers
🎥YouTube Channel: https://bit.ly/3eQOgwP
Silvia Sellan:
The key idea is we extend this, very well known algorithm called PO surface reconstruction. We give it a statistical formalism and study the space of possible surfaces that are reconstructed from a.
Welcome to Talking Papers, the podcast where we talk about papers and let the papers do the talking. We host early career academics and PhD students to share their cutting edge research in computer
vision, machine learning, and everything in between. I'm your host it Ben Shabbat, a researcher by day and podcaster by night. Let's get started.
Happy woman greeting and talking on video call - May 9, 2022:
Hello and welcome to Talking Papers, the podcast where we talk about papers and let the papers do the talking. Today we'll be talking about the paper Stochastic for San Surface Reconstruction,
published at Sea Graph Asia 2022. I am happy to host the first author of the paper, Sylvia Cian. Hello and welcome to the podcast.
Silvia Sellan:
Happy woman greeting and talking on video call - May 9, 2022:
Can you, introduce yourself?
Silvia Sellan:
Uh, yes. I'm Sylvia Sian. I'm a student, uh, a PhD student at the University of Toronto. I'll be finishing up in one year.
Happy woman greeting and talking on video call - May 9, 2022:
Excellent. And who are the co-authors of the
Silvia Sellan:
this is a joint work with my advisor, professor Alec Jacobson from the University of Toronto. And that's,
Happy woman greeting and talking on video call - May 9, 2022:
All right, so let's get started in a TLDR kind of format, two, three sentences. What is This paper about?
Silvia Sellan:
This is about? Quantifying the uncertainty of surface reconstruction from point land. So there's this classic algorithm called on surface reconstruction, that, takes a point set and puts an implicit
representation of a surface. We take that algorithm and generalize it to give it a full statistical formalizm.
Happy woman greeting and talking on video call - May 9, 2022:
So what is the problem that the paper's addressing?
Silvia Sellan:
the the overarching problem is surface reconstruction, so you get a point cloud as input, which can be. they output up a 3D scanner or a lighter scanner or something like that, and you want to
recover a fully determined surface. so you can imagine that you're a car driving down the street, an autonomous car driving down the street. You scan your surroundings using some lighter scanner, and
you want to know what they look like so that you know that you're not crashing into anything. traditionally if you ask any computer graphics researchers, they'll tell you the easy way of doing it is
by using a thing called put on surface recon. Uh, This was an algorithm published in, uh, 2006 that takes that point loud and outputs an implicit distribution. So something that tells you in or out
for any point in space. however it does it, it only gives you one possible implicit distribution. And of course, recovering a fully determined surface from a point cloud is an under determined
problem, right? There are many possible surfaces that could interpolate the points in the point cloud. instead of just outputting one, we extend personal surface reconstruction and we output every
possible surface with a specific probability. So every possible surface that could be reconstructed from a given point cloud with different probabilities.
Happy woman greeting and talking on video call - May 9, 2022:
Okay, so essentially given a point cloud as input, You could find, Multiple ways to connect between the points, right? So finding the surface that, that these points were sampled on, that's the big
question that everybody wants to solve. And you're saying, well, there's an infinite number of surfaces that could theoretically go through these points, especially if there's like a gap in the
Silvia Sellan:
That's right. That's right.
Happy woman greeting and talking on video call - May 9, 2022:
And, and the on surface construction method basically says, well, there's only one. Here you go. That's what my output is. And your method is saying, well, there could be other options.
Silvia Sellan:
That's exactly it. That's exactly it. And, and in a way we interpreted on reconstruction as giving you the most likely output under some, constraint under some, prior, uh,, conditions. But, Sometimes
you don't want just the most likely, right? You can imagine if you're the one that's driving the car, that's doing the point, you want to know, okay, I won't crash into anything. Not just under the
most likely reconstruction, but under 99% of the reconstruction so that you keep driving, right? So we quantify that uncertainty of of the reconstruction.
Right. So, so this kind of ties to the question. So, so why is this problem important? And I think the example of autonomous driving is one of these amazing examples where you say, well, I don't
wanna found out the collision after I collided. I wanna know beforehand.
Silvia Sellan:
Uh, that, that, that's right. That's a great example. I, I was recently talking to, to some people about this work and, and they work in, Automated surgery. So they were also telling me that
sometimes you want to be very, very sure that you're not cutting through a nerve. So you want to be absolutely sure of what your nerve looks like. And apparently in some software they do use a point
leverage reconstruction algorithm. So, so this is that, that would be yet another example of a situation where you want, you really want to quantify the uncertainty, cuz you don't wanna paralyze
Happy woman greeting and talking on video call - May 9, 2022:
Super interesting. So now we know why this is useful, but what are the main challenges in this domain?
Silvia Sellan:
Well, the main challenge is, is that, uh, it's not specially hard to quantify the uncertainty of reconstruction, in general to go to, to device an algorithm that takes a pointent and would give you
an uncertain, surface. The problem is that we already have this other algorithm called on surface reconstruction, that combines many of the good things we would want in a surface reconstruction
algorithm, and they also have very good, efficient code online, so it. almost everyone who is doing point cloud reconstruction is using on surface reconstruction. So the challenge was not justise
some other algorithm, but generalize this one. So we needed to really understand on surface reconstruction, and give it a new statistical formalism. That meant for me, the main challenge was that I'm
a graphics or a geometry researcher. I'm not a statist researcher. So it meant familiarity myself with a lot of. Statistical learning literature that I would understand where the statistical
formalism come in. And uh, that was mostly the theoretical. Our, our paper is mainly theoretical, and that was the main theoretical challenge that, that we struggle with. It took a couple of weeks
over, uh, last year's winter, uh, Christmas break to really understand where did the, the statistical formalism, where, where can we plug that in to? Plus on surface reconstruction,
Happy woman greeting and talking on video call - May 9, 2022:
Okay. Can't wait to hear more about that in the approach section, but before we go down to that, let's talk a little bit about the contributions. So what are the main contributions of the paper?
Silvia Sellan:
Well, the main contribution, like I said, is we, we give a statistical formalism to put on surface reconstruction. that's the one sentence version, uh, that's two sentence version would be that
usually put on surface reconstruction gives you just one value of an implicit function. We extend that and instead of one value we give. Uh, a mean and a variance that fully determine a ga and
distribution of what the value at that point is. I know that like people in, in, in your field use this term, coordinate network. This is not a network, but it's kind of a, a coordinate function that
implicitly defines a surface. Think of it as quantifying the variance of the output of a coordinate.
Happy woman greeting and talking on video call - May 9, 2022:
So I'm super excited to get down to what the approach is doing, but before we do, let's talk a little bit about the related works. So if you had to name two or three, works that are crucial for
anyone coming to read your paper, which ones would those
Silvia Sellan:
Well, the most obvious one is on surface reconstruction, uh, by capstone at all. That's, uh, 2000. six, symposium and geometry processing paper. That's the main work that we're extending. So, you
know, we, we give a summary in our paper, so you could read our paper without reading on surface reconstruction, but that's the main work that, that we build on. So definitely that's, that's the most
important one. Uh, then we use GA processes, I'll explain this later, but we'll use GA processes to formalize this statistical understanding. So, two of the ones I've most, I would recommend someone
read are, uh, gian processes for machine learning. This is a book, and also there's, uh, geometric priors for Gian Process Implicit Surfaces by Martins at all. This is. A paper that, that basically
uses gotcha processes for the specific case of recon, of surface reconstruction. and it's written more for a graphics audience. So it's, it might, it was easier to understand them for me than one of
these gotcha processes for machine learning. More general papers. So if you come from a graphics, background, geometric priors for calcium process, implicit surface,
Happy woman greeting and talking on video call - May 9, 2022:
Okay, excellent. I will be sure to put links to all of those relevant references in the episode's description. Personally, I think that any researcher working on surface reconstruction, has to read
for on surface reconstruction, like it's a must read. so it's time to dive deep into the approach. So tell us what did you do and how did you do it?
Silvia Sellan:
Uh, well we combine put on surface reconstruction with this concept called kan processes. Uh, I'll be careful on how to explain this cuz I know that most of your a is from machine learning, not
necessarily graphics. So I'll. Basically you need to understand both things to understand our approach. And our approach relies on one very specific interpretation of on surface reconstruction, on
one very specific interpretation of gian processes. And then we put those together. so what we did is we went through on surface reconstruction and we interpreted to work in two steps. So basically
on surface reconstruction takes a point, cloud as input that point. Cloud is oriented, so it comes with a bunch of normal vector. the, the first step of on surface reconstruction is to take those
vectors and interpolate them into a full vector field that's defined for everywhere in space. So that's step one. Step two is that they then solve a partial differential equation to get an implicit
function. Whose gradient field is that vector field. So basically step one, you go from a discreet set of points to a vector field, and then step two, you go from a vector field to an implicit.
that's basically all you need to understand about on surface reconstruction. And of course, that PDE that you solve is azo, uh, equation. So that's why it's called on reconstruction. But really the
part we care about is that step where you go from an oriented point cloud to a vector field. We notice that, uh, that step can be seen a as a gian process. So what is a gauch process? Basically, a
gian process for your audience is just a way of doing supervised learning. but just, just in case someone from graphics is listening to this and is wondering what, what is that? Uh, that means that
you want to learn some function that you don't know what that looks like. Um, and you've observed it at some points, a finite, discrete set of points. So I like to think of, uh, the function being
how long does my advisor to respond take, to respond to an email, right? So, uh, the, the, the variable is the time of the data you send the. And the response or the, the function that you wanna
learn is that the hours it takes for him to respond. So, you know, maybe I send my advisor an email at noon, I send my advisor an email at 2:00 PM and I get two data points, right? But then I ask
myself, well, what, what would it look like if I sent him an email at 1:00 PM Right? That's a new point that I haven't considered. Uh, we call that the test point. And the cool thing about gotcha
processes is that they tell me, well, if it. Two hours for him to respond at noon, and it took five hours for him to respond at 2:00 PM Then at 1:00 PM it'll take something like three hours plus
minus two. Right? So it, it will not just tell me a guess for how long it'll take. It'll tell me sort of an error bar, a variance for how long it'll take. And you, we can compete those, um, that,
that mean, and that variance with simple matrix. So we do some, uh, assumptions that I'm not gonna get into until a gallian process. And we notice that that step from on surface reconstruction, that
going from a discreet set of oriented points to a vector field, that step is a supervised learning step. So like the, the vector field that was on reconstruction outputs is the mean of a Gaussian
process where you're trying to learn the fun that vector field as the. So as the major, it bears stopping there for a second. So we notice that the vector field from reconstruction could be
understood as the mean of a Gaussian process. So then we wonder, well, what, what would it look like if reconstruction had wanted to do a Gaussian process from the start? So we reinterpret
reconstruction, that's what we call stochastic surface. So instead of just this mean, we wondered, well, if we wanted to get to, to do a gar process from the start, we would not just get the mean, we
would get a variance too. So we get this sort of sarcastic vector field instead of just a vector field, and we can solve the same equation that we solved earlier to go from vector field to implicit
function. We can solve it again now in the space of statistical distributions to go from stochastic vector field to stochastic scaler. What does this give us? This means that at the end we get for
each point in space, not just an, not just a value. So reconstruction, traditional reconstruction would give you, for this point in space, the value is 0.2. And since that's bigger than zero, that
means outside. Instead we would give you a full distribution, so, so we would tell you 0.2 plus minus 0.3, and that'll give you an idea of how sure you are of that point being inside or. So that's
our approach. We take reconstruction and we reinterpret it as a GA process, and we can output a fully stochastic scaler field as the output.
Happy woman greeting and talking on video call - May 9, 2022:
Uh, this is such an interesting approach. It's really not like all of those new papers coming out that, oh, yeah, we. Switch some block and now it works better. It's actually like looking at the
problem from a different perspective, right? Like looking at it in the way that you, you now have like this go process, which gives you this stochastic properties you can, you can utilize to, to do
so much more than you could before with the traditional or classic for surface reconstruction.
Silvia Sellan:
I'm glad you enjoy our first. I can shout out, uh, Derek Lou, who I think was on this podcast a few months. Uh, I asked him, before I started to work on anything related to machine learning, I asked
him, how do you work on machine learning such that you're not waking up every Monday? Check, checking archive to see if you still have a project? Cuz that feels too stressful for me. Uh, I don't
wanna panic. I already panic enough in my life. I don't, I don't want that to have that. and he said you just need to work on something so fundamentally different from everything. That no one's gonna
scoop you, which is like a classic Derek advice where you're like, well, obviously if I could, I would work at something revolutionary and fundamentally different. Right? Uh, the problem is I don't,
but, but this paper felt like that in the sense that, you know, for some reconstruction has been around for 15, 16, 17 years. There aren't many people that are working on, uh, statistically
formalizing post reconstruction. So it felt like a nice newspaper to write that. Wasn't gon that. I didn't need to be anxious about being skipped on. So that's kind of the reason.
Happy woman greeting and talking on video call - May 9, 2022:
Yeah, and I think this message is like super important because most of the audience of this podcast are early career academics or PhD students and, and I think the message of. not, you know,
crunching the parameters all day and trying to really find something that's fundamentally different than what everybody else is doing, I think is a super important message to convey. So thank you for
Silvia Sellan:
To be clear, that's Derek Lou's message. No, not mine.
Happy woman greeting and talking on video call - May 9, 2022:
Yeah. That fits for you. And it fits for me too. I mean, um, I, I think that what research should be about, right? It shouldn't be. It's unfortunate that the, that the, a lot of fields are now in the
place where it's about tuning parameters and rather than coming up with new and interesting approaches and perspective. on this podcast, I try to bring all those that do, do that. Like they give a
new perspective on a field, solve a problem, and seems like I've got two for tune for now. Um,
Silvia Sellan:
okay. Maybe I'll, I'll ask you a question then I'll change the format a little bit and you can cut it if you don't want it. Uh, but this, you are, you're focused around papers a lot in this, uh,
podcast, obviously. Uh, and I wonder if part of the problem is that we're, we're using this academic currency, so, you know, obviously if we have an incentive, we, we, if we, if we create an
incentive, People and I include myself, are gonna look for, you know, what is the idea that most quickly and concisely resembles a neuros paper or a cvpr paper, a cigarette paper? Um, I wonder how
much our current, like scientific publication process encourages those types of works that are, I changed the parameter a little bit and I got like this bold face number at the end of the table, and
that's a CPR paper, like, which are definitely important work works of research. But currently we have no way of distinguishing, putting fundamentally different approaches and those types of works.
So, um, you know, how much of it, how much of it is our fault for focusing on papers too? Is there gonna be a talking blog post podcast send?
Happy woman greeting and talking on video call - May 9, 2022:
Well, actually that's a great question. Um, and well, Personally, I try to bring those papers that do the extra mile as well. So usually papers that have a project website and a blog post and they
try to convey it and teach it. And not only, okay, here are the ball numbers, we, we were the best. Right? Where some of the, an idea, but I think you touched on a very important point where you
said, well, our incentive system. Not good. I'm not sure I have a good idea for what is a good incentive system, but the current one is not good. We're judged by the number of papers that we shoot
out and the more they get accepted to high venues, the the better. And that helps you secure funding and securing funding helps you get students and that's your academic career and. There's nothing
that looks at, and you can actually even see it on Twitter, right? Every other academic kind of post, oh, we had seven papers accepted to
Silvia Sellan:
Yeah, exactly.
Happy woman greeting and talking on video call - May 9, 2022:
and now is this out of how many? Right? It's it's not just about the successes, right? It's also about the, all of the times that you tried something new and risky and novel and failed because you're
doomed to fail, right? That's research. If we knew the answer before we started the project, then it's not a very interesting question to work on. And yeah, I agree. It's a big problem in, in, in
multiple fields at the moment. Um, but the upside is that it pushes everyone to the limit and it pushes the field forward much faster than than any other field. And I think there are some, Things
that people now do in addition to papers that even further promote that, right? Publishing the code, that wasn't a huge thing. I don't know, 10 years ago, like who would've put the code online and
made sure that it can run on multiple platforms. Today it's almost standard to put your code on GitHub, right? Um, so bidder, right? You can't have the one without the outer. Okay. But back to the
episode, that was a great, uh, question. Thank you. and by the way, I think this is one of the thing that, that this podcast is trying to do, right? It's, it tries to, to kind of have a little look
like a peak inside the mind, behind the paper. It tries to see the way of thought, not just the results,
Silvia Sellan:
that's great. I just wonder, I just wonder, you know, there are papers that I've, and you know, I don't mean this as a compliment necessarily, but take it as that if you want. There are papers that
I've. uh, listen to this podcast on and, and looked at the project page and I feel like I understand them. You know, like I feel like the actual pdf, I've never opened it and I feel like I understand
that paper well enough to, to be inspired by it and like work on on future works. So like at some point, yeah, the whole format of a paper is being rewritten.
Happy woman greeting and talking on video call - May 9, 2022:
Yeah, this is part of the reason I started a new medium for sharing research. Uh, But yeah, it would be interesting to see where we are in a few years. I know that, um, it used to be only about
citations, but now there's this whole line of, they call it alt matrix. So all these kind of different ways of measuring the impact of the paper, which are not necessarily influenced by citations,
but it's not as widely adopted as citations Back to the episode structure. So we talked about the approach, super interesting. let's talk a little bit about results and applications. So in which
situation did you, apply your stochastic for surface reconstruction and how did that work?
Silvia Sellan:
Right. So as I was telling you, by using, uh, by combining on surface reconstruction with aian process, we had this. Um, uncertainty map for every point in space that told us how likely that point
was of being in the reconstructed surface. Instead of just, is it in or is it out? We got something like, oh, it has a 60% chance of being inside the reconstructive surface. This map, we can ask for
every point in space, and that's actually very useful. The main use is, for example, that car example I said at the beginning. So like, you know, we, we had a very toy example of a car that's driving
in 3d. Takes a scan of its surroundings and it, you know, through a trajectory, it can ask, how likely am I to intersect any of the other shapes in this scene? And you can see that like, it's like
30%. You know, 30% means that probably what on surface reconstruction, the traditional method would've told you no, there's no intersection and that's it. Right? But 30% chance of crashing your car.
It's you, it probably means that you want to take another trajectory, right? You, you can only do so many of those chances before you break your car. Uh, so we do examples like that. another thing we
do is, um, you know, if you think about it, the more these probabilities are closer to either a hundred or zero, the more certain you are of what the shape looks like. So, you know, if I give you an
uncertainty map that just looks. It's zero in all this part and a hundred in all of this part. It means we're, we're very sure of what the shape looks like, because for every point of space we can
ask, we can very confidently say if it's in or out. But if it's mostly 50%, then we don't have a lot of idea of what the shape looks like. So another thing that we do is we can introduce a thing
called Integrated Uncertainty that just measures how far this probability is from zero point. How close this probability is to 0.5. So the, the higher, the more uncertain you are about what the shape
looks like. And that's something that we can use, uh, for example as a reconstruction threshold. So if we're scanning something from different angles, we can compute this integrated uncertainty and
say, you know, keep scanning from random angles until you reach 0.1. Integrated uncertainty. And this is something that's agnostic on the shape that you're actually reconstructing. So you can use it
as a t. For scanning unknown shapes so that you get a similar reconstruction quality. So this is something that, that we output.
Happy woman greeting and talking on video call - May 9, 2022:
So this is something that's very interesting for, I guess kind of like robotics applications, right? You have a robot walking around the house, it sees a bunch of things. It's not sure what it's
seeing, so it want, it should get a better look. And this is what we do as human, right, humans, right? We see something we've never seen before. The first thing we would do is like,
Silvia Sellan:
Right. The, the, the only difference is that, we humans would have an intuitive feeling for where we should look as the next point. Right. Whereas what I'm saying is just like, oh, I would tell the
robot to like, keep scanning randomly until it, it, it figures out what the shape is, which isn't exactly what we as humans into, right? Like, if you see something, you would turn it around because
you know that it's the back part that you haven't seen yet. Um, so this is actually something that we looked at further. Um, so we, we, we have an example in the paper where, Uh, have an incomplete
point cloud and we set different cameras around it and we simulate Ray from those cameras onto the point cloud. So this is something that we can do. We call it Ray casting on uncertain geometry, but
basically we can use the same statistical formalism to cast ray from a hypothetical scan position on the surface that that means that for each possible camera we can simulate which points would this
scanning position add to the. We can add those points. I see if our integrated uncertainty got better, right? So we can ask, you know, by adding this new scanning position, did I actually gain any
knowledge or not? And that's kind of closer to what the human is doing, which is identifying the next best view position. Um, and that's kind of a further example that we have, so we can also do it
with our statistical formal. There's also, there's also, so I, I always make this joke that like if you, you might have heard that actors have some movies that they do one for them, one for me. So
like they have one movie that that sells. So they'll do Avengers so that it allow of tickets, but then they'll use that money to fund their small project that won't sell as much, but they really,
really want. So sort of these like next week planning, collision detection, all of these applications are the ones I did for the reviewers, right? So this is for them. Uh, the application I really
liked is that by understanding post reconstruction as a, as a gaugin process in, in the process of understanding it like that, we needed to assume a certain prior, right? Because a gotcha process
start by assuming a prior, but now that we understand reconstruction like this, we. Does that prior make sense for every reconstruction test? So like, can we change that prior? So can we use this
statistical understanding, not just for new applications, but to improve the application of reconstruction, which is just straight up surface reconstruction. So the main result application I'm
interested in is using different priors. So for example, we show examples in the paper where we enforce that the reconstructive surface has to be. This is something that's a known problem with
reconstruction. They some sometimes outputs open surfaces. We solve that with like less than a line of code with half a line of code. We we solve that. We have a similar one where we, we, we close a
car reconstruction that on reconstruction would've, um, would've given an open output. So basically changing these priors to, uh, improve the reconstruction is, are some of the most exciting results
that we have and some of the most exciting future work directions that, that we have, that I guess we'll talk about.
Happy woman greeting and talking on video call - May 9, 2022:
Yeah. And I think like it really makes sense to, to have that like dependency on some prior, right? Because a lot of, I don't know, classification networks, uh, I mean, many people can say that it's
already solved, right? So if you knew that you're looking at a car, And you have like, I don't know, a very noisy one, directional scan of that car. It would be really good for the reconstruction
process to say, well, that's a car now. You know, that's a car. Use that information to improve the
Silvia Sellan:
right. That's right, that's right. And, and there are, to be clear, like, you know, point completion or, or, or surface reconstruction algorithms that use, database knowledge, the problem is that
those don't leverage all the good things about Pozo reconstruction. So Pozo reconstruction is extremely fast, it's extremely efficient. It has. These local global dual steps that, that make it fast
and robust, but also noise resilient. So it has, it, it some reconstruction is the best of the best that we have for surface reconstruction. So like, I think there, thanks to our paper, hopefully
there's a very clear follow up of data driven on surface reconstruction. I guess I'll, I'll pitch it to your audience. If you wanna write that paper, send me an email cuz we can write it together
and, and I know how to use the code so I'll do that. But you do the data driven part. I, I think that there's an opportunity for an immediate follow up there that's very easy and, and could be revol
Happy woman greeting and talking on video call - May 9, 2022:
Low hanging fruit.
Silvia Sellan:
Or rather, we, we built a ladder that tells, takes you very far, very near the fruit. Right? It wasn't low hanging five months ago.
Happy woman greeting and talking on video call - May 9, 2022:
Okay. Were there any like fail cases or any unexpected, results that you encountered?
Silvia Sellan:
Well. Hmm, good question. The main drawback of our algorithm is the speed. So our algorithm is slow. This isn't really like a failure. It's more like computing. The variance of our estimation is very
slow. This doesn't affect the project. I was just pitch. Because, uh, that would just be comput in the reconstruction, that that is fast. But computing the variance is, is slow. So we, uh, found
that, you know, when we jumped from 2D to 3d, we straight up couldn't do what we were doing in 2d, in 3d, in a reasonable time, in a reasonable computer. So we had to use, uh, you know, a space
decomposition, uh, a space reduction trick to, to make the solver manageable. So that was a bit disappointing. Um, another. Failure case, we, maybe not a failure case, but uh, something in our paper
that didn't go as planned is that we have this step where we basically lump a ma, we make a matrix diagonal so that it's easier to invert. Uh, basically. And uh, this is based on something that I
know from finite element analysis where people usually, uh, make a matrix diagonal we show that it's valid and there's some assumptions, but it's not entire. Accurate under all assumptions. Um, this
is not our, like we had to do this lumping so that we recover post on reconstruction. So this is, it's not that we proposed this, it's as we explained post on reconstruction as having done this. but
recently there's been, a new paper by Alex Turnin at all called Numerical numerically stable sparse SCO from processes via minimum Separation using cover trees. And this basically shows you a better
way of doing what we did. This is a, this is a paper that was posted on archive two weeks ago, so, so there's no way we could have used it for our, work. But, this is a very, I recommend this to
anyone working on gian processes and thinking about applying gian process. At scale, because this basically gives you, I, I don't think Alex would agree with this, uh, interpretation, but it
basically gives you a smart way of turning that ma of, of making that matrix smaller. So, uh, this is one thing that I wish, I wish this other paper had come out a year ago, and then we would've, we
would've used it.
Happy woman greeting and talking on video call - May 9, 2022:
Yeah, these kind of like the field is going forward and you never know which block you can swap into another block and something that's a challenge today that you had to circumvent in some way at
some point would turn up to be solved. but, but this is great. It means we're working in a very, productive and high paced field.
Silvia Sellan:
Happy woman greeting and talking on video call - May 9, 2022:
Moving on to the conclusions and future work section. So how do you see the impact of the paper going forward?
Silvia Sellan:
So I look at this paper as a, as a computer graphics and geometry processing researcher, and the part that excites me the most about this paper is that it's a, it's a way of quantifying uncertainty
of a process that we use in geometry processing. So namely surface reconstruction from point light. So, You know, one thing I would like to work on in the future and I would like to encourage people
to work on, cuz I think it's a, a very, can be a very promising field, is a fully uncertain geometry processing pipeline. So there are all these works like ours that quantify the uncertainty of the
capture steps. So like going from a real world object to an uncertain surface. There are several works like that, like that hours among them. But we sort of stopped. And I would like, I would like us
to do things to that uncertain surface, right? So, you know, geometry processing doesn't stop at capture. We then solve partial differential equations on that geometry. We then compute differential
quantities on that geometry. We then deformed that geometry. We do physics simulation on that geometry, right? There's a lot of things that that geometry processing does, but we're not doing it for
those uncertain shapes. So the next steps, the ones I'm interested by. Okay. Now that, now that I've given, I've scanned the thing and I've given you the different possibilities of surfaces that it
could be, now tell me the different possibilities of curvatures that it could have. Right? That extra step. It, I don't think it has been done before and I think that's very exciting cuz then we can
inform the scanning, right? We can go back and. Well, I want, I know that the shape has certain maximum curvature, so if I take it all the way, I know where I should scan next because there are
regions where my curvature is more uncertain or something like that. Right. So this is something that, this is a direction I think is very promising. We already talked about, uh, task specific or
data driven on reconstruction that I think is an extremely promising avenue for future work. Low hanging fruit, like you said.
Happy woman greeting and talking on video call - May 9, 2022:
No, uh, a tall
Silvia Sellan:
A tall ladder. Exactly. Uh, and, uh, you know, there, yeah, there, there are many applica, many ways of quantifying uncertainty that, that, uh, we could do in geometry processing. And I hope that
this is just a first step, uh, to that vision. There are some steps that I think we will take, and there are some steps that I hope the community also.
Happy woman greeting and talking on video call - May 9, 2022:
Yeah, this one of the things I really liked about this paper. From the first read, I could totally see that it kind of opens up this whole branch of sarcastically informed, inspired, or motivated
further steps in the pipeline that you can use this work with. And yeah, it's, it's exciting to see what what will come up next. now to my favorite part of the podcast, what did reviewer two say?
Please share some of the insightful comments that Reviewer had through the review process.
Silvia Sellan:
Okay. So this was a very, so anyway, the whole story of how we published this paper was, was really fun. Uh, I, I. I, I am one of those people that doesn't like crunching for a deadline. So I work on
something steadily, consistently, several months instead of one week where I don't sleep. There are two kinds of people. I'm one kind, my advisor is mostly the other kind. Uh, but this time I got c
four days before the deadline, so I couldn't crunch. So I just sent basically my current draft to my advisor and. Look, here's the victor. If you wanna change anything, change it. But I'm not working
on it cuz I have like a fever and I'm just gonna lay on the couch for, for five days. So, we basically submitted our draft without a lot of the things that we would've liked to do in those four days.
So I was a bit disappointed that maybe we would get rejected because I couldn't do, for example, the data driven part that was like a plan that I wanted to do in the days before the submission
deadline. So that was a bit sad. but we actually got very positive reviews. We got seven review. Uh, which is un unheard of for sra. Like SRA usually has five reviews. Sometimes they bring in a sixth
one, but I had never had seven. I don't know anyone who has had seven reviewers. So that was very, surprising. Most of the reviews were positive. I think six outta seven or five outta seven were
positive, except that we had one, I don't know if it was strong reject or reject. So like the work, the way it works at Segra is you have like strong reject. We reject, we accept, accept and strong
accept. So six. And, uh, we had one that was either strong rejector or, or reject that, that basically tanks your paper if you have one of those usually. Uh, and the, the review was very surprising.
It said, Basically, you know, I like the paper on a first read. I loved it. But then on a second read, I started realizing that none of the quantities that the authors are introducing make any sense.
So if you look at the variance maps, so these are the maps of like where we are most confident of the reconstruction. The variance is higher near the sample points. That shouldn't be like that. The
variant should be lower near the sample points, right? Because we are more. of what the value is near the data, right? So it doesn't make any sense. So then I realized that nothing in the paper makes
sense, so now I want to reject it. The problem was as simple as that, that revere was misreading our color map. So the color map was yellow, yellow meant high, and purple meant low. It was this
plasma, uh, matte plot lib color bar that, that you may have used or your, your. Viewers might be familiar with. So it was just a matter that we did not include a color bar saying like, this is low,
this is high. We did not include a color bar in all our pictures. This is basically, our figures are full of these color images. So if we added color bars, we would have 200 color bars in the paper.
Uh, but this reviewer misunderstood it. So, you know, it was a very interesting rebuttal to write where we had to. You know, we're, we will add color bars to every figure and they will show that
unlike reviewer two or three is interpretation variances indeed lower near the data points. That's, you know, it is what you expected it to be. Uh, so it was very scary cuz I, for some seconds there,
I thought we might just get a paper rejected because we didn't add color bars to the plot. So, you know, under every, you know, behind every sign there's a story. Always add color bars to your plots.
You never know. If we had had one other negative review, it might have, it might have tagged the paper completely. So that was a lesson I will never forget. I'm sorry, review two that we didn't add
color Barss. It's not your fault. You, you, you get, you. There were two possible interpretations and you took one of them. Uh, we should have added it. And now if you look at our favorite, it has a
lot of color bars. Because we're not making that mistake again. So that, that's my Revere two story,
Happy woman greeting and talking on video call - May 9, 2022:
Oh wow. I absolutely love those kind of paper war stories, the whole C submission deadline, and then the coba and, and yeah, I think it's an amazing lesson. And I, and I know that every paper from
now on you or any one of your future collaborators, you would never forget to put the coba.
Silvia Sellan:
right. That's
Happy woman greeting and talking on video call - May 9, 2022:
Uh, yeah. So don't forget the color about everyone. That was a great story. alright, anything else before we wrap up?
Silvia Sellan:
I guess, if any of what I sent sounds interesting. I don't have enough time to do all the project ideas. I, I. So, uh, definitely email me if you wanna work on anything related to what I just said,
and we can work on it together. So I'm sure its, it will put my website somewhere in the episode notes. Uh, you can go there, find my email and send me an email. I'm always open to, to getting random
emails from people.
Happy woman greeting and talking on video call - May 9, 2022:
Yeah, excellent. I'll be sure to put all of the contact information for Sylvia in the project description, and I should also probably mention to all of the more senior listeners that we have that
Sylvia is looking for postdoc or faculty position starting fall 2024. So don't miss out on this amazing opportu. and All right, Sylvia, thank you very much for being a part of the podcast, and until
next time that your papers due to talking.
Thank you for listening. That's it for this episode of Talking Papers. Please subscribe to the podcast Feed on your favorite podcast app. All links are available in this episode description and on
the Talking Papers website. If you would like to be a guest on the podcast, sponsor it, or just share your thoughts with us, feel free to email talking papers dot podcast gmail.com. Be sure to tune
in every week for the latest episodes, and until then, let your papers do the talking.
People on this episode | {"url":"https://talking.papers.podcast.itzikbs.com/1914034/episodes/11823154","timestamp":"2024-11-07T23:34:38Z","content_type":"text/html","content_length":"122832","record_id":"<urn:uuid:422264d0-3bf1-4c73-9cb9-a8482666b998>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00510.warc.gz"} |
Question & Answer: You have written a stock trading book that guarantees vast riches for anyone who utilizes its…..
You have written a stock trading book that guarantees vast riches for anyone who utilizes its strategies. To publish the book you can either use ABC publishing or you can self-publish.
If the book is published with ABC, there may be an opportunity to go on a book tour. If you go on the book tour and if the book is successful, it is estimated that it will sell 30,000 copies. If you
go on the book tour and the book is not successful, sales are estimated at 2000 copies. If you do not go on a book tour (no matter if you self-publish or use ABC), then the estimated sales for a
successful book are 20,000 copies and the estimated sales for an unsuccessful book are 1000 copies.
If you choose ABC as your publisher you will receive a $2000 signing bonus and $1 for each book sold. The first thing ABC will do is send the book out for review. Due to their clout in the industry,
the book will receive generally positive reviews with a probability of 0.7. With a probability of 0.3, the book will receive generally negative reviews. Given that the book has received negative
reviews, the probability that the book is a success is only 0.2 and the probability that the book is not successful is 0.8. If you receive positive reviews, then you may decide if you would like to
go on a book tour. If you go on the book tour, the probability of a successful book is 0.85, while if you do not go on the book tour, the probability of a successful book is 0.75. Because you dislike
travel, you assess a cost of $2000 to the book tour.
If you self-publish you will earn $2 for each book sold, however, there is very little impactful marketing that you can do. Based on the evidence at your disposal, you estimate a probability of the
book being successful at 0.585 and the probability of the book not being successful at 0.415.
You are strongly considering using ABC to publish the book, but would like to investigate the situation more carefully before committing. In particular, you wish to simulate the process of book
review, book success, and book sales. Note, you have decided that you will definitely go on the book tour if the review is positive, so there is no need to simulate that decision. Develop a
simulation of 100 instances and determine the average number of books sold. In particular, your simulation should:
i. Determine if the book receives positive or negative reviews (use the probabilities above).
ii. Determine if the book will be successful or not, based on the book’s reviews (use the probabilities above).
iii. Determine the book sales, based on the books success (or lack thereof). Assume that book sales are normally distributed. In cases where the reviews were negative, assume that the mean sales are
20,000 with a standard deviation of 2000 for a successful book; for an unsuccessful book, assume that mean sales are 1000 with a standard deviation of 200. In cases where the reviews are positive,
assume that the mean sales are 30,000 with a standard deviation of 2000 for a successful book; for an unsuccessful book, assume that the mean sales are 2000 with a standard deviation of 300.
Please provide a simulation in Excel and show all formulas used.
Expert Answer
B20 =IF(RAND()<=$D$7,$C$7,$C$13)
C20 =IF(B20=$C$7,IF($C$17=$E$7,IF(RAND()<=$G$7,$F$7,$F$8),IF(RAND()<=$G$10,$F$10,$F$11)),IF(RAND()<=$G$13,$F$13,$F$14))
D20 =IF(B20=$C$7,IF($C$17=$E$7,IF(C20=$F$7,NORMINV(RAND(),$H$7,$I$7),NORMINV(RAND(),$H$8,$I$8)),IF(C20=$F$10,NORMINV(RAND(),$H$10,$I$10),NORMINV(RAND(),$H$11,$I$11))),IF(C20=$F$13,NORMINV(RAND
J19 =COUNTIF(B20:B119,C7)/COUNT(A20:A119)
J22 =COUNTIF(B20:B119,C13)/COUNT(A20:A119)
L19 =COUNTIF(E20:E119,I19&” and “&K19)/COUNT(A20:A119)/J19
L20 =COUNTIF(E20:E119,I19&” and “&K20)/COUNT(A20:A119)/J19
L22 =COUNTIF(E20:E119,I22&” and “&K22)/COUNT(A20:A119)/J22
L23 =COUNTIF(E20:E119,I22&” and “&K23)/COUNT(A20:A119)/J22 | {"url":"https://grandpaperwriters.com/question-answer-you-have-written-a-stock-trading-book-that-guarantees-vast-riches-for-anyone-who-utilizes-its/","timestamp":"2024-11-08T02:52:00Z","content_type":"text/html","content_length":"45452","record_id":"<urn:uuid:5ef1ae9f-b5b2-4927-b2d9-7a31b322b6a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00559.warc.gz"} |
Evaluate lim x → 0 e^x-1-x/x² by L’Hospital’s Rule | L’Hôpital’s Rule
Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1-x}{x^2}}$ by L’Hospital’s rule
According to the direct substitution method, the limit of natural exponential function in $x$ minus one minus $x$ divided by square of $x$ is indeterminate as the value of $x$ approaches zero.
$\implies$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1-x}{x^2}}$ $\,=\,$ $\dfrac{0}{0}$
The given function is basically a rational expression and its limit is indeterminate. The two qualities direct us to use the l’hôpital’s rule to find the limit of the given function in rational form.
Try to find the Limit by L’Hospital’s rule
The expression in the numerator is a trinomial and the expression in the denominator is a monomial but they are defined in terms of $x$. So, the expressions in both numerator and denominator should
be differentiated with respect to $x$ to use the l’hospital’s rule.
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}(e^x-1-x)}{\dfrac{d}{dx}{(x^2)}}}$
The terms of the expression in the numerator are connected by a minus sign. So, the derivative can be distributed to all the terms in the numerator by the subtraction rule of the differentiation.
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}(e^x)-\dfrac{d}{dx}(1)-\dfrac{d}{dx}(x)}{\dfrac{d}{dx}{(x^2)}}}$
In the numerator, the derivative of natural exponential function can be evaluated by the derivative rule of exponential function, the derivative of one can be calculated by the derivative rule of a
constant and the derivative of variable $x$ can be evaluated by the derivative rule of a variable. Similarly, the derivative of $x$ square can be calculated by the power rule of derivatives.
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-0-1}{2 \times x^{2-1}}}$
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1}{2 \times x^1}}$
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1}{2 \times x}}$
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1}{2x}}$
Now, evaluate the limit of the rational function by the direct substitution method.
$\,\,=\,\,\,$ $\dfrac{e^0-1}{2(0)}$
According to the zero power rule, the mathematical constant $e$ raised to the power of zero is equal to one.
$\,\,=\,\,\,$ $\dfrac{1-1}{2 \times 0}$
$\,\,=\,\,\,$ $\dfrac{0}{0}$
Use the L’Hôpital’s rule one more time
According to the direct substitution, it is evaluated that the limit of natural exponential function in $x$ minus one divided by two times variable $x$ is indeterminate as the $x$ tends to $0$. So,
the l’hospital’s rule should be used one more time to avoid the indeterminate form.
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}(e^x-1)}{\dfrac{d}{dx}(2x)}}$
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}(e^x-1)}{\dfrac{d}{dx}(2 \times x)}}$
The number $2$ is a factor and it multiplies the variable $x$ in denominator of the rational function. The constant number $2$ can be released from the differentiation by the constant multiple
derivative rule.
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}(e^x-1)}{2 \times \dfrac{d}{dx}(x)}}$
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}(e^x)-\dfrac{d}{dx}(1)}{2 \times \dfrac{d}{dx}(x)}}$
The derivatives of natural exponential function in $x$, one and variable $x$ can be evaluated by the corresponding differentiation formulas.
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-0}{2 \times 1}}$
$\,\,=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x}{2}}$
Evaluate the Limit by Direct substitution
Now, find the limit of the mathematical constant $e$ raised to the power of $x$ divided by two as the value of $x$ approaches $0$, by the direct substitution.
$\,\,=\,\,\,$ $\dfrac{e^0}{2}$
$\,\,=\,\,\,$ $\dfrac{1}{2}$ | {"url":"https://www.mathdoubts.com/evaluate-limit-e-power-x-1-x-divided-by-x-square-as-x-tends-to-0-l-hospitals-rule/","timestamp":"2024-11-02T14:33:53Z","content_type":"text/html","content_length":"32810","record_id":"<urn:uuid:47652930-8422-44c5-87e6-29f56c88e75b>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00311.warc.gz"} |
Arithmetic constraints
class Test::Int::Arithmetic::MultXYZ
Test for multiplication constraint More...
class Test::Int::Arithmetic::MultXXY
Test for multiplication constraint with shared variables More...
class Test::Int::Arithmetic::MultXYX
Test for multiplication constraint with shared variables More...
class Test::Int::Arithmetic::MultXYY
Test for multiplication constraint with shared variables More...
class Test::Int::Arithmetic::MultXXX
Test for multiplication constraint with shared variables More...
class Test::Int::Arithmetic::SqrXY
Test for squaring constraint More...
class Test::Int::Arithmetic::SqrXX
Test for squaring constraint with shared variables More...
class Test::Int::Arithmetic::SqrtXY
Test for square root constraint More...
class Test::Int::Arithmetic::SqrtXX
Test for square root constraint with shared variables More...
class Test::Int::Arithmetic::DivMod
Test for division/modulo constraint More...
class Test::Int::Arithmetic::Div
Test for division constraint More...
class Test::Int::Arithmetic::Mod
Test for modulo constraint More...
class Test::Int::Arithmetic::AbsXY
Test for absolute value constraint More...
class Test::Int::Arithmetic::AbsXX
Test for absolute value constraint with shared variables More...
class Test::Int::Arithmetic::MinXYZ
Test for binary minimum constraint More...
class Test::Int::Arithmetic::MinXXY
Test for binary minimum constraint with shared variables More...
class Test::Int::Arithmetic::MinXYX
Test for binary minimum constraint with shared variables More...
class Test::Int::Arithmetic::MinXYY
Test for binary minimum constraint with shared variables More...
class Test::Int::Arithmetic::MinXXX
Test for binary minimum constraint with shared variables More...
class Test::Int::Arithmetic::MaxXYZ
Test for binary maximum constraint More...
class Test::Int::Arithmetic::MaxXXY
Test for binary maximum constraint with shared variables More...
class Test::Int::Arithmetic::MaxXYX
Test for binary maximum constraint with shared variables More...
class Test::Int::Arithmetic::MaxXYY
Test for binary maximum constraint with shared variables More...
class Test::Int::Arithmetic::MaxXXX
Test for binary maximum constraint with shared variables More...
class Test::Int::Arithmetic::MinNary
Test for n-ary minimmum constraint More...
class Test::Int::Arithmetic::MinNaryShared
Test for n-ary minimmum constraint with shared variables More...
class Test::Int::Arithmetic::MaxNary
Test for n-ary maximum constraint More...
class Test::Int::Arithmetic::MaxNaryShared
Test for n-ary maximum constraint with shared variables More...
const int Test::Int::Arithmetic::va [7]
const int Test::Int::Arithmetic::vb [9]
Gecode::IntSet Test::Int::Arithmetic::a (va, 7)
Gecode::IntSet Test::Int::Arithmetic::b (vb, 9)
Gecode::IntSet Test::Int::Arithmetic::c (-8, 8)
MultXYZ Test::Int::Arithmetic::mult_xyz_b_a ("A", a, Gecode::ICL_BND)
MultXYZ Test::Int::Arithmetic::mult_xyz_b_b ("B", b, Gecode::ICL_BND)
MultXYZ Test::Int::Arithmetic::mult_xyz_b_c ("C", c, Gecode::ICL_BND)
MultXXY Test::Int::Arithmetic::mult_xxy_b_a ("A", a, Gecode::ICL_BND)
MultXXY Test::Int::Arithmetic::mult_xxy_b_b ("B", b, Gecode::ICL_BND)
MultXXY Test::Int::Arithmetic::mult_xxy_b_c ("C", c, Gecode::ICL_BND)
MultXYX Test::Int::Arithmetic::mult_xyx_b_a ("A", a, Gecode::ICL_BND)
MultXYX Test::Int::Arithmetic::mult_xyx_b_b ("B", b, Gecode::ICL_BND)
MultXYX Test::Int::Arithmetic::mult_xyx_b_c ("C", c, Gecode::ICL_BND)
MultXYY Test::Int::Arithmetic::mult_xyy_b_a ("A", a, Gecode::ICL_BND)
MultXYY Test::Int::Arithmetic::mult_xyy_b_b ("B", b, Gecode::ICL_BND)
MultXYY Test::Int::Arithmetic::mult_xyy_b_c ("C", c, Gecode::ICL_BND)
MultXXX Test::Int::Arithmetic::mult_xxx_b_a ("A", a, Gecode::ICL_BND)
MultXXX Test::Int::Arithmetic::mult_xxx_b_b ("B", b, Gecode::ICL_BND)
MultXXX Test::Int::Arithmetic::mult_xxx_b_c ("C", c, Gecode::ICL_BND)
MultXYZ Test::Int::Arithmetic::mult_xyz_d_a ("A", a, Gecode::ICL_DOM)
MultXYZ Test::Int::Arithmetic::mult_xyz_d_b ("B", b, Gecode::ICL_DOM)
MultXYZ Test::Int::Arithmetic::mult_xyz_d_c ("C", c, Gecode::ICL_DOM)
MultXXY Test::Int::Arithmetic::mult_xxy_d_a ("A", a, Gecode::ICL_DOM)
MultXXY Test::Int::Arithmetic::mult_xxy_d_b ("B", b, Gecode::ICL_DOM)
MultXXY Test::Int::Arithmetic::mult_xxy_d_c ("C", c, Gecode::ICL_DOM)
MultXYX Test::Int::Arithmetic::mult_xyx_d_a ("A", a, Gecode::ICL_DOM)
MultXYX Test::Int::Arithmetic::mult_xyx_d_b ("B", b, Gecode::ICL_DOM)
MultXYX Test::Int::Arithmetic::mult_xyx_d_c ("C", c, Gecode::ICL_DOM)
MultXYY Test::Int::Arithmetic::mult_xyy_d_a ("A", a, Gecode::ICL_DOM)
MultXYY Test::Int::Arithmetic::mult_xyy_d_b ("B", b, Gecode::ICL_DOM)
MultXYY Test::Int::Arithmetic::mult_xyy_d_c ("C", c, Gecode::ICL_DOM)
MultXXX Test::Int::Arithmetic::mult_xxx_d_a ("A", a, Gecode::ICL_DOM)
MultXXX Test::Int::Arithmetic::mult_xxx_d_b ("B", b, Gecode::ICL_DOM)
MultXXX Test::Int::Arithmetic::mult_xxx_d_c ("C", c, Gecode::ICL_DOM)
SqrXY Test::Int::Arithmetic::sqr_xy_b_a ("A", a, Gecode::ICL_BND)
SqrXY Test::Int::Arithmetic::sqr_xy_b_b ("B", b, Gecode::ICL_BND)
SqrXY Test::Int::Arithmetic::sqr_xy_b_c ("C", c, Gecode::ICL_BND)
SqrXY Test::Int::Arithmetic::sqr_xy_d_a ("A", a, Gecode::ICL_DOM)
SqrXY Test::Int::Arithmetic::sqr_xy_d_b ("B", b, Gecode::ICL_DOM)
SqrXY Test::Int::Arithmetic::sqr_xy_d_c ("C", c, Gecode::ICL_DOM)
SqrXX Test::Int::Arithmetic::sqr_xx_b_a ("A", a, Gecode::ICL_BND)
SqrXX Test::Int::Arithmetic::sqr_xx_b_b ("B", b, Gecode::ICL_BND)
SqrXX Test::Int::Arithmetic::sqr_xx_b_c ("C", c, Gecode::ICL_BND)
SqrXX Test::Int::Arithmetic::sqr_xx_d_a ("A", a, Gecode::ICL_DOM)
SqrXX Test::Int::Arithmetic::sqr_xx_d_b ("B", b, Gecode::ICL_DOM)
SqrXX Test::Int::Arithmetic::sqr_xx_d_c ("C", c, Gecode::ICL_DOM)
SqrtXY Test::Int::Arithmetic::sqrt_xy_b_a ("A", a, Gecode::ICL_BND)
SqrtXY Test::Int::Arithmetic::sqrt_xy_b_b ("B", b, Gecode::ICL_BND)
SqrtXY Test::Int::Arithmetic::sqrt_xy_b_c ("C", c, Gecode::ICL_BND)
SqrtXY Test::Int::Arithmetic::sqrt_xy_d_a ("A", a, Gecode::ICL_DOM)
SqrtXY Test::Int::Arithmetic::sqrt_xy_d_b ("B", b, Gecode::ICL_DOM)
SqrtXY Test::Int::Arithmetic::sqrt_xy_d_c ("C", c, Gecode::ICL_DOM)
SqrtXX Test::Int::Arithmetic::sqrt_xx_b_a ("A", a, Gecode::ICL_BND)
SqrtXX Test::Int::Arithmetic::sqrt_xx_b_b ("B", b, Gecode::ICL_BND)
SqrtXX Test::Int::Arithmetic::sqrt_xx_b_c ("C", c, Gecode::ICL_BND)
SqrtXX Test::Int::Arithmetic::sqrt_xx_d_a ("A", a, Gecode::ICL_DOM)
SqrtXX Test::Int::Arithmetic::sqrt_xx_d_b ("B", b, Gecode::ICL_DOM)
SqrtXX Test::Int::Arithmetic::sqrt_xx_d_c ("C", c, Gecode::ICL_DOM)
DivMod Test::Int::Arithmetic::divmod_a_bnd ("A", a)
DivMod Test::Int::Arithmetic::divmod_b_bnd ("B", b)
DivMod Test::Int::Arithmetic::divmod_c_bnd ("C", c)
Div Test::Int::Arithmetic::div_a_bnd ("A", a)
Div Test::Int::Arithmetic::div_b_bnd ("B", b)
Div Test::Int::Arithmetic::div_c_bnd ("C", c)
Mod Test::Int::Arithmetic::mod_a_bnd ("A", a)
Mod Test::Int::Arithmetic::mod_b_bnd ("B", b)
Mod Test::Int::Arithmetic::mod_c_bnd ("C", c)
AbsXY Test::Int::Arithmetic::abs_xy_b_a ("A", a, Gecode::ICL_BND)
AbsXY Test::Int::Arithmetic::abs_xy_b_b ("B", b, Gecode::ICL_BND)
AbsXY Test::Int::Arithmetic::abs_xy_b_c ("C", c, Gecode::ICL_BND)
AbsXY Test::Int::Arithmetic::abs_xy_d_a ("A", a, Gecode::ICL_DOM)
AbsXY Test::Int::Arithmetic::abs_xy_d_b ("B", b, Gecode::ICL_DOM)
AbsXY Test::Int::Arithmetic::abs_xy_d_c ("C", c, Gecode::ICL_DOM)
AbsXX Test::Int::Arithmetic::abs_xx_b_a ("A", a, Gecode::ICL_BND)
AbsXX Test::Int::Arithmetic::abs_xx_b_b ("B", b, Gecode::ICL_BND)
AbsXX Test::Int::Arithmetic::abs_xx_b_c ("C", c, Gecode::ICL_BND)
AbsXX Test::Int::Arithmetic::abs_xx_d_a ("A", a, Gecode::ICL_DOM)
AbsXX Test::Int::Arithmetic::abs_xx_d_b ("B", b, Gecode::ICL_DOM)
AbsXX Test::Int::Arithmetic::abs_xx_d_c ("C", c, Gecode::ICL_DOM)
MinXYZ Test::Int::Arithmetic::min_xyz_b_a ("A", a, Gecode::ICL_BND)
MinXYZ Test::Int::Arithmetic::min_xyz_b_b ("B", b, Gecode::ICL_BND)
MinXYZ Test::Int::Arithmetic::min_xyz_b_c ("C", c, Gecode::ICL_BND)
MinXYZ Test::Int::Arithmetic::min_xyz_d_a ("A", a, Gecode::ICL_DOM)
MinXYZ Test::Int::Arithmetic::min_xyz_d_b ("B", b, Gecode::ICL_DOM)
MinXYZ Test::Int::Arithmetic::min_xyz_d_c ("C", c, Gecode::ICL_DOM)
MinXXY Test::Int::Arithmetic::min_xxy_b_a ("A", a, Gecode::ICL_BND)
MinXXY Test::Int::Arithmetic::min_xxy_b_b ("B", b, Gecode::ICL_BND)
MinXXY Test::Int::Arithmetic::min_xxy_b_c ("C", c, Gecode::ICL_BND)
MinXXY Test::Int::Arithmetic::min_xxy_d_a ("A", a, Gecode::ICL_DOM)
MinXXY Test::Int::Arithmetic::min_xxy_d_b ("B", b, Gecode::ICL_DOM)
MinXXY Test::Int::Arithmetic::min_xxy_d_c ("C", c, Gecode::ICL_DOM)
MinXYX Test::Int::Arithmetic::min_xyx_b_a ("A", a, Gecode::ICL_BND)
MinXYX Test::Int::Arithmetic::min_xyx_b_b ("B", b, Gecode::ICL_BND)
MinXYX Test::Int::Arithmetic::min_xyx_b_c ("C", c, Gecode::ICL_BND)
MinXYX Test::Int::Arithmetic::min_xyx_d_a ("A", a, Gecode::ICL_DOM)
MinXYX Test::Int::Arithmetic::min_xyx_d_b ("B", b, Gecode::ICL_DOM)
MinXYX Test::Int::Arithmetic::min_xyx_d_c ("C", c, Gecode::ICL_DOM)
MinXYY Test::Int::Arithmetic::min_xyy_b_a ("A", a, Gecode::ICL_BND)
MinXYY Test::Int::Arithmetic::min_xyy_b_b ("B", b, Gecode::ICL_BND)
MinXYY Test::Int::Arithmetic::min_xyy_b_c ("C", c, Gecode::ICL_BND)
MinXYY Test::Int::Arithmetic::min_xyy_d_a ("A", a, Gecode::ICL_DOM)
MinXYY Test::Int::Arithmetic::min_xyy_d_b ("B", b, Gecode::ICL_DOM)
MinXYY Test::Int::Arithmetic::min_xyy_d_c ("C", c, Gecode::ICL_DOM)
MinXXX Test::Int::Arithmetic::min_xxx_b_a ("A", a, Gecode::ICL_BND)
MinXXX Test::Int::Arithmetic::min_xxx_b_b ("B", b, Gecode::ICL_BND)
MinXXX Test::Int::Arithmetic::min_xxx_b_c ("C", c, Gecode::ICL_BND)
MinXXX Test::Int::Arithmetic::min_xxx_d_a ("A", a, Gecode::ICL_DOM)
MinXXX Test::Int::Arithmetic::min_xxx_d_b ("B", b, Gecode::ICL_DOM)
MinXXX Test::Int::Arithmetic::min_xxx_d_c ("C", c, Gecode::ICL_DOM)
MaxXYZ Test::Int::Arithmetic::max_xyz_b_a ("A", a, Gecode::ICL_BND)
MaxXYZ Test::Int::Arithmetic::max_xyz_b_b ("B", b, Gecode::ICL_BND)
MaxXYZ Test::Int::Arithmetic::max_xyz_b_c ("C", c, Gecode::ICL_BND)
MaxXYZ Test::Int::Arithmetic::max_xyz_d_a ("A", a, Gecode::ICL_DOM)
MaxXYZ Test::Int::Arithmetic::max_xyz_d_b ("B", b, Gecode::ICL_DOM)
MaxXYZ Test::Int::Arithmetic::max_xyz_d_c ("C", c, Gecode::ICL_DOM)
MaxXXY Test::Int::Arithmetic::max_xxy_b_a ("A", a, Gecode::ICL_BND)
MaxXXY Test::Int::Arithmetic::max_xxy_b_b ("B", b, Gecode::ICL_BND)
MaxXXY Test::Int::Arithmetic::max_xxy_b_c ("C", c, Gecode::ICL_BND)
MaxXXY Test::Int::Arithmetic::max_xxy_d_a ("A", a, Gecode::ICL_DOM)
MaxXXY Test::Int::Arithmetic::max_xxy_d_b ("B", b, Gecode::ICL_DOM)
MaxXXY Test::Int::Arithmetic::max_xxy_d_c ("C", c, Gecode::ICL_DOM)
MaxXYX Test::Int::Arithmetic::max_xyx_b_a ("A", a, Gecode::ICL_BND)
MaxXYX Test::Int::Arithmetic::max_xyx_b_b ("B", b, Gecode::ICL_BND)
MaxXYX Test::Int::Arithmetic::max_xyx_b_c ("C", c, Gecode::ICL_BND)
MaxXYX Test::Int::Arithmetic::max_xyx_d_a ("A", a, Gecode::ICL_DOM)
MaxXYX Test::Int::Arithmetic::max_xyx_d_b ("B", b, Gecode::ICL_DOM)
MaxXYX Test::Int::Arithmetic::max_xyx_d_c ("C", c, Gecode::ICL_DOM)
MaxXYY Test::Int::Arithmetic::max_xyy_b_a ("A", a, Gecode::ICL_BND)
MaxXYY Test::Int::Arithmetic::max_xyy_b_b ("B", b, Gecode::ICL_BND)
MaxXYY Test::Int::Arithmetic::max_xyy_b_c ("C", c, Gecode::ICL_BND)
MaxXYY Test::Int::Arithmetic::max_xyy_d_a ("A", a, Gecode::ICL_DOM)
MaxXYY Test::Int::Arithmetic::max_xyy_d_b ("B", b, Gecode::ICL_DOM)
MaxXYY Test::Int::Arithmetic::max_xyy_d_c ("C", c, Gecode::ICL_DOM)
MaxXXX Test::Int::Arithmetic::max_xxx_b_a ("A", a, Gecode::ICL_BND)
MaxXXX Test::Int::Arithmetic::max_xxx_b_b ("B", b, Gecode::ICL_BND)
MaxXXX Test::Int::Arithmetic::max_xxx_b_c ("C", c, Gecode::ICL_BND)
MaxXXX Test::Int::Arithmetic::max_xxx_d_a ("A", a, Gecode::ICL_DOM)
MaxXXX Test::Int::Arithmetic::max_xxx_d_b ("B", b, Gecode::ICL_DOM)
MaxXXX Test::Int::Arithmetic::max_xxx_d_c ("C", c, Gecode::ICL_DOM)
MinNary Test::Int::Arithmetic::min_nary_b (Gecode::ICL_BND)
MinNary Test::Int::Arithmetic::min_nary_d (Gecode::ICL_DOM)
MinNaryShared Test::Int::Arithmetic::min_s_nary_b (Gecode::ICL_BND)
MinNaryShared Test::Int::Arithmetic::min_s_nary_d (Gecode::ICL_DOM)
MaxNary Test::Int::Arithmetic::max_nary_b (Gecode::ICL_BND)
MaxNary Test::Int::Arithmetic::max_nary_d (Gecode::ICL_DOM)
MaxNaryShared Test::Int::Arithmetic::max_s_nary_b (Gecode::ICL_BND)
MaxNaryShared Test::Int::Arithmetic::max_s_nary_d (Gecode::ICL_DOM)
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Manufacturing-Cost-Driven Topology Optimization of Welded Frame Structures
This work presents a method for the topology optimization of welded frame structures to minimize the manufacturing cost. The structures considered here consist of assemblies of geometric primitives
such as bars and plates that are common in welded frame construction. A geometry projection technique is used to map the primitives onto a continuous density field that is subsequently used to
interpolate material properties. As in density-based topology optimization techniques, the ensuing ersatz material is used to perform the structural analysis on a fixed mesh, thereby circumventing
the need for re-meshing upon design changes. The distinct advantage of the representation by geometric primitives is the ease of computation of the manufacturing cost in terms of the design
parameters, while the geometry projection facilitates the analysis within a continuous design region. The proposed method is demonstrated via the manufacturing-cost-minimization subject to a
displacement constraint of 2D bar, 3D bar, and plate structures.
1 Introduction
Manufacturing cost is seldom considered in the topology optimization of structures, which is often based solely on structural criteria and weight. Consequently, the optimal design may exhibit
superior mechanical performance but be costly to manufacture. While the amount of material in a structure may contribute significantly to its cost, there are also other costs associated with the
fabrication process. In the case of the welded frames considered in this work, the manufacturing cost is also influenced by the cost of cutting, welding, and painting frame members.
The main obstacle to incorporating manufacturing cost in topology optimization lies in the difficulty to express it in terms of the design parameters. Conventional density-based and level-set
topology optimization techniques employ a field representation of the structure, which endows the optimizer with substantial design freedom but makes it difficult to compute manufacturing cost.
Ground structure methods have been used to minimize the manufacturing cost of truss structures [1]. In these methods, which use 1D-elements to represent the structure and for analysis, it is easy to
compute costs associated with, for example, the length of the truss elements and the number of struts. However, other cost components, such as the cost of welding, cannot be readily computed because
they require a calculation of the welding length, which is difficult to compute from the 1D-representation. Moreover, the structure must be a topological subset of the initial design (the ground
structure), thus ground structure methods cannot model arbitrary topologies. Also, a ground structure approach does not accommodate primitives like plates.
Some density-based topology optimization techniques for continua consider the manufacturing cost of additively manufactured single components [2–5] or assemblies of 2-dimensional stamped and
spot-welded components [6,7]. For these manufacturing processes, it is possible to express the manufacturing cost in terms of quantities that can be computed from the field representation, such as
surface area, bounding box, and the volume of the support material. Some topology optimization techniques (e.g., Ref. [8]) design multi-component structures in which the boundaries among components
in the optimal design are intended to be joined via continuous welds. However, in these techniques, there is no computation of the weld length between components and no consideration for the welding
It should be noted that some methods have incorporated various mechanisms to limit the number of structural members or holes in the optimization, which is an indirect way of controlling manufacturing
cost. These works include the ground structure approach of Ref. [9] and the moving morphable components method of Ref. [10], which impose a constraint on the maximum number of members in the
structure; and the work of Ref. [11], where a limit is imposed on the maximum number of holes in density-based topology optimization.
In this paper, we propose a method for the topology optimization of welded frame structures with regard to manufacturing cost. The frame is modeled as the union of bar primitives or plate primitives.
The analysis is performed on a fixed finite element mesh as in density-based methods. To employ the primitive-based representation of the frame while performing the analysis on a fixed mesh, we
employ the geometry projection (GP) method [12], whereby the geometric parameters of the primitives are smoothly mapped onto a density field that is subsequently used to interpolate material
properties, just as in density-based methods. The GP mapping is differentiable, hence we can use efficient gradient-based methods for the optimization.
The proposed approach has several advantages. Each term of the manufacturing-cost function can be computed directly in terms of the geometric parameters of the primitives, the projected density
field, or both. For example, the geometry representation allows us to determine the weld location among components and approximate the weld length. Re-meshing is circumvented as in conventional
topology optimization techniques. Compared to ground structure methods that use 1D-elements for the analysis, the bars need not be connected during the optimization and the optimal design is not a
subset of the ground structure, leading to increased design freedom and thus efficient structures that can be generated with only a few bars (the trade-off, however, is that the analysis in the GP
method discretizes a continuum with 2D- and 3D-elements and thus is more computationally expensive). Moreover, the GP method captures intersections between 3D-primitives in a way that is not possible
using 1D-elements. For instance, in methods that use truss elements to model cylindrical bars, two elements that come very close to each other may not intersect even if the corresponding cylindrical
bars in 3D do intersect because of their dimensions. Finally, although stress constraints are out of the scope of the current work, we note that the continuum 2D- and 3D-representation of the
structure can capture stress concentrations arising from the intersection of primitives, which cannot be captured by 1D-elements.
The remainder of the paper is organized as follows: in Sec. 2, we briefly introduce the GP method. Section 3 formulates the manufacturing-cost function based on the GP method for both bar and plate
primitives. The optimization problem is stated in Sec. 4. A brief description of design sensitivities is given in Sec. 5. In Sec. 6, we demonstrate the proposed method in the design of 2D and 3D
cantilever beams with bar primitives and of a Messerschmitt–Bölkow–Blohm (MBB) beam in 3D with plate primitives. Finally, we draw conclusions in Sec. 7.
2 Geometry Projection
The proposed technique employs the GP method to map the geometric primitives that make up the structure onto a fixed finite element mesh for analysis, thus avoiding re-meshing upon design changes [12
,13]. The GP method consists of a differentiable map between the high-level parameters that describe the geometric primitives and a density field defined over the design region, which is subsequently
discretized via a finite element mesh for analysis. The differentiability of the map ensures that efficient gradient-based techniques can be used for the optimization.
In this work, the projected density at point
corresponding to component
is computed as
is a
regularized Heaviside function,
) is the signed distance from
to the boundary of
is the vector of design parameters defining component
, and
is the radius of the transition region of the Heaviside. In all the examples shown in Sec.
is equal to the diagonal of the element, i.e., twice the size of the radius of the ball that circumscribes the element. We note that, as demonstrated in Refs. [
must be strictly smaller than the radius of the primitive to ensure well-defined sensitivities. Correspondingly, the element size must be at most half of the primitive radius.
The formulation of the signed distance
depends on the specific representation of the geometric components. For the frame structures considered in this paper, we consider bar and plate primitives represented by offset solids. In the case
of bars, the solid corresponds to all points within a distance
of a line segment (the bar’s medial axis), and the corresponding vector of design parameters is
= {
}, where
denote the endpoints of the medial axis and
is a membership variable that will be introduced later in this section. In the case of plates, we use quaternions to represent the plate orientation to avoid issues like gimbal lock and 2
-periodicity that arise when using Euler angles [
]. The vector of quaternion components describing the orientation in space of the plate is denoted as
. The rotation matrix of plate
is given as
are the basis vectors of plate
in the local coordinate system. Therefore, the vector of design parameters of the plate primitive is
, where
is the location of the center of plate,
are the dimensions of the rectangular medial surface, and
is the semi-thickness of the plate. The design parameters are illustrated in Fig.
The convenience of offset solids is that the signed distance from any point x to the boundary of the primitive can be simply computed as ϕ[c](x) = r[c] − d[c](x), where d[c](x) is the distance from x
to the medial axis/surface. Moreover, d[c] can be computed in closed form as a function of z[c]. For details on the computation of the signed distance for offset bar and plate primitives and the
corresponding design sensitivities, the reader is referred to Refs. [14–16]. It should be noted, however, that the geometry projection is not limited to offset solids and can be used with any
geometric representation, provided a signed distance to the boundary of the primitive can be computed.
In addition to the geometric parameters that determine the dimensions, position, and orientation of the component, we also ascribe a continuous membership variable
to each primitive. When
= 1, it signifies the component is part of the design, and an interior point of the component will have the elastic properties of the material that the component is made of. Conversely, when
= 0, the component is removed from the structure, and the elastic properties at an interior point of the component will not be influenced by the component, regardless of the component’s dimensions.
This membership variable enables the optimizer to remove geometric components by penalization. The
penalized density$ρ^c(zc,x)$
that incorporates with the membership variable
is given by
is a penalization function that renders intermediate values of
structurally inefficient. This penalization ensures that the optimal design has bars with 0/1 membership variables and is essentially the same penalization used by density-based topology optimization
technique. For instance, it may correspond to the power law of the solid isotropic material with penalization interpolation scheme (cf. Ref. [
]) given by
) =
, with
≥ 3.
We combine multiple primitives via their Boolean union. Since the projected and penalized densities are ultimately implicit representations of the primitives, the Boolean union corresponds to their
maximum (cf. Ref. [
]). The maximum function, however, is not differentiable, which precludes the use of efficient gradient-based optimizers. To circumvent this, we use a softmax differentiable approximation in which
the interpolated material properties are given by [
is a
combined density
that corresponds to the Boolean union of all components, and the weights
are given by
In these expressions,
denotes the elasticity tensor of solid material that each component is made of,
is the vector of penalized densities for all the components, and
is the elasticity tensor of a relatively weak material to prevent an ill-posed analysis.
is the number of components.
denotes the vector of design parameters for all components. As the parameter
→ ∞, the weights
in the softargmax function approach a one-hot vector, i.e.,
→ 1 for the component with the largest penalized density and
→ 0 for all other components. In the finite element analysis, we assume for simplicity an element-uniform combined density
, which is computed at the element centroid; consequently the ersatz elasticity tensor of
is also element-uniform.
While the softmax material interpolation of (5) admits a different elasticity tensor for each component, which accommodates multi-material structures and anisotropic materials (cf. Ref. [16]), in
this work we consider that all components are made of a single material with elasticity tensor $Csolid$.
3 Manufacturing-Cost Function
We formulate the manufacturing-cost function based on the fabrication cost presented in Ref. [
]. The manufacturing-cost function consists of two parts: material cost
and fabrication cost
. We take four stages of fabrication into consideration, namely preparation, cutting, welding, and painting. The relative weights of the terms making up this function correspond to monetary cost per
unit (e.g., length, area) for each stage. Different supply chains would have different values of these weights and therefore may lead to different designs. The manufacturing cost of the structure is
defined as
∈ {1, 2, 3, 4}, is the cost for each manufacturing stage. It is important to note that
is an estimate, since an accurate assessment of the manufacturing cost of a structure can only be made once a detailed design of the structure and detailed knowledge of the cost structure for the
material and the manufacturing process are available. Therefore, it is essential to emphasize that the place of the proposed technique in the structure’s design workflow is in the conceptual design
stage, and that its aim is to produce designs with significantly improved manufacturing cost as compared to that of an optimal structure driven purely by mechanical criteria.
In order to tie the manufacturing cost to the geometric description of the primitives, each of the terms in (8) must be expressed as a function of the geometric parameters of the components. The
remainder of this section details these terms. Some of them are computed directly in terms of the geometric parameters, while other terms are more easily computed in terms of the projected densities.
In this sense, the manufacturing-cost function introduced here takes advantage of the dual representation (geometric parameters/densities). It is worth noting that even for terms that are computed in
terms of projected densities, their calculation is possible because there is a projected density field ρ[c] associated with each component c and ρ[c] serves as a (fuzzy) point classification (i.e.,
an inside/outside test), which makes it possible to compute, for example, the weld length of Sec. 3.4.
3.1 Material Cost.
The material cost
is simply computed as the weight of the material cost
times the mass of the whole structure ϱ
depends on the type of the material and the detailed supply chain, ϱ is the material density (which, in this work, we assume to be 1.0 for all the examples), and
is the volume of the structure.
can be computed as the sum of the volumes of all the elements in the mesh:
is the number of elements in the mesh,
is the volume of a single element
, and
is the projected density of element
3.2 Preparation Cost.
Before components are fully welded, they usually need to be prepared, assembled, and pre-positioned by tack welding. The cost of preparation stage mainly depends on the number of the components and
the weight of each component. According to Ref. [
], the cost for preparation, assembly, and tacking is defined as
is the corresponding weight of
is the number of existing components in the structure. In the GP method,
can be expressed as the sum of the membership variable
of all the components:
It should be noted that κ ≤ N[c]. While N[c] corresponds to all the components available to design the structure, κ represents the components actually used for the design.
3.3 Cutting Cost.
Cutting and edge grinding can be made with different technologies, such as flame cutting, plasma cutting, and laser cutting. The corresponding cost
is simply proportional to the cutting area of the component:
where the weight
of the cutting cost depends on the detailed manufacturing condition, and
is the total cross-sectional area to be cut for all the components. By using the GP method, the area of cutting for different components is computed as
$Acut={αc(2rctc)for 2D bar primitivesαc(πrc2)for 3D bar primitivesαc(2rc(lc1+lc2))for 3D plate primitives$
is the out-of-plane thickness of bar
. These expressions assume each bar component only needs to be cut at one end, while plate components are cut at all edges. The presence of the membership variable
ensures no cutting area is computed for components that have been removed from the design. Also, we note that this is an estimation, as an accurate calculation of cutting cost requires a detailed
3.4 Welding Cost.
The cost of welding
incorporates not only the welding process itself but also all the additional fabrication steps associated with welding. These additional steps include changing the electrode, deslagging, and
depends on the welding length
of the whole structure:
where the weight
of the welding cost depends on, e.g., the technology of welding. The length of welding can be computed as follows: the norm of the gradient of the projected density
is nonzero for a point
on the boundary of the component and zero elsewhere. For two components
, the quantity
$lij(x)={‖∇ρi(x)‖ρj(x)in 2D‖∇ρi(x)‖‖∇ρj(x)‖in 3D$
is 1 if
belongs to the weld between the two components and 0 otherwise. The total weld length is subsequently given by
where Ω denotes the design region. The condition
for the innermost sum of
ensures that the intersection between two components is counted only once, and it also prevents the situation in which
, for which
would render the perimeter in 2D and the surface area in 3D for component
. One could alternatively use the condition
to achieve the same purpose. Since we assume uniform projected densities and gradients within each element, the integral in
is computed as a sum over the elements (with the integrand multiplied by the element volume). Figure
shows an example of bar intersections plotting the integrand of
We note that from
and using the chain rule, we have that
denotes the first derivative of
. Since the distance function satisfies the eikonal equation
, the norm of the gradient of the projected density for component
can simply be computed as
which can be readily obtained from
Note that even though the weld length is computed in terms of the projected densities and not directly in terms of the geometric parameters of the components, this calculation is only possible
because we have distinct structural members and can compute ‖ρ[c]‖ and α[c] for each component.
3.5 Painting Cost.
The fourth term of the fabrication cost in
includes the cost of painting and surface preparation. The surface preparation includes cleaning (e.g., sand-spray), grinding, and application of a top coat. This cost of painting
is proportional to the surface area of the whole structure
with the proportionality constant weight
. For frames made of 2D bars, the surface area is computed as
For structures consisting of 3D bars, the surface area is given by
and for plates it is given by
4 Optimization Problem
We consider minimization of the manufacturing cost subject to a constraint that the displacement at a point
does not exceed a specified value:
The constraint of
represents the discretized finite element equations for the linear elasticity problem, with
, and
the global stiffness matrix, displacement vector, and force vector, respectively. We assume that the applied force is design-independent.
In the displacement constraint of
= (
is the magnitude of the displacement at a specified point
, with
a square matrix such that
= 1 for the degrees-of-freedom
corresponding to point
, and
= 0 for every other component. For all the examples, the function
provides scaling to aid convergence. Since the displacement magnitude can attain extremely large values for disconnected design during the optimization, the following log-scale version of the
constraint ensures that values in consecutive iterations do not change drastically:
is the maximum allowable displacement at
. For a positive displacement in double precision, we can assume that
∈ [0, 10
]. Therefore, when
= 0,
) = −1; when
) = 0; when
= 10
. Thus, the displacement at a point constraint is bounded as
, which is in the range of values recommended for the method of moving asymptotes (MMA). For the same reason, we linearly scale down the value of the manufacturing cost by 100 times. For all the
examples in this paper,
is the point at which the load is located.
We impose lower and upper bounds,
, on the individual design variables
, with
the number of design variables per component (six for 2D bars, eight for 3D bars, and 11 for 3D plates). The endpoints
of the medial axes of bars and the center points
of plates are bound to lie inside the design region. The membership variables are bound as
∈ [0, 1], and the quaternion components as
∈ [−1, 1]. The dimensions of the components are given different bounds for the examples in Sec.
. As in prior implementations of the GP method (cf. Ref. [
]), all design variables are scaled as
and a move limit
is imposed at each iteration
on the scaled variables to improve convergence:
5 Sensitivity Analysis
To employ efficient gradient-based optimizers, we compute design sensitivities of the manufacturing-cost function and the displacement constraint. For a given function
, its sensitivities with respect to a design variable
are computed as
is the combined density for element
. For functions that depend on the solution
to the equilibrium equation of
is computed using adjoint differentiation and is similar to the sensitivity computed in density-based topology optimization. For functions that do not depend on the analysis,
can be computed directly, as is the case for all the terms of the manufacturing-cost function. The term
is obtained from the geometry projection; its derivation is here omitted for brevity, and the reader is referred to Refs. [
] for details. However, it is worth noting that the sensitivity of the norm of the projected density gradient is computed from
denotes the second derivative of
which is readily obtained from
. The term ∂
can be found in the aforementioned references. The derivatives of all the terms in the manufacturing-cost function can be obtained from their respective expressions.
6 Numerical Examples
In this section, we present three examples to demonstrate the proposed method. The computer implementation of the method is done in matlab. The finite element analysis is performed using a regular
mesh of bilinear quadrilateral elements. The linear system of equations arising from the finite element discretization is solved using the conjugate gradient method with a multigrid preconditioner. A
modulus of E[min] = 10^−6 is used for the weak material with elasticity tensor $Cvoid$ of (5). The solid material in all examples (i.e., $Csolid$ in (5)) has a Young’s modulus of 1 and a Poisson’s
ratio of 0.3
Each design update in the optimization is obtained as follows. An arbitrary initial design composed of a finite number of primitives is specified. For this initial design or any given design Z, we
compute the projected densities of (1) for each component at the centroid of each element in the mesh. The combined density of (6) and the ersatz elasticity tensor of (5) are subsequently computed
for each element. This is followed by the usual finite element assembly and solution, where each element stiffness matrix is computed using the corresponding ersatz material. The solution of the
finite element analysis is used to compute the relevant objective and constraint functions in the optimization. Where applicable (e.g., when a displacement constraint is imposed), adjoint analysis is
also performed to compute design sensitivities. The objective and constraints, together with their sensitivities, are passed to a gradient-based optimization algorithm, which produces a new design.
The foregoing process is repeated until a stopping criterion is satisfied. For simplicity, we stop the optimization when a maximum number of iterations has been completed. The optimization problem is
solved using MMA with the default parameters suggested in Ref. [20].
To demonstrate the effectiveness of the proposed approach, the examples considered in this work compare the manufacturing-cost-minimization problem of
to a compliance-minimization problem subject to a volume constraint. The latter problem can be stated as
$minZC(Z):=u(Z)TfSubject to:v(Z)≤v¯K(Z)u(Z)=fzi_≤zi≤zi¯,i=1,…,NdNc$
is the compliance of the structure,
is the volume fraction, and
is the maximum allowable volume fraction.
6.1 Two-Dimensional Cantilever Beam With Bar Primitives.
In this first example, we design a cantilever beam with 2D bar primitives. The initial design, loading and boundary conditions are depicted in the top portion of Fig. 3. The dimensions of the design
region are 60 × 10. The mesh size is 360 × 60. The left-hand edge of the design domain is fixed, and the load F = 0.1 is applied at the middle point of the right-hand edge. The initial design
consists of 42 bars with radius and membership variables of 0.5. The bottom portion of Fig. 3 shows the weld locations (i.e., the integrand of α[i]α[j]l[ij] in (17)) in the initial design. We bound
the radius of each bar to [0.25, 2], and prescribe out-of-plane thickness of the 2D bar as t[c] = 1. The move limit is m = 0.025.
To obtain the maximum allowable displacement $u¯$ for the displacement constraint in the manufacturing-cost-minimization problem of (23), we first perform the compliance minimization problem of (32)
with a maximum material volume fraction $v¯=0.3$. The final design and its weld locations are shown in Fig. 4. We deem this design as a reference for comparison, and we choose the point where the
load is applied as point p for the displacement constraint in the manufacturing-cost-minimization problems. Therefore, we subsequently assign $u¯=uref$, where u[ref] denotes the displacement at
point p in the reference minimum-compliance design.
In practice, the weights for the terms in the manufacturing-cost function of (8) depend on many factors, including the size of the structure, the fabrication capabilities of the manufacturer, and the
number of structures to be fabricated. It also depends on the type of structure; for instance, the cutting and preparation for a tubular frame for a racing car, which requires careful coping of the
tubes, may be more expensive than the welding; on the other hand, the opposite is generally the case for a truss made of structural profiles joined to plate gussets. Here, we assign weights that are
realistic for the manufacture of a steel frame truss whose dimensions are in inches. We assign w[m] = 1.3333 $/in^3, w[1] = 0.05 $/in^3/2, w[2] = 0.344 $/in^2, w[3] = 0.2143 $/in, and w[4] = 0.0104 $
/in^2. These values are estimated based on some typical costs for preparation, cutting, welding, and painting in the United States. The manufacturing cost is minimized with these weights, and the
resulting optimized design is shown in Fig. 5. We refer to this design as the baseline design.
The cost corresponding to each term in the manufacturing-cost function and other relevant measures for the reference and baseline designs are listed in Table 1. The comparison between the reference
(minimum-compliance) and baseline (minimum-manufacturing-cost) designs clearly demonstrates that the manufacturing-cost function can effectively reduce every cost term compared to the reference
design. The manufacturing cost of the baseline design is approximately $59%$ of that of the reference design, with a similar compliance C. We also observe that the manufacturing-cost-driven design
employs significantly fewer bars (as indicated by κ in Table 1) and it substantially reduces the welding length l[w] and surface area A[s] compared to the reference design.
Table 1
Design M C[m] C[1] C[2] C[3] C[4] C g κ A[cut] l[w] A[s]
Reference 468.73 239.99 3.74 9.51 189.72 25.76 17.02 170.21 31.23 27.65 885.30 2476.97
Baseline 278.28 253.71 3.00 6.88 5.63 9.04 17.10 171.06 18.96 20.00 26.29 869.73
$wm=5wm0$ 289.09 255.31 3.16 7.07 13.06 10.46 17.10 171.00 20.98 20.56 60.98 1006.42
$w1=5w10$ 282.96 253.46 3.28 6.73 9.56 9.91 17.10 171.04 22.66 19.58 44.63 953.62
$w2=5w20$ 274.30 249.46 3.00 6.02 6.72 9.07 17.10 171.01 19.35 17.51 31.39 873.06
$w3=5w30$ 303.57 275.77 3.07 8.17 6.68 9.86 17.10 171.04 18.27 23.75 31.21 948.19
$w4=5w40$ 304.29 276.39 3.50 9.10 6.19 9.10 17.10 171.00 23.64 26.46 28.91 875.29
Design M C[m] C[1] C[2] C[3] C[4] C g κ A[cut] l[w] A[s]
Reference 468.73 239.99 3.74 9.51 189.72 25.76 17.02 170.21 31.23 27.65 885.30 2476.97
Baseline 278.28 253.71 3.00 6.88 5.63 9.04 17.10 171.06 18.96 20.00 26.29 869.73
$wm=5wm0$ 289.09 255.31 3.16 7.07 13.06 10.46 17.10 171.00 20.98 20.56 60.98 1006.42
$w1=5w10$ 282.96 253.46 3.28 6.73 9.56 9.91 17.10 171.04 22.66 19.58 44.63 953.62
$w2=5w20$ 274.30 249.46 3.00 6.02 6.72 9.07 17.10 171.01 19.35 17.51 31.39 873.06
$w3=5w30$ 303.57 275.77 3.07 8.17 6.68 9.86 17.10 171.04 18.27 23.75 31.21 948.19
$w4=5w40$ 304.29 276.39 3.50 9.10 6.19 9.10 17.10 171.00 23.64 26.46 28.91 875.29
Note: The weights for the manufacturing-cost function in the baseline design are denoted by $wi0$.
We also note that there are co-linear bars in the optimal design of Fig. 5 that could be converted into a single bar (as long as they have the same radius), which would remove the weld between them
and consequently decrease the welding cost. This could be achieved by a heuristic strategy that detects this situation and replaces the two bars with a single one, such as the one presented in Ref. [
10]. However, generalizing this strategy to other geometric primitives, such as the plates shown in Sec. 6.3, is not straightforward. Therefore, this possibility is deferred to future work.
An interesting and important aspect of considering manufacturing cost as an optimization function is that we expect different supply chains (i.e., different values of the weights for the cost terms)
to render different designs. In other words, the optimal design is a function of the supply chain. To demonstrate this, we repeat the manufacturing-cost-minimization with different weights for the
terms of the cost function. For each optimization run, we assign the weight for one term to be five times the corresponding weight for the baseline design, keeping all other weights the same. The
resulting designs are shown in the radar plot of Fig. 6 and the values of each cost terms and other measures are listed in Table 1.
It is worth noting that all these designs exhibit the same structural performance, as the displacement constraint g is active in all these results. As expected, assigning different weights to the
individual terms of the manufacturing-cost function renders different designs. Although the results corresponding to the increased weights for individual terms tend to render a lower value of the
corresponding term, this is not necessarily the case because the entire manufacturing-cost function is minimized. In other words, if we view the manufacturing-cost-minimization problem as a
multi-objective optimization, whereby each term of M is an objective, then we expect that each of the optimal designs with these modified weights corresponds to a non-dominated design in the Pareto
It should also be mentioned that the values of these weights cannot be arbitrary. If the weight for a particular term is too large, then the optimization can render designs that are poor in terms of
their structural performance.
6.2 Three-Dimensional Cantilever Beam With Bar Primitives.
The second example is the design of a 3D cantilever beam with bar primitives. The initial design, depicted in Fig. 7, consists of 43 bars with a unit radius and membership variable of 0.5. The
loading and boundary conditions are depicted in Fig. 8. To reduce the analysis time, we exploit the problem’s symmetry with respect to the x–y plane and model only half of the domain. We note,
however, that it is possible that an asymmetric design that is better than the design obtained with symmetry boundary conditions may be attained, as has been shown for frame structures in Refs. [21,
22] and as is discussed in Ref. [23] for feature-mapping methods. The dimensions of the design domain are 60 × 10 × 5 and the mesh size is 180 × 30 × 15 elements. A load F = 0.1 is applied at the
point (60, 0, 0). We bound the radius of each bar to [0.5, 1]. The move limit for this example is m = 0.05.
Similar to the previous example, we minimize the compliance subject to a maximum material volume fraction constraint limit of $v¯=0.15$. As before, we use the displacement at the point of
application of the load for this minimum-compliance design to define the maximum allowable displacement $u¯$ for the displacement constraint of the manufacturing-cost-minimization problem. The final
design and its corresponding welding locations are shown in Fig. 9. The values of different terms for the optimal design are listed in Table 2.
Table 2
Design M C[m] C[1] C[2] C[3] C[4] C g κ A[cut] l[w] A[s]
Min C 1214.19 600.61 6.28 22.83 550.99 33.46 8.43 85.08 35.11 66.38 2571.12 3217.61
Min M 766.18 656.74 5.85 21.21 64.52 17.83 8.42 84.99 27.84 61.67 301.11 1715.17
Design M C[m] C[1] C[2] C[3] C[4] C g κ A[cut] l[w] A[s]
Min C 1214.19 600.61 6.28 22.83 550.99 33.46 8.43 85.08 35.11 66.38 2571.12 3217.61
Min M 766.18 656.74 5.85 21.21 64.52 17.83 8.42 84.99 27.84 61.67 301.11 1715.17
We perform the manufacturing-cost-minimization, using the same weights as the previous example. The final design is shown in Fig. 10. The cost corresponding to each term in the manufacturing cost and
other relevant measures for the reference design and the minimum-manufacturing-cost design are listed in Table 2. The history of the manufacturing-cost objective and displacement constraint for this
problem are shown in Fig. 11.
As shown in Table 2, the material cost is slightly increased in the minimum-manufacturing-cost design, but the fabrications costs are drastically decreased. Also, the number of existing components κ
decreases from 35 to 27. For this example, the cost of welding dominates the optimization. As can be observed in Fig. 9, there is substantial overlap between bars in the minimum-compliance design,
leading to a large weld length. In contrast, the weld length of the minimum-manufacturing-cost design of Fig. 10 is substantially shorter. The history plots of Fig. 11 show that the optimization
exhibits good convergence towards the optimum. This example demonstrates that the proposed methodology can be readily applied to 3D bar structures.
6.3 Three-Dimensional Messerschmitt–Bölkow–Blohm Beam With Plate Primitives.
Our last example corresponds to an MBB beam design with the loading and supports depicted in Fig. 12. This example demonstrates the proposed method that can be readily applied with plate geometric
primitives—something that cannot be easily done with ground structure approaches. Similar to the 3D cantilever beam example, we take advantage of symmetry and only model a quarter of the design
region, with dimensions 60 × 10 × 5. The mesh size is 180 × 30 × 15 elements. The corners of the design domain are fixed only in the y-direction. The load of magnitude F = 0.1 is applied at the top
center point along the negative z-direction. The initial design, consisting of 18 plates of dimension 5 × 5 with a fixed (non-designable) semi-thickness r[c] = 1.2R = 0.6928, is depicted in Fig. 13.
The membership variable of all plates in the initial design is 0.5. The lengths of the plates are bounded to [0, 20]. The move limit for this example is m = 0.05. It is worth noting that a large
plate semi-thickness r[c] may cause the optimizer to render designs in which plates collapse into bars (i.e., one of the dimensions of the rectangular medial surface approaches zero), as this
minimizes the surface area of the structure.
As before, we first solve the reference minimum-compliance problem and set the corresponding displacement of the point where the load located as the displacement constraint limit for the
minimum-manufacturing-cost problem. The weight of material cost for plate primitives is set to $wm=1.45/in3$ based on a typical unit price of plate stock material in United States. The
minimum-compliance and minimum-manufacturing-cost designs are shown in Figs. 14 and 15, and the corresponding cost of each term in M and other relevant measures are listed in Table 3.
Table 3
Design M C[m] C[1] C[2] C[3] C[4] C g κ A[cut] l[w] A[s]
Min C 5631.83 599.44 4.49 4841.67 180.14 6.07 10.47 99.07 17.97 14074.64 840.60 584.47
Min M 3584.14 849.41 4.95 2682.23 43.91 3.61 10.02 100.26 16.77 7797.20 204.92 347.73
Design M C[m] C[1] C[2] C[3] C[4] C g κ A[cut] l[w] A[s]
Min C 5631.83 599.44 4.49 4841.67 180.14 6.07 10.47 99.07 17.97 14074.64 840.60 584.47
Min M 3584.14 849.41 4.95 2682.23 43.91 3.61 10.02 100.26 16.77 7797.20 204.92 347.73
As we can see from Table 3, the manufacturing cost of the minimum-manufacturing-cost design is only $63.54%$ of the cost of the reference design. The cost of cutting dominates the manufacturing cost
of structure constructed with plates. From Fig. 15, we observe that the optimal design looks like a boxed beam, which helps reduce the cutting cost by reducing the perimeter of the plates.
To visualize designs with their welding locations for the two 3D problems, we employ the visualization software paraview [24,25]. The structure in the figures corresponds to a combined density value
above 0.4, and the welding locations correspond to a value of α[i]α[j]l[ij] above 0.05.
7 Conclusions
This work introduced a geometry projection technique for the topology optimization of welded frame structures made of bar or plate components with regard to manufacturing cost. The examples
demonstrate the effectiveness of the proposed method for 2D and 3D problems. In particular, the manufacturing-cost-driven designs attain a manufacturing cost that is significantly lower than that of
the minimum-compliance designs.
It is important to recall that the manufacturing-cost function employed in this work is a rough estimate to be used in the concept design stage, as an accurate estimation of cost requires a
significant amount of additional information that is only available when a detailed design of the structure is available. Nevertheless, the proposed method is useful in rendering designs that are
more manufacturing-cost effective and is a tool to incorporate manufacturing-cost considerations early in the design. In particular, this technique makes it possible to explore different topologies
that may be better for different manufacturing supply chains with different cost structures, i.e., for which different components of the manufacturing cost have different costs.
The dual representation of high-level geometric parameters and projected densities for each primitive makes it possible to express the components of the manufacturing-cost function in terms of design
variables, something which is not possible with density-based and level-set representations of the entire structure. The geometry projection method, in which each individual primitive is first mapped
onto its own density field and all component densities are subsequently combined for the material interpolation, makes it possible to capture, for instance, the welding length between components.
Moreover, the geometry projection enables the analysis with a fixed grid throughout the optimization, and the components need not be connected in a predefined ground structure. Since the geometry
projection is differentiable, it is possible to use efficient gradient-based methods to perform the optimization.
We note that, as in any gradient-based technique, the proposed method will in general converge to a local minimum. While we did not observe poor local minima in our experiments, it is possible that
they may occur, particularly since the design representation is more compact, as demonstrated in Ref. [23]. Finally, we note that stress constraints and other structural criteria are important in the
design of frame structures and are deferred to future work.
The authors thank the United States National Science Foundation, Award CMMI-1751211, for support of this work. We are also grateful to Prof. Krister Svanberg for providing his MMA matlab optimizer to
perform the optimization.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The datasets generated and supporting the findings of this article may be obtained from the corresponding author upon reasonable request.
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