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The given figure is a combined solid made up of a cylinder and a
cone. The diameter of the base of the solid object is 6cm, length
of the cylindrical part is 116 cm and the total length of the solid
object is 120cm. Find the total surface area of the solid object.
1 thought on “The given figure is a combined solid made up of a cylinder and a<br />cone. The diameter of the base of the solid object is 6cm, l”
1. Answer:
Step-by-step explanation:
We have radius of cylinder and radius of cone both are same i.e 3 cm.height of cylinder is 116cm then height of cone will be 4cm then slant height of cone is 5cm. Then total surface area of solid
will be the sum of curved surface area of cone &curved surface area of cylinder& also the area of circular base.i.e Πrl+2Πrh+Πr² after calculating
we get 2251.58 cm²
Leave a Comment | {"url":"https://wiki-helper.com/the-given-figure-is-a-combined-solid-made-up-of-a-cylinder-and-a-cone-the-diameter-of-the-bas-37395299-58/","timestamp":"2024-11-12T20:06:34Z","content_type":"text/html","content_length":"128954","record_id":"<urn:uuid:4f472c2c-28fd-4642-8c00-474131b5486a>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00187.warc.gz"} |
Integer roots and perfect powers
Version on this page: 1.0
LTS Haskell 22.40: 1.0.2.0@rev:1
Stackage Nightly 2024-11-02: 1.0.2.0@rev:1
Latest on Hackage: 1.0.2.0@rev:1
Maintained by Andrew Lelechenko andrew dot lelechenko at gmail dot com
This version can be pinned in stack with:integer-roots-1.0@sha256:5ea7ecb80633fb05a8a36acd40e9da6fe22a6cef320b1310cb9f0330801ac124,2271
Module documentation for 1.0
Calculating integer roots and testing perfect powers of arbitrary precision.
Integer square root
The integer square root (integerSquareRoot) of a non-negative integer n is the greatest integer m such that . Alternatively, in terms of the floor function, .
For example,
> integerSquareRoot 99
> integerSquareRoot 101
It is tempting to implement integerSquareRoot via sqrt :: Double -> Double:
integerSquareRoot :: Integer -> Integer
integerSquareRoot = truncate . sqrt . fromInteger
However, this implementation is faulty:
> integerSquareRoot (3037000502^2)
> isqrt (2^1024) == 2^1024
The problem here is that Double can represent only a limited subset of integers without precision loss. Once we encounter larger integers, we lose precision and obtain all kinds of wrong results.
This library features a polymorphic, efficient and robust routine integerSquareRoot :: Integral a => a -> a, which computes integer square roots by Karatsuba square root algorithm without
intermediate Doubles.
Integer cube roots
The integer cube root (integerCubeRoot) of an integer n equals to .
Again, a naive approach is to implement integerCubeRoot via Double-typed computations:
integerCubeRoot :: Integer -> Integer
integerCubeRoot = truncate . (** (1/3)) . fromInteger
Here the precision loss is even worse than for integerSquareRoot:
> integerCubeRoot (4^3)
> integerCubeRoot (5^3)
That is why we provide a robust implementation of integerCubeRoot :: Integral a => a -> a, which computes roots by generalized Heron algorithm.
Higher powers
In spirit of integerSquareRoot and integerCubeRoot this library covers the general case as well, providing integerRoot :: (Integral a, Integral b) => b -> a -> a to compute integer k-th roots of
arbitrary precision.
There is also highestPower routine, which tries hard to represent its input as a power with as large exponent as possible. This is a useful function in number theory, e. g., elliptic curve
> map highestPower [2..10] | {"url":"https://www.stackage.org/lts-15.3/package/integer-roots-1.0","timestamp":"2024-11-03T13:44:16Z","content_type":"text/html","content_length":"17506","record_id":"<urn:uuid:7ae14fb8-52ea-49e3-a48e-7226fe48172c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00746.warc.gz"} |
Math, Grade 7, Working With Rational Numbers, Model Integers Multiplication
Hot Air Balloon
Hot Air Balloon
Think back to the Hot Air Balloon simulation you worked with earlier in this unit. Then consider the following situation:
Suppose the balloon starts at 0, and you add 2 units of weight to the balloon every minute for 4 minutes. Where will the balloon be after 4 minutes?
Represent this situation in each of the following ways.
• On the number line
• As an addition equation
• As a multiplication equation
• Remember that weight represents a negative number.
• How many units of –2 are added together?
• How can you represent repeated addition as multiplication?
• Each time 2 units of weight are added to the hot air balloon, the balloon's altitude decreases by 2 units. | {"url":"https://goopennc.oercommons.org/courseware/lesson/5094/student/?section=1","timestamp":"2024-11-09T01:41:26Z","content_type":"text/html","content_length":"30796","record_id":"<urn:uuid:33b487a1-39b7-47ad-942a-2b8912c78d3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00473.warc.gz"} |
20++ Triangle Congruence Proofs Worksheet Answer Key
2 min read
20++ Triangle Congruence Proofs Worksheet Answer Key. Δabc ≅ δa'b'c', ab is 2x + 5 and a'b' is x + 9. Triangle congruence worksheet 2 answer key awesome geometry proofs from triangle congruence.
Proving Triangles Congruent Worksheet Answer Key g2 topic 9 6 more from lbartman.com
The two pairs of proofs worksheet explains how. Open it up with online editor and start editing. Congruent triangles packet 2013 with correct answers
Denton Independent School District / Overview
Open it up with online editor and start editing. 8 images about proofs involving congruent triangles worksheet answer key congruent : ∠ ≅ ∠t ___ 1.
Triangles Similar Right Worksheet Key Answer Geometry Figures Visual Sheets.
Save time with this no. 15 best images of health. The two pairs of proofs worksheet explains how.
Triangle Congruence Proofs Foldable Practice Booklet Geometry Lessons Proof Writing Practices.
Triangles worksheet answers answer triangle congruent isosceles congruence practice key proving proofs unit homework similar geometry math worksheets grade asa pin on fabric. Get the triangle
congruence worksheet 1 answer key you require. Triangle congruence proofs worksheet from triangle congruence worksheet 2 answer key , source:shanepaulneil.com the end result is at the right time of
evaluation, there’s a great deal.
Congruent Triangles Packet 2013 With Correct Answers
Worksheet congruence triangle answer key. Triangle congruence worksheet 2 answer key awesome geometry proofs from triangle congruence. Answer key similarity congruence and proofs answer key this is
likewise one of the factors by obtaining the soft documents of this similarity congruence and proofs answer key.
Triangle Congruence Worksheet Answer Key Pdf.
Some of the worksheets for this concept are using cpctc with triangle congruence, proving triangles congruent, 4 congruence and triangles, congruent triangles work 1, 4 s sas asa and. Guides students
through finding the basic premise of triangle congruence. Find the value of x. | {"url":"https://worksheets.decoomo.com/triangle-congruence-proofs-worksheet-answer-key/","timestamp":"2024-11-12T02:41:02Z","content_type":"text/html","content_length":"199660","record_id":"<urn:uuid:027d43c3-7d20-4e1c-95b7-7a49b098ebfd>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00016.warc.gz"} |
Asymptotic Analysis Basics | CS61B Guide
This concept is a big reason why a strong math background is helpful for computer science, even when it's not obvious that there are connections! Make sure you're comfortable with Calculus concepts
up to power series.
An Abstract Introduction to Asymptotic Analysis
The term asymptotics, or asymptotic analysis, refers to the idea of analyzing functions when their inputs get really big. This is like the asymptotes you might remember learning in math classes,
where functions approach a value when they get very large inputs.
That's cool, but how is it useful?
Graphs and functions are great and all, but at this point it's still a mystery as to how we can use these concepts for more practical uses. Now, we'll see how we can represent programs as
mathematical functions so that we can do cool things like:
Figure out how much time or space a program will use
Objectively tell how one program is better than another program
Choose the optimal data structures for a specific purpose
As you can see, this concept is absolutely fundamental to ensuring that you write efficient algorithms and choose the correct data structures. With the power of asymptotics, you can figure out if a
program will take 100 seconds or 100 years to run without actually running it!
How to Measure Programs
In order to convert your public static void Main(String[] args) or whatever into y = log(x), we need to figure out what x and y even represent!
TLDR: It depends, but the three most common measurements are time, space, and complexity.
Time is almost always useful to minimize because it could mean the difference between a program being able to run on a smartphone and needing a supercomputer. Time usually increases with the number
of operations being run. Loops and recursion will increase this metric substantially. On Linux, the time command can be used for measuring this.
Space is also often nice to reduce, but has become a smaller concern now that we can get terabytes (or even petabytes) of storage pretty easily! Usually, the things that take up lots of space are big
lists and a very large number of individual objects. Reducing the size of lists to hold only what you need will be very helpful for this metric!
There is another common metric, which is known as complexity or computational cost. This is a less concrete concept compared to time or space, and cannot be measured easily; however, it is highly
generalized and usually easier to think about. For complexity, we can simply assign basic operations (like println, adding, absolute value) a complexity of 1 and add up how many basic operation calls
there are in a program.
Simplifying Functions
Since we only care about the general shape of the function, we can keep things as simple as possible! Here are the main rules:
There are two cases where we can't remove other variables and constants though, and they are:
The Big Bounds
There are three important types of runtime bounds that can be used to describe functions. These bounds put restrictions on how slow or fast we can expect that function to grow!
Orders of Growth
There are some common functions that many runtimes will simply into. Here they are, from fastest to slowest:
Don't worry about the examples you aren't familiar with- I will go into much more detail on their respective pages.
Asymptotic Analysis: Step by Step
Identify the function that needs to be analyzed.
Identify the measurement that needs to be taken. (Time, space, etc.)
Generate a function that represents the complexity. If you need help with this step, try some problems!
Simplify the function (remove constants, smaller terms, and other variables).
Select the correct bounds (O, Omega, Theta) for particular cases (best, worst, overall). | {"url":"https://cs61b.bencuan.me/asymptotics/asymptotics","timestamp":"2024-11-13T10:44:21Z","content_type":"text/html","content_length":"519497","record_id":"<urn:uuid:06797e00-8e08-4ddb-9a98-514539ed3d03>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00405.warc.gz"} |
2nd Prize approach of The 4th Tellus Satellite Challenge-2位手法解説 | 宙畑
2nd Prize approach of The 4th Tellus Satellite Challenge-2位手法解説
The 2nd place winner of The 4th Tellus Satellite Challenge explains his approach.
Reading of images:
The provided images are in tiff format with the pixel-values range in [0,65535]. There are two ways how images were pre-processed into [0,1] pixel-values range. The first one is based on linear
mapping, i.e., f’=f/65535, where f is the original image, and f’ is the output one. The second one reflects the physical properties of satellites (noise ratio) as a non-linear transformation f’ =
10log(f^2+c), where c is a noise reduction coefficient, namely c=-83 in our case. We involved both models based on linear and non-linear input in our ensemble.
Normalizing images into the model input resolution:
The additional manipulation can involve image rescale and crop according to the model resolution. There are two main reasons to avoid rescale: Firstly, the training dataset is small; it has 25
images. So, there is a need to use as much information as possible. If we rescale the images to the resolution of the model, for some cases we use less than one percent of the original data.
Secondly, all the images have the same physical resolution, i.e., one pixel in the image always represents the same size in meters, but they have different resolution, i.e., their capture different
area of the landscape. By rescaling them into the same resolution, we will create an additional distortion. Therefore, we used cropping in our pipeline. Because it is useless to realize cropping in
advance, we postponed it to online augmentation.
Handling labels:
We have used two kinds of labels. At the beginning, the labels are represented in {0,1}: the pixel has value 1 if it is a coastline, 0 otherwise. The classes are highly imbalanced, i.e., the total
number of coastline pixels is tiny compared to the rest. To postpone the problem of such imbalance, we applied a form of label smoothing technique and obtained labels in the interval [0,1] – the
first kind of labels. We also manually notated original labels into {sea, land, no-data} classes (the no-data class is for image areas without missing information) – the second kind. Such label
classes are more balanced and include more information. In our pipeline, some models were trained on the original labels (smoothed coastline), and some were trained on the modified version (labels as
classes) to secure the diversity of the models in the ensemble.
Data augmentation:
Our intensity augmentation involves multiplicative intensity shift and cutout. Our spatial augmentation includes flips, rotation, rescale, creating random crops, and multi-sample mosaicing. In our
pipeline, we took an original image, created a crop at a random position with a side-size in the interval [1024,1536] and downscaled it into our model’s side-size, 512. Then we applied the other
augmentations. Finally, we realized our own custom augmentation, multi-sample mosaicing. It means that we split the data sample into n rectangular parts and replaced some of them with a same-sized
data area from a different image from the training set. The main advantage is that such a composed image includes multiple characteristics, which simulate a bigger batch size and, therefore, can
postpone overfitting.
Main architecture:
The segmentation architectures are based on an encoder-decoder scheme, i.e., it decodes the input representation into a latent one, which is then projected back (decoded) to the original resolution.
The spatial dimension is reduced during the encoding while the feature dimension is increased, and vice-versa for the decoding phase. The typical representative is U-Net, with main advantages of the
simplicity of coding and low memory requirements compared to the other architectures. We have selected U-Net in the competition because it allowed us to use a bigger batch size than the other
architectures with the same setting. Compared to FPN, it also yields a lower error, namely 16.2 vs. 17.2.
In U-Net, the encoder’s ability to extract features is limited, so it is beneficial to replace the default one with some of the SOTA networks known, e.g., from the classification problem. These
networks are called backbones and can be pre-trained on ImageNet to converge faster. The most powerful are ResNet, SE-ResNet, ResNeXt, Inception-ResNet, Xception, or ResNeSt, to name a few. In our
pipeline, we have selected EfficientNet, or EffNet for short, a fully convolutional architecture based on compound scaling, that allows easy control of the trade-off between the network capacity and
memory requirements.
Loss function:
Because we planned to create an ensemble of different models, we trained several models based on U-Net with EffNet backbone. Regarding loss function, the commons are based on group-characteristic
such as IOU, Tversky loss, or Sorensen-Dice loss, or pixel-characteristic, like Binary cross-entropy (BCE) or Focal loss. Each of them has pros and cons. Sorensen-Dice considers spatial dimension but
generally does not lead to the best performance; Focal loss can partially solve the class imbalance problem but may overfit; BCE can be marked as a good and universal baseline. In our pipeline, we
combined Dice with Focal for two models and BCE for the other two models.
The first choice is generally Adam, a combination of AdaGrad and RMSProp, which has been several times marked as one of the best optimizers. On the other hand, there are known problems (such as
CIFAR-10 classification) where it yields sub-optimal performance. In our experience, we have confirmed the behavior. Therefore, we used Adam optimizers for two models and AdaDelta for the next two
based on the knowledge.
The rest of the training setting is as follows. We use the resolution of the models equal to 512x512px and as big batch size as possible, varying from 3 to 12. The models were trained for 100 epochs
with reducing the learning rate on a plateau and with saving the best model according to the validation dataset. The models with sea/land/no-data labels have in the last layer softmax (a smooth
approximation of one-hot argmax); the models with only coastline class have in the last layer sigmoid (a smooth logistic function). It means the former creates a decision between the classes, and the
latter produces the probability of being coastline.
To produce predictions, each model creates its own ensemble. We used the technique of floating window, where we created overlapping crops in a multi-scale resolution equal to the conditions we had
during the training phase. Because the inference is significantly less demanding for memory than the training, we were able to process hundreds of crops at once, so the process was fast. When the
predictions were projected back into the original image, the overlapping parts were aggregated by summation, because the process of extracting the coastline described above does not depend on
absolute values. These produced predictions were smoothed by Gaussian filter to decrease impact of noisy outliers.
Extraction coastline from predictions:
The process of extracting coastline differs for models with softmax in the last layer and for models with sigmoid in the last layer.
Softmax models: These models use the following label encoding: 0=sea, 1=no-data, 2=land. The models produce a structure f where each pixel (x,y) is a vector of three values with a probability of the
certain class. Firstly we create f'(x, y) = argmax(f(x,y)), so it holds that f'(x,y) in {0,1,2}. From it, we create the final ‘coastline’ image f” as f”(x,y) = 1 if ma(x,y) – mi(x,y) = 2; 0 else.
Here, ma and mi extract the maximum and minimum value of f’ in 3×3 neighborhood of (x,y). For f” holds that f”(x,y)=1 marks presence of coastline and f”(x,y)=0 no coastline. In other words, we say
that there is a coastline if some area contains both ‘sea’ and ‘land’ classes regardless the ‘no-data’ class.
Sigmoid models: Firstly, we initialize f”(x, y) = 0 for all (x,y), and then check if the image has a landscape or portrait orientation. For landscape, we browse all x coordinates and for each of them
we set f”(x, argmax(f(x,_)) = 1$ where by _ we mean all y coordinates. In other words, we are searching for the maximum probability of being a coastline in each column for all rows. The process is
the same for the landscape, but we search for each column’s maximum in a row. The advantage of the process is the absence of a threshold, so we are able to extract even the most uncertain coastlines.
The disadvantage is we can miss coastline points if the coastline slope is stronger than diagonal or miss a coastline if there are two coastlines in a row/column. We suppress the disadvantages in the
postprocessing later.
Ensemble of coastlines:
The output image functions $f”$ of the particular models have been taken and processed in the following way. We browsed the images column/row-wise, as same as when we made predictions for models with
sigmoid. If we browse rows, then for each of the rows, we find coordinates of coastline in column a create a final prediction as the weighted average of the four predictions. The models’ weights have
been set according to a particular model evaluation in the public competition’s leaderboard.
Satellite imagery is one of many areas where artificial intelligence can be applied. This one is interesting because it is connected with spatial hardware orbiting the earth. The problem is the
accessibility of the data. There are not many public available datasets as in the case of general object classification or detection. The available datasets are usually in a special format connected
with GIS. So, there is a big opportunity to work in a team consisting of geographers with GIS knowledge, image processing stuff, and guys focusing on artificial intelligence.
Books, websites, etc. that helped you to participate in the competition
The first starting point for us was the organizers’ website, namely https://sorabatake.jp/14130, from which we continued papers that the website referenced. During the competition, we read an
uncountable number of scientific papers about segmentations, see benchmarks on https://paperswithcode.com/task/semantic-segmentation, and examined old winning solutions on https://kaggle.com/c/
To see the solutions of other winners, please see this article. | {"url":"https://sorabatake.jp/18109/","timestamp":"2024-11-11T05:23:05Z","content_type":"text/html","content_length":"65443","record_id":"<urn:uuid:507b859b-8be7-4849-bd0f-0994ecea94b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00012.warc.gz"} |
Adjusted level of significance
adjusted_alpha {blindrecalc} R Documentation
Adjusted level of significance
This method returns an adjusted significance level that can be used such that the actual type I error rate is preserved.
adjusted_alpha(design, n1, nuisance, ...)
design object of class TestStatistic created by setup
n1 total number of patients that are recruited before the sample size is recalculated
nuisance nuisance parameter that is estimated at the interim analysis
... Further optional arguments.
The method is only vectorized in either nuisance or n1.
The method is implemented for the classes Student, ChiSquare, and FarringtonManning. Check the class-specific documentation for further parameters that have to be specified.
Value of the adjusted significance level for every nuisance parameter and every value of n1.
d <- setupStudent(alpha = .025, beta = .2, r = 1, delta = 0, delta_NI = 1.5, n_max = 848)
sigma <- c(2, 5.5, 9)
adjusted_alpha(design = d, n1 = 20, nuisance = sigma, tol = 1e-4, iters = 1e3)
version 1.0.1 | {"url":"https://search.r-project.org/CRAN/refmans/blindrecalc/html/adjusted_alpha.html","timestamp":"2024-11-12T20:08:13Z","content_type":"text/html","content_length":"3214","record_id":"<urn:uuid:6fcaaa87-cef0-4730-8b51-5357ae7802ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00423.warc.gz"} |
Polish Puzzle Championships 2014: Individual Round
Last weekend were the Polish Sudoku and Puzzle Championships. The Championship was also open to International solvers this year. The playoffs would feature the best 4 solvers, while having at least
two Polish solvers.
The Sudoku playoffs featured Tiit Vunk, Jakub Ondrousek, Jan Mrozowski and Krystian Swiderski. Jakub Ondrousek finish first in the playoffs, followed by Krystian Swiderski, Tiit Vunk and Jan
Mrozowski, making Krystian Swiderski the new Polish Sudoku champion. Full results can be found
The Puzzle playoffs featured Przemysław Dębiak, Matus Demiger, Zoltan Horvath and Tomasz Stróżak. The final results remained almost the same with Przemyslaw Dębiak finishing first, followed by Zoltan
Horvath, Matus Demiger and Tomasz Stróżak. Full results can be found
I contributed a set for the Puzzle Championships for the individual round and wrote a team round together with Zoltan Horvath.
You can find all puzzles of the Championships in the following link:
Sudoku Rounds + Team Round
Puzzle Rounds
. My puzzle set is round 4. This post will feature all the puzzles from my individual round. Tomorrow I will post the Team round puzzles with a special surprise.
Last year's set
was a bit on the difficult side, so I tried to think of a way to rectify that this year. I decided to write 2 puzzles per type, one smaller/easier one and a larger/harder one. I didn't want to make
any too difficult. I selected 10 varying types of puzzles. I was hoping to average about 1.5 minutes per puzzle on the smaller ones and about 4.5 minutes per puzzle on the larger ones. I had the set
tested by Prasanna Seshadri, James McGowan and Stefan Gaspar. The smaller puzzles made that average pretty well, but the larger puzzles were more inching towards 5-5.5 average and all of them had
some outliers. I couldn't really decide well which puzzles to cut so I sent in the whole set and let them know that they could leave out a puzzle type if necessary. During their testing 3 of the
larger puzzles seem to have cause some problems as they went up in score. I think the set will have worked well none-the-less for all solvers with the easier puzzles to work with as well, but I'd
love to hear some feedback form those who were there.
Puzzles can be found below.
1. Anglers (15 + 35 points)
The smaller one is a really small and sweet puzzle. The larger one requires a bit of counting to get the larger numbers right, especially in the end. But it shouldn't cause too many problems.
Rules: Draw a line from each Fisherman (number) to a fish by moving horizontally and vertically form cell to cell. Each fish is caught by one fisherman. The numbers indicate the length of the line of
the fisherman.
Puzzle #1
Puzzle #2
2. Castle Wall (25 + 50 points)
I like this genre, but hadn't done anything with it for a while. As it's not as common a genre, I figured I could make easier puzzles that would still take a bit more time to solve. The opening of
the smaller one is easy, but one nice step in the middle. I am really happy with how the larger one turned out and think it's one of the better puzzles in the set.
Draw a single closed loop through the grid by connecting the centres of cell. The loop can't go through any cells with a bold black border. If such a cell is coloured black, it will be outside the
loop; if such a cell is coloured white, it will be inside the loop. A number with a horizontal arrow indicates the number of horizontal line segments in the direction of the arrow; a number with a
vertical arrow indicates the number of vertical line segments in the direction of the arrow.
Puzzle #1
Puzzle #2
3. Compass (35 + 50 points)
I like writing these puzzles. It sometimes gets a bit tricky to get them unique without ruining the solving path. I've seen some of Naoki Inaba's puzzles for Japanese competitions and they're
generally a bit more tricky than mine and show well what can be done with it.
The small one is not too hard, but you have to know where to start. The larger one has an easier opening and shouldn't cause too much problems. I like how they both turned out.
Rules: Colour some squares so that all remaining white squares form a single contiguous polyomino. Black squares can't touch each other by a side. There can't be any 2x2 area of white cells anywhere.
Squares with arrows or stars can't be coloured. Arrows indicate that this direction is the only way you can travel to the star over the white cells without backtracking.
Puzzle #1
Puzzle #2
4. Capsules (25 + 55 points)
I like puzzles with simple rules and this is a pretty simple number placement puzzle. I tried writing one for the championship last time, but that just tested horribly. So this time I made them with
many more givens. Although the small one was too hard at first, so I rewrote it to this one.
Rules: Place the digits 1~5 once in every blackbordered region. No two equal digits can touch eachother, not even diagonally.
Puzzle #1
Puzzle #2
5. Star Battle (20 + 60 points)
Star Battle has become one of the standard genres and I like writing them. I couldn't really write an 8x8 puzzle for this genre as there are only two solutions for that grid size. So I wrote a very
easy 10x10 puzzle instead. The easy one has 6 really quick placements, which gives you a lot to work with from the start. The harder one has a trickier opening to eliminate a few cells. After that it
gets going.
Rules: Place 2 stars in every row, column and black bordered area. The stars can't touch eachother, not even diagonally.
Puzzle #1
Puzzle #2
6. Tapa (35 + 60 points)
I hadn't written any Tapa puzzles in a while. I tend to write less puzzles of types I have written many of before. I find it hard to think of fun new ways to start a puzzle. I have solved so many
Tapas already with the TVC, CTC, Serkan Yurekli's Tapa book and the upcoming GM Puzzles book and all the others available online. But as it's become such an established genre I gave it another try.
The small one was just all about interaction between adjacent clues. It took a while to get it correct, so that the middle turned out unique. It's one of the harder smaller puzzles in the set. The
larger one was all about finding an opening. I don't normally put clues next to eachother much, so I gave that a try. Originally I was going to have 16 clue in squares, but there were just too many
clues, so make the other two corners triangles instead.
Rules: Colour some cells to create a single contiguous shape. The shape can't have any 2 by 2 coloured areas. The clues in the grid tell you how many consecutive cells around it have to be coloured.
If there's more than one digit in a cell, the groups of cells have to be separated by at least one empty cell. Cells with clues remain empty.
Puzzle #1
Puzzle #2
7. Pentopia (25 + 60 points) This was the most successful creation for my LMI test. It's the only one I've really kept making. I think it works well in smaller sizes as well. I made these 10x10 and
12x12 as 12x12 is the normal size for pentomino placement puzzles. The two + clues set up the easy one really quickly. The larger one is a bit more tricky but at least has an easier opening. Rules:
Place pantominos in the grid without repeating any shape. Rotations and reflections are considered the same shape. The pentominos are not allowed to touch, not even at the corners. The lines in the
grid indicate all direction(s) in which the pentominos is/are closest when looking from that square.
Puzzle #1
Puzzle #2
8. Araf (45 + 100 points)
These were some of the first Araf's I had written. I was asked to write an Araf for the LMI Puzzle Marathon, but as I had never even attempted it, I wasn't sure how that would turn out. I normally go
a bit too hard in my first puzzles, which isn't the best for marathon size. I rewrote both these puzzles, which you'd know if you'd been following my blog. The smaller one is not too hard. I'm a bit
surprised with the points value. If you understand the rules, it should solve pretty quickly once you get the right pairings figured out. Most force themselves quickly though. The harder one has no
bordering clues, which I thought worked out well. The opening is tricky, but should still be able to figured out logically.
Rules: Divide the grid into some regions, formed of adjacent squares. Each region should contain exactly two given numbers. The size of each region should be a value (in unit square) between the two
numbers inside that region.
Puzzle #1
Puzzle #2
9. Hexa Skyscrapers (20 + 135 points) I thought I should put in a type using a hexagonal grid. I hadn't written these before, but I think they turned out well. The smaller one was almost accidental.
I put down the 2 clues to create an opening and then started solving to see where I needed to put down another clue. except that turned out to not be necessary. The larger one was the hardest puzzle
in the set in my opinion. It took some time to create as I regularly ran into no solution. There's one tricky step in the opening, but seeing it will give you all fours quickly. Rules: Place a digit
1-4 in some cells so that each digit appears once in each row and each diagonal line. Each digit represents a skyscraper of that height. Clues on the outside indicate how many digits are visible in
that direction from that side. Larger digits hide behind smaller digits.
Puzzle #1
Puzzle #2
10. Nurikabe (20 + 135 points)
This was the last type I had added. I hadn't included a Nikoli type, so I figured I should included at least one. It should make the set more complete. The smaller one was an adjustment of the puzzle
posted before. I was looking at it and went over it in my mind and thought that it seemed to be unique with pairs of 2's and 5's. So I jotted it on paper and checked and it worked. The larger one has
a simple opening but the ending is quite tricky. It took me a while to draw up the last digits, which all turned out to be 3's to be unique.
Rules: Determine for each cell if it's part of the stream or an island. Each number is part of a single island of horizontally and vertically connected cells, which size is equal to that number.
Islands can't touch eachother horizontally or vertically. The cells not part of an island form the stream. The stream is a single connected area, which doesn't cover any 2x2 areas anywhere.
Puzzle #1
Puzzle #2
10 comments:
1. In Pentopia ,is it necessary that all 12 pentominoes need to be placed in the grid?I could not place the X and the T pentominoes but still ended with a solution that looks valid.
1. No, you don't have to place them all. I never have had that as a rule. You have the correct solution.
2. Thank you.I will solve the rest and see if I can complete them.Great puzzles.
3. Feedbacks:
Anglers: Nice, easy puzzles for start
Castle Wall: I haven't solved this kind of puzzle for a while. I like the way you use the black and white clues. But sometimes I was stucked when I didn't see some trivials.
Compass: Unfortunately, I couldn't make too much progress in this puzzles.
Capsules: First was ok, but the second was too hard for me. I could solve it by uniqueness trick.
Star Battle: First was really easy. The second was a bit harder but solvable.
Tapa: Second one was harder thank I expected. I didn't find a logical find easily, so I just started guessing. It worked.
Pentopia: Common nice puzzles.
Araf: First was very smooth. I liked the second one. I relatively fast realised that it is needed to find a "snake".
Hexa skyscrapers: I solved a lot hexa skyscrapers. I like that it has a strong latin square property. Just some numbers can determine the grid. They were nice, but nothing extra for an
experienced Hexa sky solver :)
Nurikabe: OK :)
1. For the Tapa, start in the right bottom corner. How does the 5 get out?
2. I started drawning in top-left corner. I think there was only one way to satisfy all clues there. But it was a bit hard to find it :)
4. Compass: Much easier if I don't forget 2x2 rule.
1. Yeah, puzzles become easier when you know the rules ;)
5. ..StarBattle : 2:36,7:55 ..
6. Nurikabe : 2:26,15:00 | {"url":"https://puzzleparasite.blogspot.com/2014/04/polish-puzzle-championships-2014.html","timestamp":"2024-11-11T07:35:40Z","content_type":"text/html","content_length":"142806","record_id":"<urn:uuid:2383fdba-93c6-434b-b72b-fded725a34ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00826.warc.gz"} |
Events and Types of Events in Probability
What are Events in Probability?
A probability event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space. So, what is sample space?
The entire possible set of outcomes of a random experiment is the sample space or the individual space of that experiment. The likelihood of occurrence of an event is known as probability. The
probability of occurrence of any event lies between 0 and 1.
The sample space for the tossing of three coins simultaneously is given by:
S = {(T , T , T) , (T , T , H) , (T , H , T) , (T , H , H ) , (H , T , T ) , (H , T , H) , (H , H, T) ,(H , H , H)}
Suppose, if we want to find only the outcomes which have at least two heads; then the set of all such possibilities can be given as:
E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}
Thus, an event is a subset of the sample space, i.e., E is a subset of S.
There could be a lot of events associated with a given sample space. For any event to occur, the outcome of the experiment must be an element of the set of event E.
What is the Probability of Occurrence of an Event?
The number of favourable outcomes to the total number of outcomes is defined as the probability of occurrence of any event. So, the probability that an event will occur is given as:
P(E) = Number of Favourable Outcomes/ Total Number of Outcomes
Types of Events in Probability:
Some of the important probability events are:
Impossible and Sure Events
If the probability of occurrence of an event is 0, such an event is called an impossible event and if the probability of occurrence of an event is 1, it is called a sure event. In other words, the
empty set ϕ is an impossible event and the sample space S is a sure event.
Simple Events
Any event consisting of a single point of the sample space is known as a simple event in probability. For example, if S = {56 , 78 , 96 , 54 , 89} and E = {78} then E is a simple event.
Compound Events
Contrary to the simple event, if any event consists of more than one single point of the sample space then such an event is called a compound event. Considering the same example again, if S = {56 ,78
,96 ,54 ,89}, E[1] = {56 ,54 }, E[2] = {78 ,56 ,89 } then, E[1] and E[2] represent two compound events.
Independent Events and Dependent Events
If the occurrence of any event is completely unaffected by the occurrence of any other event, such events are known as an independent event in probability and the events which are affected by other
events are known as dependent events.
Mutually Exclusive Events
If the occurrence of one event excludes the occurrence of another event, such events are mutually exclusive events i.e. two events don’t have any common point. For example, if S = {1 , 2 , 3 , 4 , 5
, 6} and E[1], E[2] are two events such that E[1] consists of numbers less than 3 and E[2] consists of numbers greater than 4.
So, E1 = {1,2} and E2 = {5,6} .
Then, E1 and E2 are mutually exclusive.
Exhaustive Events
A set of events is called exhaustive if all the events together consume the entire sample space.
Complementary Events
For any event E[1] there exists another event E[1]‘ which represents the remaining elements of the sample space S.
E[1] = S − E[1]‘
If a dice is rolled then the sample space S is given as S = {1 , 2 , 3 , 4 , 5 , 6 }. If event E[1] represents all the outcomes which is greater than 4, then E[1] = {5, 6} and E[1]‘ = {1, 2, 3, 4}.
Thus E[1]‘ is the complement of the event E[1].
Similarly, the complement of E[1], E[2], E[3]……….E[n ]will be represented as E[1]‘, E[2]‘, E[3]‘……….E[n]‘
Events Associated with “OR”
If two events E[1] and E[2] are associated with OR then it means that either E[1] or E[2] or both. The union symbol (∪) is used to represent OR in probability.
Thus, the event E[1]U E[2] denotes E[1] OR E[2].
If we have mutually exhaustive events E[1], E[2], E[3 ]………E[n] associated with sample space S then,
E[1] U E[2] U E[3]U ………E[n] = S
Events Associated with “AND”
If two events E[1] and E[2] are associated with AND then it means the intersection of elements which is common to both the events. The intersection symbol (∩) is used to represent AND in probability.
Thus, the event E[1] ∩ E[2] denotes E[1] and E[2].
Event E[1] but not E[2]
It represents the difference between both the events. Event E[1 ]but not E[2] represents all the outcomes which are present in E[1] but not in E[2]. Thus, the event E[1] but not E[2] is represented
E[1], E[2] = E[1] – E[2]
Example Question on Probability of Events
Question: In the game of snakes and ladders, a fair die is thrown. If event E[1] represents all the events of getting a natural number less than 4, event E[2] consists of all the events of getting an
even number and E[3] denotes all the events of getting an odd number. List the sets representing the following:
i)E[1] or E[2] or E[3]
ii)E[1] and E[2] and E[3]
iii)E[1] but not E[3]
The sample space is given as S = {1 , 2 , 3 , 4 , 5 , 6}
E[1] = {1,2,3}
E[2] = {2,4,6}
E[3] = {1,3,5}
i)E[1] or E[2] or E[3]= E[1] E[2] E[3]= {1, 2, 3, 4, 5, 6}
ii)E[1] and E[2] and E[3] = E[1] E[2] E[3] = ∅
iii)E[1] but not E[3] = {2}
Probability Related Video:
More Topics Related to Probability Events
Probability and Statistics Probability Formulas
Multiplication Rule Probability Bayes Theorem
Bernoulli Trials Binomial Distribution Independent Events And Probability
Frequently Asked Questions
What are Events in Probability?
In probability, events are the outcomes of an experiment. The probability of an event is the measure of the chance that the event will occur as a result of an experiment.
What is the Difference Between Sample Space and Event?
A sample space is a collection or a set of possible outcomes of a random experiment while an event is the subset of sample space. For example, if a die is rolled, the sample space will be {1, 2, 3,
4, 5, 6} and the event of getting an even number will be {2, 4, 6}.
What is the Probability of an Impossible Event and a Sure Event?
The probability of a sure event is always 1 while the probability of an impossible event is always 0.
What is an Example of an Impossible Event?
An example of an impossible event will be getting a number greater than 6 when a die is rolled. | {"url":"https://mathlake.com/Events-and-Types-of-Events-in-Probability","timestamp":"2024-11-13T07:51:40Z","content_type":"text/html","content_length":"18965","record_id":"<urn:uuid:9864b032-9463-4e6e-8fed-0bb34eff87c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00200.warc.gz"} |
Atanh: Excel Formulae Explained - ExcelAdept
Key Takeaway:
• ATANH is a mathematical function in Excel that calculates the inverse hyperbolic tangent of a given value. It is useful for finding the angle whose hyperbolic tangent is equal to a given number.
• To use the ATANH formula in Excel, use the syntax =ATANH(number) where “number” is the value for which you want to calculate the inverse hyperbolic tangent.
• The benefits of using ATANH formula include obtaining precise results and its ease of use. However, it also has limitations such as domain and range restrictions and error messages that can
Are you stuck trying to understand the ATANH Excel formula? This article provides an easy-to-follow guide for understanding and utilizing this formulae, making complex computations easier than ever.
You can quickly get the results you need, leaving you more time for the important tasks.
What is ATANH in Excel?
ATANH, short for ArcTanh, is an Excel function that returns the inverse hyperbolic tangent of a given number. It is a mathematical formula used to calculate the angle whose hyperbolic tangent is the
specified number. ATANH is commonly used in statistics and engineering to process large datasets and analyze trends. It is a powerful tool that helps in measuring the degree of association between
variables. By using this formula, we can determine the strength and direction of the correlation between two variables. It is important to note that the output of the ATANH function is always in
In addition to its mathematical applications, ATANH is also used to solve problems in physics, astronomy, and economics. It can be used to calculate the distance between two points in space or to
analyze the behavior of financial markets. One unique characteristic of the ATANH formula is that it is capable of handling both positive and negative values, making it ideal for a wide range of
To ensure accurate results when using the ATANH formula, it is important to pay close attention to the input values. It is recommended to use a calculator or a spreadsheet to perform the
calculations, as manual calculations can lead to errors. Additionally, it is important to use the proper syntax when entering the formula into Excel. The syntax for the ATANH function is "ATANH
(number)", where "number" is the value for which to calculate the inverse hyperbolic tangent.
How to use ATANH Formula in Excel:
To work out the ATANH Formula in Excel, two bits of knowledge are needed: syntax and practical application. Syntax will show how input values should be used, and practical application will show how
to use it in Excel. In this part of “ATANH: Excel Formulae Explained,” we’ll go over both syntax and an example of how to apply it to data analysis.
Syntax of ATANH Formula
The ATANH formula in Excel is a powerful statistical tool that calculates the hyperbolic arctangent of a given number. Its syntax is =ATANH(number), where ‘number’ refers to the numerical value for
which you want to find the inverse hyperbolic tangent.
The result of this formula ranges from negative infinity to positive infinity, with zero at the midpoint.
When using ATANH formula in Excel, you must keep in mind that its input range should be between -1 and +1. If the range exceeds these limits, it returns an error value like “#NUM!” or “#VALUE!”.
Besides, you can also use this formula in combination with other functions like SUMIF, AVERAGEIF, or COUNTIFS to perform complex calculations and analysis.
One crucial aspect of using ATANH formula is understanding how it works and what its output means. As mentioned earlier, ATANH produces values between negative and positive infinity. Therefore, it
can be challenging to interpret them without some prior knowledge or context about your data. Still, they are useful for performing various statistical analyses such as regression analysis or
hypothesis testing.
Interestingly, The history of mathematical functions dates back centuries ago when mathematicians first began using logarithms to simplify complex calculations. This curiosity led them to discover
more sophisticated functions that play an essential role in modern-day scientific research and analysis today – including ATANH!
Not sure if I’m calculating ATANH correctly or just using some advanced form of counting sheep to fall asleep at my desk.
Example of ATANH Formula
To compute the inverse hyperbolic tangent or ATANH for any given number in Excel, a user can employ the ATANH formula. This formula returns the inverse hyperbolic tangent of a given number, which is
expressed in radians. The output can be converted to degrees by multiplying it by 180/PI(). By using this formula, users can solve problems involving hyperbolic functions and calculus with ease.
One scenario where ATANH is particularly useful is when users have to calculate the area under a special curve called the catenary curve. This curve arises naturally in architecture and engineering
when modeling hanging cables or arches held by cables. Calculating the area under the catenary curve requires computing its derivative, which involves inverse hyperbolic functions like ATANH.
When a civil engineer named John used Excel to design an arch bridge for a city using the catenary curve as a baseline for mathematical computations, he realized that he needed to apply the ATANH
formula to calculate critical values accurately. Armed with his knowledge of advanced Excel techniques, John was able to deliver accurate and stable designs of arch bridges measuring up-to 100 meters
Using ATANH Formula in Excel: Because who doesn’t want to calculate the hyperbolic arctangent of their data?
Benefits of ATANH Formula:
Use the ATANH formula for precise results and ease in your Excel sheet. It can bring many benefits to enhance your experience. We will look at how ATANH can offer accuracy and how easy it is to use.
Its precision and ease make it a great choice!
Precise results
The ATANH formula provides unparalleled accuracy in calculating results. It is a robust and reliable tool that uses complex mathematical calculations to give precise outcomes. Through the integration
of high-level algorithms, this formula accurately represents data patterns and provides accurate information for decision-making.
With ATANH, users can retrieve accurate results with minimal effort. The formula reduces the time needed for manual calculations, making it an efficient method to analyze data sets. Precision is
achieved through logarithmic functions that help eliminate errors caused by practical limitations.
Beyond this, ATANH is highly versatile and can be applied to various types of mathematical expressions. It adapts to different input values, ensuring unmatched precision in all types of computations.
Through its flexibility and precision, it plays a critical role in scientific and mathematical research applications.
One instance where ATANH came into play was during NASA’s lunar missions in the 1960s. Computational inconsistencies were causing issues with spacecraft navigation systems. The ATANH formula was
employed to resolve these irregularities allowing for better accuracy in space exploration.
In summary, the ATANH formula offers precise results for complex computations with minimal manual effort. Its versatility makes it ideal for a range of applications from science experimentation to
financial analysis. With its exceptional accuracy record and history of resolving computational issues, we can trust ATANH as a reliable method across numerous industries and situations. ATANH may be
a mouthful, but using it is a breeze – Excel just couldn’t make it any easier.
Ease of use
The ATANH formula offers simplicity and convenience to users, making it easy for them to calculate the hyperbolic arctangent of a real number. This formula can be used in a variety of applications,
including finance and statistics.
With its straightforward syntax, the ATANH formula is easy to implement in Microsoft Excel spreadsheets, offering fast calculations with just a few keystrokes. Users can also take advantage of
Excel’s built-in functions to make computations using the ATANH formula more precise and efficient.
Furthermore, since many professionals rely on Excel for financial analysis and other crucial tasks, knowing how to use the ATANH formula can make all the difference in streamlining workflows and
increasing productivity.
Moreover, it is worth noting that using the ATANH formula requires only basic mathematical knowledge. This makes it accessible even to those who may not have extensive experience with advanced
mathematics or data analysis techniques.
In fact, one finance professional was able to save hours of time each week by mastering the ATANH formula in Excel. By automating complex calculations related to portfolio management, he was able to
improve his efficiency and productivity dramatically – all without sacrificing accuracy or precision.
However, it is important to remember that while the ATANH formula may solve your problem, it won’t solve your trust issues.
Limitations of ATANH Formula:
To surpass the ATANH formula’s boundaries, the answer is to comprehend its sub-sections. These are “Domain and Range” and “Error Messages”.
Domain and Range
ATANH Formula: Understanding its limits in terms of input and output values is crucial to avoid incorrect data interpretation. The ATANH Formula works on a specific range of input values, having a
defined domain, and produces results within a certain output range known as its range.
Domain Range
-1<x<1 -∞<y<∞
The formula’s output goes beyond the range ‘[-1, 1]’ of hyperbolic tangents (tanh) and has discontinued values for extreme input values reaching negative infinity or positive infinity. Hence, the
users must ensure they provide valid inputs since invalid inputs will result in NaN(#NAME?) errors.
It is vital to ensure proper input validation before utilizing ATANH in your analysis. A single wrong value could produce inaccurate misleading results during data representation.
According to Microsoft Excel Support, “If we calculate atanh(2), it fails because this value only exists between (-1 to +1). However, Excel still calculates it with the answer “NaN”.
Why have an error message when you can just blame it on the user’s lack of Excel skills?
Error messages
When using the ATANH formula in Excel, it is possible to encounter error messages. These messages are indications that the formula cannot perform the calculation due to certain limitations and
The error messages that can appear when using ATANH include “#NUM!”, which occurs when the input argument falls outside the range of -1 to 1; “#DIV/0!” when the input value equals 1 or -1; and “#
VALUE!” when the input argument is not recognized as a numeric value.
It is crucial to be aware of these error messages as they can negatively impact your calculations and result in erroneous data. It is highly recommended to review your inputs carefully and ensure
their compatibility with the formula before proceeding with any further analysis.
In addition, it is essential to note that while ATANH does have its applications, it also has limitations in terms of accuracy and precision. As a result, it may not be suitable for all types of
calculations or data sets.
To ensure accurate results, it is strongly advised to explore alternative formulas and functions available in Excel that may better suit your specific needs. Don’t let these limitations and error
messages prevent you from delivering reliable analyses. Take the necessary precautions and never hesitate to explore alternative methods for greater precision and accuracy.
Five Facts About “ATANH: Excel Formulae Explained”:
• ✅ ATANH is an Excel formula that returns the hyperbolic arctangent of a number. (Source: Excel Easy)
• ✅ The ATANH function can be used for calculating the inverse hyperbolic tangent of a number, that is, finding the value whose hyperbolic tangent is a certain number. (Source: Ablebits)
• ✅ ATANH is useful for engineering, mathematics, and other scientific applications that involve calculations of hyperbolic functions. (Source: Excel Jet)
• ✅ The ATANH function is similar to the ATAN function, but it returns a hyperbolic result instead of a trigonometric result. (Source: Vertex42)
• ✅ The ATANH function can be used in combination with other Excel functions to perform complex calculations, such as calculating the average of hyperbolic tangents. (Source: Excel Campus)
FAQs about Atanh: Excel Formulae Explained
What is ATANH: Excel Formulae Explained?
ATANH is an Excel formula that returns the inverse hyperbolic tangent of a number. The ATANH function calculates the inverse of the hyperbolic tangent for a given number, which is expressed in
How do I use the ATANH function in Excel?
To use the ATANH function in Excel, simply enter “=ATANH(number)” in a cell and replace “number” with the actual value you want to find the inverse of the hyperbolic tangent for. Press Enter and the
calculation will be displayed in the cell.
What is the syntax of the ATANH formula?
The syntax of the ATANH formula is “=ATANH(number)”, where “number” is the value for which you want to calculate the inverse hyperbolic tangent. The result of the calculation is displayed in the cell
where the formula is entered.
Can the ATANH function be used with multiple values?
No, the ATANH function can only be used to calculate the inverse hyperbolic tangent for one value at a time. If you need to find the inverse hyperbolic tangent for multiple values, you will need to
enter the ATANH formula into each cell separately.
What is the range of values for the ATANH function?
The range of values for the ATANH function is from -1 to 1. Any value outside of this range will result in a #VALUE! error.
Can the ATANH function be used in combination with other Excel functions?
Yes, the ATANH function can be used in combination with other Excel functions to perform more complex calculations. Some examples of functions that could be used with ATANH include SUM, AVERAGE, and | {"url":"https://exceladept.com/atanh-excel-formulae-explained/","timestamp":"2024-11-07T06:07:53Z","content_type":"text/html","content_length":"70183","record_id":"<urn:uuid:a50c23bb-9ae7-420b-bfb8-b702569e4fb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00751.warc.gz"} |
A Powerful Subvector Anderson Rubin Test in Linear Instrumental Variables Regression with Conditional Heteroskedasticity
Authors: Patrik Guggenberger, Frank Kleibergen (University of Amsterdam), Sophocles Mavroeidis (University of Oxford)
Abstract: We introduce a new test for a two-sided hypothesis involving a subset of the structural parameter vector in the linear instrumental variables (IVs) model. Guggenberger, Kleibergen and
Mavroeidis (2019), GKM19 from now on, introduce a subvector Anderson-Rubin (AR) test with data-dependent critical values that has asymptotic size equal to nominal size for a parameter space that
allows for arbitrary strength or weakness of the IVs and has uniformly nonsmaller power than the projected AR test studied in Guggenberger, Kleibergen, Mavroeidis and Chen (2012). However, GKM19
imposes the restrictive assumption of conditional homoskedasticity. The main contribution here is to robustify the procedure in GKM19 to arbitrary forms of conditional heteroskedasticity. We first
adapt the method in GKM19 to a setup where a certain covariance matrix has an approximate Kronecker product (AKP) structure which nests conditional homoskedasticity. The new test equals this adaption
when the data is consistent with AKP structure as decided by a model selection procedure. Otherwise the test equals the AR/AR test in Andrews (2017) that is fully robust to conditional
heteroskedasticity but less powerful than the adapted method. We show theoretically that the new test has asymptotic size bounded by the nominal size and document improved power relative to the AR/AR
test in a wide array of Monte Carlo simulations when the covariance matrix is not too far from AKP.
Link to work
Presentation slides | {"url":"https://ceba-lab.org/tpost/a6pe0llcn1-a-powerful-subvector-anderson-rubin-test","timestamp":"2024-11-12T20:29:52Z","content_type":"text/html","content_length":"40285","record_id":"<urn:uuid:d28754c9-2069-4c1a-8ca1-25f8763658ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00089.warc.gz"} |
An interference pattern is produced by light with a wavelength
600 nm from a distant source...
An interference pattern is produced by light with a wavelength 600 nm from a distant source...
An interference pattern is produced by light with a wavelength 600 nm from a distant source incident on two identical parallel slits separated by a distance (between centers) of 0.500 mm .
Part A
If the slits are very narrow, what would be the angular position of the first-order, two-slit, interference maxima?
Part B
What would be the angular position of the second-order, two-slit, interference maxima in this case?
Part C
Let the slits have a width 0.330 mm . In terms of the intensity I0 at the center of the central maximum, what is the intensity at the angular position of ?1?
Part D
What is the intensity at the angular position of ?2?
Part A
To determine the angular position of the interference maxima we use the following equation:
solving for
as it is for first order then m = 1
transformed d = 0.500mm to nm (d = 500000nm)
Part b
it is done the same as in part A but now m = 2
Part C
In this case d = 0.330mm.
The intensity for any
d=0.330mm=330000nm and
Part D
In this case d = 0.330mm.
The intensity for any
d=0.330mm=330000nm and | {"url":"https://justaaa.com/physics/990892-an-interference-pattern-is-produced-by-light-with","timestamp":"2024-11-11T20:19:55Z","content_type":"text/html","content_length":"47424","record_id":"<urn:uuid:fc9b88af-040a-4dd4-b1a7-c71f8876e1aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00640.warc.gz"} |
User Manual
SCR Calculator User Manual
Version 1.15.2.0 Last modified 2024-9-9
Menu Items in the Main Form
• Data and Parameters
Items in this menu item include:
□ Click Reference Data & Parameters, the following form would appear:
Get/set the following application-wide constant parameters:
☆ Solvency II - Data Source choice between EIOPA and PRA - is explained in the "Matching Adjustment" chapter.
☆ Solvency II - Symmetric Adjustment Current value of the Equity Symmetric Adjustment under Solvency II; updated monthly.
☆ LAGIC - ASX200 Dividend Yield This determines the Australian LAGIC equity stress percentage. The higher the dividend yield, the lower the equity stress.
☆ S&P Insurer Target Rating The higher the rating, the heavier the capital factors.
☆ S&P and LAGIC - Bond Portfolio Type You must choose from: life, non-life and shareholder portfolios. Each has a different set of capital charges.
☆ Insurance Capital Standard - Regime Choice between the ICS and its local variations - namely Korean ICS and Japan ESR.
☆ ICS - Concentration Risk Threshold determines the concentration risk calculation in ICS.
☆ ICS - Number of Interest Rate Simulationsself-explanatory.
☆ Random Simulation Seeds Sequence this is a series of seeds that drive random simulations such as the ICS interest rate calculation.
☆ Taiwan RBC - DM and EM equity counter-cyclical Adjustments similar to Solvency II Symmetric Adjustment, updated quarterly by the Taiwan insurance regulator, but the SCR Calculator does
not maintain their values in its database. The user needs to manually enter every time.
☆ Hong Kong RBC - Equity counter-cyclical adjustment similar to the Taiwan case above. The HK RBC has not been finalised, so this item defaults to zero.
☆ Bermuda Solvency Discount Curve Chooses between 'RFR' and 'CORP'. The Bermuda Monetary Authority publishes two sets of curves, with the 'CORP' set incorporating an illiquidity premium.
Note that what you 'get' are QUARTERLY officially published curves. The user can upload own-calibrated, more timely curves if needed.
□ Solvency II Fundamental Spread The user can get/set these parameters using the template spreadsheet provided.
☆ ‘get’ generates a spreadsheet containing the tabs below; and ‘set’ allows you to upload such a spreadsheet (with possibly modified values and/or added rows) to customize the parameters:
☆ “GovFS”: government bond fundamental spreads
☆ “CorpFS”: corporate bond fundamental spreads
☆ “CorpPDprob”: corporate bond probability of default suitable for use in cashflow de-risking
☆ “CorpPDpct”: corporate bond probability of default expressed in absolute value (bps) terms
☆ “CorpCOD”: corporate bond cost of downgrade
☆ “GovLTAS”: government bond long-term average spread
☆ “CorpLTASbasic”: corporate bond long-term average spread for EUR, GBP and USD
☆ “CorpLTASoverEURO”: corporate bond long-term average spread for other countries
□ Solvency II Risk Free Curves
‘Get’ generates a spreadsheet with 3 tabs - “BaseCurve”, “UpCurve” and “DownCurve”, containing curves used for interest rate SCR calculation. ‘Set’ allows you to upload such a spreadsheet
(with possibly modified values and/or added rows) to customize the parameters. ‘Set Base’ allows you to update a spreadsheet containing only the “BaseCurve” tab, and the up/down stressed
curves will be generated according to your supplied base curves.
If you wish to add more curves, you must modify the "BaseCurve", "UpCurve" and "DownCurve" tabs simultaneously and in a consistent format to the original. The country name you add must be of
a real country; otherwise the calculator will prompt you to correct.
□ Bermuda discount curves Similar to the Solvency II case.
□ Australian LAGIC curves Similar to the Solvency II case.
□ ICS risk free curves Similar to the Solvency II case. However, what you “get” is a static set of curves as of Mar 22; and there is not a “set base” option, because the official curve
stressing parameters are not disclosed yet and the Calculator cannot perform the stresses.
□ Singapore RBC risk free curves Similar to the Solvency II case.
• Open Subforms
Each option under this section opens the corresponding Sub Form, which we discussed in more detail in previous chapters.
• Input Templates → Basic/Bloomberg Inputsheets
Each item is a downloadable spreadsheet. You can combine rows to form a multi-asset portfolio inputsheet.
The Bloomberg Barclays inputsheet has a distinct layout that is different from the "Basic" inputsheets and should be used separately.
• Misc. Tools → Kill Hidden Excel
Sometimes there are hidden, read-only Excel instances that create difficultites for further processing. This button solves the problem.
• Misc. Tools → FX Rates as of Valuation Date
The SCR Calculator's database contains daily FX rates between all EIOPA and ICS currencies since 1 Jan 2014: AUD, BGN, BRL, CAD, CHF, CLP, CNY, COP, CZK, DKK, EUR, GBP, HKD, HUF, IDR, ILS, INR,
ISK, JPY, KRW, MXN, MYR, NOK, NZD, PEN, PHP, PLN, RON, RUB, SAR, SEK, SGD, THB, TRY, TWD, USD, ZAR.
These FX rates are useful when you work with a multi-currency portfolio. The calculator can convert between local currency and portfolio currency market values of the assets behind the scene; You
do not need to manually look up FX rates, unless it is not among the above currencies, in which case the SCR Calculator will prompt for your input. The local currency values are useful for
cashflow projection. The portfolio currency values are useful for calibrating weights and total SCR. This menu button is provided for reference, such that you have transparency what FX rates are
being used.
• Misc. Tools → Bond Credit Rating Convertor
Solvency II and a few other regimes require the use of the "second best" rating out of 3 major rating agencies' ratings. This facility is provided for users to conveniently derive such ratings.
Click the "Download Template" button and obtain the spreadsheet like following:
In the first three columns, you need to fill in the three ratings from your chosen rating agencies. The SCR calculator can digest a wide range of input rating formats. The example screenshot has
one row but you can fill in multiple rows of course. Then use the "Upload & Process" button, the calculator will give you a new spreadsheet containing a range of ratings outputs such as below
□ Rating Index is a numerical index representing notch-level average credit quality of the portfolio: 1=Aaa, 2=Aa1, 3=Aa2, 4=Aa3, 5=A1, 6=A2, 7=A3, 8=Baa1, 9=Baa2, 10=Baa3, 11=Ba1, 12=Ba2, 13=
Ba3, etc. The "RatIdx1","RatIdx2", "RatIdx3" columns are these index values corresponding to your inputs.
□ Simple Rating means ratings such as these: "AAA", "AA1", "AA2", "AA3", "A1", "A2", "A3", "BBB1", "BBB2", "BBB3", "BB1", "BB2", "BB3", "B1", "B2", "B3", "CCC1", "CCC2", "CCC3", "CC1", "CC2",
"CC3", "C", "D". These are the preferred rating format used by the SCR Calculator during portfolio data import.
□ CQS means "Credit Quality Step". Their values range from 0 (AAA), 1 (AA), 2 (A), 3 (BBB), 4 (BB), 5 (B) and 6 (CCC and below). These are the EIOPA-defined rating grades used to calculate
spread SCR.
□ Second Best and Worst columns are self-explanatory.
□ BSCR Rating is 0-8. 0 is for AAA-AA government bonds. 1 (AAA), 2 (AA), 3 (A), 4 (BBB), 5 (BB) , 6 (B) , 7 (CCC) and 8 (below and non-rated).
□ SST Rating is the Swiss Solvency rating mapping and is defined similarly to BSCR Rating.
• Misc. Tools → NSS Many-Point Single-Curve Fitting
This is a tool for fitting a Nelson-Siegal-Svensson curve out of many curve data points.
□ Download the template and fill the “Input” tab with your own curve points data.
□ In the “Fitted” tab, enter the tenors for which you wish to obtain the fitted curve points.
□ Upload the template. It will be regenerated for you with the Nelson-Siegal-Svensson parameters populated and the fitted curve points populated.
• Misc. Tools → Smith-Wilson Batch Curve Generation
This is a convenient tool where you can generate a large number of Solvency II-compliant curves using the Smith-Wilson algorithm
□ Download the template and fill in the “Params” tab with Smith-Wilson parameters. The table column names must correspond to the rest tab names.
□ Each of the rest tabs contains a set of curves with the same tenor data.
□ Upload the template and a complete set of Smith-Wilson interpolated/extrapolated curves each with 150 years of length will be generated.
• Help & Demo Inputs → Download Demo Inputs
Offers downloadable input spreadsheet samples for each of the subforms. | {"url":"https://scrcalculator.com/usermanual.php?page=appendix_menu_items","timestamp":"2024-11-01T20:34:39Z","content_type":"text/html","content_length":"47981","record_id":"<urn:uuid:c42c770f-8f0d-416b-a646-54ec44705fea>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00676.warc.gz"} |
Why would one want to generalize notions such as convergence and continuity to a setting even more abstract than metric spaces?
Topology deals with the relative position of objects to each other and their features. It is not about their concrete length, volume, and so on. Hence, topological features will not change if
continuous transformations are applied to these objects. That is, topological features are preserved under stretching, squeezing, bending, and so on but they are not preserved under non-continuous
transformations such as tearing apart, cutting and so on. Objects such as a circle, a rectangle and a triangle are from a topological point of view “equal” / homeomorphic even though the shapes are
geometrically rather different.
What features are therefore of interest such that it is worth studying topology?
Assume that
Hence, the generalization of continuity and the concept of convergence (i.e. points being ‘close to each other’) are the two most characterizing features in topological spaces.
However, convergence and continuity in a metric space closeness“.
This post is based on the literature [1] to [5]. For English-speaking beginners, [5] is recommended. The German lecture notes [3] is also a good introduction to topology.
Topology on a Set
The term “topology on a set” is based on an axiomatic description of so-called “open sets” with respect to some set-theoretic operators. It will turn out, that a topology is a set that has just
enough structure to meaningful speak of convergence and continuous functions on it.
Open Sets
Definition 1.1 (Topological Space)
A topological space is a pair
Let topological space in this post.
The following video provides a rather unorthodox way of thinking about a topology. However, it might help to get a heuristic understanding. The connection between metrics and topologies is also
Topology vs. “a” Topology by PBS Infinite Series
Some examples will further support the understanding.
Example 1.1 (Topologies)
(a) Let trivial, chaotic or indiscrete topology. The only open sets of the trivial topology are
(b) The power set discrete topology. In this topology every subset is open.
(c) There are four topologies on the set
It is still an open problem, which is related to combinatorics and lattice theory, to find a simple formula for the number of topologies on a finite set.
(d) Let
of subsets of subspace of
(e) Let finite complement topology. First note, that an element
The union of arbitrary open as well as the intersection of a finitely many open sets are open again. The corresponding proof employs De Morgan’s Laws.
(f) Let countable complement topology.
Apparently, proofs of the union and intersection of open sets also employs De Morgan’s Laws.
According to Proposition 1.1 – Fundamentals of Topology & Metric Spaces, the set
Definition 1.2 (Induced & Equivalent Topology)
A topology induced by a metric space
Two metrics topologically equivalent if
Let us define a very basic but important term.
Definition 1.3 (Neighborhood)
A neighborhood
Some examples will improve the understanding.
Example 1.2 (Metrizable and Equivalent Topologies)
(a) If metrizable.
(b) Let us consider natural topology by taking the topology Euclidean metric
All three induced topologies would be equivalent, i.e.
for all
Fig. 1: Unit balls centered at (0,0) by three equivalent metrics
An open ball
From a topological point of view the shapes in Fig. 1 are all equivalent.
Closed Sets
Directly linked via the definition to open sets and equivalent in their explanatory power are closed sets.
A set
However, the open ball
Let us now study sets
Definition 1.4 (Closed Sets)
A set closed in
The definition actually tells us that one just needs to consider the complement set
Example 1.3 (Closed Sets)
(a) Let
(b) The sets
The topological space is trivial / indiscrete if and only if these two sets are the only closed sets in
(c) The topology
(d) The subset
The subsets neither open nor closed.
(e) In the finite complement topology on an infinite set
Let us characterize closed sets.
Proposition 1.1: (Characterization of Closed Sets)
(1) The set
(2) Let
(1) This follows directly from the definitions and applying the rules
(2) The family of closed sets
The family of closed sets of a topology could also be used to define a topological space, i.e. the set of all closed sets contains exactly the same information as the set of all open sets that
actually define the topology.
Points, that lie at the boundary between
Interior, Closure & Boundary
The points that lie close to both the “inside” and the “outside” of the set
Definition 1.5 (Interior & Closure)
Let interior of
The closure of
Apparently, the interior of
The following theorem provides some useful relationships. However, we will not prove all of the statements and refer to section 2.1 in [5], for instance.
Theorem 1.1 (Properties of Closure and Interior)
For sets
(i) If
(ii) If
(iii) If
Proof. (i) Since Int
(ii) Since Cl
(iii) Since
(v) If
Example 1.4 (Closure and Interior)
(a) Consider
(b) Consider
(c) Consider
The last example highlights that not only the actual sets matter but also the sourounding topology. The next theorem provides a simple means for determining when a particular point
Theorem 1.2 (Closure, Interior and Open Sets)
(i) Then
(ii) Then
Proof. (i) First, suppose that there exists an open set
Next, if
(ii) Considering the contrapositive:
Conversely, if there is a neighborhood
Theorem 1.3 (Relations of Closure and Interior)
For sets
(i) Int
(ii) Cl
(iii) Int
(iv) Int
Proof. Refer to Theorem 2.6 in [5] and to Folgerung 1.2.25 in [3].
Now, we have all ingredients to define the so-called boundary.
Definition 1.6 (Boundary Set & Points)
Let boundary of boundary point.
There are situations that challenge or defy our intuitive understanding of boundary sets. For example, what is the boundary of
Example 1.4 (Boundary Sets & Points)
(a) Consider
(b) Consider
(c) Let
In a metric space, a point
for all
Let us now prove the generalized statement about boundary sets.
Proposition 1.2 (Characterization of Boundary Points)
Proof. Suppose
Now suppose that every neighborhood of
Theorem 1.4 (Open Sets and Neighborhoods)
Proof. First, suppose that
Now suppose that for every
The set
• If all boundary points
• If all boundary points are contained within the set, then it is a closed set.
Proposition 1.2 (Closure and Open Sets)
Proof. The closure
The property of the closure hitting an open set is so important that usually a new term is defined. Note, however, that we will not further use it in this post.
Definition 1.7 (Adherent Point)
Let adherent point if every open
Let us come back to Example 1.2 (a) – considering this example we could ask whether every topological space is metrizable?
The answer is no, and the root-cause is that topological spaces have different types of separation properties.
Separation Properties
A metric enables us to separate points in a metric space since any two distinct points have a strictly positive distance. In general topological spaces, separating points from each other is more
Hausdorff Space
Hausdorff spaces and the Hausdorff condition are named after Felix Hausdorff, one of the founders of topology. Let us first check out the formal definition.
Definition 2.1: (Hausdorff Space,
A topological space Hausdorff or
Every Euclidean space is Hausdorff since we can use the Euclidean metric to separate two distinct points. The following video outlines the Hausdorff condition and it provides a simple example of a
Hausdorff space.
Hausdorff Condition incl. an Example and Counterexample by DanielChanMaths
Example 2.1 (Metric Space is Hausdorff)
(a) Let
(b) Consider the indiscrete / trivial topology
In a Hausdorff space, distinct points can be separated by open sets.
The situation in
Proposition 2.1: (Subset of
Proof. Let
Spaces with Weaker Separation Property
The following separation properties are weaker than the Hausdorff (
Definition 2.2. (
A topological space or Kolmogorov space if, for any
The most striking difference between a Hausdorff /
Example 2.2:
(a) Let
(b) Let
In a
Definition 2.3. (
A topological space -space if, for any
The main difference between a
Hence, every
Proposition 2.2: (Characterization of
(a) Let
(b) Each Hausdorff space is a
Proof. (a) Suppose
Conversely, suppose that all singleton subsets of
(b) Let
Convergent Sequences
One of the key features of topological spaces is the generalization of the convergence concept.
A sequence in a (metric) space
We say that a sequence converges to
In other words, for all
Definition 3.1. (Convergent Sequence)
Let sequence converges to
The point limit of the sequence
Note that the set of
Lemma 3.1. (Limit of a sequence is unique)
The limit
Proof. Assume that this is not the case and
Lemma 1.1 is false in arbitrary topological spaces.
Example 3.1:
(a) Let
(b) Let
Closely related to converging sequences and their limits are accumulation points.
Definition 3.2. (Accumulation Point)
An element accumulation point (sometimes also cluster or limit point) if
The subtle but important difference between an accumulation point and a limit is that the complement set of
Example 3.2:
Let us consider the sequence
A sub-sequence
Due to the fact that the finiteness of Example 3.2.
Theorem 3.1 (Convergence in Topological Spaces)
(i) Every limit of a convergent sequence is also the limit of any sub-sequence.
(ii) Every accumulation point of any sub-sequence
(iii) Every accumulation point of a sequence
Proof. (i) If
(ii) Let
(iii) Let
Let us now assume that
The concept of compactness is not as intuitive as others topics such as continuity. In compact sets are the closed and bounded sets, but in a general topology compact sets are not as simple to
Compact sets are so important since they possess important properties, that are known from finite sets:
The famous Heine-Borel Theorem shows that compact sets in metric spaces do indeed have these properties. This analogy is also outlined in this really nice video (in German only) by Prof. Dr. Edmund
Definition 4.1 (Cover)
The collection cover a set
If open cover of
A sub-collection subcover of
A cover of
Example 4.1 (Real Line)
Let us consider the set finite or countable union of open mutually disjoint intervals.
Now, we have the ingredients for the central definition of this section.
Definition 4.2 (Compact Set)
A subset compact if every open cover finite (open) subcover.
A topological space compact if
It is clear that every finite set is compact and the following example is going to illustrate that.
Example 4.2 (Finite Set & Compactness)
Let us consider another simple example.
Example 4.3 (Real Line)
The real line
The last example directly used the definition of a compact set to show that it is not compact since every open cover needs to have a finite sub-cover such that the set can be compact.
Even though all open covers (including the one defined in Example 5.2/5.3) would have to have a finite sub-cover.
Example 4.4 (Converging Sequence & Compact Set)
(a) The subset
Given any open cover
(b) The compact sets of the discrete topology formal proof.
Let us now extend the definition of compactness to subsets of topological spaces.
Definition 4.1 (Subspace Topology)
is a topology on subspace topology. With this topology,
Let us check that
Lemma 4.1 (Compactness & Subspaces)
Proof. Suppose that
Suppose that every open cover of
A closed subspace is a subspace
Theorem 4.1 (Compactness & Closed Spaces)
Every closed subspace
Proof. Let
Theorem 4.2 (Compactness in Hausdorff Spaces)
Every compact subspace
Proof. Let
Example 4.5 (Theorems 4.1 and 4.2)
(a) Once we prove that the interval
(b) The Hausdorff condition in Theorem 4.2 is necessary. Consider the finite complement topology on the real line. The only proper subsets of
(c) The interval
Continuous Functions
Topological spaces have been introduced because they are the natural habitat for continuous functions. These spaces have been built such that the topological structure is respected. Continuous
functions therefore take the same role on topological spaces as linear maps within vector spaces.
The notion of continuity is particularly easy to formulate in terms of open (and closed) sets and the following version is called the open set definition of continuity.
Definition 5.1: (Continuous Function on Topological Space)
Let function continuous if
Before we will illustrate this definition let us recall the definitions of an image preimage
The condition that
Example 5.1: (Simple Continuous Function)
The functions
Let us now study how the functions map closed sets. By definitions these are the complements of
Recognize that the image of a closed set in
The last example made the definition of a continuous function between simple topological spaces rather clear.
Sometimes it is also helpful to study the properties that will not be preserved: a continuous function does not necessarily map open sets to open sets.
For example, the function
Now, let us have a look at more general examples.
Example 5.2: (Identity and Constant Function)
(a) The identity function id:
(b) The constant function
Continuous functions preserve proximity as we can see in the next theorem. Also refer to Example 4.1.
Theorem 5.1 (Continuous Functions & Closeness)
Proof. Suppose that
Hence, suppose that
The next theorem translates the well-known
Theorem 5.2 (Continuity &
A function
Proof. First, suppose that the open set definition holds for functions
Now assume that for every
Theorem 4.2 generalizes this idea of continuity in metrizable topological spaces to general topological spaces. In a metric space, we can consider an open ball as an open set and therefore as a
neighborhood. That is, for each deep-mind.org post and this Wikipedia article.
The second important property that is preserved by continuous functions is the concept of convergence.
Theorem 5.3 (Continuity & Convergent Sequences)
Assume that
Proof. Let
Another important property of continuous functions is directly linked to the actual definition.
Theorem 5.4 (Continuity & Pre-Image of Closed Sets)
Proof. Let
Theorem 5.5 (Continuous Functions & Compact Sets)
Proof. Let
A really nice introduction to the abstract concept of a topology, however, in German language only.
4421349 {4421349:MQPZQJTB} 1 harvard1 50 default 5315
Kuratowski, K. (1966) Topology: Transl. from French. New York: Academic Press [u.a.] (1).
4421349 {4421349:R7XT3GES} 1 harvard1 50 default 5315
Runde, V. (2005) A taste of topology. New York: Springer (Universitext).
4421349 {4421349:MI5DALFS} 1 harvard1 50 default 5315
Horst Herrlich (no date) Topologische Räume. FernUniversität in Hagen.
4421349 {4421349:6JLY6M4D} 1 harvard1 50 default 5315
Munkres, J.R. (2014) Topology. 2. ed., Pearson new internat. ed. Harlow: Pearson.
4421349 {4421349:5K9US5SW} 1 harvard1 50 default 5315
Adams, C.C., Franzosa, Robert and Dorling Kindersley (2009) Introduction to topology: pure and applied. Uttar Pradesh: Dorling Kindersley. | {"url":"https://www.deep-mind.org/tag/metricspace/","timestamp":"2024-11-12T10:44:01Z","content_type":"text/html","content_length":"554731","record_id":"<urn:uuid:49d4769f-44b7-4b58-9fea-c820574ea24a>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00495.warc.gz"} |
Propositions - The Next Generation Logic Framework
A Proposition is a formula with additional information like a textual description or a user-provided object.
Application Insight
Storing this extra information with the formulas can be useful in various application contexts. For example, we use propositions when we know that we will not only be interested in the result of an
algorithm, but also the explanation of its result. For more information see the chapter on explanations. The big advantage here is that the original formula context is maintained. When adding a
formula to the SAT solver, internally the formula is transformed to CNF and single clauses are added to the solver. When you now extract an explanation for an unsatisfiable formula from the solver,
the result will contain these single clauses which often are hard to map to the original input formulas. By using propositions, the result will be in terms of the original prospotision which makes
understanding the explanation much easier in practise.
The abstract class Proposition has the single abstract method formula() which returns the formula of the proposition. LogicNG provides two implementations of a Proposition:
Standard Proposition
The StandardProposition holds a formula and a textual description. You can configure a proposition with and without a description:
1. With a description:
generates the proposition StandardProposition{formula=A | ~B & C, description=my formula}
2. Without a description:
generates the proposition StandardProposition{formula=A | ~B & C, description=}
Extended Proposition
The idea from extended propositions is to store additional domain-specific information with a formula. This information is not used for any algorithms in LogicNG - however, it can be useful to "drag
it along" during your application.
An ExtendedProposition is a formula with additional information provided in a user-defined object which implements the empty (marker-) interface PropositionBackpack.
In your implementation of the PropositionBackpack you can store all sorts of information which you want to keep to your formula. Some examples are: An ID from the respective rule system the formula
is from, the person who is responsible for the formula, the origin or the type of the formula.
You can think of this information as literally the "backpack" of the formula. No algorithm in LogicNG looks "inside" this backpack, but the backpack is always kept. For example, if LogicNG performs
algorithms on the formula, the result still holds the backpack, and maybe this helps you to understand the result better.
Let us consider an example of using the extended proposition with an own backpack MyBackpack. Our backpack stores an ID, a person responsible for this formula and the rule type:
class MyBackpack implements PropositionBackpack {
private final long id;
private final String responsiblePerson;
private final RuleType ruleType;
MyBackpack(final long id, final String responsiblePerson,
final RuleType ruleType) {
this.id = id;
this.responsiblePerson = responsiblePerson;
this.ruleType = ruleType;
public boolean equals(final Object o) {
if (this == o) {
return true;
if (o == null || getClass() != o.getClass()) {
return false;
final MyBackpack that = (MyBackpack) o;
return id == that.id &&
Objects.equals(responsiblePerson, that.responsiblePerson) &&
ruleType == that.ruleType;
public int hashCode() {
return Objects.hash(id, responsiblePerson, ruleType);
public String toString() {
return "MyBackpack{" +
"id=" + id +
", responsiblePerson='" + responsiblePerson + '\'' +
", ruleType=" + ruleType +
enum RuleType {
Let's generate some propositions:
new ExtendedProposition<>(new MyBackpack(1, "Rouven", RuleType.EQUIV),
f.equivalence(p.parse("A & B"), p.parse("C")));
new ExtendedProposition<>(new MyBackpack(2, "Verena", RuleType.IMPL),
f.implication(p.parse("A"), p.parse("C | D")));
new ExtendedProposition<>(new MyBackpack(3, "Martin", RuleType.CC),
f.amo(f.variable("A"), f.variable("C")));
The resulting propositions are:
ExtendedProposition{formula=A & B <=> C, backpack=MyBackpack{id=1, responsiblePerson='Rouven', ruleType=EQUIV}}
ExtendedProposition{formula=A => C | D, backpack=MyBackpack{id=2, responsiblePerson='Verena', ruleType=IMPL}}
ExtendedProposition{formula=A + C <= 1, backpack=MyBackpack{id=3, responsiblePerson='Martin', ruleType=CC}} | {"url":"https://logicng.org/documentation/propositions/","timestamp":"2024-11-09T21:49:29Z","content_type":"text/html","content_length":"81685","record_id":"<urn:uuid:d58a8e07-36a5-45c3-bd9b-5221a5dac5b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00339.warc.gz"} |
Unraveling Key Concepts Of Similar Triangles - Dive In!
Exploring Similar Triangles: Unraveling the Key Concepts and Properties
Welcome to Warren Institute! In this article, we will explore the concept of similar triangles and their significance in Mathematics education. Similar triangles are polygons that have the same shape
but may differ in size. Understanding their properties is crucial for various geometric applications. Join us as we delve into the world of similar triangles and uncover their remarkable properties.
Stay tuned!
Understanding Similar Triangles
Similar triangles are an important concept in geometry that play a significant role in mathematics education. In this section, we will explore what similar triangles are and why they are relevant to
the study of mathematics.
Similar triangles are triangles that have the same shape but may differ in size. The corresponding angles of similar triangles are equal, and the corresponding sides are proportional. This means that
if we have two triangles that are similar, we can compare the lengths of their sides by setting up ratios.
For example, consider two triangles ABC and DEF. If angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F, then the triangles are similar. We can also
state that the ratio of the lengths of corresponding sides of these triangles, AB/DE, BC/EF, and AC/DF, will be equal.
Understanding similar triangles is essential in mathematics education as it helps students develop geometric reasoning skills and enables them to solve problems involving proportions and scaling. By
recognizing and applying the properties of similar triangles, students can solve various real-world problems, such as determining the height of a building or finding the distance between two objects.
Applications of Similar Triangles
Similar triangles find numerous applications in various fields, including architecture, engineering, physics, and art. In this section, we will delve into some practical applications of similar
triangles and how they are used in real-life scenarios.
One common application of similar triangles is in map scaling. Maps are scaled-down representations of real-world locations, and accurate scaling is crucial for proper navigation. By using similar
triangles, cartographers can determine distances, heights, and angles on a map based on known measurements in the real world.
Another application is in the field of photography. Similar triangles help photographers determine the optimal distance and height from which to take a picture to capture the desired composition. By
understanding the relationship between similar triangles and perspective, photographers can achieve visually pleasing images.
In architecture and engineering, similar triangles play a vital role in designing and constructing structures. Architects and engineers use the principles of similarity to scale and proportion
buildings accurately. Similar triangles allow them to calculate the dimensions of various elements, such as windows, doors, and roof slopes, ensuring structural integrity.
Proving Similarity of Triangles
Establishing the similarity of triangles is an essential step when working with geometric figures. In this section, we will discuss different methods to prove the similarity of triangles and explore
the theorems associated with it.
One method to prove triangle similarity is by using the Angle-Angle (AA) criterion. If two triangles have two pairs of congruent angles, then they are similar. This criterion is based on the fact
that if two angles of one triangle are equal to two angles of another triangle, the third angles must also be congruent.
Another way to prove similarity is by using the Side-Angle-Side (SAS) criterion. If two triangles have two pairs of proportional sides and the included angle between them is congruent, then the
triangles are similar. This criterion ensures that both the ratios of corresponding sides and the congruence of the included angle are satisfied.
The Side-Side-Side (SSS) criterion can also be used to prove similarity. If the corresponding sides of two triangles are proportional, then the triangles are similar. This criterion is based on the
fact that if all three sides of one triangle are proportional to the corresponding sides of another triangle, the angles must also be congruent.
Solving Problems with Similar Triangles
Solving problems involving similar triangles relies on applying the properties and theorems associated with them. In this section, we will explore some common problem-solving techniques that involve
similar triangles.
One commonly used technique is solving for unknown side lengths. By setting up ratios using corresponding sides of similar triangles, we can determine the length of an unknown side. For example, if
we know the lengths of two sides of a triangle and the corresponding lengths of a similar triangle, we can use cross-multiplication to find the missing length.
Another technique is solving for unknown angles. Since corresponding angles of similar triangles are congruent, we can use this property to find unknown angles. By knowing the values of some angles
in one triangle and the corresponding angles in a similar triangle, we can find the measure of the unknown angle using subtraction or addition.
Students can also solve problems involving similar triangles by using proportionality. If we have a known ratio between the lengths of sides in one triangle and the lengths of corresponding sides in
a similar triangle, we can set up a proportion and solve for the unknown length.
By utilizing these problem-solving techniques, students can apply the concept of similar triangles to various mathematical and real-world scenarios, aiding their mathematical understanding and
critical thinking skills.
frequently asked questions
What are similar triangles and how are they defined in mathematics education?
Similar triangles are geometric figures that have the same shape but may differ in size. In mathematics education, they are defined as triangles that have proportional sides and equal corresponding
angles. This means that if two triangles are similar, their corresponding angles are congruent and the ratios between the lengths of their corresponding sides are equal.
What are the properties of similar triangles and why are they important in mathematics education?
The properties of similar triangles include having corresponding angles that are congruent and corresponding sides that are in proportion. These properties are important in mathematics education
because they enable students to solve problems involving proportional reasoning and geometric relationships. Understanding and applying the properties of similar triangles also serve as a foundation
for more advanced topics like trigonometry and geometry proofs.
How can we use similar triangles to solve problems and improve understanding in mathematics education?
We can use similar triangles to solve problems and improve understanding in mathematics education by:
• Identifying and applying the properties of similar triangles to find missing side lengths or angles.
• Using the concept of proportionality to set up and solve equations involving similar triangles.
• Applying the knowledge of similar triangles to solve real-life problems, such as determining heights or distances.
• Recognizing and utilizing the relationship between corresponding sides and angles of similar triangles.
• Applying the concept of similarity to other geometric figures, such as polygons or circles.
In conclusion, understanding the concept of similar triangles is crucial in Mathematics education. By identifying and working with similar triangles, students can unlock a wide range of geometric
principles and applications. It allows them to solve problems involving proportions, congruence, and trigonometry. Moreover, recognizing similar triangles plays a significant role in geometry
proofs and in real-life situations such as map scaling and architectural design. By grasping this fundamental concept, students can build a strong foundation for further mathematical exploration
and problem-solving skills. Therefore, educators should emphasize the importance of similar triangles in the curriculum and provide adequate opportunities for students to practice and apply their
If you want to know other articles similar to Exploring Similar Triangles: Unraveling the Key Concepts and Properties you can visit the category Geometry. | {"url":"https://warreninstitute.org/what-are-similar-triangles-2/","timestamp":"2024-11-06T01:08:19Z","content_type":"text/html","content_length":"106805","record_id":"<urn:uuid:9c2dc564-aee4-4586-8b26-7114e01bfc43>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00748.warc.gz"} |
What is the assumption of no omitted variables bias in panel data modeling? | Hire Some To Take My Statistics Exam
What is the assumption of no omitted variables bias in panel data modeling? \[sec:hypo\_misparamd\].1\] In Hypothesis \[hypo\_misparamd\], the assumptions of no omitted variables bias in the
analysis. The null hypothesis is assumed to be null. We can use panel data to account for missing data or observations and the Null Hypothesis \[hypo\_misparamd\] to justify the more common
assumption of no omitted variables bias as well as null hypothesis. Simulation studies ================== In the simulation study, we study the time series of temperature data and several other time
series as a continuous process. Specifically, we represent both data and thermodynamics in a single “model” parameter space. To simulate data, we assume the same “model” for thermodynamics as the
data and represent them as a continuous superlinear distribution consisting of mixed-asymptotic (MAs), bicubic (BC), and unimodal (MN) parts. For each simulation study, we study the simulation
points-to-result plot (SPR) with data. We obtain the parameter values and their regression coefficients for each variable as a covariate. The linear sum of coefficients for each variable leads to the
regression coefficients. We then perform univariate least-squares regression (LS-RM) to obtain the regression coefficients for individual variables as an outcome. While multivariate regression is
more accurate in more complex settings, in the real data, our interest to study the effects of bias and other imprecision becomes less significant because of the high complexity of the true data
space and the difficulty in doing multiple different equations to estimate it. To facilitate the simulation study, the remaining data provides a variety of additional observations while minimizing
the effect of imprecision. [^1]: Reviewing for obviousness, we include the heat model as a joint continuous process model. [^2]: Simulated: 1.44 (see Discussion below). [^3]: Here and in the
following, the set click over here observed and expectation values is a ‘template’ for the data. We fix all exponents other than zero to avoid unwanted deviation from the same. We set each variable’s
magnitude as $1$, while the zero ones give a distribution model. What is the assumption of no omitted variables bias in panel data modeling? A few years ago, I wrote an article, entitled “Webbist’s
Box: A Case Study of the Negligible Effects of Self-Confidentity on the Risk of Disease.
Sites That Do Your Homework
” To understand why the paper’s title attracted so many criticisms, I looked at it from several perspectives: 1) How bias in the association of these variables with disease risks really matters, and
how do they account for any inherent discrepancy in the results of systematic studies? 2) The impact of “obvious explanations” in a model in which the person is an individual, but not a collective
“self”, on “outcomes” of people who share the same symptoms/anatomical characteristics that are described(i.e. a similar symptom was occurring in people who had been diagnosed differently as a
condition of the same condition) A few papers, by a few authors, have shown that the tendency to inflate the associations of self-reporting with other symptoms of general (neuropsychiatric or
autoimmune) and/or high(f), but not with disease-related symptoms, when studied as an outcome measures; the authors concluded that it was because their data explicitly show that people with less
frequent reporting criteria, without extra-exhibitors of self-investigation, suffered from a disproportionate burden of “obvious explanations” (i.e. a similar symptom was occurring in more than them)
and that people who reported health conditions that resulted in the disease benefit least from health examination in the evaluation prior to their treatment and, worse be estimated to show increased
disease burden which is actually causal if self-reports were based on clinical criteria as compared to clinical self-report. I have been speaking at conferences, such as Good Morning America or
Mezzabrry, where I had the opportunity to present my presentation last week at the American College of Medical Genetics. Many of the authors were there to argue about whether those studies were in
the best interest and to understand the most realistic limitations to the association results, and their responses (and the implications for our methods). On the above, the author argued that the
association of self-reported symptoms, ie: “lack”, “lack”, “flimceasy” in general for every condition, to any given individual will indicate a significant difference between the sample population,
and perhaps to a degree, at least during follow-up, as the general effect of the association (and the sample size) are known. The study also suggests that some people may get more or less precise
reporting criteria and their symptoms are more likely to be associated with their symptoms, were they gone, so that a lower proportion of the population at risk of being diagnosed with a possible
condition (such as Alzheimer’s) is more or less the same as the population in the sample of peopleWhat is the assumption of no omitted variables bias in panel data modeling? The statement: “‘Each
month if time spent taking corrective action was small approximately 100 times’” seems bit harsh but I want to draw a straight line. Does anyone know why it happened? A couple years ago, there was a
paper at TPS titled “One Foot in the City of Shippers” [www.newtvsus.org/news/286842-203837/single-foot-in-the-city-of-shippers], where some (1 in 3) people say that taking corrective action appears
to be one of the biggest mistakes made by the city. The headline states: “‘Everyone who sits at his computer looks like their computer is getting information that it never would have been made to
function normally. Like most everything else you can imagine, a computer is just a few steps away from its nearest living, working, office.” And from any given year, if its about “the day” rather
than “the week”, how deep does this error in writing fall? It is a mistake like an application never existed and all i know is that since early systems never worked, i would go to these guys that
they do. The year after that, is it the end of the year when the errors that hit the day hit, or are we still missing the story? About Me Before i got started taking care of myself or anyone i’m
trying to get out of my world, the love for computer education and knowledge kept me going with my eyes to. Today as a small business, i’ve been teaching of some sort in the shops on the east end of
Yucatán (seated like a bulldog) but i’ve not been able to make myself out there. | {"url":"https://hireforstatisticsexam.com/what-is-the-assumption-of-no-omitted-variables-bias-in-panel-data-modeling-2","timestamp":"2024-11-07T04:02:03Z","content_type":"text/html","content_length":"169018","record_id":"<urn:uuid:fc71ddcc-fdc2-4b5e-afa0-45cd75f5a1d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00366.warc.gz"} |
How to Use Options Greeks? | Learn with Upsurge.club
Chapter 10: How to Use Options Greeks to Trade Better?
Options trading can be a powerful tool to manage risk, enhance returns, or speculate on market movements.
However, to become a successful options trader, it’s essential to grasp the concept of “Options Greeks.”
These Greek letters represent a set of metrics that help traders better understand and manage their positions.
In this Chapter, we will break down what Options Greeks are and how they can be used to trade options more effectively.
What are Options Greeks?
Options Greeks are a group of risk metrics that quantify various aspects of an options contract.
They help traders evaluate and predict potential changes in an option’s price with factors such as underlying asset price movements, time decay, implied volatility changes, and interest rates.
Each Greek letter corresponds to a different aspect of options pricing and risk management:
• Delta
• Gamma
• Theta
• Vega
• Rho
1. Delta
Delta measures how much an option’s price will change in response to a 1 Rupee change in the underlying asset’s price.
Put options have negative delta whereas call options have positive delta and It ranges from -1 to 1 for put and call options, respectively.
A higher delta means the option’s price moves more closely in line with the underlying asset.
For example, if you have a call option with a delta of 0.70 and the underlying stock increases by Rs. 10, the option’s price would rise by ~ 7 rupees.
2. Gamma
Gamma measures the rate of change of an option’s delta concerning changes in the underlying asset’s price.
It tells you how delta itself changes as the stock price moves. Gamma is highest for options that are near the money and close to expiration.
For example, if your option has a gamma of 0.05, its delta will change by 0.05 for every 1 rupee move in the underlying asset’s price.
3. Theta
Theta quantifies the rate at which an option’s value decreases with the passage of time, also known as time decay.
It’s particularly crucial for traders holding options contracts, as time decay can erode the value of the option.
For example, if your option has a theta of -0.50, its value will decrease by Rupees 0.50 (50 paise) per day, all else being equal.
4. Vega
Vega measures how much an option’s price will change for each percentage point change in implied volatility.
It reflects sensitivity to changes in market sentiment and can be crucial during volatile times.
For example, if your option has a vega of 0.5, it should increase by Rupees 0.50 (50 paise) for every 1% increase in implied volatility.
5. Rho
Rho indicates how much an option’s price will change for a 1% change in interest rates.
This Greek is less critical for short-term traders but can be relevant for longer-term options.
For example, if your option has a rho of 0.05, its price should increase by 0.05 rupees Or 5 paise for every 1% increase in interest rates.
How Can Options Greeks Help in Options Trading?
One can argue over the fact that option Greeks are quite different in theory than in practice.
However, it’s really important to understand these concepts in theory and then apply the logic behind these concepts to improve your trading.
Having said that, option Greeks can have multiple applications and can be used to design various option trading strategies too.
The most common usecase by understanding and applying Options Greeks like Delta and Gamma is – you may be able to protect your portfolio during a market downturn and capitalize on the changing
dynamics to enhance your hedging strategy.
Here are some examples on how each one can be used to enhance your options trading strategies:
1. Delta for Directional Trading Or Option Selling
Delta can help you select options that match your market outlook. For bullish views, choose call options with high positive delta values. For bearish views, select put options with high negative
delta values.
You also need to keep in mind that higher delta values could have higher fluctuations in your MTMs.
If the underlying asset is volatile, the higher delta value option premiums will also be volatile compared to the lower delta options.
So it’s important to choose a right strike price which is aligned to your risk reward matrix.
Delta can also help you in strike selection. For example, if you are an option writer, selling call or put options but your strategy is designed to handle minimum risk, then you might want to sell
OTM options with lower delta values.
That’s because there is lesser volatility and thus, you could have a higher probability of making profits in your strategy versus selling higher delta value options.
Options with higher delta values indicate that the option’s price is more sensitive to changes in the underlying asset’s price, meaning it will move more in sync with the stock’s movements.
2. Gamma for Risk Management
Gamma is crucial for managing your delta risk. If you want to keep a specific delta, you’ll need to adjust your position regularly as gamma changes. This is especially important when hedging or
managing a portfolio of options.
For example, if you know the gamma values of your positions, it’s easier to predict how fast the option prices can move incase of a sharp move in the prices of the underlying asset
3. Theta for Time Decay Strategies
Theta can guide you in selecting the right time horizon for your trades.
If you’re trading options with limited time to expiration, you need to be aware of theta’s impact.
Options with high theta can be suitable for short-term trades, while those with low theta might be better for longer-term strategies.
4. Vega for Volatility Trading
Vega can help you gauge market sentiment and adapt your strategy accordingly.** In times of expected volatility, you might favour options with higher vega to capitalise on potential price swings.
5. Rho for Interest Rate Sensitivity
Rho is most relevant when interest rates are expected to change significantly. If you anticipate interest rate movements, consider options with higher rho values to potentially benefit from these
rate changes.
Let’s put it all together in an example
Option Greeks in Practice
Successful options trading involves a combination of these Greeks, depending on your strategy and market conditions.
Let’s explore another example of how to use Options Greeks to trade better. Imagine you’re a trader expecting high volatility in Bajaj Finance stock due to an upcoming earnings report.
This quarter has been good for the company and you are expecting that the results to be exceptionally good. The CMP of Bajaj Finance is 7400 and you are expecting a sharp move in the coming days
before the quarterly results.
To navigate this, you analyze the Options Greeks.
1. Using Delta for Guidance
By looking at the option strikes available, you plan to choose a call option having a Delta of 0.70.
This implies that for every 10 Rupee increase in the stock, your option’s value should go up by around 7 Rupee.
Thus, it aligns with your bullish outlook as if the spot price of the stock will see a spike, the option you choose will see some great momentum.
2. Analysing Option Time Sensitivity by Theta
Recognizing that Theta of the same call option -0.03, and you are okay with this theta since you are anyway expecting some momentum in the shorter term.
A higher theta value would mean that time decay may impact your option premium prices in case there is no momentum during the holding period.
But in our case, since we are banking on the price of Bajaj finance to increase quickly (within some days), the above theta value seems fair to us and probably your call option will be less impacted
by time decay.
3. Gauging Volatility with Vega
Seeing a Vega of 0.15, you anticipate a potential spike in implied volatility around the earnings report. This insight encourages you to hold onto the option, expecting an increase in its value.
4. Hedging with Gamma
Later, the stock starts moving. Keeping the Gamma of 0.07 in mind , you adjust your position as the stock price shifts. This helps maintain your desired Delta, preventing overexposure.
By incorporating Options Greeks into your strategy, you’ve strategically chosen an option that aligns with your outlook and risk tolerance.
As the stock behaves, you use Gamma to fine-tune your position, maximizing potential gains while managing risk.
This example showcases how traders use Options Greeks to make informed decisions and adapt to market dynamics. | {"url":"https://learn.upsurge.club/guides/stock-market-guides/options-trading-guide/how-to-use-options-greeks/","timestamp":"2024-11-03T01:19:40Z","content_type":"text/html","content_length":"126054","record_id":"<urn:uuid:c7fef58f-fad8-4d68-82e2-2b52b61cc004>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00106.warc.gz"} |
t-Test, Chi-Square, ANOVA, Regression, Correlation...
Friedman Test
The Friedman test is a non-parametric statistical test used for analyzing repeated measures data. It is mainly used when the assumptions of normality and homogeneity of variances are not met, making
it a robust alternative to repeated measures ANOVA.
What is a dependent sample (repeated measure)? In a dependent sample, the measured values are connected. For example, if a sample is drawn of people who have knee surgery and these people are each
surveyed before the surgery and one and two weeks after the surgery, it is a dependent sample. This is the case because the same person was interviewed at multiple time points.
Friedman Test vs. ANOVA with repeated measures
You might rightly say that the analysis of variance with repeated measures tests exactly the same thing, since it also tests whether there is a difference between three or more dependent samples?
That is correct, the Friedman test is the non-parametric counterpart of the analysis of variance with repeated measures. But what is the difference between the two tests?
The analysis of variance tests the extent to which the measured values of the dependent sample differ. The Friedman test, meanwhile, uses ranks rather than the actual measured values.
The point in time where a person has the highest value gets rank 1, the point in time with the second highest value gets rank 2 and the point in time with the smallest value gets rank 3. This is now
done for all persons or for all rows. Afterwards the ranks of the single points of time are added up.
At the first time we get a sum of 7, at the second time we get a sum of 8 and at the third time we get a sum of 9. Now we can check how much these rank sums differ.
Why are ranks used? The big advantage is that if you don't look at the mean difference, but at the rank sum, the data doesn't have to be normally distributed.
Simplified, if your data are normally distributed, parametric tests are used. For more than two dependent samples, this is ANOVA with repeated measures.
If your data are not normally distributed, non-parametric tests are used. For more than two dependent samples, this is the Friedman test.
Hypotheses in the Friedman test
This brings us to the research question, which you can answer with the Friedman test. The research question is, is there a significant difference between more than two dependent groups? The null and
alternative hypothesis are therefore:
• Null hypothesis: there is no significant difference between the dependent groups.
• Alternative hypothesis: there is a significant difference between the dependent groups.
Of course, as already mentioned, the Friedman test does not use the true values, but the ranks.
Friedman test example
You might be interested to know whether therapy after a herniated disc has an influence on the patient's perception of pain. For this purpose, you measure the pain sensation before the therapy, in
the middle of the therapy and at the end of the therapy. Now you want to know if there is a difference between the different time points.
So, your independent variable is time, or the progress of the therapy over time. Your dependent variable is the perception of pain. You now have a progression of pain perception from each person over
time and now you want to know if the therapy has an effect on the pain perception.
Put simply, in this one case the therapy has an influence and in this other case the therapy has no influence on the pain perception. In the course of time, the pain perception does not change in
this case, and it does in that other one.
Calculate Friedman test
Let's say you want to investigate whether there is a difference in the responsiveness of people in the morning, at noon and in the evening. For this purpose, you measured the reactivity of 7 people
in the morning, at noon and in the evening.
In the first step we have to assign ranks to the values. For this we look at each row separately.
In the first row, or in the first person, 45 is the largest value, this gets rank 1, then comes 36 with rank 2 and 34 with rank 3. We now do the same for the second row. Here 36 is the largest value
and gets rank 1, then comes 33 with rank 2 and 31 with rank 3. We now do this for each row.
Afterwards we can calculate the rank sum for each time of the day, so we simply sum up all ranks at each column. In the morning we get 17, at noon 11 and in the evening 14.
If there were no difference between the different time points in terms of reaction time, we would expect the expected value at all time points. The expected value is obtained with the first equation
on the image and in this case it is 14. So if there is no difference between morning noon and evening, we would actually expect a rank sum of 14 at all 3 time points.
Next we can calculate the Chi^2 value, we get it with the second ecuation on the image. N is the number of persons, i.e. 7, k is the number of time points, i.e. 3 and the sum of R^2 is 17^2 + 11^2 +
14^2. Thus we get a Chi^2 value of 2.57.
Now we need the number of degrees of freedom. This is given by the number of time points minus 1, so in our case 2.
At this point we can read the critical Chi^2 value in the critical values table. For this we take the predefined significance level, let's say it is 0.05 and the number of degrees of freedom. We can
read that the critical Chi^2 value is 5.99. This is greater than our calculated value. Thus, the null hypothesis is not rejected and based on this data, there is no difference between the
responsiveness at the different time points. If the calculated Chi^2 value were greater than the critical one, we would reject the null hypothesis.
Calculate Friedman test with DATAtab
For the calculation of the Friedman test you can simply use DATAtab. To do this, simply go to the Friedman test calculator on DATAtab and copy your own data into the table.
Now we get the results for the Friedman test.
First you get the descriptive statistics. Then you can read the p-value. If you don't know exactly how to interpret the p-value, you can simply look at the interpretation in words. A Friedman test
showed that there is no significant difference between the variables. Chi^2 = 2.57, p = 0.276
If your p-value is greater than your set significance level, then your null hypothesis is not rejected. The null hypothesis is that there is no difference between the groups. Usually, a significance
level of 0.05 is used, so this p-value is greater.
Post-Hoc Test
In addition, DATAtab provides you the post-hoc test. If your p-value is smaller than 0.05 you can examine here which of the groups really differ!
Here, two groups are considered in each row and the null hypothesis is tested whether both samples are the same, the "Adjusted p-value" is obtained by multiplying the p-value by the number of tests.
If the post-hoc test indicates that the p-value is less than 0.05, it is assumed that these groups are different.
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• ideal for exams and theses
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"It could not be simpler"
"So many helpful examples" | {"url":"https://datatab.net/tutorial/friedman-test","timestamp":"2024-11-05T01:11:34Z","content_type":"text/html","content_length":"63860","record_id":"<urn:uuid:c1063d28-6ab5-444d-9e03-c190d7c6a928>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00769.warc.gz"} |
I got asked for the design of my academic posters. Indeed I have templates in landscape and portrait and I’m happy to share them. In addition I can recommend the blog better-posters which has
regularily features and link-roundups on poster-design related things.
In my newest poster (landscape below) I tried to move as much text to the side, so that people can still understand the poster, but it does not obscure the content. I also really like the 15s
summary, an easy way to see whether you will like the poster, or you can simply move on. Maybe it even needs to be a 5s summary!
These are two examples posters based on my template.
Neat Features
Titles’ backgrounds follow along
This is useful because you do not manually need to resize the white background of the text that overlays on the borders
Borders are effects, easy resizing
The corners are based on illustrator effects, thus resizing the containers does not change the curvature. Before I often had very strange curvatures in my boxes. No more!
Download here
Portrait Equal Columns (ai-template, 0.3mb)
Portrait Unequal Columns (ai-template, 0.3mb)
Landscape (ai-template, 0.4mb)
Licence is CC-4.0, you can aknowledge me if you want, but no need if you don’t 🙂
Layman Paper Summary: Humans treat unreliable filled-in percepts as more real than veridical ones
We recently published an article (free to read): “Humans treat unreliable filled-in percepts as more real than veridical ones”. Inspired by Selim Onat and many others, I try to to explain the main
findings in plain language. First let me give you some background:
To make sense of the world around us, we must combine information from multiple sources while taking into account how reliable they are. When crossing the street, for example, we usually rely more on
input from our eyes than our ears. However we can reassess our reliability estimate: on a foggy day with poor visibility, we might prioritize listening for traffic instead.
The human blind spots
But how do we assess the reliability of information generated within the brain itself? We are able to see because the brain constructs an image based on the patterns of activity of light-sensitive
proteins in a part of the eye called the retina. However, there is a point on the retina where the presence of the optic nerve leaves no space for light-sensitive receptors. This means there is a
corresponding point in our visual field where the brain receives no visual input from the outside world. To prevent us from perceiving this gap, known as the visual blind spot, the brain fills in the
blank space based on the contents of the surrounding areas. While this is usually accurate enough, it means that our perception in the blind spot is objectively unreliable.
You can try it out by using this simple test (click the image to enlarge)
What we wanted to find out
To find out whether we are aware of the unreliable nature of stimuli in the blind spot we presented volunteers with two striped stimuli, one on each side of the screen. The center of some of the
stimuli were covered by a patch that broke up the stripes. The volunteers’ task was to select the stimulus with uninterrupted stripes. The key to the experiment is that if the central patch appears
in the blind spot, the brain will fill in the stripes so that they appear to be continuous. This means that the volunteers will have to choose between two stimuli that both appear to have continuous
What we thought we would find
If subjects have no awareness of their blind spot, we might expect them to simply guess. Alternatively, if they are subconsciously aware that the stimulus in the blind spot is unreliable, they should
choose the other one.
In reality, exactly the opposite happened:
The volunteers chose the blind spot stimulus more often than not. This suggests that information generated by the brain itself is sometimes treated as more reliable than sensory information from the
outside world. Future experiments should examine whether the tendency to favor information generated within the brain over external sensory inputs is unique to the visual blind spot, or whether it
also occurs elsewhere.
All images are released under CC-BY 4.0.
Cite as: Ehinger et al. “Humans treat unreliable filled-in percepts as more real than veridical ones”, eLife, doi: 10.7554/eLife.21761
EEGlab: Gracefully overwrite the default colormap
EEGlab has ‘jet’ as the default colormap. But jet is pretty terrible
You see structure where there is none (e.g. rings in the third example).
The problem:
Eeglabs sets the default colormap to ‘jet’, thus overwriting a system wide default set e.g. by
It does so by calling “`icadefs.m “` in various functions (e.g. topoplot, erpimage) and defining:
DEFAULT_COLORMAP = ‘jet’
We want to overwrite the one line, but keep it forward compatible i.e. we do not want to copy the whole icadefs file, but just replace the single line whenever icadefs is called.
Overwrite the line in icadefs.m default
This has the benefit that it will always work irrespective of your path-ordering. The con is, you will loose the change if you switch eeglab versions or update eeglab.
Change/create your eeglab “`eeg_options.txt“`.
This has the benefit that it will carry over to the next version of eeglab, but it is an extra file you need to have somewhere completly different than your project-folder (your user-folder ~/
eeg_options.txt). It is thereby hard to make selfcontained code.
Make a new icadefs.m
Make a file called icadefs.m (this script will be called instead of the eeglab “`icadef“`) and add the following code:
run([fileparts(which(‘eegrej’)) filesep ‘icadefs.m’]);
DEFAULT_COLORMAP = ‘parula’;
This will call the original “`icadef“` (in the same folder as “`eegrej.m“` and then overwrite the eeglab-default
Important: The folder to your icadef file must be above eeglab in your path.
Try this: “`edit(‘icadefs.m’)“` to see which function comes up. If the eeglab-one comes up you have a path conflict here. Your own “`icadefs“` has to be above the eeglab one.
In my “`project_init.m“` where I add all paths, I make sure that eeglab is started before adding the path to the new “`icadefs.m“`
ICA – Topoplots of a single subject
Single component of an IC-Decomposition that included noisy data portions (and thus, I would say, is not usable)
Simple Filter Generation
I sometimes explain concepts to my students. Then I forget the code and the next time round, I have to redo it. Take this post as an extended memory-post. In this case I showed what filter-ringing
artefacts could look like. This is especially important for EEG preprocessing where filtering is a standard procedure.
A good introduction to filtering can be found in this slides by andreas widmann or this paper by andreas widmann
Impulse with noise
I simulated as simple impulse response with some additional noise. The idea is to show the student that big spikes in the EEG-data could result in filter-ringing that is quite substantial and far
away from the spike.
The filter
This is the filter I generated. It is a BAD filter. It shows huge passband ripples. But for educational purposes it suits me nice. I usually explain what passbands, transitionband, stopband, ripples
and attenuation means.
The code
fs = 1000;
T = 2;
time = 0:1/fs:T-1/fs;
data = zeros(length(time),1);
% data(end/2:end) = data(end/2:end) + 1;
data(end/2) = data(end/2) + 1;
data = data + rand(size(data))*0.02;
filtLow = designfilt(‘lowpassfir’,’PassbandFrequency’,100, …
‘StopbandFrequency’,110,’PassbandRipple’,0.5, …
% 0-padding to get the borders right
data = padarray(data,round(filtLow.filtord));
% Filter twice to get the same phase(non-causal)
a = filter(filtLow,data);
b = filter(filtLow,a(end:-1:1));
b = b(round(filtLow.filtord)+1:end – round(filtLow.filtord));
fvtool(filtLow) % to look at the filter
Logistic Regression: Will it rain in Osnabrück tomorrow?
52% of days it rained (Precipitation>0)
Is it sunny? 2x as likely that it is sunny tomorrow as well.
Is it rainy? 2.3x as likely that it is rainy tomorrow as well
Predicting rainfall using logistic regression
I’m playing around with some analyses for an upcoming course. This follows loosely the example of “Advanced Data Analysis from an Elemental Point of View” Chapter 12.7
It is a somewhat naive analysis. Further improvements are discussed at the end.
We load the data and change some of the German variables
# I downloaded the data from here:
# http://www.dwd.de/DE/leistungen/klimadatendeutschland/klimadatendeutschland.html
weather = data.frame(read.csv(file='produkt_klima_Tageswerte_19891001_20151231_01766.txt',sep=';'))
weather$date = as.Date.character(x=weather$MESS_DATUM,format="%Y%m%d")
weather = rename(weather,replace = c("NIEDERSCHLAGSHOEHE"="rain"))
weather$rain_tomorrow = c(tail(weather$rain,-1),NA)
weather <- weather[-nrow(weather),] # remove NA row
## date rain rain_tomorrow
## Min. :1989-10-01 Min. : 0.000 Min. : 0.000
## 1st Qu.:1996-04-23 1st Qu.: 0.000 1st Qu.: 0.000
## Median :2002-11-15 Median : 0.100 Median : 0.100
## Mean :2002-11-15 Mean : 2.055 Mean : 2.055
## 3rd Qu.:2009-06-07 3rd Qu.: 2.200 3rd Qu.: 2.200
## Max. :2015-12-30 Max. :140.100 Max. :140.100
The data range from 1989 up until the end of 2015.
p1= ggplot(weather,aes(x=rain))+geom_density()
Notice that the plot on the left is in native scale, the one on the right in x-axis-log-scale.
We take the mean of rainy days (ml/day > 0): There is a 52% probability it will rain on a given day (what everyone living in Osnabrück already knew, it rains more often than not).
Predicting rain from the day before
For exercise reasons I made several logistic regressions. I try to predict whether there will be rain on the day afterwards, based on the amount of rain on the current day.
m.weather.1 = glm(formula = rain_tomorrow>0 ~ rain>0,data=weather,family = "binomial")
## Call:
## glm(formula = rain_tomorrow > 0 ~ rain > 0, family = "binomial",
## data = weather)
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5431 -0.8889 0.8514 0.8514 1.4965
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.72478 0.03136 -23.11 <2e-16 ***
## rain > 0TRUE 1.55287 0.04400 35.29 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Dispersion parameter for binomial family taken to be 1)
## Null deviance: 13278 on 9586 degrees of freedom
## Residual deviance: 11938 on 9585 degrees of freedom
## AIC: 11942
## Number of Fisher Scoring iterations: 4
Whoop Whoop, Wald’s t-value of ~35! Keep in mind that the predictor values are in logit-space. That means, a predictor value of -0.72 is a log-odd value. In order to calculate this back to an easier
interpreted format, we can simply take the exponential and receive the odds-ratio.
Do the calculations:
## (Intercept)
## 0.4844302
exp(sum(coef(m.weather.1))) # the odds for rain tomorrow if it is rainy (2 : 1)
## [1] 2.288933
Now we can interprete the odds:
The odds of rain tomorrow if there was a sunny day are 0.5:1, i.e. it is double as likely that the next day is sunny, if it was sunny today
The odds of rain tomorrow if it was a rainy day are 2.3:1, i.e. it is more than double as likely that the next day is rainy, if it was rainy today.
Can we get better?
We could try to predict rain based on the daily continuous precipitation (in ml).
We will compare this with the previous model using BIC (smaller number => better model).
m.weather.2 = glm(formula = rain_tomorrow>0 ~ rain,data=weather,family="binomial")
## df BIC
## m.weather.1 2 11956.09
## m.weather.2 2 12505.85
Turns out: No, a categorical model (is there rain today, or not?) beats the continuous one. But why?
d.predNew = data.frame(rain=seq(0,50));
d.pred= cbind(d.predNew,pred=arm::invlogit(predict(m.weather.2,newdata = d.predNew)),model='2')
d.pred=rbind(d.pred,cbind(d.predNew,pred=arm::invlogit(predict(m.weather.1,newdata = d.predNew)),model='1'))
p1 = ggplot()+geom_point(data=d.pred,aes(x=rain,y=pred,color=model) )+
We plot the predictions model 1 does in green, model 2 in red and the smoothed data in blue. On the left we have the x-axis in native units [ml?] on the right on log-scale. It is very clear, that the
red dots do not match the empirical data (blue) at all. The green dots (model 1) are better. My guess is, this is due to some outlier influencing the slope, but also due to the non-homogenious
distribution of rain-values as seen i the first figure
We will try two transformations off X (a log effect seems possible?)
• sqrt(rain)
• log(rain+0.001)
m.weather.3 = glm(formula = rain_tomorrow>0 ~ sqrt(rain),data=weather,family="binomial")
# Box-Cox transform with lambda2 = 0.001 http://robjhyndman.com/hyndsight/transformations/
m.weather.4 = glm(formula = rain_tomorrow>0 ~ log(rain+0.001),data=weather,family="binomial")
## df BIC
## m.weather.1 2 11956.09
## m.weather.2 2 12505.85
## m.weather.3 2 12020.09
## m.weather.4 2 11795.15
It is surprisingly hard to beat the simple first model, but in the end – we did it! The log(rain) model seems to capture the data best.
d.predNew = data.frame(rain=seq(0,50));
d.pred= cbind(d.predNew,pred=arm::invlogit(predict(m.weather.4,newdata = d.predNew)),model='2')
d.pred=rbind(d.pred,cbind(d.predNew,pred=arm::invlogit(predict(m.weather.1,newdata = d.predNew)),model='1'))
d.pred=rbind(d.pred,cbind(d.predNew,pred=arm::invlogit(predict(m.weather.3,newdata = d.predNew)),model='3'))
d.pred=rbind(d.pred,cbind(d.predNew,pred=arm::invlogit(predict(m.weather.4,newdata = d.predNew)),model='4'))
p1 = ggplot()+geom_point(data=d.pred,aes(x=rain,y=pred,color=model) )+
Visually we can see that model 4 comes the non-linear smoother the closest.
• use nonlinear effects (GAM) as done in the book
• multivariate (make use of the plentitude of other information as well)
• use the months as a cyclical predictor, i.e. seasonal trends will be clearly present in the data
This post was directly exported using knit2wp and R-Markdown. It works kind of okay, but clearly the figures are not wide enough, even though I specify the correct width in the markdown. I might
upload them later manually.
How to use bimodal priors for bayesian data analysis in STAN
I tried adding a bi-modal prior in STAN for a homework exercise on Bayesian Data Analysis. At first, I thought this could work:
mu ~ normal(-0.5,0.3) + normal(0.5,0.3);
But it doesn’t. I had to dig deeper and I thought I could simply add multiple times to the log-posterior due to a sideremark of Bob Carpenter:
target += normal_lpdf(mu|.5,0.3);
target += normal_lpdf(mu|-.5,0.3);
Which also does not work. Finally, the solution is akin to the mixture model in the STAN manual:
target += log_sum_exp(normal_lpdf(mu|.5,0.3),normal_lpdf(mu|-.5,0.3));
This results in beautiful bi-modal priors:
I did not find anything on google or the manual of how to do this. If there is a smarter way to do it, please leave a comment.
model <- " data { } parameters { real mu; } transformed parameters { } model { //mu ~ normal(10,1); //mu ~ normal(-10,1); target += log_sum_exp(normal_lpdf(mu|-.5,.3),normal_lpdf(mu|.5,.3)); }"
samples <- stan(model_code=model, iter=2000, chains=4, thin=1, # seed=123 # Setting seed; Default is random seed ) ggmcmc::ggs_density(ggmcmc::ggs(samples))+theme_minimal() ```
Scatterplots, regression lines and the first principal component
I made some graphs that show the relation between X1~X2 (X2 predicts X1), X2~X1 (X1 predicts X2) and the first principal component (direction with highest variance, also called total least squares).
The line you fit with a principal component is not the same line as in a regression (either predicting X2 by X1 [X2~X1] or X1 by X2 [X1~X2]. This is quite well known (see references below).
With regression one predicts X2 based on X1 (X2~X1 in R-Formula writing) or vice versa. With principal component (or total least squares) one tries to quantify the relation between the two. To
completely understand the difference, image what quantity is reduced in the three cases.
In regression, we reduce the residuals in direction of the dependent variable. With principal components, we find the line, that has the smallest error orthogonal to the regression line. See the
following image for a visual illustration.
For me it becomes interesting if you plot a scatter plot of two independent variables, i.e. you would usually report the correlation coefficient. The ‘correct’ line accompaniyng the correlation
coefficient would be the principal component (‘correct’ as it is also agnostic to the order of the signals).
Further information:
How to draw the line, eLife 2013
Gelman, Hill – Data Analysis using Regression, p.58
Also check out the nice blogpost from Martin Johnsson doing practically the same thing but three years earlier 😉
corrCoef = 0.5 # sample from a multivariate normal, 10 datapoints
dat = MASS::mvrnorm(10,c(0,0),Sigma = matrix(c(1,corrCoef,2,corrCoef),2,2))
dat[,1] = dat[,1] – mean(dat[,1]) # it makes life easier for the princomp
dat[,2] = dat[,2] – mean(dat[,2])
dat = data.frame(x1 = dat[,1],x2 = dat[,2])
# Calculate the first principle component
# see http://stats.stackexchange.com/questions/13152/how-to-perform-orthogonal-regression-total-least-squares-via-pca
v = dat%>%prcomp%$%rotation
x1x2cor = bCor = v[2,1]/v[1,1]
x1tox2 = coef(lm(x1~x2,dat))
x2tox1 = coef(lm(x2~x1,dat))
slopeData =data.frame(slope = c(x1x2cor,1/x1tox2[2],x2tox1[2]),type=c(‘Principal Component’,’X1~X2′,’X2~X1′))
# We want this to draw the neat orthogonal lines.
pointOnLine = function(inp){
# y = a*x + c (c=0)
# yOrth = -(1/a)*x + d
# yOrth = b*x + d
x0 = inp[1] y0 = inp[2] a = x1x2cor
b = -(1/a)
c = 0
d = y0 – b*x0
x = (d-c)/(a-b)
y = -(1/a)*x+d
points = apply(dat,1,FUN=pointOnLine)
segmeData = rbind(data.frame(x=dat[,1],y=dat[,2],xend=points[1,],yend=points[2,],type = ‘Principal Component’),
geom_abline( data=slopeData,aes(slope = slope,intercept=0,color=type))+
geom_abline( data=slopeData,aes(slope = slope,intercept=0,color=type))+
EEG: Contours in multiple topoplots
The problem
It is commonly accepted that axes of plots should align if data needs to be compared between subplots. But the default way on how multiple topoplots are plotted violates this principle. While usually
the limits of the colormap are kept constant for all colormaps, the contours rarely are. This default plot looks similar to this one:
The solution
It is simple, keep the contours constant!
In eeglab this is done using the topoplot function with the argument 'numcontours', linspace(-scale,scale,n_contours) or similar. You can also use my new plotting script available here on github
So if we would keep the values constant at which contours are generated it looks like this:
More ideas
Each topoplot has its individual color-limits. While the local (in a single topoplot) extremata a clearly visible, not much to compare between topoplots
Individual contours improve the readability of each map, but they do not add anything that eases the comparison.
Forcing the same color-limits in the colormap allows for direct comparison between topoplots. But whether the white of the 9th’s or the 12th’s topoplot is bigger is hard to tell.
Going back to individual colormaps, but keeping the same contours: This helps already a lot, I seem to abstract the colormap away a bit and use the contours for comparison
The opposite way, same color-limits but individual contours. Again I seem to rely more on the contours, in this case this is more confusing than before.
In the final plot colormap and contour are aligned. This enhances comparison between topoplots.
One problem with the same color-limits or the same contour lines between topoplots is, that large deflections could hide small ones. As in many cases, it depends on what features of the data you want
to highlight. I recommend the final plot where contour and colormap align as the default.
If you are plotting multiple topoplots, try to keep the color-limits of the colormap as well as the contour levels constant
Multitaper Time vs Frequency Resolution
Give a graphical representation of multitaper time frequency parameters.
Frequency Resolution and sFFT
In M/EEG analysis we are often interested in oscillations. One tool to analyse oscillations is by using time-frequency methods. In principle they work by segmenting the signal of interest in small
windows and calculating a spectrum for each window. A tutorial can be found on the fieldtrip page. There exists a time-frequency tradeof: $\delta f = \frac{1}{T}$ with T the timewindow (in s) and $\
delta f$ the frequency resolution. The percentage-difference of two neighbouring frequency resolution gets smaller the higher the frequency goes, e.g. a $\delta f$ of 1 at 2Hz and 3Hz is 50%, but the
same difference at 100Hz and 101Hz is only 1%. Thus usually we try to decrease the frequency resolution the higher we go. A popular examples are wavelets, they inherently do this. One problem with
wavelets is, that the parameters are not perfectly intuitive (I find). I prefer to use short-time FFT, and a special flavour the Multitaper.
We can sacrifice time or frequency resolution for SNR. I.e. one can average estimates at neighbouring timewindows or estimates at neighbouring frequencies. A smart way to do this is to use
multitapers, I will not go into details here. In the end, they allow us to average smartly over time or frequency.
Parameter one usually specifies
• foi: The frequencies of interest. Note that you can use arbitrary high resolution but if you have $\delta foi < \delta f$ then information will be displayed redundantly (but it looks smoother).
• tapsmofrq: How much frequency smoothing should happen at each foi.
• t_ftimwin: The timewindow for each foi
Plotting the parameters
With the matlab-function attached to the end of the post one can see the effective frequency-resolution of the parameters in a single plot. For this example I use the frequency parameters by Hipp
2011. They logarithmically scale up the frequencybandwith. In order to do so for the low frequencies (1-16Hz) they change the size of the timewindow, for the higher frequencies they use multitaper
with a constant window size of 250ms.
This is the theoretical resolution (interactive graph), i.e. a plot of the parameters you put in. foi +- tapsmofrq scaled by each timewindow size. Note: The individual bar size depicts the
window-size of the respective frequency-bin.
This is the numeric resolution (interactice graph). I used the frequency response of 100 repetitions of white noise (flat spectrum over repetitions) and calculated the frequency response of the
multitaper filters. This is scaled in the x-axis by the time-window used.
Check out the interactive graphs to see how the time-windows change as intendet with lower frequencies.
The colorbar depict the number of tapers used. As intended, the number of tapers do not change from 1-16Hz.
function [] = plot_taper(inpStruct,cfg)
%% plot_taper
% inpStruct: All fields need to have the same length
% foi : The frequencies of interest.
% tapsmofrq : How much frequency smoothing should happen at each foi (+-tapsmofrq)
% t_ftimwin : The timewindow for each foi
% You can also directly input a fieldtrip time-freq object
% cfg:
% logy : plot the y-axis as log10
% type : "theoretical" : This plots the foi+-tapsmofrq
% "numerical" : Simulation of white noise to approx the
% effective resolution
if nargin == 1
cfg = struct();
if isfield(cfg,'logy')
assert(ismember(cfg.type,[0,1]),'logy has to be 0 or 1')
logy = cfg.logy;
logy = 0;
if isfield(cfg,'type')
assert(ismember(cfg.type,{'numerical','theoretical'}),'type has to be ''numerical'' or ''theoretical''')
type = cfg.type;
type = 'numerical';
nRep = 100; % repetitions of the numeric condition
xscaling = 0.5;
if ~isempty(inpStruct) && isfield(inpStruct,'t_ftimwin')
fprintf('cfg detected\n')
cfg = inpStruct;
elseif ~isempty(inpStruct) && isfield(inpStruct,'cfg')
fprintf('fieldtrip structure detected\n')
cfg = inpStruct.cfg;
fprintf('using parameters from Hipp 2010 Neuron \n')
cfg =[];
% we use logspaced frequencies from 4 to 181 Hz
cfg.foi = logspace(log10(4),log10(181),23);
% The windows should have 3/4 octave smoothing in frequency domain
cfg.tapsmofrq = (cfg.foi*2^(3/4/2) - cfg.foi*2^(-3/4/2)) /2; % /2 because fieldtrip takes +- tapsmofrq
% The timewindow should be so, that for freqs below 16, it results in n=1
% Taper used, but for frequencies higher, it should be a constant 250ms.
% To get the number of tapers we use: round(cfg.tapsmofrq*2.*cfg.t_ftimwin-1)
cfg.t_ftimwin = [2./(cfg.tapsmofrq(cfg.foi<16)*2),repmat(0.25,1,length(cfg.foi(cfg.foi>=16)))];
max_tapers = ceil(max(cfg.t_ftimwin.*cfg.tapsmofrq*2));
color_tapers = parula(max_tapers);
for fIdx = 1:length(cfg.t_ftimwin)
timwin = cfg.t_ftimwin(fIdx);
tapsmofrq = cfg.tapsmofrq(fIdx); % this is the fieldtrip standard, +- the taper frequency
freqoi = cfg.foi(fIdx);
switch type
case 'theoretical'
% This part simply plots the parameters as requested by the
% user.
y = ([freqoi-tapsmofrq freqoi+tapsmofrq freqoi+tapsmofrq freqoi-tapsmofrq]);
ds = 0.3;
ys = ([freqoi-ds freqoi+ds freqoi+ds freqoi-ds]);
if logy
y = log10(y);
ys = log10(ys);
x = [-timwin/2 -timwin/2 +timwin/2 +timwin/2]+fIdx*xscaling;
patch(x,ys,[0.3 1 0.3],'FaceAlpha',.5)
case 'numerical'
fsample = 1000; % sample Fs
% This part is copied together from ft_specest_mtmconvol
timwinsample = round(timwin .* fsample); % the number of samples in the timewin
% get the tapers
tap = dpss(timwinsample, timwinsample .* (tapsmofrq ./ fsample))';
tap = tap(1:(end-1), :); % the kill the last one because 'it is weird'
% don't fully understand, something about keeping the phase
anglein = (-(timwinsample-1)/2 : (timwinsample-1)/2)' .* ((2.*pi./fsample) .* freqoi);
% The general idea (I think) is that instead of windowing the
% time-signal and shifting the wavelet per timestep, we can multiply the frequency representation of
% wavelet and the frequency representation of the data. this
% masks all non-interesting frequencies and we can go back to
% timespace and select the portions of interest. In formula:
% ifft(fft(signal).*fft(wavelet))
% here we generate the wavelets by windowing sin/cos with the
% tapers. We then calculate the spectrum
wltspctrm = [];
for itap = 1:size(tap,1)
coswav = tap(itap,:) .* cos(anglein)';
sinwav = tap(itap,:) .* sin(anglein)';
wavelet = complex(coswav, sinwav);
% store the fft of the complex wavelet
wltspctrm(itap,:) = fft(wavelet,[],2);
% % We do this nRep times because a single random vector does not
% have a flat spectrum.
datAll = nan(nRep,timwinsample);
for rep = 1:nRep
sig = randi([0 1],timwinsample,1)*2-1; %http://dsp.stackexchange.com/questions/13194/matlab-white-noise-with-flat-constant-power-spectrum
spec = bsxfun(@times,fft(sig),wltspctrm'); % multiply the spectra
datAll(rep,:) = sum(abs(spec),2); %save only the power
dat = mean(datAll); %average over reps
% normalize the power
dat = dat- min(dat);
dat = dat./max(dat);
t0 = 0 + fIdx*xscaling;
t = t0 + dat*timwin/2;
tNeg = t0 - dat*timwin/2;
f = linspace(0,fsample-1/timwin,timwinsample);%0:1/timwin:fsample-1;
if logy
f = log10(f);
% this is a bit hacky violin plot, but looks good.
patch([t tNeg(end:-1:1)],[f f(end:-1:1)],color_tapers(2*round(timwin*tapsmofrq),:),'FaceAlpha',.7)
c= colorbar;colormap('parula')
c.Label.String = '# Tapers Used';
caxis([1 size(color_tapers,1)])
xlabel('width of plot depicts window size in [s]')
if logy
set(gca,'ylim',[0 log(max(1.2*(cfg.foi+cfg.tapsmofrq)))]);
ylabel('frequency [log(Hz)]')
set(gca,'ylim',[0 max(1.2*(cfg.foi+cfg.tapsmofrq))]);
ylabel('frequency [Hz]')
Latency Measurements in a Vision Lab
We recently moved to a new building and the labs moved with us. We did not only use this opportunity to get rid of old computers and monitors, but also to thoroughly check that our timing is correct.
This work was mostly done by Ule Diallo, an intern from Marburg University. We were very lucky to have him!
We searched for fixed and jittered delays in our setup. We measured many monitors with two luminance sensors. We also checked our parallel-ports trigger-timing. We did not find jittered delays, the
fixed delays are, as expected, only by the monitors.
Why is timing important?
For EEG, but also Eye-Tracking or Reaction Time measurements, effects can be quite small (for some paradigms \<10ms effects can be important. Measurement noise can be an issue (even though I recently
read a paper that jitters of 10-15ms do not matter because subject variance is so high, but I can’t find it right now). For computer mice you can see an example of high jitter here and mice are
supposed to be faster/less variable than keyboards!
In EEG, a P100 component is maybe 30ms broad. You don’t necessarily want to jitter it even further.
For most purposes, a fixed delay does not matter, you could simply recalculate your EEG or your reaction times. But a jitter is really the bad thing, very difficult to compensate for!
Where does jitter come from?
1. Presentation programs
2. Synchronization delays between computers/systems/amplifiers/eyetrackers/parallelports
3. Lag of LCD/CRT Monitors
Left PC is the eyetracking computer (eyelink/SR-Research). Middle computer is the stimulus computer, right is the monitor with a custom made luminance sensor and bottom the EEG amplifier. As you
can see from the graph, the monitor (measured by a photosensor) is really the biggest source of delay (if not strictly controlled).
Parallel Port Timing
Our stimulus host PC communicates with the EEG using parallel port electrical triggers. We first of all confirmed that we can trigger with submillisecond speed (see table here). Also sometimes we use
an eye-tracker (ET) in the loop. What we do is send over LAN a command to set the parallel port of the ET host which goes to the EEG. By that we have all three machines synchronized by the event.
Again we find only some glitches in less than 0.0002% of cases (all glory details here).
Monitor timing
We mostly use LCD monitors in our lab, and I have the feeling that many labs move towards those monitors (away from CRT technology) anyway. We measured the input lag (the time needed to detect the
first luminance-change) and the raise-time (the time needed until the actual color/luminance we wanted was reached). The sum of the two are called responsetime
We displayed the stimuli using psychtoolbox-3, we waited after each stimulus draw for the vertical-refresh (in addition to V-Sync). Ule wrote some scripts to automatically extract the relevant
numbers. The summary plot can be seen here:
X-Axes are different monitors, Y-Axes are the measured delays, lower is better. Top-Row is the raisetime, middle row the input lag and last row is the summed responsetime. Colors depict different
changes i.e. from white to black, white to gray etc. Columns depict two different sensors, sensor one on the top, sensor two at the bottom of the screen. The monitors seem to respond with a low
variance (errorbars are SD). Good news for us!
We see that gray to white and black to white are several milliseconds slower than switches from black to gray or gray to black.
We see that our old Apple Cinema 30″ monitor has some trouble (and also in generally is slow). Likely cause: old IPS Panel and PMD (see below) for backlight.
Luckily we recently replaced it with an Asus 32″ inch, the second in line. Our BenQ Monitors (2 identical ones and one 144Hz) seem to run just fine, with a minimal total input lag of ~8ms to 14ms
(the right panel shows the luminance in the lower right corner, thus a delay of one refresh is expected, with 144Hz that’s 7ms).
Lessons Learned
• Pulse-Width-Modulation ruins your day
Some monitors change the brightness by turning on and off the backlight with a given frequency. This is called PWM. The frequency can be as low as 200Hz. You clearly don’t want to have additional
strong modulation in your monitor. Either check that your monitor does not do PWM or turn the brightness completely up
• The bottom of the monitor is one refreshtime slower than the top.
The refresh of a LCD monitor starts on the top and ends at the bottom, thus with 120Hz you have a 8ms time delay between top and bottom. With 60Hz a 16ms delay. This is also true for CRTs. I like
this video
Bonus: G-Sync/FreeSync vs. V-Sync
Disclaimer: We don’t own a FreeSync/G-Sync system, but this is what I understood while reading a lot about monitors. check out this longer guide
A screen is refreshed every 8.33ms (@120Hz). It sometimes happen, that your frame has been calculated faster than the 8ms. In principle the graphicscard can force the monitor to restart its updating
from the top with the new frame. This results in ugly “tearing” of the monitor. To get rid of it, V-Sync forces the graphicscard to wait until the refresh of the monitor is finished.
FreeSync/GSync is a different thing: Usually the monitor starts a new refresh immediately after it finished the old one. If no new frame arrived it will update the old one. With G-Sync it simply
waits until a new frame has been produced and thus possible is faster, without any tearing.
Thus for only reducing input-lag, i.e. for gaze-dependent display, g-sync does not really help – if your calculations are all well below ~8ms (which they usually are for our experiments)
All in all I’m very happy with our setup. We don’t have mayor lags or jitters somewhere, we now know exactly how much to correct and we have a nice infrastructure to remeasure the monitors for each
experiment. I also learned quite a bit about monitors and latency measurements.
Many thanks to Ule Diallo for doing most of the work. Also thanks to Silja Timm and Anna Gert
The icons were taken from the awesome Noun Projekt
by Edward Boatman,by Iconic,by Ji Sub Jeong, by Hans, by Vaibhav Radhakrishnan, by ProSymbols | {"url":"https://benediktehinger.de/blog/science/page/5/","timestamp":"2024-11-04T01:13:46Z","content_type":"text/html","content_length":"108910","record_id":"<urn:uuid:cd9a1c1c-44bf-402e-bed6-43b304347362>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00114.warc.gz"} |
Orient BlackSwan
This text emphasizes rigorous mathematical techniques for the analysis of boundary value problems for ODEs arising in applications. The emphasis is on proving existence of solutions, but there is
also a substantial chapter on uniqueness and multiplicity questions and several chapters which deal with the asymptotic behavior of solutions with respect to either the independent variable or some
parameter. These equations may give special solutions of important PDEs, such as steady state or traveling wave solutions. Often two, or even three, approaches to the same problem are described. The
advantages and disadvantages of different methods are discussed.
The book gives complete classical proofs, while also emphasizing the importance of modern methods, especially when extensions to infinite dimensional settings are needed. There are some new results
as well as new and improved proofs of known theorems. The final chapter presents three unsolved problems which have received much attention over the years.
Both graduate students and more experienced researchers will be interested in the power of classical methods for problems which have also been studied with more abstract techniques. The presentation
should be more accessible to mathematically inclined researchers from other areas of science and engineering than most graduate texts in mathematics..
Stuart P. Hastings, University of Pittsburgh, PA, and J. Bryce McLeod, Oxford University, England, and University of Pittsburgh, PA
Chapter 1. Introduction
1.1. What are classical methods?
1.2. Exercises
Chapter 2. An introduction to shooting methods
2.1. Introduction
2.2. A first order example
2.3. Some second order examples
2.4. Heteroclinic orbits and the FitzHugh-Nagumo equations
2.5. Shooting when there are oscillations: A third order problem
2.6. Boundedness on (-8,8) and two-parameter shooting
2.7. Waz?ewski's principle, Conley index, and an n-dimensional lemma
2.8. Exercises
Chapter 3. Some boundary value problems for the Painlev´e transcendents
3.1. Introduction
3.2. A boundary value problem for Painlev´e
3.3. Painlev´e II—shooting from infinity
3.4. Some interesting consequences
3.5. Exercises
Chapter 4. Periodic solutions of a higher order system
4.1. Introduction, Hopf bifurcation approach
4.2. A global approach via the Brouwer fixed point theorem
4.3. Subsequent developments
4.4. Exercises
Chapter 5. A linear example
5.1. Statement of the problem and a basic lemma
5.2. Uniqueness
5.3. Existence using Schauder's fixed point theorem
5.4. Existence using a continuation method
5.5. Existence using linear algebra and finite dimensional continuation
5.6. A fourth proof
5.7. Exercises
Chapter 6. Homoclinic orbits of the FitzHugh-Nagumo equations
6.1. Introduction
6.2. Existence of two bounded solutions
6.3. Existence of homoclinic orbits using geometric perturbation theory
6.4. Existence of homoclinic orbits by shooting
6.5. Advantages of the two methods
6.6. Exercises
Chapter 7. Singular perturbation problems—rigorous matching
7.1. Introduction to the method of matched asymptotic expansions
7.2. A problem of Kaplun and Lagerstrom
7.3. A geometric approach
7.4. A classical approach
7.5. The case n = 3
7.6. The case n = 2
7.7. A second application of the method
7.8. A brief discussion of blow-up in two dimensions
7.9. Exercises
Chapter 8. Asymptotics beyond all orders
8.1. Introduction
8.2. Proof of nonexistence
8.3. Exercises
Chapter 9. Some solutions of the Falkner-Skan equation
9.1. Introduction
9.2. Periodic solutions
9.3. Further periodic and other oscillatory solutions
9.4. Exercises
Chapter 10. Poiseuille flow: Perturbation and decay
10.1. Introduction
10.2. Solutions for small data
10.3. Some details
10.4. A classical eigenvalue approach
10.5. On the spectrum of D?,R? for large R
10.6. Exercises
Chapter 11. Bending of a tapered rod; variational methods and shooting
11.1. Introduction
11.2. A calculus of variations approach in Hilbert space
11.3. Existence by shooting for p > 2
11.4. Proof using Nehari's method
11.5. More about the case p = 2
11.6. Exercises
Chapter 12. Uniqueness and multiplicity
12.1. Introduction
12.2. Uniqueness for a third order problem
12.3. A problem with exactly two solutions
12.4. A problem with exactly three solutions
12.5. The Gelfand and perturbed Gelfand equations in three dimensions
12.6. Uniqueness of the ground state for ?u - u + u3 = 0
12.7. Exercises
Chapter 13. Shooting with more parameters
13.1. A problem from the theory of compressible flow
13.2. A result of Y.-H. Wan
13.3. Exercise
13.4. Appendix: Proof of Wan's theorem
Chapter 14. Some problems of A. C. Lazer
14.1. Introduction
14.2. First Lazer-Leach problem
14.3. The pde result of Landesman and Lazer
14.4. Second Lazer-Leach problem
14.5. Second Landesman-Lazer problem
14.6. A problem of Littlewood, and the Moser twist technique
14.7. Exercises
Chapter 15. Chaotic motion of a pendulum
15.1. Introduction
15.2. Dynamical systems
15.3. Melnikov's method
15.4. Application to a forced pendulum
15.5. Proof of Theorem 15.3 when d = 0
15.6. Damped pendulum with nonperiodic forcing
15.7. Final remarks
15.8. Exercises
Chapter 16. Layers and spikes in reaction-diffusion equations, I
16.1. Introduction
16.2. A model of shallow water sloshing
16.3. Proofs
16.4. Complicated solutions ("chaos")
16.5. Other approaches
16.6. Exercises
Chapter 17. Uniform expansions for a class of second order problems
17.1. Introduction
17.2. Motivation
17.3. Asymptotic expansion
17.4. Exercise
Chapter 18. Layers and spikes in reaction-diffusion equations, II
18.1. A basic existence result
18.2. Variational approach to layers
18.3. Three different existence proofs for a single layer in asimple case
18.4. Uniqueness and stability of a single layer
18.5. Further stable and unstable solutions, including multiple layers
18.6. Single and multiple spikes
18.7. A different type of result for the layer model
18.8. Exercises
Chapter 19. Three unsolved problems
19.1. Homoclinic orbit for the equation of a suspension bridge
19.2. The nonlinear Schr¨odinger equation
19.3. Uniqueness of radial solutions for an elliptic problem
19.4. Comments on the suspension bridge problem
19.5. Comments on the nonlinear Schr¨odinger equation
19.6. Comments on the elliptic problem and a new existence proof
19.7. Exercises | {"url":"https://orientblackswan.com/details?id=9781470409241","timestamp":"2024-11-04T11:39:35Z","content_type":"text/html","content_length":"117173","record_id":"<urn:uuid:72084cbe-1bbb-41f1-ad06-6757d8b1b159>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00024.warc.gz"} |
Method for Determining Flow Measurement Values of a Coriolis Mass Flowmeter in the Presence of a of a Two-phase Flow
Patent application title: Method for Determining Flow Measurement Values of a Coriolis Mass Flowmeter in the Presence of a of a Two-phase Flow
Inventors: Tao Wang (Rough Common, GB) Xue Wang (Rough Common, GB) Yong Yan (Sevenoaks, GB) Jinyu Liu (Wellingborough, GB)
IPC8 Class: AG01F184FI
USPC Class: 1 1
Class name:
Publication date: 2021-12-09
Patent application number: 20210381868
A method is disclosed for determining flow measurement values of a Coriolis mass flowmeter in the presence of a two-phase flow of a two-phase medium having a gas phase and the subsequent presence of
a single-phase flow of a single-phase medium not having a gas phase. The method includes: detecting a start time of a two-phase measurement interval at an onset of the two-phase flow; detecting an
end time of the two-phase measurement interval at an end of the presence of the two-phase flow; determining and at least partially storing two-phase flow measurement values of the two-phase flow;
determining at least one state variable of the single-phase medium; determining subsequently corrected two-phase flow measurement values as at least indirect input variables of a correction
calculation; and outputting the corrected two-phase flow measurement values as individual values or as part of a cumulative flow measurement value.
A method for determining flow measurement values of a Coriolis mass flowmeter in the presence of a two-phase flow of a two-phase medium having a gas phase in a two-phase measurement interval and a
subsequent presence of a single-phase flow of a single-phase medium not having a gas phase in a single-phase measurement interval, comprising: detecting the start time of the two-phase measurement
interval at the onset of the two-phase flow; detecting the end time of the two-phase measurement interval at the end of the presence of the two-phase flow; in the two-phase measurement interval,
determining and at least partially storing two-phase flow measurement values of the two-phase flow; in the single-phase measurement interval, determining at least one state variable of the
single-phase medium; from the stored two-phase flow measurement values and from the at least one state variable of the single-phase medium determined in the single-phase measurement interval,
determining subsequently corrected two-phase flow measurement values as at least indirect input variables of a correction calculation; and outputting the corrected two-phase flow measurement values
as individual values or as part of a cumulative flow measurement value.
The method according to claim 1, further comprising: determining at least the density of the single-phase medium as state variable of the single-phase medium; and using at least the density of the
single-phase medium as at least an indirect input variable of the correction calculation.
The method according to claim 2, further comprising: calculating the gas-volume fraction of the two-phase medium using the density of the single-phase medium; and using the calculated gas-volume
fraction of the two-phase medium as a direct input variable of the correction calculation; wherein the gas volume fraction of the two-phase medium is calculated by forming the quotient of the
difference between the density of the single-phase medium and the density of the two-phase medium and the density of the single-phase medium; and wherein the quotient is formed from the difference
between the density of the single-phase medium and the density of the two-phase medium and the difference between the density of the single-phase medium and the density of the gas phase of the
two-phase medium.
The method according to claim 3, further comprising: determining the density of the gas phase of the two-phase medium by measuring the temperature of the two-phase medium and measuring the pressure
at the outflow side of the Coriolis mass flowmeter; and determining the density of the gas phase of the two-phase medium based on the measured temperature of the two-phase medium and based on the
measured pressure at the outflow side of the Coriolis mass flowmeter.
The method according to claim 1, further comprising: using the viscosity of the single-phase medium as a further input variable of the correction calculation; and determining the viscosity of the
single-phase medium from a temperature-dependent viscosity curve using the temperature of the two-phase medium.
The method according to claim 1, further comprising: using the differential pressure over the inflow side and the outflow side of the Coriolis mass flowmeter as a further input variable of the
correction calculation; at least partially storing the differential pressures determined in the two-phase measurement interval; and storing a differential pressure for each two-phase flow measurement
The method according to claim 1, further comprising: implementing the correction calculation with an approximate solution method in which at least the stored two-phase flow measurement values and the
at least one state variable of the single-phase medium determined in the single-phase measurement interval as at least indirect input variables, are approximately mapped onto the corrected two-phase
flow measurement values.
The method according to claim 7, wherein the correction calculation is implemented by an artificial neural network having an input layer with at least two input neurons for supply of the stored
two-phase flow measurement values to be corrected and for supply of the state variable of the single-phase medium determined in the single-phase measurement interval or a variable derived therefrom
as at least indirect input variables, having an output layer with an output neuron for output of the subsequently corrected two-phase flow measurement values, and having at least one intermediate
layer with at least two neurons, wherein each input neuron is connected to each neuron of the intermediate layer via directed and weighted signal paths and wherein each neuron of the intermediate
layer is connected to the output neuron of the output layer via a directed and weighted signal path.
The method according to claim 8, wherein the artificial neural network comprises: at least four input neurons in the input layer for supply of the stored two-phase flow measurement values to be
corrected, the gas-volume fraction of the two-phase medium, the viscosity of the single-phase medium and the differential pressure via the inflow side and the outflow side of the Coriolis mass
flowmeter; an output neuron for output of the subsequently corrected two-phase flow measurement values; and four neurons in an intermediate layer, wherein each input neuron is connected to each
neuron of the intermediate layer via directed and weighted signal paths and wherein each neuron of the intermediate layer is connected to the output neuron of the output layer via a directed and
weighted signal path.
The method according to claim 8, wherein the artificial neural network is trained with a training data set; and wherein the training data set is collected for one design of a Coriolis mass flowmeter
and the training data set includes value tuples from the used input variables of the artificial neural network and the output variable of the artificial neural network.
The method according to claim 10, wherein a training data set is collected for each two-phase medium and a separate artificial neural network is trained for each two-phase medium.
The method according to claim 1, further comprising: storing all flow measurement values during a measurement operation; after the completed measurement operation, determining the single-phase
measurement interval and the two-phase measurement interval from the stored flow measurement values or other recorded data; and carrying out the correction calculation.
A Coriolis mass flowmeter, comprising: at least one measuring tube through which a medium can flow, at least one oscillation generator; at least two oscillation sensors; and at least one control and
evaluation unit; wherein the control and evaluation unit is designed such that, in the presence of a two-phase flow of a two-phase medium having a gas phase in a two-phase measurement interval and a
subsequent presence of a single-phase flow of a single-phase medium not having a gas phase in a single-phase measurement interval, the starting time of the two-phase measurement interval is detected
at the onset of the two-phase flow; wherein the end time of the two-phase measurement interval is detected at the end of the presence of the two-phase flow; wherein, in the two-phase measurement
interval, two-phase flow measurement values of the two-phase flow are determined and at least partially stored; wherein, in the single-phase measurement interval, at least one state variable of the
single-phase medium is determined; wherein, from the stored two-phase flow measurement values and from the at least one state variable of the single-phase medium determined in the single-phase
measurement interval, subsequently corrected two-phase flow measurement values are determined as at least indirect input variables of a correction calculation; and wherein the corrected two-phase
flow measurement values are output as individual values or are output as part of a cumulative flow measurement value.
The Coriolis mass flowmeter according to claim 13, wherein the control and evaluation unit is designed such that: at least the density of the single-phase medium is determined as state variable of
the single-phase medium; and at least the density of the single-phase medium is used as at least an indirect input variable of the correction calculation.
The Coriolis mass flowmeter according to claim 13, wherein the start time of the two-phase measurement interval is detected at the onset of the two-phase flow and/or that the end time of the
two-phase measurement interval is detected at the end of the presence of the two-phase flow by evaluating the level of the excitation signal of the oscillation generator and/or by evaluating the
level of the sensor signal of the oscillation sensor; wherein the start time of the two-phase measurement interval is detected when a limit height of the excitation signal and/or of the sensor signal
is exceeded; and wherein the end time of the two-phase measurement interval is detected when the excitation signal and/or the sensor signal falls below a limit height.
TECHNICAL FIELD [0001]
The invention relates to a method for determining flow measurement values of a Coriolis mass flowmeter in the presence of a two-phase flow of a two-phase medium having a gas phase in a two-phase
measurement interval and a subsequent presence of a single-phase flow of a single-phase medium not having gas phase in a single-phase measurement interval. Furthermore, the invention also relates to
a Coriolis mass flowmeter having at least one measuring tube through which a medium can flow, at least one oscillation generator, at least two oscillation sensors and at least one control and
evaluation unit, wherein the Coriolis mass flowmeter is designed to carry out the aforementioned method.
BACKGROUND [0002]
Coriolis mass flowmeters and methods for determining flow measurement values of a Coriolis mass flowmeter have been known for many decades in quite different designs. High measuring accuracy can be
achieved with Coriolis mass flowmeters which, in the case of single-phase flow, is sometimes better than 0.1% of the measured value, so that Coriolis mass flowmeters can also be used, for example, in
custody transfer applications. In the presence of a two-phase flow, the measuring accuracy can be massively negatively affected and then, for example, only amount to a few percent or even only a few
tens of percent of the measured value. As a rule, the higher the gas volume fraction of the two-phase medium, the greater the impairment.
When the term two-phase flow is used here, it generally refers to a multi-phase flow that has a gaseous component. Thus, the term two-phase flow is based on the understanding that this flow has a
gaseous phase and a non-gaseous phase, which in turn has liquid components, but may also have solid components. Thus, the term "two-phase flow" is also used here. Correspondingly, the term
single-phase flow is based on the understanding that the single-phase medium has no gaseous phase in any case, but may contain liquid constituents and also solid constituents.
The occurrence of a two-phase flow is problematic as already explained because gas inclusions can strongly influence the measuring accuracy and also make the safe operation of a Coriolis mass
flowmeter more difficult or even impossible. In the present case, the issue is the aspect of measurement accuracy; the aspect of maintaining reliable operation of a Coriolis mass flowmeter is
described in a large number of applications filed by the patent applicant; an overview with various cross-references is provided, for example, in DE 10 2018 123 534 A1.
The problem of greatly reduced measurement accuracy in a two-phase flow compared to the presence of a single-phase flow is due to the measurement principle in Coriolis mass flowmeters being based on
mechanical interaction with the flowing medium. Coriolis mass flowmeters belong to the class of oscillation flowmeters. Specifically, at least one measuring tube through which the medium to be
measured flows is excited to oscillation by an oscillation generator. The oscillation generator is controlled by the control and evaluation unit, which is often implemented by an embedded computer
system based on a microcontroller, in particular based on a digital signal processor. The mass-bearing medium reacts on the wall of the measuring tube due to the Coriolis inertial force caused by two
orthogonal motions that of the flow and that of the measuring tube. This reaction of the medium on the measuring tube results in a change in the measuring tube velocity compared to the non-flowed
oscillating state of the measuring tube. By capturing these features of the oscillations of the Coriolis measuring tube with flow (phase shift between inlet-side and outlet-side measuring tube
oscillations) by the control and evaluation unit, the mass flow through the measuring tube can be detected with high accuracy.
Regardless of whether media are intentionally (e.g. in the food industry in the case of foamy whipped media) or unintentionally (often unavoidable due to the application, as in filling operations,
e.g. in the refueling of ships, also known as "bunkering"), the measuring accuracy in a two-phase flow suffers under the circumstance that the entire mass of the flowing medium is no longer fully
deflected by the oscillating measuring tube, but in some cases only experiences part of the deflection. This may be due, for example, to the fact that less dense components in the medium flow around
denser components, which also applies to motion components orthogonal to the direction of flow, which are essential for Coriolis measurement. Compression or decompression of gaseous components in the
medium, caused by the inertia of heavier liquid or solid medium components, can also be of importance.
In the prior art, various attempts have been made to improve the measurement accuracy of Coriolis mass flowmeters in two-phase flows. This has included the use of so-called soft computing methods,
i.e., approximate solution methods using, for example, artificial neural networks and support vector machines (see, e.g., Wang, L. et al.: "Gas-Liquid Two-Phase Flow Measurement Using Coriolis
Flowmeters Incorporating Artificial Neural Network, Support Vector Machine, and Genetic Programming Algorithms"; IEEE Transactions On Instrumentation And Measurement; Vol. 66; No. 5; May 2017).
Although various improvements have been made, the treatment of two-phase flows remains problematic. This is especially true for those measurement operations that start with a two-phase flow and then
change to a single-phase flow.
SUMMARY [0008]
Against this background, the object is thus to provide a method with which increased measurement accuracy can be achieved in Coriolis mass flowmeters when single-phase and also two-phase flows occur.
The same applies to the design of the Coriolis mass flowmeter mentioned at the beginning.
The previously derived object is achieved in the method for determining flow measurement values of a Coriolis mass flowmeter in the presence of a two-phase flow of a two-phase medium having a gas
phase in a two-phase measurement interval and a subsequent presence of a single-phase flow of a single-phase medium not having a gas phase in a single-phase measurement interval in that the start
time of the two-phase measurement interval is detected at the onset of the two-phase flow and that the end time of the two-phase measurement interval is detected at the end of the presence of the
two-phase flow. This initially ensures that there is clarity as to the period during which a two-phase flow is present or has been present at all.
In the two-phase measurement interval, two-phase flow measurement values of the two-phase flow are determined and at least partially stored. This means that the two-phase flow measurement values are
also fundamentally available for subsequent examination and processing.
In the single-phase measurement interval, which follows the two-phase measurement interval, at least one state variable of the single-phase medium is determined. Subsequently corrected two-phase flow
measurement values are determined from the stored two-phase flow measurement values and from the at least one state variable of the single-phase medium determined in the single-phase measurement
interval as at least indirect input variables of a correction calculation.
The invention is thus based on the idea of subjecting the erroneous two-phase flow measurement values to a subsequent correction calculation, wherein at least one state variable of the single-phase
medium is used for this correction calculation, which has been determined in the single-phase measurement interval occurring only after the two-phase measurement interval, and which can be determined
there very precisely due to the presence of a single-phase flow that can be easily managed in terms of measurement. When it is said that the stored two-phase flow measurement values and the
determined state variable of the single-phase medium (or also the several determined state variables of the single-phase medium) are used as at least indirect input variables of a correction
calculation, then it is meant that, for example, the determined state variable does not have to enter directly into the correction calculation, but the state variable is first converted into another
value and then fed to the correction calculation. This will be explained by means of an embodiment described later.
In any case, the idea is essentially that the end of the two-phase flow that is problematic in terms of measurement is awaited and the single-phase flow that then sets in is used to precisely detect
state variables of the single-phase medium that are suitable for correcting the erroneous two-phase flow measurement values (at least much more precisely than would be possible in the two-phase
measurement interval). The state variable of the single-phase medium used for the correction calculation must, of course, affect the two-phase flow in some way, so that the possibility for correction
is given.
The corrected two-phase flow measurement values obtained in this way can then be output as individual values or output as part of a cumulative flow measurement value.
It goes without saying that in the single-phase measurement interval, not only is the state variable of the single-phase medium detected which is required for the correction calculation, but flow
measurement values are usually also further detected.
According to an advantageous implementation of the method, it is provided that at least the density .rho..sub.SP of the single-phase medium is determined as state variable of the single-phase medium
and is used as at least indirect input variable of the correction calculation. Examinations using various state variables of the single-phase medium have shown that the density .rho..sub.SP of the
single-phase medium is particularly suitable for influencing the correction calculation in such a way that a measurement error in the two-phase measurement interval, which may well be in the range of
several 10% of the measured value, can be reduced to the single-digit percentage range. For a typical refueling operation (two-phase measurement interval and single-phase measurement interval taken
together) of a ship, for example, this means that measurement errors are now in the range of only 0.5% of the measured value, which is a considerable improvement over the state of the art.
In a preferred implementation of the method, the density .rho..sub.SP of the single-phase medium is used to calculate the (apparent) gas-volume fraction GVF of the two-phase medium, and the
calculated gas-volume fraction GVF of the two-phase medium is used as a direct input variable to the correction calculation. Thus, this is an example where the state variable of the single-phase
medium, in this case the density of the single-phase medium .rho..sub.SP, is indirectly used as an input variable of the correction calculation. In particular, the (apparent) gas volume fraction GVF
of the two-phase medium is calculated by taking the quotient of the difference between the density .rho..sub.SP of the single-phase medium and the density .rho..sub.TP of the two-phase medium and the
density .rho..sub.SP of the single-phase medium. Since the density of the two-phase medium can only be detected with a certain inaccuracy, which is caused by the two-phase medium, we could also speak
here of an apparent density .rho..sub.TP of the two-phase medium; when using it also only the apparent gas volume fraction GVF of the two-phase medium is detected. Nevertheless, in the following we
will always speak of the gas-volume fraction GVF of the two-phase medium. Particularly preferably, the gas volume fraction GVF of the two-phase medium is calculated by forming the quotient of the
difference between the density .rho..sub.SP of the single-phase medium and the density .rho..sub.TP of the two-phase medium and the difference between the density .rho..sub.SP of the single-phase
medium and the density .rho..sub.G of the gas phase of the two-phase medium:
A further development of this method is characterized in that the density .rho..sub.G of the gas phase of the two-phase medium is detected by measuring the temperature of the two-phase medium and
measuring the pressure at the outlet of the Coriolis mass flowmeter. The density .rho.G of the gas phase of the two-phase medium is then detected based on the measured temperature of the two-phase
medium and based on the measured pressure at the outflow side of the Coriolis mass flowmeter. This has the advantage that the density .rho..sub.G of the gas phase of the two-phase medium does not
have to be specified entirely as a parameter, but is determined based on the specific operating conditions of the Coriolis mass flowmeter. The fact that both pressure and temperature are essential
for detecting the density of a gas follows from the thermal equation of state of ideal gases, which can also be applied approximately to real gases. It can also be helpful if information about the
material composition of the gas phase is available, which is often the case, for example, in the previously mentioned example of use in the refueling of ships.
In a further development of the method described above, it is provided that the viscosity .rho..sub.SP of the single-phase medium is used as a further input variable of the correction calculation, in
particular wherein the viscosity .rho..sub.SP of the single-phase medium is determined from a temperature-dependent viscosity curve using the temperature of the two-phase medium.
Temperature-dependent viscosity curves can be recorded, for example, under laboratory conditions for different media and stored in the respective Coriolis mass flowmeter.
In a further preferred implementation of the method, it is provided that the differential pressure P.sub.D across the inflow side and the outflow side of the Coriolis mass flowmeter is used as a
further input variable of the correction calculation, in particular wherein the differential pressures P.sub.D determined in the two-phase measurement interval are at least partially stored, and
particularly preferably a differential pressure P.sub.D is also stored for each stored two-phase flow measurement value. The differential pressure can be captured very easily, for example, by means
of a differential pressure sensor.
There are quite different ways to determine the aforementioned correction calculation in its essence and also to store it in a Coriolis mass flowmeter as an algorithm. If the exact functional
relationship between the input variables and the desired output variable, i.e., the corrected two-phase flow measurement value, is not known, i.e., the relationship cannot be mapped analytically or
numerically without further ado by means of exact calculation methods, it has proved advantageous to use an approximate solution method. Solution methods of this class are technically referred to as
soft computing.
A further development of the method is therefore characterized in that the correction calculation is implemented with an approximate solution method, in which the input variables of the correction
calculation, i.e., at least the stored two-phase flow measurement values and the at least one state variable of the single-phase medium determined in the single-phase measurement interval as at least
indirect input variables, are mapped approximately to the corrected two-phase flow measurement values. Preferably, the approximate solution method is implemented by an artificial neural network or by
a support vector machine.
It has been found to be particularly suitable that the correction calculation is implemented by an artificial neural network with an input layer with at least two input neurons for supply of the
stored two-phase flow measurement values to be corrected and for supply of the state variable of the single-phase medium determined in the single-phase measurement interval or a variable derived
therefrom as at least indirect input variables, with an output layer with an output neuron for output of the subsequently corrected two-phase flow measurement values, and with at least one
intermediate layer with at least two neurons. The artificial neural network is preferably implemented as a fully interconnected network, wherein each input neuron is connected to each neuron of the
intermediate layer via directed and weighted signal paths, and wherein each neuron of the intermediate layer is connected to the output neuron of the output layer via a directed and weighted signal
path. The intermediate layer neurons sum the input values arriving via the weighted signal paths and evaluate them via an activation function. The neurons of the intermediate layer are further
applied with an offset value.
In a preferred variation of the method, the artificial neural network has at least four input neurons in the input layer for supply of the stored two-phase flow measurement values to be corrected,
the gas volume fraction GVF of the two-phase medium, the viscosity .mu..sub.SP of the single-phase medium, and the differential pressure P.sub.D across the inflow side and the outflow side of the
Coriolis mass flowmeter. Further, the artificial neural network has an output neuron for output of the subsequently corrected two-phase flow measurement values, and four neurons in an intermediate
layer, wherein each input neuron is connected to each neuron of the intermediate layer via directed and weighted signal paths, and wherein each neuron of the intermediate layer is connected to the
output neuron of the output layer via a directed and weighted signal path and is applied with an offset value.
When implementing the correction calculation as an artificial neural network, it is further provided that the artificial neural network is trained with a training data set, wherein the training data
set is collected for one design of a Coriolis mass flowmeter and the training data set comprises value tuples from the used input variables of the artificial neural network and the output variable of
the artificial neural network. Such a training data set is collected, for example, under laboratory conditions. Preferably, the flow is thereby varied over at least 80% of the measurement range, in
particular wherein the gas volume fraction GVF of the two-phase medium is varied at least in the range between 0% and 60%.
Preferably, a training data set is collected for each particular two-phase medium and a separate artificial neural network is trained for each two-phase medium. From the different trained artificial
neural networks, data sets are then obtained that contain the weights of the different signal paths and also the offset values of the neurons of the intermediate layer. This then allows a
structurally similar implemented artificial neural network to be parameterized for each two-phase medium.
All previously described variations of the method according to the invention can be carried out online. The start time of the two-phase measurement interval is then determined in real time, the
subsequent two-phase flow measurement values are stored until the end of the two-phase measurement interval is detected. The correction calculation can then be initiated and corrected flow
measurement values (individually or as a cumulative measurement result) can be output. However, it is also possible to proceed in such a way that during a measurement operation, for example during a
filling operation, all flow measurement values are stored, that after the completed measurement operation, the single-phase measurement interval and the two-phase measurement interval are determined
from the stored flow measurement values or other recorded data, and that the correction calculation is then carried out.
The described object is also achieved in the case of the Coriolis mass flowmeter mentioned at the beginning in that it is prepared in such a way that it can carry out the described method. This means
that the control and evaluation unit is designed such that, in the presence of a two-phase flow of a two-phase medium having a gas phase in a two-phase measurement interval and a subsequent presence
of a single-phase flow of a single-phase medium not having a gas phase in a single-phase measurement interval, the start time of the two-phase measurement interval is determined at the onset of the
two-phase flow, that the end time of the two-phase measurement interval is determined at the end of the presence of the two-phase flow, that the two-phase flow measurement values of the two-phase
flow determined in the two-phase measurement interval are stored at least partially, that at least one state variable of the single-phase medium is determined in the single-phase measurement
interval, that, from the stored two-phase flow measurement values and from the at least one state variable of the single-phase medium determined in the single-phase measurement interval, subsequently
corrected two-phase flow measurement values are determined as at least indirect input variables of a correction calculation, and that the corrected two-phase flow measurement values are output as
individual values or are output as part of a cumulative flow measurement value. Preferably, the control and evaluation unit is designed such that it also implements all the preferred implementations
and variations of the method described above in the operation of the Coriolis mass flowmeter.
Various possibilities are known from the prior art as to how the detection of a two-phase flow can proceed. DE 10 2006 017 676 A1 is mentioned here as an example for the pure detection of a two-phase
flow, whereby various state variables and changes in these state variables are evaluated in part statistically to obtain a two-phase signal that indicates the presence or absence of a two-phase flow
with a high degree of accuracy.
In the implementation of the Coriolis mass flowmeter described here, it is preferably provided that the start time of the two-phase measurement interval at the onset of the two-phase flow and/or that
the end time of the two-phase measurement interval at the end of the presence of the two-phase flow is detected by evaluating the level of the excitation signal of the oscillation generator and/or by
evaluating the level of the sensor signal of the oscillation sensor. In particular, the start time of the two-phase measurement interval is determined when a limit height of the excitation signal and
/or the sensor signal is exceeded, and in particular, the end time of the two-phase measurement interval is determined when the excitation signal and/or the sensor signal falls below a limit height.
Other quantities can also be used as indicators for the presence of a two-phase flow, such as the standard deviation of the mass flow (small for single-phase flow, significantly larger for two-phase
flow), the sound emission at the measurement location (small for single-phase flow, larger for two-phase flow), oscillations at the measuring device or connected lines (small for single-phase flow,
significantly larger for two-phase flow).
BRIEF DESCRIPTION OF THE DRAWINGS [0031]
In detail, there are now a multitude of possibilities for designing and further developing the method according to the invention and the Coriolis mass flowmeter according to the invention. For this,
reference is made to the following description of embodiments in connection with the drawings.
FIG. 1 schematically illustrates the design of a Coriolis mass flowmeter.
FIG. 2 illustrates the time course of various relevant physical variables during a typical refueling operation of a ship with two-phase flow and subsequent single-phase flow.
FIG. 3 schematically illustrates a method for determining corrected flow measurement values of a Coriolis mass flowmeter.
FIG. 4 schematically illustrates another embodiment of a method for determining corrected flow measurement values of a Coriolis mass flowmeter.
FIG. 5 illustrates an embodiment for the implementation of a correction calculation within the method of interest here with the aid of an artificial neural network.
FIG. 6 illustrates a further embodiment for the implementation of a correction calculation within the method of interest here with the aid of an extended artificial neural network.
DETAILED DESCRIPTION [0038]
In all figures, a method 1 for determining flow measurement values of a Coriolis mass flowmeter 2 in whole or in part is shown. FIG. 2 schematically shows a Coriolis mass flowmeter 2 in which the
method 1 for determining flow measurement values described below is implemented.
All the embodiments shown have in common that they are concerned with determining flow measurement values of a Coriolis mass flowmeter 2 in the presence of a two-phase flow of a two-phase medium
having a gas phase in a two-phase measurement interval 3 and a subsequent presence of a single-phase flow of a single-phase medium not having a gas phase in a single-phase measurement interval 4.
FIG. 1 schematically shows a Coriolis mass flowmeter 2 with a measuring tube 6 through which a medium 5 (indicated by the arrows running horizontally from left to right) can flow, an oscillation
generator 7, two oscillation sensors 8 and a control and evaluation unit 9. Flow measurement values can be displayed here on the display unit 10. The Coriolis mass flowmeter 2 shown here as an
example has a measuring tube 6 bent into a V-shape. The Coriolis mass flowmeter 2 can just as easily be designed in any other shape, for example with straight measuring tubes, with U-shaped or
omega-shaped measuring tubes, etc.; the chosen design of a Coriolis mass flowmeter is not important. The control and evaluation unit 9 is based on a small computer with a digital signal processor and
with I/O interfaces via which the oscillation generator 7 can be controlled and via which the sensor signals 24 of the oscillation sensors 8 can be captured by means of measurement. With suitable
programming of the control and evaluation unit 9, the method 1 described here for determining flow measurement values is implemented in the Coriolis mass flowmeter 2.
On the basis of the time curves of mass flow rate, density of the medium and the sensor signal 24 of an oscillation sensor, FIG. 2 shows the typical course of a filling operation using the example of
the refueling of a ship, in which both a two-phase flow and a subsequent single-phase flow are present. At the very beginning of the filling operation, the fuel is mixed with a gas phase, so that a
two-phase flow is present in the two-phase measurement interval 3. The gas content in the two-phase medium varies greatly here, and the flow measurement values and the determined density .rho..sub.TP
of the two-phase medium also vary accordingly. The amplitude of the sensor signal 24 of one of the oscillation sensors 8 is also subject to strong fluctuations, which is characteristic of a two-phase
The range of two-phase flow in the two-phase measurement interval 3 is problematic in terms of measurement, the achievable measurement accuracy is frequently worse by one or even two powers of ten
than the measurement accuracy in the range of single-phase flow in the single-phase measurement interval 4.
The method 1 according to the invention is based on the idea of initially storing the inaccurate two-phase flow measurement values q.sub.TP, meas and later supplying them to a correction calculation
in the knowledge of state variables x.sub.SP of the single-phase medium determined with high accuracy in the single-phase measurement interval 4.
FIG. 3 shows an example of an implementation of method 1 by means of a flow chart. The start time t.sub.start of the two-phase measurement interval 3 is determined at the onset of the two-phase flow
11, and the end time t.sub.end of the two-phase measurement interval 3 is determined at the end of the presence of the two-phase flow 12. In the two-phase measurement interval 3, two-phase flow
measurement values q.sub.TP, meas of the two-phase flow are determined and at least partially stored 13. Furthermore, in the single-phase measurement interval 4, at least one state variable x.sub.SP
of the single-phase medium is determined 14. From the stored two-phase flow measurement values q.sub.TP, meas and from the state variable x.sub.SP of the single-phase medium determined in the
single-phase measurement interval 4, subsequently corrected two-phase flow measurement values q.sub.TP, corr are then determined 15 as at least indirect input variables of a correction calculation
f.sub.corr. In this embodiment, the corrected two-phase flow measurement values q.sub.TP, corr are output 16 as part of a cumulative flow measurement value m.sub.tot.
Of course, the question arises which specific state variable x.sub.SP of the single-phase medium is suitable to be used meaningfully as at least indirect input variable of a correction calculation
f.sub.corr; for this, the actually present two-phase flow measurement values must be dependent on the state variable. In the embodiments shown here, the density .rho..sub.SP of the single-phase
medium is determined as the state variable of the single-phase medium--at least also--and used as at least indirect input variable of the correction calculation f.sub.corr. The density .rho..sub.SP
of the single-phase medium can be detected with a Coriolis mass flowmeter, since the density of the medium 5 flowing in the measuring tubes 6 affects the natural angular frequency of the excited
oscillation mode of the measuring tube 6 (or several measuring tubes 6).
As is indicated in FIG. 4, the density .rho..sub.SP of the single-phase medium is used to calculate 17 the gas-volume fraction GVF of the two-phase medium and the calculated gas-volume fraction GVF
of the two-phase medium is used as a direct input to the correction calculation f corr, wherein the gas volume fraction GVF of the two-phase medium is calculated by forming the quotient of the
difference between the density .rho..sub.SP of the single-phase medium and the density .rho..sub.TP of the two-phase medium and the difference between the density .rho..sub.SP of the single-phase
medium and the density .rho..sub.G of the gas phase of the two-phase medium. For example, the density .rho..sub.TP of the two-phase medium can be detected in the same way as the density .rho..sub.SP
of the single-phase medium.
In the method 1 shown in FIG. 4, the density .rho..sub.G of the gas phase of the two-phase medium is detected by measuring the temperature T.sub.TP of the two-phase medium and measuring the pressure
p at the outlet of the Coriolis mass flowmeter 2, and the density .rho..sub.G of the gas phase of the two-phase medium is detected based on the measured temperature T.sub.TP of the two-phase medium
and based on the measured pressure p at the outflow side of the Coriolis mass flowmeter.
As can also be seen in FIG. 4, the viscosity .rho..sub.SP of the single-phase medium is used as another input variable for the correction calculation f.sub.corr. The viscosity .rho..sub.SP of the
single-phase medium is determined from a temperature-dependent viscosity curve using the temperature T.sub.TP of the two-phase medium; the temperature-dependent viscosity curve is not shown
separately here.
Furthermore, in the method 1 according to FIG. 4, the differential pressure P.sub.D over the inflow side and the outflow side of the Coriolis mass flowmeter 2 is used as a further input variable of
the correction calculation f.sub.corr, wherein the differential pressures P.sub.D determined in the two-phase measurement interval 3 are also stored 13 at least in part. Presently, a differential
pressure P.sub.D is also stored for each stored two-phase flow measurement value q.sub.TP, meas.
For the illustrated embodiments for method 1, it holds true that the correction calculation f.sub.corr is implemented with an approximate solution method in which the input variables of the
correction calculation f.sub.corr, i.e., at least the stored two-phase flow measurement values q.sub.TP and the at least one state variable x.sub.SP of the single-phase medium determined in the
single-phase measurement interval 4 as at least indirect input variables (FIG. 3), are mapped approximately to the corrected two-phase flow measurement values q.sub.TP, corr. These approximate
solution methods are all implemented here by an artificial neural network 18, which is shown in more detail in FIGS. 5 and 6.
FIG. 5 shows that the correction calculation f.sub.corr is implemented by an artificial neural network 18 having an input layer with two input neurons 19 for supply of the stored two-phase flow
measurement values q.sub.TP to be corrected and for supply of the state variable x.sub.SP of the single-phase medium (density .rho..sub.SP of the single-phase medium) determined in the single-phase
measurement interval 4. In fact, it is not the density .rho..sub.SP of the single-phase medium that is supplied, but the gas volume fraction GVR as a quantity derived from the density .rho..sub.SP of
the single-phase medium. Furthermore, an output layer with an output neuron 20 for output of the subsequently corrected two-phase flow measurement values q.sub.TP, corr is provided, and an
intermediate layer with four neurons 21. Each input neuron 19 is connected to each neuron 21 of the intermediate layer via directed and weighted signal paths 22, and each neuron 21 of the
intermediate layer is connected to the output neuron 20 of the output layer via a directed and weighted signal path 23. An offset value b.sub.i is applied to each neuron 21 of the intermediate layer.
The weights of the signal paths are denoted by w, although not all weights of all signal paths 22 have been drawn for clarity.
The embodiment according to FIG. 6 is characterized in that the artificial neural network 18 has four input neurons 19 in the input layer for supply of the stored two-phase flow measurement values
q.sub.TP, meas, the gas volume fraction GVF of the two-phase medium, the viscosity .mu..sub.SP of the single-phase medium and the differential pressure P.sub.D to be corrected via the inflow side and
the outflow side of the Coriolis mass flowmeter 2. As in the previous embodiment, the artificial neural network 18 has an output neuron 20 for output of the subsequently corrected two-phase flow
measurement values q.sub.TP, corr and four neurons 21 in an intermediate layer, wherein each input neuron 19 is connected to each neuron 21 of the intermediate layer via directed and weighted signal
paths 22, and wherein each neuron 21 of the intermediate layer is connected to the output neuron 20 of the output layer via a directed and weighted signal path 23, and in particular each neuron 21 of
the intermediate layer is supplied with an offset value b.sub.i. As in FIG. 5, not all weighting w of the signal paths 22, 23 have been drawn here.
The artificial neural networks 18 according to FIGS. 5 and 6 are trained with a training data set in order to be operational, wherein the training data set is collected for one design of a Coriolis
mass flowmeter 2. In the present case, this has been done under laboratory conditions. The training data set comprises value tuples from the used input variables of the artificial neural networks 18
and the output variable of the artificial neural network 18, i.e., the--preferably error-free--value for the flow. In the example shown here, the flow has been varied over about 80% of the
measurement range to create the training data set, and in addition, the gas volume fraction GVF of the two-phase medium has been varied in the range between about 0% and 60%.
The training of the artificial neural networks 18 according to FIGS. 5 and 6 is done in an iterative optimization process. The artificial neural network 18 is repeatedly supplied with the input data
of the associated value tuples, and in each case a value results at the output of the artificial neural network 18 for the corrected two-phase flow measurement value q.sub.TP, corr. This value is
compared to the two-phase flow measurement value contained in the value tuple of the training data set. Ideally, the value calculated by the artificial neural network 18 is as close as possible to
(or as similar as possible to) the exact value from the corresponding value tuple of the training data set. When training the network, the deviation of both values from each other (difference, amount
of difference, square of error, etc.) is then minimized by suitably changing the weighting w.sub.i of the signal paths and also the offset values b.sub.i of the neural nodes 21 of the intermediate
layer (e.g., gradient descent method). The suitability of the resulting artificial neural network 18 is tested with a portion of the training data set which was not used during the training of the
artificial neural network 18. If the deviation of the corrected two-phase flow measurement values of the artificial neural network 18 from the exact results of the training data set is below a
tolerable threshold, the training is terminated and the trained artificial neural network 18 then approximately implements the correction calculation f.sub.corr.
A variation of method 1, not shown separately in the figures, is that during a measurement operation, for example during a filling operation, all flow measurement values q.sub.TP, q.sub.SP are
stored, i.e., those of the two-phase flow as well as those of the single-phase flow. After the measurement operation is completed, the single-phase measurement interval 4 and the two-phase
measurement interval 3 are determined from the stored flow measurement values or other recorded data, then the correction calculation f.sub.corr is performed.
As described above, the method 1 is implemented in the Coriolis mass flowmeter 2 in that the control and evaluation unit 9 is designed such that, when there is a two-phase flow of a two-phase medium
having gas phase in a two-phase measuring interval 3 and a subsequent presence of a single-phase flow of a single-phase medium not having a gas phase in a single-phase measuring interval 4, the start
time t.sub.start of the two-phase measurement interval 3 is detected at the onset of the two-phase flow, that the end time t.sub.end of the two-phase measurement interval 3 is detected at the end of
the presence of the two-phase flow, that in the two-phase measurement interval (3) two-phase flow measurement values q.sub.TP, meas of the two-phase flow are determined and at least partially stored
13, that in the single-phase measurement interval 4 at least one state variable x.sub.SP of the single-phase medium is determined 14, that, from the stored two-phase flow measurement values q.sub.TP,
meas and from the at least one state variable x.sub.SP of the single-phase medium determined in the single-phase measurement interval 4, subsequently corrected two-phase flow measurement values
q.sub.TP, corr are determined 15 as at least indirect input variables of a correction calculation f.sub.corr, and that the corrected two-phase flow measurement values q.sub.TP, corr are output as
individual values or are output 16 as part of a cumulative flow measurement value not.
In FIG. 1, the control and evaluation unit 9 is configured in such a way that the Coriolis mass flowmeter 2 is capable of carrying out the previously described embodiments of the method 1.
In particular, the Coriolis mass flowmeter shown in FIG. 1 is designed such that the start time t.sub.start of the two-phase measurement interval 3 is detected at the onset of the two-phase flow and
also the end time t.sub.end of the two-phase measurement interval 3 is detected at the end of the presence of the two-phase flow by evaluating the level of the sensor signal 24 of the oscillation
sensor 8. Here, the start time t.sub.start of the two-phase measurement interval 3 is determined when a limit height of the sensor signal 24 is exceeded, and the end time t.sub.end of the two-phase
measurement interval 3 is determined when the sensor signal 2 falls below a limit height, for which reference is made to FIG. 2.
User Contributions:
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Geometry Homework Help
Geometry Homework Help
I am generally good at maths. But when it comes to geometry, I often tend to slay it.
Excuse me for being self-congratulatory and trust me when I say it hasn’t always been this way. In fact, when I started, things were rather greyer with geometry than with any other chapter I had ever
In this article, I will tell you how I was able to make merry with geometry with limited geometry homework help and unlimited grit. Yes, you need that bit too if you are serious about keeping things
highly entertaining at the highest level.
Keep an eye-out for the different angles through the article!
Geometry homework: what causes the upsets?
There are not many students who feel geometry is an invincible Goliath. Almost everybody I have met likes geometry. For the most part, it is because geometry is a very agreeable subject. Once you
start liking it, it starts liking you back.
However, before that happens, some students:
• Miss out on the basics of the subject
• Tend to need greater professional help with geometry homework
• Do not follow the right sequence of chapters
These can be upsetting if you are new to the chapter. But every problem was created to be solved.
You are as good as your formulas
The formulas that you have been using all the while are the ones that you need to depend on the most. However, it is equally important that you get a hang of the new sets of formulas that need to be
learnt and revised as well.
Make sure you have the right bits of formulas to depend on every time you need to start a new chapter.
The revised applications
For every formula, there are scores of applications that you can make use of. This is where the geometric smartness comes in handy. You can know about these diverse applications once you are through
with the basics of every chapter.
Keep a tab on how many new applications you devised each week. Jot them down, lest you should forget them later.
Get to someone who has the shortcuts
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If you are successful in looking for someone who can provide handy geometry homework help online, just keep a few things in check so that it does not grow out of proportion.
Advanced geometry: merge chapters at will
There is no need to think exclusively about this part if you have not yet mastered at least a few chapters in geometry.
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To the uninitiated – geometry is still easy if I can ask myself to do my geometry homework without any frills. | {"url":"https://www.fallenblue.org/math-homework-help/geometry/","timestamp":"2024-11-14T20:38:00Z","content_type":"text/html","content_length":"13972","record_id":"<urn:uuid:304f4e9d-01b1-4600-96e3-48751c6c839b>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00319.warc.gz"} |
hyperbolic polynomial
We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of
the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we … Read more
Automorphisms of rank-one generated hyperbolicity cones and their derivative relaxations
A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect the underlying hyperbolic polynomial. This is a natural
class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of … Read more
The central curve in linear programming
The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for
minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal … Read more
Central Swaths (A Generalization of the Central Path)
We develop a natural generalization to the notion of the central path — a notion that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the
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Polynomial time algorithms to approximate mixed volumes within a simply exponential factor
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An LMI description for the cone of Lorentz-positive maps
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$ of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality (LMI) that … Read more
Hyperbolic Programs, and Their Derivative Relaxations
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The mathematics of eigenvalue optimization
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The Lax conjecture is true
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Helton and Vinnikov. Citation Department of Mathematics, Simon Fraser University, Canada Article Download View The Lax conjecture is true | {"url":"https://optimization-online.org/tag/hyperbolic-polynomial/","timestamp":"2024-11-08T11:23:29Z","content_type":"text/html","content_length":"105868","record_id":"<urn:uuid:f785bbbb-3716-4ea4-9b51-f4195f7b21ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00690.warc.gz"} |
Distributed computation of virtual coordinates
Sensor networks are emerging as a paradigm for future computing, but pose a number of challenges in the fields of networking and distributed computation. One challenge is to devise a greedy routing
protocol - one that routes messages through the network using only information available at a node or its neighbors. Modeling the connectivity graph of a sensor network as a 3-connected planar graph,
we describe how to compute on the network in a distributed and local manner a special geometric embedding of the graph. This embedding supports a geometric routing protocol based on the "virtual"
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Original language English (US)
Title of host publication Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07
Pages 210-219
Number of pages 10
State Published - 2007
Externally published Yes
Event 23rd Annual Symposium on Computational Geometry, SCG'07 - Gyeongju, Korea, Republic of
Duration: Jun 6 2007 → Jun 8 2007
Publication series
Name Proceedings of the Annual Symposium on Computational Geometry
Other 23rd Annual Symposium on Computational Geometry, SCG'07
Country/Territory Korea, Republic of
City Gyeongju
Period 6/6/07 → 6/8/07
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• Geometry and Topology
• Computational Mathematics
• Distributed computing
• Greedy routing
• Planar embedding
• Power diagrams
• Virtual coordinates
Dive into the research topics of 'Distributed computation of virtual coordinates'. Together they form a unique fingerprint. | {"url":"https://researchwith.njit.edu/en/publications/distributed-computation-of-virtual-coordinates","timestamp":"2024-11-14T14:51:13Z","content_type":"text/html","content_length":"49077","record_id":"<urn:uuid:f431f2b3-b3b0-4eba-bb12-61b982a119df>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00164.warc.gz"} |
Understanding Explore vs Exploit Dilemma in Online decision-making
Online decision-making in the context of reinforcement learning involves a fundamental choice:
Exploitation: Make the best decision given current information
Exploration: Gather more information The best long-term strategy may involve short-term sacrifices Gather enough information to make the best overall decisions
Two potential definitions of exploration problem:
• How can an agent discover high-reward strategies that require a temporally extended sequence of complex behaviors that, individually, are not rewarding?
• How can an agent decide whether to attempt new behaviors (to discover ones with higher reward) or continue to do the best thing it knows so far? (from keynote ERL Workshop @ ICML 2019)
There are a variety of exploration techniques some more principled than others.
• Naive exploration—greedy methods
• Optimistic Initialisation—A very simple idea that usually works very well
• Optimism under uncertainty—prefer actions with uncertain values eg. UCB
• Probability matching—pick action with largest probability of being optimal eg. Thompson sampling
• Information state space—construct augmented MDP and solve it, hence directly incorporating the value of information
They are studied in the context of mulit-armed bandits (because we understand these settings well). I recommend watch this lecture for more details on fundamentals
let's now ask three basic questions from a research perspective :
1. How important is RL in exploration?
2. State of the art—Where do we stand ?
3. How do we assess exploration methods? and what would it mean to solve exploration?
1. How important is RL in exploration?
In this debate, There are two camps—one thinks exploration is super important and you have to get it right—its hard to build intelligent systems without learning and getting good data is
the other camp thinks there will be so much data that getting the right data doesn’t matter—some exploration will be needed but the technique doesn’t matter.
we take the pendulum example where the task is to balance the pendulum in an upright position.
In this scenario, there is a small cost 0.1 for moving and reward when its upright and in the middle.
State of the art deep RL uses e-greedy exploration takes 1000 episodes and learns to do nothing, which means default exploration doesnt solve the problem. it requires planning in a certain sense or
‘deep exploration’ techniques, which usually help us gets there and learn to balance by the end of 1000 episodes.
key lesson: exploration is key to learning quickly(off-policy) so its a core component but some problems can be solved without exploration. e.g driverless cars
What is the state of the art?
Exploration is not solved—we don’t have good exploration algorithms for the general RL problem—we need a principled/scaleable approach towards exploration.
Theoretical analysis/tractability becomes harder and harder as we go from top to bottom of the following list
• multi-armed bandits ( multi-armed bandits with independent arms is mostly solved )
• contextual bandits
• small, finite MDPs (exploration in tabular settings is well understood )
• large, infinite MDPs, continuous spaces
In many real applications such as education, health care, or robotics, asymptotic convergence rates are not a useful metric for comparing reinforcement learning algorithms. we can’t have millions of
trails in the real world and we care about convergence and we care about quickly getting there. To achieve good real world performance, we want rapid convergence to good policies, which relies on
good, strategic exploration.
other important open research questions:
• how do we formulate the exploration problem ?
• how do we explore at different time scales?
• safe exploration is an open problem
• mechanism is not clear—planning or optimism?
• generalisation (identify and group similar states) in exploration?
• how to incorporate prior knowledge?
How do we assess exploration methods? and what would it mean to solve exploration?
In General, we care about algorithms that have nice PAC bounds and regret but they have limitations where expected regret make sense—we should aim to design algorithms with low beysien regret. We
know Bayes-optimal is the default way to asset these but that can be computationally intractable so literature on efficient exploration is all about trying to get good beyes regret but also maintain
statistical efficiency
But how do we quantify notions of prior knowledge and generalisation? Which is why some experts in the area think its hard to pin down the definition of exploration. i.e when do we know when we will
have but its not obvious
At the end of the day, we want intelligent directed exploration which means behaviour should be that of learning to explore . i.e gets better at exploring and there exists a Life long—ciricula of
Scaling RL (taken from [1] and [2]):
Real systems, where data collection is costly or constrained by the physical context, call for a focus on statistical efficiency. A key driver of statistical efficiency is how the agent explores
its environment.To date, RL’s potential has primarily been assessed through learning in simulated systems, where data generation is relatively unconstrained and algorithms are routinely trained
over millions to trillions of episodes. Real systems, where data collection is costly or constrained by the physical context, call for a focus on statistical efficiency. A key driver of
statistical efficiency is how the agent explores its environment. The design of reinforcement learning algorithms that efficiently explore intractably large state spaces remains an important
challenge. Though a substantial body of work addresses efficient exploration, most of this focusses on tabular representations in which the number of parameters learned and the quantity of data
required scale with the number of states. The design and analysis of tabular algorithms has generated valuable insights, but the resultant algorithms are of little practical importance since, for
practical problems the state space is typically enormous (due to the curse of dimensionality). Current literature on ‘efficient exploration’ broadly states that only agents that perform deep
exploration can expect polynomial sample complexity in learning(By this we mean that the exploration method does not only consider immediate information gain but also the consequences of an
action on future learning.) This literature has focused, for the most part, on bounding the scaling properties of particular algorithms in tabular MDPs through analysis. we need to complement
this understanding through a series of behavioural experiments that highlight the need for efficient exploration. There is a need for algorithms that generalize across states while exploring
intelligently to learn to make effective decisions within a reasonable time frame.
[1] arxiv.org/pdf/1908.03568.pdf
[2] Deep Exploration via Randomized Value Functions: stanford.edu/class/msande338/rvf.pdf | {"url":"https://www.asjadk.com/strategic-exploration-in-online-decision-making/","timestamp":"2024-11-06T02:34:50Z","content_type":"text/html","content_length":"25636","record_id":"<urn:uuid:28f43d56-14dc-408b-9e06-6896d9079b82>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00849.warc.gz"} |
Triangle Formula Sin How To Calculate The Sides And Angles Of s Owlcation
The sine angle formula is, \[\large sin\,\theta=\frac{opposite}{hypotenuse}\] solved examples Give your answer correct to 2 decimal places. = digit 1 2 4 6 10 f.
Solve the triangle ABC with a=123,b=224,c=28minutes and 40
Cos 2 (a) + sin 2 (a) = 1.
\[\text{area of a triangle} = \frac{1}{2} bc \sin{a}\] \[\text{area} = \frac{1}{2} \times 7.1 \times 5.2 \sin{42}\] area = 12.4 cm 2
Calculate the side of a triangle if given two other sides and the angle between them ( cosine rule) ( a ) : In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation
relating the lengths of the sides of a triangle to the sines of its angles. Remember that the given angle must be between the two given sides. If a + b = 180° then:
The following is the formula for the law of sines:
Tan (30°) = 1 / 1.732 = 0.577. Sine and cosine apply to an angle, any angle, so it's possible to have two lines meeting at a point and to evaluate sine or cosine for that angle even though there's no
triangle as such. The longest side is the hypotenuse and the opposite side of the hypotenuse is the opposite side. In a formula, it is written as 'sin' without the 'e':
Calculate the side of a triangle if given side and any two angles ( sine rule ) ( a ) :
The law of cosines generalizes the pythagorean formula to all triangles. Sin value table is given below: Sin (θ + 2nπ) = sin θ for every θ. They’re called the law of cosines and the law of sines.
What are the sine, cosine and tangent of 30° ?
Find the area of triangle pqr if p = 6.5 cm, r = 4.3 cm and ∠ q = 39˚. Here are the formulas of sin, cos, and tan. When the height and the base side of the right triangle are known, we can find the
sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. Enter sides a and b and angle c in degrees as positive real.
There are two main ways in which trigonometric functions are typically discussed:
The sine of an angle is a function that relates to the sides of a right triangle. Here c = [latex]30^ {\circ} [/latex], b = [latex]48.6^ {\circ} [/latex], a = [latex]101.4^ {\circ} [/latex]. In any
right triangle , the sine of an angle x is the length of the opposite side (o) divided by the length of the hypotenuse (h). Sin (−θ) = − sin θ.
Specifically, the sine is found by taking the side that is opposite the angle and dividing it by the hypotenuse of the triangle.
Sine is the ratio of the opposite side to the hypotenuse side of the right triangle. These formulas help in giving a name to each side of the right triangle. Let's learn the basic sin and cos
formulas. Up to 10% cash back sin ( a ) = opposite side hypotenuse = h c sin ( a ) = h c ⇒ h = c sin ( a ) substituting the value of h in the formula for the area of a triangle, you get r.
Area of triangle = ½ ab sinc.
Basic trigonometric identities for sine and cos. [latex]a=\frac {12sin\, 101.4^ {\circ}} {sin\,48.6^ {\circ}} [/latex] = 15.7 (1 d.p.) case 2. In trigonometry, sin is the shorthand of sine function.
Where, a, b, c represent the lengths of the sides of the triangle and a, b, c represent the angles of the triangle.
The area area of a triangle given two of its sides and the angle they make is given by one of these 3 formulas:
Area = (1 / 2) b c sin (a) = (1 / 2) c a sin (b) = (1 / 2) a b sin (c) how to use the calculator. Side of a triangle : Sine, written as sin(θ), is one of the six fundamental trigonometric functions.
Sin θ = perpendicular / hypotenuse.
According to the law, a sin α = b sin β = c sin γ = 2 r, {\displaystyle {\frac {a}{\sin {\alpha }}}\,=\,{\frac {b}{\sin {\beta }}}\,=\,{\frac {c}{\sin {\gamma }}}\,=\,2r,} where a, b, and c are
the lengths of the sides of a triangle, and α, β,.
Outside the triangle, the sine function can be used to find the y component of a vector that has any angle. The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an
adjacent side of √ 3: Area of triangle pqr = ½ pr sinq = ½ × 6.5 × 4.3 × sin 39˚ = 8.79 cm 2 It says that c 2, the square of one side of the triangle, is equal to a 2 + b 2, the sum of the squares of
the the other two sides, minus 2ab cos c, twice their
However, sine and cosine are derived from the sides of an imaginary right triangle superimposed on the lines.
[latex]\frac {a} {sin\, a}=\frac {b} {sin\, b} [/latex] from which. We use the sine rule in the form. Cos (30°) = 1.732 / 2 = 0.866. The law of sines formula is an equation that relates the sides of
a triangle to the sines of their respective angles.
The Sine Rule Area of Triangle YouTube
Question Video Using the Sine Rule to Calculate an
Solve the triangle ABC with a=123,b=224,c=28minutes and 40
F5 Mathematics_Ch.9 Solving Triangles_P.113 (Sine Formula
Sine Rule for the Area of a Triangle YouTube
Area of a Triangle Formula Sine Formula for the Area of
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Sorting Arrays
Now, let's dive into sorting arrays. Sorting an array involves arranging its elements in a specific order, whether it's ascending, descending, or some other sequence.
Let's try to sort the array:
import numpy as np
array = np.array([3, -4, 6, 0, 12, 499, -123])
print('unsorted array', array)
print('sorted array', np.sort(array))
The sort() function returns a copy of the array, while the original array remains unchanged.
Let's try to sort a 2-D array:
import numpy as np
array = np.array([[-2, 4, -12, -434, 62], [1, 4, 7, 93, 75]])
print('Unsorted array', array)
print('Sorted array', np.sort(array))
You have the following array: [15, -2, 33, 47, -55, 16, 0, 834]. Your task is to sort it.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Now, let's dive into sorting arrays. Sorting an array involves arranging its elements in a specific order, whether it's ascending, descending, or some other sequence.
Let's try to sort the array:
import numpy as np
array = np.array([3, -4, 6, 0, 12, 499, -123])
print('unsorted array', array)
print('sorted array', np.sort(array))
The sort() function returns a copy of the array, while the original array remains unchanged.
Let's try to sort a 2-D array:
import numpy as np
array = np.array([[-2, 4, -12, -434, 62], [1, 4, 7, 93, 75]])
print('Unsorted array', array)
print('Sorted array', np.sort(array))
You have the following array: [15, -2, 33, 47, -55, 16, 0, 834]. Your task is to sort it.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Now, let's dive into sorting arrays. Sorting an array involves arranging its elements in a specific order, whether it's ascending, descending, or some other sequence.
Let's try to sort the array:
import numpy as np
array = np.array([3, -4, 6, 0, 12, 499, -123])
print('unsorted array', array)
print('sorted array', np.sort(array))
The sort() function returns a copy of the array, while the original array remains unchanged.
Let's try to sort a 2-D array:
import numpy as np
array = np.array([[-2, 4, -12, -434, 62], [1, 4, 7, 93, 75]])
print('Unsorted array', array)
print('Sorted array', np.sort(array))
You have the following array: [15, -2, 33, 47, -55, 16, 0, 834]. Your task is to sort it.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Thanks for your feedback!
Now, let's dive into sorting arrays. Sorting an array involves arranging its elements in a specific order, whether it's ascending, descending, or some other sequence.
Let's try to sort the array:
import numpy as np
array = np.array([3, -4, 6, 0, 12, 499, -123])
print('unsorted array', array)
print('sorted array', np.sort(array))
The sort() function returns a copy of the array, while the original array remains unchanged.
Let's try to sort a 2-D array:
import numpy as np
array = np.array([[-2, 4, -12, -434, 62], [1, 4, 7, 93, 75]])
print('Unsorted array', array)
print('Sorted array', np.sort(array))
You have the following array: [15, -2, 33, 47, -55, 16, 0, 834]. Your task is to sort it.
Switch to desktop for real-world practiceContinue from where you are using one of the options below
Switch to desktop for real-world practiceContinue from where you are using one of the options below | {"url":"https://codefinity.com/courses/v2/671389bc-34ed-4de7-83cd-2d1bfcf00a76/f102fd90-f0ab-4fd5-b753-f194b043c4d3/5dc27f9b-0be1-46d3-a110-8eb8a7808dca","timestamp":"2024-11-06T14:18:30Z","content_type":"text/html","content_length":"434779","record_id":"<urn:uuid:bc4f7623-3e50-4ee3-8eac-b5be5c8b9fd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00360.warc.gz"} |
Return The Remainder From Two Numbers Javascript With Code Examples
In this article, we will look at how to get the solution for the problem, Return The Remainder From Two Numbers Javascript With Code Examples
How do you use JavaScript for division?
Division (/) The division operator ( / ) produces the quotient of its operands where the left operand is the dividend and the right operand is the divisor.
function remainder(x, y) {
return x % y;
How do you return a remainder in JavaScript?
The modulus operator ( % ) returns the division remainder.
Which operator returns remainder?
The // (remainder) operator returns the remainder from integer division and is defined as being the residue of the dividend after the operation of calculating integer division as previously
described. The sign of the remainder, if nonzero, is the same as that of the original dividend.
How do you find a remainder of two numbers?
Finding the remainder is an easy method. We need to just divide the number by another number with its multiples and get the remainder.
How do I return a remainder?
r = rem( a , b ) returns the remainder after division of a by b , where a is the dividend and b is the divisor. This function is often called the remainder operation, which can be expressed as r = a
- b. *fix(a./b) . The rem function follows the convention that rem(a,0) is NaN .
What is the formula for remainder?
One way to write the remainder is to separate the quotient and the remainder with a "R." 7/2 = Q=3 and R=1 is the formula for dividing 7 by 2. In this case, Q=3 is a quotient, and R=1 is a remainder.
Another way to represent the remainder is as a component of a mixed fraction.
How does reminder work in JavaScript?
The remainder / modulus operator ( % ) returns the remainder after (integer) division. This operator returns the remainder left over when one operand is divided by a second operand. When the first
operand is a negative value, the return value will always be negative, and vice versa for positive values.
What is the remainder theorem formula?
The remainder theorem states that when a polynomial p(x) is divided by (x - a), then the remainder = f(a). This can be proved by Euclid's Division Lemma. By using this, if q(x) is the quotient and
'r' is the remainder, then p(x) = q(x) (x - a) + r.
How do you find remainders in Java?
Get the remainder using % operator. Expressions used in program to calculate quotient and remainder: quotient = dividend / divisor; remainder = dividend % divisor; Note: The program will throw an
ArithmeticException: / by zero when divided by 0.
Settimeout En Javascript With Code Examples
In this article, we will look at how to get the solution for the problem, Settimeout En Javascript With Code Examples Is setTimeout a callback? Introduction to JavaScript setTimeout() cb is a
callback function to be executed after the timer expires. delay is the time in milliseconds that the timer should wait before executing the callback function. setTimeout(function(){ alert("Sup!"); },
2000);//wait 2 seconds setTimeout(function(){ alert("Hello"); }, 3000); setTimeout(function(){ //c
How To Search For An Item In A List In Python With Code Examples
In this article, we will look at how to get the solution for the problem, How To Search For An Item In A List In Python With Code Examples How do you find the index of a value in a list Python? The
index() method returns the index of the specified element in the list.list index() parameters element - the element to be searched. start (optional) - start searching from this index. end (optional)
- search the element up to this index. >>> ["foo", "bar", "baz"].index("bar") 1 l = [1, 2, 3, 4, 5,
Jquery Select 2 Classes With Code Examples
In this article, we will look at how to get the solution for the problem, Jquery Select 2 Classes With Code Examples Can you assign two classes to an element? HTML elements can be assigned multiple
classes by listing the classes in the class attribute, with a blank space to separate them. function checkads() { $('.someclass').each(function(){ if($(this).height() < 50) { $(this).parent
().parent().prepend('<div id="ad-notice">Please support our website</div>'); } }); } $
Ipywidegtes Dropdown With Code Examples
In this article, we will look at how to get the solution for the problem, Ipywidegtes Dropdown With Code Examples How do I get a dropdown in Jupyter? After running the cell above in Jupyter Notebook,
the dropdown menu displayed below will appear. The dropdown menu choices can be made now to display the corresponding Plotly graph. Dropdown menu demo can be found at the end of the article.
widgets.Dropdown( options=['1', '2', '3'], value='2', description
How To Make A Discord Bot In Python With Code Examples
In this article, we will look at how to get the solution for the problem, How To Make A Discord Bot In Python With Code Examples What language are Discord bots written in? In conclusion, Discord bots
are coded in Python. However, if you don't know Python, you can consider a chatbot building platform such as Appy Pie Chatbot. import discord from discord.ext import commands client =
commands.Bot(command_prefix=".") @client.event async def on_ready(): print("Ready!") bot.run("TOKEN") import | {"url":"https://www.isnt.org.in/return-the-remainder-from-two-numbers-javascript-with-code-examples.html","timestamp":"2024-11-07T20:44:03Z","content_type":"text/html","content_length":"149337","record_id":"<urn:uuid:7269efb7-de9d-4771-a2a2-5cf7d01542dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00794.warc.gz"} |
cramp twins theory
The angles ADC and BDA make a linear pair which and hence called as adjacent supplementary angles. If YS is 5, what is ZS? Earn Transferable Credit & Get your Degree, Angle Bisector Theorem:
Definition and Example, Perpendicular Bisector Theorem: Proof and Example, Congruence Proofs: Corresponding Parts of Congruent Triangles, Congruency of Isosceles Triangles: Proving the Theorem,
Properties of Right Triangles: Theorems & Proofs, The AAS (Angle-Angle-Side) Theorem: Proof and Examples, What is a Paragraph Proof? If this equation were in a line-up, it'd be like our theorem, but
maybe it's wearing a fake mustache. Similar triangles are in proportion to one another. In this lesson, we set out to prove the theorem and then look at a few examples of how it's used. Okay, time to
start putting the pieces together. It's time to play detective. Angle ADB is congruent to angle CDF. What Will I Need for the SAT Registration Form? How about an angle-bisector problem? But it
sounded too good to be true. Find: 1.) lessons in math, English, science, history, and more. Things to know about an angle bisector. Let's do some investigating and see what we can find. How Long
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Philosophy? Applied Chemistry, Flashcards - Real Estate Marketing Basics, Flashcards - Promotional Marketing in Real Estate, What is Common Core? Divide that by 10 to get 6. Oh, just BCUZ. courses
that prepare you to earn Did you know We have over 220 college AC! Anyone can earn Tech and Engineering - Questions & Answers, Health and Medicine - Questions & Answers, A triangle has vertices A =
(1, 2, 3), B = (2, 4, 5), and C = (3, 2, 3). Already registered? imaginable degree, area of Select a subject to preview related courses: AB/BD = FC/CDthat looks sort of familiar, doesn't it? We'll
need to get our hands a little dirty to find out. If we cross-multiply, we have 21 * 20 = 15 * 28. In the figure above, point D lies on bisector BD of angle ABC. Their relevant lengths are equated to
relevant lengths of the other two sides. Currently voted the best answer. How can you prove that the angle bisector of an angle of a triangle must intersect the opposite side? The following figure
illustrates this. One day, it got itself mixed up with an angle bisector. We'll label this point F. We can hardly recognize poor old triangle ABC anymore. Create your account. |
{{course.flashcardSetCount}} In summary, we did some good detective work here. What does it look like in practice? How can I prove that any point on the bisector of an angle is equidistant from the
arms of the angle? Line jk bisects mn at point j, find mn if jm = 6 \frac{3}{4} feet. Originally posted Jan 06 2009 1:42 PM. But is it? We know angle BAD equals angle DFC. GIVEN: A triangle ABC, AP
is the bisector of angle A. AP intersects triangle's circumcircle with centre O at P. TO PROVE THAT: OP is the perpendicular bisector of BC. Here's triangle XYZ with angle bisector XS: Let's say we
know that XY is 10 and XZ is 12. If we look at triangle ACF below, we have two equal angles, which makes this an isosceles triangle. Why? We also used the theorem to determine if a line in a triangle
is or isn't an angle bisector. - Definition & Examples, The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples, How to Find the Circumradius of a Triangle, The Parallel Postulate:
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Resource, Introduction to Statistics: Help and Review. Since the sines of supplementary angles are equal. If a point lies anywhere on an angle bisector, it is equidistant from the 2 sides of the
bisected angle; this will be referred to as the equidistance theorem of angle bisectors, or equidistance theorem, for short. flashcard set{{course.flashcardSetCoun > 1 ? It's sad, I know. Are we
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The Fingerprints of G-d
by Jeffrey Meiliken | Apr 4, 2018 | Kabbalah Secrets, Revelations | 29 comments
There is a place with G-d’s fingerprints all over it. And they form a pattern, two hand prints, 10 fingers and a date.
Last week, I was wrapping one of our Pesach cabinets, when I discovered 3 bags of Israeli cous cous (chametz gamor) in the cabinet we were selling beside it. My ego swelled and I felt like a hero. In
the very next second, the door swung shut on my pinky, crushing it. I carefully removed the chametz and paused to ponder the relationship between the two events, which by the way is the real
kabbalah—examining the causes and effects in our lives, not reading people’s astrological signs. I realized G-d was not allowing me the slightest ego and was crushing it out of me like oil from an
olive. I was so grateful. The pain and bruising went away instantaneously and I saw G-d’s fingerprint in the event.
I had no trouble seeing G-d’s hand in the cause-and-effect event. Thanks to the wave of Binah energy that is blanketing us, shortening the time frame between cause and effect, many like-minded
individuals can see it too. But what of the skeptics and doubters? At the time of the incident I had been working on an answer for them, a place where G-d’s fingerprints would be unmistakable and
unquestionable. Moreover, these fingerprints tell a story, tell us about the opening of the 50 Gates of Binah and the onset of the 1000 years of Moshiach Ben David that are upon us.
If G-d were to have carved His Designs, Plans and Name in a 1000-year-old tree would people believe it was real? No, anyone could have carved them there. If He engraved them in large letters high up
on a giant rock ledge, would they be heeded? Probably not, again someone could conceivably carved them. If He gave us specific instructions in the Torah would Man follow them? Again, as is evidenced,
most would not, and even many of those that do do not know what to make of them. Total certainty, the necessary password for the Gates of Moshiach, is so difficult to achieve.
So where could G-d place these designs and plans of His so that we can be sure that only He could have written them, and none other? There is a place and it is hard to reach, in fact, it was only
reachable in the last few decades and only to the people least likely to have such faith in G-d. Even though we can prove beyond the slightest doubt that Man could not have written the Torah and that
much of the Torah is designed to connect us with the various Names of G-d we cannot prove to you that G-d wrote it; after all, a hyper intelligent life force from outer space or from the future could
have done it too. Nevertheless, the omniscient G-d redundantly placed those designs and proofs for us in a place that aliens could not have created. G-d hardwired them into the fabric of the
universe, into the physics and mathematics that have existed since the dawn of time. G-d made them tamperproof. G-d made them immutable and he made them readily accessible to us only as a last
resort. We had thousands of years to find G-d on our own. With no time left, we demand proof.
Before the bundled 7 dimensions of Zein Anpin, the stairway to Heaven represented by the Torah, there existed the realm of Binah. Binah is characterized by the Primordial Hebrew letter Aleph (א),
whose numerical value is both 1 and also 1000. The spiritual dimension of Binah (Understanding) is the World to Come. Binah is the 1000-year long Shabbat; it is the 7th millennium. It is
represented by both the 1000 letters in the Shema prayer and by the first 1000 digits in the mathematical constant Pi. It is the place of the primordial letter Aleph (א). It is the place of
infinite understanding and limitless possibilities. It is upon us and the 50 Gates of Binah are wide open.
Given that the standard mathematics in our universe is base 10 and that our modern scientists and ancient kabbalists agree that the 10 sefirot (dimensions) form a whole or unity/oneness in both our
physical and spiritual universe, base 10 logarithms are the equalizers that make all numbers equate to 10. The standard mathematics in our universe is base 10, meaning we count from 1 to 10, then 10
to 100, and 100 to 1000 in cycles of 10. Not coincidentally, the standard gematria (numerical valuation) of the Hebrew letters are counted the same way. Logarithms sound complicated, but the
logarithm (log) of a number is the exponent (power) to which another fixed number, called the base, must be raised to produce that number. If we wanted to know the base 10 logarithm of 1000 we would
need to calculate to what power 10 must be raised to equal 1000 and since 1000 = 10×10×10 = 10^3, the “logarithm to base 10” of the number 1000 is 3.
This simple schematic of 10 x 10 x 10 equaling 1000 and thus 10^3, mimicking the spiritual structure of the universe in 3 columns and 3 iterations of the 10 sefirot (dimensions) whereby the 1 st
iteration is equivalent to our world, Malchut , the next one to Zeir Anpin and the 3 rd to Binah.
This unity/oneness in our base 10 mathematical system is why the logarithm of the number 10 is 1 (0ne). That same system dictates that the sum of the logarithms of the integers from 1 to 70 is
100.078, in order words if we take all the resultant logs for each number from 1, 2, 3… 70 and add them all together they would give us 100.078, metaphorically 1000 and 78. Kabbalistically, the
decimal place is irrelevant, as it is just a multiple of 10. Also 78 is the value of the Hebrew initials of the 3 column Tree-0f-Life (חיים עצ) which is the choice we must make in 5778. We must
choose the tree-of-life over the tree-of-knowledge of good and evil and yet G-d purposely embedded his signature within the boundaries of our knowledge. And while 78 is 3 times the value (26) of the
Tetragrammaton (יהוה), 1000 is 3 iterations of the 10 sefirot of the tree-of-life (10 x 10 x 10).
Adding up 70 irrational numbers and coming up with a whole round number like 100.0 to 3 decimal places is pretty special, but the number 70 itself has a very special place in G-d’s Divine plans.
There were 70 members of Jacob’s family that entered Egypt; 70 members of the Sanhedrin that were formed at Mt Sinai; the 10 Commandments were found in the 70 th chapter in the Torah; and the many
all-important 70-year periods throughout Biblical history from Adam to Abraham to David to the exiles and all 3 Holy Temples, culminating in the 70 years since the birth of the State of Israel to the
prophesied coming of Moshiach this year in 5778. This deep impression connecting the 1000 sub-sefirot (dimensions) of Binah with the 70 sub-sefirot of Zeir Anpin is but a single fingerprint that G-d
left for us within the logarithmic scale. It is also the connection we need to make to jump from Zeir Anpin, the heavenly ladder, to Binah.
The logs of a number are immutable, and man has no say in them. The results are unique. After noting that within the endless stream of logs, the number 70 stands out starkly, as in the 70 members of
Jacob’s family that entered Egypt, we can see yet another fingerprint of G-d in the unity and oneness of the 600,000 souls that received those 10 Commandments at Mt Sinai: the base 10 log of 600,000
is 5.778. It is because all is One and in G-d’s plans everything is One, everything ties together. There were about 3 million Israelites at Mt Sinai, but the Torah specifically singled out and
mentioned the 600,000 men who were there, just as it specifically mentioned the 70 family members. Man barely knew how to count 3330 years ago when the Torah was given at Sinai, so there is no way
he could have calculated the logarithms to choose these specific numbers. As the Arizal pointed out, those 600,000 souls at Sinai divided by the 70 family members and again by the 12 sons of Israel
and the 3 Patriarchs gives us 10000/42, representing the 42-Letter Name of G-d capable of unifying us.
Another thing that Man could not have known about 3330 years ago is that there are 3 strings of the digit sequence “…5778…” within the first 1000 digits of Pi (3.14159265358…) and also 3 strings of
the sequence of “…2019…”, giving us consecutive back-to-back years 2018 and 2019 in the Gregorian calendar. Two of those “…2019…” numerical strings are actually matching 5-digit strings of “…42019…”
each, coupling the year 2019 with the number 42. Also, when Man was told at Mt Sinai in the year 2448 to start counting jubilee years (50-year periods), he could not have known that he would have to
wait 3330 or 66.6 jubilee years, until the year 5778 for his next opportunity to reach Binah.
You can find 1000s of explanations of how the mathematical constant Pi is anything but random in the books, Nothing is Random in the Universe, and some illustrations of how standard gematria
interprets them such as (3.14159265358…) breaking down to the Names of G-d 31 (אל); 15 (יה); 26 (יהוה) and at the 10th decimal place 358, Moshiach. The World to Come, Binah, associated with the
number 1000 is called the great Shabbat, one of 1000 years so how telling is it that the first string of “…42019…” within those 1000 digits of Pi is located at digit #702, the numerical value of
Shabbat, or that the final 42019 is the last 5 digits of the 1000 and that the sum of the 995 digits until it is 4446, when 44 x 46 equals 57.78 yet again. Another thing about 4446 is that it is 1332
or 2 x 666 less than 5778.
Let us be crystal clear, Man can mess with his DNA; Man can mess with nuclear atoms, but Man can not shape of influence which digit in Pi is where or how often it reoccurs. Only G-d could do that.
And while Man had a hand in warming up the Earth a little, it was G-d’s prerogative to make the surface temperature of the Sun, 5778K.
The fingerprints of G-d are all over the mathematical constants Pi and Phi. For example, intercalated amongst those first 13 digits of Pi with the Names El, Yah, the Tetragrammaton (יהוה) and
Moshiach are the 3 digits 4, 9 and 5, who add up to 18, whose square root is 4.24 (Moshiach Ben David). When we raise the other primordial mathematical constant Phi (1.61803399…) to 18, we get
5778.000 yet again.
By the way, those 3 intercalated numbers (4, 9, and 5) were the 3 rd, 6 th and 9 th digits, which also sum to 18, and (4 x 3 + 9 x 6 + 5 x 9) = 111 or Aleph (אלף) representing Binah. Nevertheless,
this article is mostly about G-d’s fingerprints found in His system of Logarithms and not so much which we have already written about extensively. Atheists would say it is nature that determines the
physical laws that created mathematics and our solar system, and believers would say G-d (Elohim) is nature. While believers can easily see G-d’s plans being unveiled, atheists probably still see it
all as mere coincidence, impossible 1 in a trillion coincidences, but coincidences nonetheless. Believing in G-d is all about letting go of our ego, and the belief that we are in control is the
hardest aspect of ego to let go of. Denial is the purest form of ego.
Let us examine some of those so-called coincidences. While the log of 9 is .95424, spelling out in gematria “The King Moshiach Ben David (95 and 424),” the number 954 also represents the digit #954
within Pi, where the 3rd and final 5778 is located within those first 1000 digits. The final one of the 3 sequences of “…2019…” comprise the absolute last 4 digits in the first 1000 digits of Pi,
metaphorically the year that the Gates of Binah and sealed. The final “…5778..” is part of a 10 digit sequence “…5778185778…”that starts at digit #948 and the sum of the digits in Pi up to that point
is 4240 or 10 x 424.
Interestingly, there are 42 digits between the last “…5778…” at digit #954 and the last “…2019…” at digit #996. This is like the prophecies of the prophets calling for a period of 42 months until the
final redemption. The log of 10 is Unity (1); the log of 9 spells out “King Moshiach Ben David;” and the Log of 8 is .90309, which hints at the 42-Letter Name of G-d since 903 is the sum of the
integers from 1 to 42.
The 42-Letter Name of G-d is the ladder to Binah built into the Torah’s first verse and according to Rav Brandwein of blessed memory, “The Understanding of the 42-Letter Name will bring about the
geulah (final redemption). Of ALL the endless sums of the logs from the bases 1 – 10, the ONE number that is closest to a whole number is 42, or most precisely 41.99997. This makes the number 42
unique in the universe. This is because the 42-Letter Name is the ladder that we can actually climb that bridges our world with heaven and eventually with Moshiach Consciousness (Binah). This is
also why we use the 42-Letter Name matrix to count the Omer every year and why it is especially important this year.
Adding up the logs of all the bases from 1 to 10 is not arbitrary, as in our base 10 mathematical and physical world, 10 represents the completion of a level. It does as well in the spiritual realm.
The Zohar explains nothing happens below that does not happen above. When we do add them up, we are adding up the resultant numbers of calculating each power exponent for each of the 10 bases,
If the log of 6 being .778 and that of the integers 8, 9, and 10 are also telling, then what of the sum of the logs from 1 to 11, whose value when squared is 57.78. Similarly, the sum of the squares
of the integers from 1 to 11 is 506, which is not only the value of the first line of the 42-Letter Name matrix and also the number of the Torah occurrences of the 14 triplets in 42-Letter Name
matrix, but it is also the complete gematria value of “Moshiach Ben David.”
We have been advising about the convergence of the Gregorian and Western Calendars upon the year 5778 for 20 years now so we would be remiss to point out that the ONLY log whose Hebrew Calendar year
aligns with its Gregorian one is the year 2000 or 5760 in the Hebrew Calendar. Through the Log of 5760, which is 3.76042248, the year 3760 HC, the year “0” in the Western Calendar is eternally
connected. 2000 years separate the two dates and there are no other dates that line up separated as whole numbers at all. Ironically, the date 3760 HC is followed by 422, the year the 1 st Holy
Temple was destroyed, -422 BCE. The 2 nd Holy Temple was destroyed in the year 70 CE, and “422” is the gematria value for the Hebrew word “seventy.” How is it possible that the only date this
happens with within the entire logarithmic system is the year 3760 HC (0 CE), the date the 2 calendars pivot on? And given 3.760 42 248 whether G-d intended this or not there are 42 rows in every
Torah scroll and 248 columns. Perhaps it is a reminder that it always comes back to the 42-Letter Name and to the Torah.
Taking it all a giant step further and knowing that Binah, the 1000-year period of Moshiach, is most represented by the number 1000, if we were to add the logarithms of all the bases from 1-10 for
the number 1000, they would sum to 42.3994 or 42.4, the numerical value of Moshiach Ben David. That must be telling us something. The 10 logs of 1000 could have added up to any other number. No
number could have added up to anything close to 42.4. Yet these 10 for the number 1000 did. Given the nature of logarithms and their very flat curve most resultant numbers fall into the double-digit
category. To reach a result above 100, even with summing all the logs of bases 1 – 10, our starting number would need to be about 12,000,000 and it goes up exponentially and unfathomably higher from
there. What we are saying is that 42.4 is the only number that could be representative of 424, especially since the sum of all the logs from bases 1 -10 for even the lowest whole number, 2, is
4.25. Please note that all the logs from bases 1 -10 for the number 1 are “0,” which if we wanted to extract meaning from, we could say that in Unity (Oneness) there is zero (“0”) space between us.
If we wanted to get closer to 4.24, we would have to forsake whole numbers and could go with 1.998 or 3 x .666 which would give us 4.248 as the sum of the logs for the bases 1-10, with 4248 being the
sum of the 22 spelled out Names of 22 Hebrew letters.
Yes another fingerprint is found on the Natural Log of 70 or LN (70), which is the log to the base of e (2.718…). The Natural Log of 70 is 4.248.
Given that all the summary logs from 1 to 12 million must fall between 4.25 and 100 and given that each one is the sum of 10 different mostly irrational numbers, the odds of one being so close to
anything meaningful are astronomical, yet there is a second number that is always associated with Binah. The kabbalists have been writing about the 50 Gates of Binah for thousands of years—it is
even in our Passover Haggadahs. Since we know that we must reach the 50 Gates of Binah in order to enter Binah, and that we last reached the 50 th Gate when Moses led the Israelites out of Egypt,
how incredible is it that the sum of the logarithms of all the bases from 1-10 for the number 3450, representing Moses (משה 345) and also Hashem (השם 345), are precisely 50.000. According to the
Zohar, it is because of this connection to the 50 Gates that we were first told to count the 50-year jubilee cycles, starting at Mt Sinai in 2448 HC, corresponding to 24.48, which is the sum of logs
of bases 1 -10 for the number 54. The number 54 stands out because 54 x 107 = 5778 and the sum of all the integers from 1 to 107 is 5778.
Within those 12 million numbers there are only a few that meet our criteria for precision and one of them is the sum of the logs for all the bases from 1-10 for the number 26, representing the
Tetragrammaton (יהוה), whose resultant value is 20 (19.99799 to be more precise). The value of 20 represents Keter, the highest dimension, which is why the 10 Commandments are found in the 20th
chapter of the Book of Exodus (Shmot) and why there are 620 letters in the 10 Commandments, like the Hebrew word for 20, Esrim (עשרים) of numerical value 620. Also, for what it is worth, 20 x 50 =
There are a few others, though one number that keeps coming up for us this year, the number 58, and as we explained in our previous article there is a strong correlation between the Megillah Ester,
Ester, Mordechai and the Biblical story of Noach and the Flood. They share mutual gematriot of 58, 78 and the 127 combinations of 7 words, amongst other things. As if to make sure we did not miss
this important correlation, G-d made sure that when we add the logarithms of all the bases from 1-10 for the number 12700, representing the 127 provinces times the 100 days in the Megillah Ester; the
127 years of Sarah, the Matriarch; and the 127 possible combinations of the 7 words is the Torah’s first verse, corresponding to the 7 sefirot of Zeir Anpin beneath Binah, we would get 58, or
57.99957 to be exact.
We would be remiss if we did not mention the further correlation between word Megillah (מגלה) with a standard value of 78 and an ordinal value of 33 for a complete value of 111, and the Aleph (אלף),
which is also eleph (אלף) meaning 1000, also of spelled out value 111. The reason we bring this up is that 78, as in 5778, is also the value for Mabul, the Biblical flood (מבול) whose ordinal value
is likewise 33, which is the sum of the logs of the bases 1 – 10 for the number 216, representing Gevurah (strong judgement) and the 216 letters in the 72 Names of G-d used to split the Red Sea. To
think that this is coincidence is to also ignore that the first 33 letters of the Torah add up to a gematria value of 3003, which is the sum of all the integers from 1 to 77.
G-d’s fingerprints are all over this. Even today, it is impossible for a team of engineers with a powerful super computer to create a document with anywhere near the complexity of the Torah, let
alone align it, as G-d did, with the mathematics that underlay the framework of all the physics of our universe. G-d also chose to design our solar system to that same alignment. He did not have to
do this. He did not have to make the Earth orbit the Sun at 66,600 mph. He could have created a system without fingerprints for us to trace; He could have left us with only one option: truly blind
faith. He tried to make it easy for us to find faith and certainty. Even in the final year 5778, with the floodwaters of Binah already upon us since Purim and the Gates of Moshiach Ben David opening
wider every day since Pesach, He wants us to find our way in, to find certainty anyway we can.
Of the fingerprints we highlighted above we can connect the dots: after 70 years since the founding of the State of Israel in 1948, the 50 Gates of Binah will swing open ushering in the Great Shabbat
(702), the 1000 years of Moshiach Ben David (424) and in accordance with the hints given to us by the Arizal, the crucial moment can happen on the 18th day of Iyar, on the 33rd day of the 50-day Omer
period, or .666% of the way through that period which began 100 days or .666% of the way through the period from Zot Chanukah to Shavuot, or .333% of the way through the period from Sukkot to Zot
Chanukah in the year 5778 (2018 CE).
Yes, L’ag B’Omer 5778.
B”H it can all unfold as written in the numbers. It depends on us.
Chag Sameach | {"url":"http://kabbalahsecrets.com/the-fingerprints-of-g-d/","timestamp":"2024-11-06T14:43:41Z","content_type":"text/html","content_length":"188289","record_id":"<urn:uuid:21ab1f8f-d2e0-400e-a58f-38d3e9f64aea>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00371.warc.gz"} |
Integrated Rate Equation - Explanation, Laws, Reactions, and FAQs
What is an Integrated Rate Equation Mean?
An equation that represents the dependence of the reaction rate on the concentration of reacting species is called the differential rate equation. The instantaneous rate of reaction can be expressed
as the tangent slope at any instant of time in the graph of concentration-time type. Therefore, it is more difficult to define the rate of reaction from the concentration-time graph. Hence, we
integrate the differential rate equation to get a relation between the rate constant and the concentration at various points. This resultant equation is called the integrated rate equation. For
different order reactions, we can notice different integrated rate equations.
Integrated Rate Law for a Zero-order Reaction
In a zero-order reaction, the rate of reaction completely depends upon the zeroth power of the concentration of reactants. The zero-order reactions are noticed very rarely. A few examples of
zero-order reactions can be given as decomposition of gaseous ammonia on a hot platinum surface, thermal decomposition of HI on a gold surface, and more. A general equation for a zero-order reaction
including the rate constant k is derived below.
A → B
Rate is given by = - \[\frac{d[A]}{d}\] = k[A]⁰
⇒ - \[\frac{d[A]}{d}\] = k
⇒ - d[A] = -k dt
Integrating on both sides, we get:
⇒ [A] = - kt + c - (1)
Where c is given as the constant of integration,
At time t=0, [A] = [A]₀
Substituting the limits in equation (1) we get the value of c as follows,
⇒ [A]₀ = c
Using the resultant value of c in the equation (1), we get as follows,
⇒ [A] = - kt + [A]₀
The above-derived equation is referred to as an integrated rate equation for the zero-order reactions. We can also observe the above equation as a straight line with the concentration of reactant on
the y-axis and time on the x-axis. And, the slope of the straight line signifies the value of the rate constant, k.
[Image will be Uploaded Soon]
Integrated Rate Law for a First-order Reaction
In the first-order reaction, the rate of reaction depends on the first power of the reactant’s concentration. Artificial and Natural radioactive decay of the unstable nuclei is a few examples of the
first-order reaction. A general equation for a first-order reaction including the rate constant k is derived below:
A → B
Rate is given by = - \[\frac{d[A]}{dt}\] = k[A]
⇒ \[\frac{d[A]}{[A]}\] = - k dt
Integrating on both sides:
⇒ ln[A] = - kt + c ----(2)
Where c is given as the constant of integration,
At time t=0, [A] = [A]₀
Substituting the limits in equation (2) we get the value of c, as given below.
⇒ ln[A]₀ = c
By using the value of c in the above equation we get,
⇒ ln[A] = - kt + ln [A]₀
We can notice that the above-derived equation can be plotted as a straight line including the ln[A] on the y-axis and time (t) on the x-axis. The negative slope of this straight line provides us with
the value of the rate constant, k.
[Image will be Uploaded Soon]
We can also define the value of the rate constant, k from the above-given equation as:
ln \[\frac{[A]}{[A]_{0}}\] = -kt
⇒ k = - \[\frac{ln\frac{[A]}{[A]_{0}}}{t}\]
So, the concentration at any time moment can be given as,
[A] = [A]0\[^{e^{-kt}}\]
Hence, we can now define the concentration and the rate of reaction at any moment by the help of the integrated rate equation for zero and the first-order reaction.
Integrated Rate Law
The mathematical relationship of the reaction rate including reactant concentrations is referred to as the rate law. This relationship can rely more heavily on the concentration of one specific
reactant, whereas, the resulting rate law can include either none, some, or all of the reactant species that are involved in the reaction.
Consider the following hypothetical reaction:
a A + b B → c C
For this, the rate law can be expressed as:
Rate = k[A]y[B]z
Here, ‘k’ is The proportionality constant, which is known as the rate constant and also specific for the reaction, represented at a specific temperature. And, the rate constant changes with the
temperature, whereas, its units depends on the sum of the concentration term exponents present in the rate law. The exponents of y and z must be experimentally determined and they do not correspond
require mentally to the coefficients in the balanced chemical equation.
Factors Affecting the Rate of Reaction
There are primarily 5 factors that affect the rate of reaction, which are listed as follows:
• Temperature
• Pressure
• Presence of catalyst
• Concentration of mixture
• The surface area of mixture molecules
For a reaction to take place/occur, as per the Collision Theory, the collisions between the 2 molecules of the 2 various mixtures must possess a degree of energy, which is called the ACTIVATION
ENERGY. Only when the energy reaches this threshold can new bonds be formed after their original bonds have been broken.
FAQs on Integrated Rate Equation
Q1. Give an Example of a Zero Order Reaction?
Ans. The zero-order reactions are very uncommon, whereas, they occur under some certain conditions. One of the examples of a zero-order reaction is the decomposition of ammonia, which is given as
2NH₃ → N₂ + 3H₂
Rate = k[NH₃]0 = k.
Q2. Given an Example of First Order Reaction?
Ans. One of the examples of a first-order reaction is the ethene hydrogenation, which is chemically equated as follows.
C₂H₄ + H₂ → C₂H₆
Thus, the rate of reaction for the above equation can be given as k [C₂H₄]. We can find the rate constants, initial and final concentrations, and the time taken for the reaction to take place using
these reactions.
Q3. How the Order of Reaction Could be Determined?
Ans. The order of reactions can be either zero or a whole number or a fraction. It also falls on the dependency of the reactant’s rate of reaction. Whereas, if the rate is independent of the
reactants, then the order of reaction can be given as zero. Thus, the rate law of a zero-order reaction would be the rate α [R]0 where [R] is the concentration of the reactant.
Q4. Mention the Factors Affecting the Chemical Reaction Rate?
Ans. There are several factors that affect the rate of a chemical reaction. A few of these factors listed below. Let us look at them.
• Nature of the solvent,
• Intensity of the light,
• Surface area,
• Catalyst,
• Temperature,
• Pressure. | {"url":"https://www.vedantu.com/chemistry/integrated-rate-equation","timestamp":"2024-11-08T14:49:42Z","content_type":"text/html","content_length":"246852","record_id":"<urn:uuid:3625de83-8353-4600-8123-66cd2b15b5a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00282.warc.gz"} |
Lambda the Ultimate
What is a Type?
After going through both of the Types and Programming Languages books, I am starting to feel confident in my understanding of Type Systems in terms of how they would be implemented. However, I still
feel uncomfortable with the theoretical foundations; even after going through the proofs it still seems like a lot of hand-waving when defining what is a type. So to try to find a good mental model
of what a type is I thought I would ask LtU: what is a type?
The reason for my uneasiness probably stems from my background programming in both static and dynamic type systems. The static definition of types seems to tend more towards what can be proven at
compile time. This definition seems to put more emphasis on creating type systems which are strongly normalizing. However, in dynamic type systems, types are often manipulated as terms. Trying to
find a consistent mental model for these two usages of the term (especially after the introduction of subtyping) I am currently stuck with defining them as propositions which map terms to truth
values. The intuition is that a type is a proposition, nothing more. I can't seem to find much theory for such a model and that prompted this post. Is this highly inconsistent with current theory?
More formally, what would your initial reaction be to a system where:
λt:T.x → λt. if T(t) then x else TypeError
The reason answers to 'What is a Type' are 'hand-wavy' is that it depends on the 'type system'.
Soundness of a type-system is well understood to mean 'progress' and 'preservation'. Progress means that any well-typed program in the type-system must always have a valid next step until reaching a
final form (i.e. no undefined behavior, no getting stuck). Preservation means that predicted types are preserved across compute steps (i.e. no bad guesses).
So long as you don't violate soundness in your type-system, you can make anything you want be a 'type'. Not all potential types are useful, of course. But there is a very wide variety of known useful
types: data types, phantom types (used many ways, though my favorite is for zero-knowledge proofs of security protocols), constraint types, dependent types, information bearing types (as utilized for
type classes, meta-programming), sub-structural typing (most commonly linear typing and region inferences, aimed at garbage collection), effect-typing, etc.
So there is good reason people who have wide experience in the field often choose not to pin down 'What is a Type' except in rather generic terms that seem hand-wavy to others. That reason: "It
But 'types as propositions', as you suggest, is actually well known. I'm blanking on the name - Curry Howard isomorphism, IIRC.
The use of first-class objects as types, as in Ruby and Smalltalk, is more part of the program expression itself than of any meaningful type-system.
The use of first-class objects as types, as in Ruby and Smalltalk, is more part of the program expression itself than of any meaningful type-system.
I think this should be emphasized. The way the term "type" is used in the static and dynamic worlds is just different. I doubt you'll find any definition that encompasses both of those usages without
becoming absurdly general (like, "a classification system"), so I suggest simply looking for two distinct definitions.
Very roughly, static type systems tend to classify terms, while dynamic types (note that I do not say type "systems") tend to classify values. In this sense they have something in common, but really
I think it's better to regard them as completely different.
Type theory means static type theory. There is really no useful theory of what dynamic languages call "types," although there are design cultures for specific languages, e.g, Smalltalk program design
I think this should be emphasized. The way the term "type" is used in the static and dynamic worlds is just different.
I agree on the emphasis, and add that types also mean something different in modeling languages. In particular, UML classes are often referred to as types, but we're talking about problem domain
analysis and identifying real world entities; there is no math involved in doing so, other than relational modeling to ensure these entities pass the Indivisibility Argument in modeling the real
Short answer: beware and aware of the context of a vocabulary word.
A different approach to the problem would be to ask yourself: what is a value?
If you think about it a bit, you would have to answer: it depends on the formal definition of the language you are working with.
Normally people don't ask themselves this question: they assume that they already understand what a value is and don't think more about it.
When you sit down and really think about ANY abstract concept, it starts to get slippery, and that's OK: that's the beginning of really coming to terms with it. ;-)
Given a domain D, the functions from D to the truth values are isomorphic to the subsets of D. So we can just as well say that types are subsets of values. For tractability, we can require that each
type has a denumerable number of values, which assures that there are only a denumerable number of types. (In practice, of course, only finitely many values are representable, which means there are
only a finite number of types.)
... each type has a denumerable number of values, which assures that there are only a denumerable number of types.
If a set is denumerable (size aleph null), the set of its subsets is not (size aleph one.) It only makes a difference in theory, since our computers all have finite memories. (Which is not to say
that in theory is unimportant. It's just different from in practice in at least this one way.) (Posts correcting other posts always introduce new errors, and I'm sure this one does, too.)
In practice, of course, only finitely many values are representable, which means there are only a finite number of types.
I'm puzzled by the statement that only finitely many values are representable. Could you give a definition of this concept? In my naïve understanding, the following Haskell code:
data Nat = Zero | Succ Nat
describes the classical type with countably infinitely many elements; and, while, on the one hand, we can certainly only write out explicitly a finite number of its elements in any given program, on
the other hand, there is no element that can't be so described ^*. To take it further, there's a single program with the property that, for any given element ^* of Nat, it will ‘represent’ it in
finite time:
let nats = Zero : map Succ nats in show nats
It thus seems to me that all elements of the type are representable. Of course, though, you know all this, which makes me think that I must not have the right definition of ‘representable’.
When I first tried to learn about a rigorous foundation for the theory of types (which is not too far away from when I last tried to learn about it, so take this with a grain of thought), I was
immediately drawn, as a mathematician, to the idea of types as sets—but then I saw that types are not sets. Of course, the abstract immediately says “[t]he title is not a statement of fact”, and
moves on to less provocative waters, but it has left me puzzled. For example, a term in most type systems has only one type (or, at best, a polymorphic type like Num a => a); but, in real life, an
element of an infinite set will belong to infinitely many of its subsets (so we would need, in pseudo-Haskell, some sort of type constraint like term :: type `Contains` term => type, which seems so
vague as to be useless). Again, this is all just coming from my attempts to generalise my mathematical understanding way too far into computer science. Could you clarify the meaning of the equation
in your title?
^* I'm intentionally ignoring the (non-)distinction between data and co-data that leads to constructions like
inf = Succ inf
because I don't understand it well enough to feel confident of not saying something foolish.
For example, a term in most type systems has only one type ... but, in real life, an element of an infinite set will belong to infinitely many of its subsets
The root of the difficulties with the "type = set" definition that others are pointing out is that that a set is a first order entity in the universe of sets, whereas a type is a second order entity
defined over some other universe of terms or values, and is intended to partition those terms or values in some restricting way.
This pretty much precludes it being closed under all the same operations that sets are closed under.
PS I got a smile from your implied definition of "real life". ;-)
The 'types = subsets' notion too easily confuses readers, who take that and tweak it ever so slightly to 'subtypes = subsets', which fails to capture covariance vs. contravariance and the resulting
multi-dimensional data-type lattices.
The 'subsets' notion also doesn't generalize well to types on the structure of the program or the relationships between values, such as dependent typing, constraint typing, linear typing, uniqueness
I've abandoned use of 'subsets' in attempting to explain types for these reasons.
More formally, what would your initial reaction be to a system where:
λt:T.x → λt. if T(t) then x else TypeError
Well, if T is an arrow type T1->T2, then what does T(t) mean without static typing of the function body?
The answer is probably something along the lines of higher-order contracts. It seems to me that contracts come closest to being a proper theory of what is often called "dynamic typing".
That paper was pretty much exactly what I was looking for. It is nice to see that typing is also being explored through dynamic semantics as well.
Section 3 of the lecture notes
is entitled `What is a type?' and tries to answer the question. The
section shows four definitions, from semantic (a type is a set of
values) to syntactic (a type system as a syntactic discipline). It
seems the original notion of types, by Bertrand Russell is the most
unifying. Type is defined to be a subset of values (semantically, so to
speak) but it should be determined just from the form of an expression
(or, syntactically).
BTW, the Russell's definition is 101 years old.
Perhaps it's worth pointing to this and this very old thread on the subject. In particular, I still feel, intuitively, that Ontic has important things to teach us in this regard. | {"url":"http://lambda-the-ultimate.org/node/3658","timestamp":"2024-11-02T14:38:23Z","content_type":"application/xhtml+xml","content_length":"28497","record_id":"<urn:uuid:55ebe0e2-a6d8-47f6-9e31-ee85ecff5ece>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00848.warc.gz"} |
Celebrating Einstein's Special Theory of Relativity
Dr. Minardi
Continuing with the historical theme of these blogs, but also the countdown to the 2023 Nobel awards, I was notified of an anniversary today of Einstein's seminal paper establishing the Theory of
Relativity in 1905.
On September 26, 1905, Einstein's paper "On the Electrodynamics of Mobing Bodies" was published. Classical Einstein put on proverbial blast Maxwell, Newton, and Michelson-Morley and in turn, defined
his own theory later coined as the "Special Theory of Relativity."
Now I was never a physics expert, but I spent time today thinking through what this theory was and what Einstein concluded to be a failure of the present-day (at the time) equations and theorems.
What is the Special Theory?
The special theory, in a nutshell, is an explanation of the relationship between time and space. It is termed "special" due to the circumstances by which Einstein concluded his findings in which the
singular use of space and time is referred to as "spacetime" and is in turn "flat" (or that the impact from gravity is negligible).
In this seminal paper of 1905, Einstein presented two postulates:
1. The laws of physics are the same for all frames of reference.
2. In a vacuum, the speed of light is the same for all observers.
Why is this important?
The power of this special theory and the newest finding from Einstein is that it countered much of the traditional understanding that other physicists had emphasized. Following these postulates and
this publication, it took Einstein nearly a decade to publish his follow-up to "special relativity" as "Theory of General Relativity."
Additionally, when being discovered in concert with Quantum mechanics, the theories were used to extrapolate electron spin and antiparticles thus more accurately describing their interactions. This
would be the foundational work of Quantum Field Theory and most appropriate to Oppenheimer's work on the Manhattan Project.
To consider
If you would like to read Einstein's original manuscript please see here. While I am always a fan of reading original manuscripts and the scientist's thought process, this is quite the paper. Granted
it was written in 1905, I'm blaming the difference in eras. I would love to know your thoughts on this, let me know in the comments!
Avaliado com 0 de 5 estrelas.
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Poisson s ratio of iron ore
Numerical study of the dynamic behaviour of iron ore
— The principal mechanism of iron ore granulation can be described as the layering of adhering fine particles onto the coarse nuclei particles under the action of water as shown in Fig this mechanism
is commonly known as auto layering [5 6] A typical iron ore granule is generally comprised of an inner seed coarse particle which
Poisson s ratio in cubic materials Proceedings of the Royal
— Expressions are given for the maximum and minimum values of Poisson s ratio ν for materials with cubic symmetry Values less than −1 occur if and only if the maximum shear modulus is associated
Login to your account On the non hookean elastic behavior of iron whiskers at high strain Materials Letters
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Mathematics in Cryptography
by cunliffe | Apr 28, 2020 | Spring 2020 | 10 comments
Imagine: You are sitting in a coffee shop on your phone when all of a sudden you get the urge to check the balance of your bank account. You go to settings in your phone and connect to the free WiFi
at the coffee shop. Then you goon the bank’s website or in their app and GAME OVER. Well, it would be if not for cryptography.
There are two forms of cryptography, public and private. Both help in internet security and keep other people from getting your information. It would be possible to go online without this technology,
however, why would you? If there was no way to secure data and communication across the internet, no one would want to go on it. Since the beginning of human communication, humans have always wanted
privacy. There are many things people do not want blurted out so the whole world knows. Privacy is a human right and their data and communication should be protected as such.
In this presentation, I will explore the two different methods of cryptography and give examples of each. These two methods need to be used in unison because private cryptography, although fast, is
not secure. Public key cryptography is secure, but extremely slow. I will compare two examples of public key cryptography, explaining how it is evolving. Although still slow, cyber security experts
have shortened the key length, making one form of public key faster than the other.
10 Comments
thorne on April 29, 2020 at 5:34 am
Great presentation, I wish we could have heard it in person. I really liked your abstract, plus the history behind cryptography I found fascinating. I did have a question about one of the
“problems” brought up; it mentioned how the RSA Algorithm isn’t completely private because it would take much too long to accomplish anything. When was this a problem? ie have computers developed
so much since then that it wouldn’t be a problem anymore? Nice job!
cunliffe on April 29, 2020 at 4:52 pm
The RSA Algorithm is not private key cryptography because of the keys. Private key cryptography involves two identical keys that both the sender and receiver have. It is a form of public key
cryptography because the public key is known to everyone, whereas the private key is only known to the receiver. Therefore, there are two different keys, they are not identical. With that
being said, public key cryptography takes a very long time to complete. This is because the keys are longer than those in private key cryptography. In the example of the Caesar Cipher (an
example of private key cryptography) I spoke of, the key was three bits long (because that is how far each letter was moved over by). In the RSA algorithm (an example of public key
cryptography) the keys can be of varying lengths. The most popular today is 2048 bits. This means the key is 2048 long. This is a very large key, so it takes more time to encrypt and decrypt
the data it is trying to protect. Elliptic curve cryptography (another example of public key cryptography) is a faster method. This is because a 256 bit key in elliptic curve cryptography is
as secure as a 2048 bit key in the RSA Algorithm. Length of keys in public key cryptography has always been a problem. Elliptic curve cryptography is the cyber security experts’ way of trying
to shorten the key so the process becomes quicker. It still takes a while but it takes less time than elliptic curve and it is just as safe. I hope this helped explain it better for you.
Please let me know if you have anymore questions.
spinosa on April 29, 2020 at 3:12 pm
Excellent job with a very interesting and difficult topic! I have a good amount of experience with cryptography (I took the upper level Web Development course at Canisius) and never quite got the
breadth of understanding to the topic as you illustrated here. Since i learned it in the computer science department, I never really got to see how abstract algebra plays a role in it, and you
explained it very well! I really liked how you pulled in discussions on Fields and Abelian groups because it does show that something that was once thought of as purely theoretical, can later be
found to be incredibly applicable. You did a really good job of giving both a breadth of knowledge on the subject while also going into the specifics just enough to help the reader learn what is
really going on mathematically. What a way to finish the math major!
lundy on April 30, 2020 at 12:55 am
Hackers beware! Mathematics is here to protect your data! Very well done presentation! I liked the part where a professor printed off all of the passwords just so that he could have more time to
work on his research. It is scary to think that high school students were able to crash computers back before all of this cryptography. I like how you slowly evolved from simple ciphers to more
complex ciphers. I built a Caesar cipher machine in computer science and it was pretty easy so I now understand why it is so easy to decipher. I really like how you didn’t say that public keys or
private keys were better. Rather, the optimal encryption is a blend of the two keys. I have never heard of the elliptic curve as a secure method for cryptography and I think that you did a great
job explaining such a concept. Well done and I’m glad that you did not encrypt this whole presentation with a Caesar cipher although that would be an interesting project if you were looking for
something to do on a rainy day!
okon on April 30, 2020 at 1:19 am
Hello! I thought your presentation was excellent! I am completely new to cryptography, so I really appreciated all of the background information you gave to explain what cryptography is and why
it is so important. You explained things in a way so that even someone completely unfamiliar to cryptography was still able to follow. Also, I think it’s amazing that you were able to cover so
much information in one presentation! It’s very obvious that you are incredibly knowledgeable about this topic, and that you put in a significant amount of work to get this presentation to be so
impressive. Also, I wanted to commend you on your abstract. Talk about attention-grabbing! I’m sorry you didn’t get to present this in person, but trust me when I say that everyone who reads it
will know and appreciate all of your hard work! Great job!
sugiyama on May 1, 2020 at 1:10 pm
Now many devices and users are connected with internet. In this meaning, cryptography is important role for protecting our data. Your introduction is really attracting because I think many people
do not care too much about using free Wi-Fi at the coffee shop even if they tapped “I agree” when the message about possibility of insecure connection showed up before using Wi-Fi. Public and
private keys are interesting concepts. I think they are also used at block chain.
faracca on May 1, 2020 at 4:02 pm
I thoroughly enjoyed this presentation! Really well done on a complex topic. I know almost nothing about cryptography, so this was a good lesson for me personally. I always love learning about
new avenues where math is essential. I also think this topic will be really useful in potential job opportunities as this is such an impressive subject to learn about. Great job Danielle, you
should be really proud!
niland on May 1, 2020 at 5:12 pm
Definitely show a strong knowledge of the subject. As always I can tell you payed a lot of attention to the formatting and layout of the presentation. I’ve noticed that cryptography is just one
of those topics that needs to be set up in a logical and systematic order or it will just confuse everyone and make the presentation go off course but this hit the nail on the head. Very good all
gordner on May 1, 2020 at 7:53 pm
Great job! Cryptography is one of the most interesting fields of mathematics and public key encryption is perhaps one of the most useful things human beings have ever created. I’m always
intrigued when I get to hear about this stuff, but it’s usually quite confusing. You did a good job of gradually increasing the complexity as the presentation went on, so we didn’t get lost.
cherven on May 3, 2020 at 3:27 am
Neat discussion on cryptography; I’m currently in the midst of developing a new website and chest-deep in hashing and encryption to maintain user security, thankfully, most of the difficult RSA
stuff is done in the background with certificate signing authorities. It’s very neat to see what is going on behind the scenes with these algorithms! I had never heard about an alternative method
to encryption involving elliptic curves!
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Fit binary Gaussian kernel classifier using random feature expansion
fitckernel trains or cross-validates a binary Gaussian kernel classification model for nonlinear classification. fitckernel is more practical for big data applications that have large training sets
but can also be applied to smaller data sets that fit in memory.
fitckernel maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space by minimizing the regularized objective function. Obtaining the
linear model in the high-dimensional space is equivalent to applying the Gaussian kernel to the model in the low-dimensional space. Available linear classification models include regularized support
vector machine (SVM) and logistic regression models.
To train a nonlinear SVM model for binary classification of in-memory data, see fitcsvm.
Mdl = fitckernel(X,Y) returns a binary Gaussian kernel classification model trained using the predictor data in X and the corresponding class labels in Y. The fitckernel function maps the predictors
in a low-dimensional space into a high-dimensional space, then fits a binary SVM model to the transformed predictors and class labels. This linear model is equivalent to the Gaussian kernel
classification model in the low-dimensional space.
Mdl = fitckernel(Tbl,ResponseVarName) returns a kernel classification model Mdl trained using the predictor variables contained in the table Tbl and the class labels in Tbl.ResponseVarName.
Mdl = fitckernel(Tbl,formula) returns a kernel classification model trained using the sample data in the table Tbl. The input argument formula is an explanatory model of the response and a subset of
predictor variables in Tbl used to fit Mdl.
Mdl = fitckernel(Tbl,Y) returns a kernel classification model using the predictor variables in the table Tbl and the class labels in vector Y.
Mdl = fitckernel(___,Name,Value) specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can
implement logistic regression, specify the number of dimensions of the expanded space, or specify to cross-validate.
[Mdl,FitInfo] = fitckernel(___) also returns the fit information in the structure array FitInfo using any of the input arguments in the previous syntaxes. You cannot request FitInfo for
cross-validated models.
[Mdl,AggregateOptimizationResults] = fitckernel(___) also returns AggregateOptimizationResults, which contains hyperparameter optimization results when you specify the OptimizeHyperparameters and
HyperparameterOptimizationOptions name-value arguments. You must also specify the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions. You can use this syntax to optimize
on compact model size instead of cross-validation loss, and to perform a set of multiple optimization problems that have the same options but different constraint bounds.
Train Kernel Classification Model
Train a binary kernel classification model using SVM.
Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').
load ionosphere
[n,p] = size(X)
resp = 2x1 cell
Train a binary kernel classification model that identifies whether the radar return is bad ('b') or good ('g'). Extract a fit summary to determine how well the optimization algorithm fits the model
to the data.
rng('default') % For reproducibility
[Mdl,FitInfo] = fitckernel(X,Y)
Mdl =
ResponseName: 'Y'
ClassNames: {'b' 'g'}
Learner: 'svm'
NumExpansionDimensions: 2048
KernelScale: 1
Lambda: 0.0028
BoxConstraint: 1
FitInfo = struct with fields:
Solver: 'LBFGS-fast'
LossFunction: 'hinge'
Lambda: 0.0028
BetaTolerance: 1.0000e-04
GradientTolerance: 1.0000e-06
ObjectiveValue: 0.2604
GradientMagnitude: 0.0028
RelativeChangeInBeta: 8.2512e-05
FitTime: 0.0960
History: []
Mdl is a ClassificationKernel model. To inspect the in-sample classification error, you can pass Mdl and the training data or new data to the loss function. Or, you can pass Mdl and new predictor
data to the predict function to predict class labels for new observations. You can also pass Mdl and the training data to the resume function to continue training.
FitInfo is a structure array containing optimization information. Use FitInfo to determine whether optimization termination measurements are satisfactory.
For better accuracy, you can increase the maximum number of optimization iterations ('IterationLimit') and decrease the tolerance values ('BetaTolerance' and 'GradientTolerance') by using the
name-value pair arguments. Doing so can improve measures like ObjectiveValue and RelativeChangeInBeta in FitInfo. You can also optimize model parameters by using the 'OptimizeHyperparameters'
name-value pair argument.
Cross-Validate Kernel Classification Model
Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').
load ionosphere
rng('default') % For reproducibility
Cross-validate a binary kernel classification model. By default, the software uses 10-fold cross-validation.
CVMdl = fitckernel(X,Y,'CrossVal','on')
CVMdl =
CrossValidatedModel: 'Kernel'
ResponseName: 'Y'
NumObservations: 351
KFold: 10
Partition: [1x1 cvpartition]
ClassNames: {'b' 'g'}
ScoreTransform: 'none'
CVMdl is a ClassificationPartitionedKernel model. Because fitckernel implements 10-fold cross-validation, CVMdl contains 10 ClassificationKernel models that the software trains on training-fold
(in-fold) observations.
Estimate the cross-validated classification error.
The classification error rate is approximately 9%.
Optimize Kernel Classifier
Optimize hyperparameters automatically using the OptimizeHyperparameters name-value argument.
Load the ionosphere data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad ('b') or good ('g').
Find hyperparameters that minimize five-fold cross-validation loss by using automatic hyperparameter optimization. Specify OptimizeHyperparameters as 'auto' so that fitckernel finds optimal values of
the KernelScale, Lambda, and Standardize name-value arguments. For reproducibility, set the random seed and use the 'expected-improvement-plus' acquisition function.
[Mdl,FitInfo,HyperparameterOptimizationResults] = fitckernel(X,Y,'OptimizeHyperparameters','auto',...
| Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | KernelScale | Lambda | Standardize |
| | result | | runtime | (observed) | (estim.) | | | |
| 1 | Best | 0.35897 | 2.3903 | 0.35897 | 0.35897 | 3.8653 | 2.7394 | true |
| 2 | Accept | 0.35897 | 0.40434 | 0.35897 | 0.35897 | 429.99 | 0.0006775 | false |
| 3 | Accept | 0.35897 | 0.57429 | 0.35897 | 0.35897 | 0.11801 | 0.025493 | false |
| 4 | Accept | 0.41311 | 0.50318 | 0.35897 | 0.35898 | 0.0010694 | 9.1346e-06 | true |
| 5 | Accept | 0.4245 | 0.68218 | 0.35897 | 0.35898 | 0.0093918 | 2.8526e-06 | false |
| 6 | Best | 0.17094 | 0.39654 | 0.17094 | 0.17102 | 15.285 | 0.0038931 | false |
| 7 | Accept | 0.18234 | 0.47982 | 0.17094 | 0.17099 | 9.9078 | 0.0090818 | false |
| 8 | Accept | 0.35897 | 0.35852 | 0.17094 | 0.17097 | 26.961 | 0.46727 | false |
| 9 | Best | 0.082621 | 0.4906 | 0.082621 | 0.082677 | 7.7184 | 0.0025676 | false |
| 10 | Best | 0.059829 | 0.62044 | 0.059829 | 0.059839 | 5.6125 | 0.0013416 | false |
| 11 | Accept | 0.062678 | 0.55309 | 0.059829 | 0.059793 | 7.3294 | 0.00062394 | false |
| 12 | Best | 0.048433 | 0.93641 | 0.048433 | 0.050198 | 3.7772 | 0.00032964 | false |
| 13 | Accept | 0.051282 | 0.95284 | 0.048433 | 0.049662 | 3.4417 | 0.00077524 | false |
| 14 | Accept | 0.054131 | 1.0627 | 0.048433 | 0.051494 | 4.3694 | 0.00055199 | false |
| 15 | Accept | 0.051282 | 1.225 | 0.048433 | 0.04872 | 1.7463 | 0.00012886 | false |
| 16 | Accept | 0.048433 | 0.88001 | 0.048433 | 0.048475 | 3.9086 | 3.1147e-05 | false |
| 17 | Accept | 0.054131 | 0.98807 | 0.048433 | 0.050325 | 3.1489 | 9.1315e-05 | false |
| 18 | Accept | 0.051282 | 0.73412 | 0.048433 | 0.049131 | 2.3414 | 4.8238e-06 | false |
| 19 | Accept | 0.062678 | 1.0564 | 0.048433 | 0.049062 | 7.2203 | 3.2694e-06 | false |
| 20 | Accept | 0.054131 | 0.58096 | 0.048433 | 0.051225 | 3.5381 | 1.0341e-05 | false |
| Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | KernelScale | Lambda | Standardize |
| | result | | runtime | (observed) | (estim.) | | | |
| 21 | Accept | 0.068376 | 0.58595 | 0.048433 | 0.05111 | 1.4267 | 1.7614e-05 | false |
| 22 | Accept | 0.054131 | 0.81825 | 0.048433 | 0.05127 | 3.2173 | 2.9573e-06 | false |
| 23 | Accept | 0.05698 | 1.0053 | 0.048433 | 0.051187 | 2.4241 | 0.0003272 | false |
| 24 | Accept | 0.059829 | 1.4387 | 0.048433 | 0.051097 | 2.5948 | 4.5059e-05 | false |
| 25 | Accept | 0.059829 | 0.95757 | 0.048433 | 0.051018 | 7.2989 | 2.6908e-05 | false |
| 26 | Accept | 0.068376 | 1.261 | 0.048433 | 0.048938 | 3.9585 | 6.9173e-06 | false |
| 27 | Accept | 0.05698 | 0.98149 | 0.048433 | 0.051222 | 4.2751 | 0.0002231 | false |
| 28 | Accept | 0.062678 | 0.54509 | 0.048433 | 0.051232 | 1.4533 | 2.8533e-06 | false |
| 29 | Accept | 0.051282 | 0.90554 | 0.048433 | 0.051122 | 3.8449 | 0.00059747 | false |
| 30 | Accept | 0.21083 | 1.0227 | 0.048433 | 0.0512 | 45.588 | 3.056e-06 | false |
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 40.3496 seconds
Total objective function evaluation time: 25.3913
Best observed feasible point:
KernelScale Lambda Standardize
___________ __________ ___________
3.7772 0.00032964 false
Observed objective function value = 0.048433
Estimated objective function value = 0.05162
Function evaluation time = 0.93641
Best estimated feasible point (according to models):
KernelScale Lambda Standardize
___________ __________ ___________
3.8449 0.00059747 false
Estimated objective function value = 0.0512
Estimated function evaluation time = 0.89718
Mdl =
ResponseName: 'Y'
ClassNames: {'b' 'g'}
Learner: 'svm'
NumExpansionDimensions: 2048
KernelScale: 3.8449
Lambda: 5.9747e-04
BoxConstraint: 4.7684
FitInfo = struct with fields:
Solver: 'LBFGS-fast'
LossFunction: 'hinge'
Lambda: 5.9747e-04
BetaTolerance: 1.0000e-04
GradientTolerance: 1.0000e-06
ObjectiveValue: 0.1006
GradientMagnitude: 0.0114
RelativeChangeInBeta: 9.3027e-05
FitTime: 0.1755
History: []
HyperparameterOptimizationResults =
BayesianOptimization with properties:
ObjectiveFcn: @createObjFcn/inMemoryObjFcn
VariableDescriptions: [5x1 optimizableVariable]
Options: [1x1 struct]
MinObjective: 0.0484
XAtMinObjective: [1x3 table]
MinEstimatedObjective: 0.0512
XAtMinEstimatedObjective: [1x3 table]
NumObjectiveEvaluations: 30
TotalElapsedTime: 40.3496
NextPoint: [1x3 table]
XTrace: [30x3 table]
ObjectiveTrace: [30x1 double]
ConstraintsTrace: []
UserDataTrace: {30x1 cell}
ObjectiveEvaluationTimeTrace: [30x1 double]
IterationTimeTrace: [30x1 double]
ErrorTrace: [30x1 double]
FeasibilityTrace: [30x1 logical]
FeasibilityProbabilityTrace: [30x1 double]
IndexOfMinimumTrace: [30x1 double]
ObjectiveMinimumTrace: [30x1 double]
EstimatedObjectiveMinimumTrace: [30x1 double]
For big data, the optimization procedure can take a long time. If the data set is too large to run the optimization procedure, you can try to optimize the parameters using only partial data. Use the
datasample function and specify 'Replace','false' to sample data without replacement.
Input Arguments
X — Predictor data
numeric matrix
Predictor data, specified as an n-by-p numeric matrix, where n is the number of observations and p is the number of predictors.
The length of Y and the number of observations in X must be equal.
Data Types: single | double
Y — Class labels
categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors
Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.
• fitckernel supports only binary classification. Either Y must contain exactly two distinct classes, or you must specify two classes for training by using the ClassNames name-value pair argument.
For multiclass learning, see fitcecoc.
• The length of Y must be equal to the number of observations in X or Tbl.
• If Y is a character array, then each label must correspond to one row of the array.
• A good practice is to specify the class order by using the ClassNames name-value pair argument.
Data Types: categorical | char | string | logical | single | double | cell
Tbl — Sample data
Sample data used to train the model, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Multicolumn variables and cell arrays
other than cell arrays of character vectors are not allowed.
Optionally, Tbl can contain a column for the response variable and a column for the observation weights.
• The response variable must be a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors.
□ fitckernel supports only binary classification. Either the response variable must contain exactly two distinct classes, or you must specify two classes for training by using the ClassNames
name-value argument. For multiclass learning, see fitcecoc.
□ A good practice is to specify the order of the classes in the response variable by using the ClassNames name-value argument.
• The column for the weights must be a numeric vector.
• You must specify the response variable in Tbl by using ResponseVarName or formula and specify the observation weights in Tbl by using Weights.
□ Specify the response variable by using ResponseVarName — fitckernel uses the remaining variables as predictors. To use a subset of the remaining variables in Tbl as predictors, specify
predictor variables by using PredictorNames.
□ Define a model specification by using formula — fitckernel uses a subset of the variables in Tbl as predictor variables and the response variable, as specified in formula.
If Tbl does not contain the response variable, then specify a response variable by using Y. The length of the response variable Y and the number of rows in Tbl must be equal. To use a subset of the
variables in Tbl as predictors, specify predictor variables by using PredictorNames.
Data Types: table
The software treats NaN, empty character vector (''), empty string (""), <missing>, and <undefined> elements as missing values, and removes observations with any of these characteristics:
• Missing value in the response variable
• At least one missing value in a predictor observation (row in X or Tbl)
• NaN value or 0 weight ('Weights')
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: Mdl = fitckernel(X,Y,'Learner','logistic','NumExpansionDimensions',2^15,'KernelScale','auto') implements logistic regression after mapping the predictor data to the 2^15 dimensional space
using feature expansion with a kernel scale parameter selected by a heuristic procedure.
You cannot use any cross-validation name-value argument together with the OptimizeHyperparameters name-value argument. You can modify the cross-validation for OptimizeHyperparameters only by using
the HyperparameterOptimizationOptions name-value argument.
Kernel Classification Options
Standardize — Flag to standardize predictor data
false or 0 (default) | true or 1
Since R2023b
Flag to standardize the predictor data, specified as a numeric or logical 0 (false) or 1 (true). If you set Standardize to true, then the software centers and scales each numeric predictor variable
by the corresponding column mean and standard deviation. The software does not standardize the categorical predictors.
Example: "Standardize",true
Data Types: single | double | logical
Cross-Validation Options
CrossVal — Flag to train cross-validated classifier
'off' (default) | 'on'
Flag to train a cross-validated classifier, specified as the comma-separated pair consisting of 'Crossval' and 'on' or 'off'.
If you specify 'on', then the software trains a cross-validated classifier with 10 folds.
You can override this cross-validation setting using the CVPartition, Holdout, KFold, or Leaveout name-value pair argument. You can use only one cross-validation name-value pair argument at a time to
create a cross-validated model.
Example: 'Crossval','on'
Convergence Controls
Other Kernel Classification Options
Verbose — Verbosity level
0 (default) | 1
Verbosity level, specified as the comma-separated pair consisting of 'Verbose' and either 0 or 1. Verbose controls the display of diagnostic information at the command line.
Value Description
0 fitckernel does not display diagnostic information.
1 fitckernel displays and stores the value of the objective function, gradient magnitude, and other diagnostic information. FitInfo.History contains the diagnostic information.
Example: 'Verbose',1
Data Types: single | double
Other Classification Options
Hyperparameter Optimization Options
OptimizeHyperparameters — Parameters to optimize
'none' (default) | 'auto' | 'all' | string array or cell array of eligible parameter names | vector of optimizableVariable objects
Parameters to optimize, specified as the comma-separated pair consisting of 'OptimizeHyperparameters' and one of these values:
• 'none' — Do not optimize.
• 'auto' — Use {'KernelScale','Lambda','Standardize'}.
• 'all' — Optimize all eligible parameters.
• Cell array of eligible parameter names.
• Vector of optimizableVariable objects, typically the output of hyperparameters.
The optimization attempts to minimize the cross-validation loss (error) for fitckernel by varying the parameters. To control the cross-validation type and other aspects of the optimization, use the
HyperparameterOptimizationOptions name-value argument. When you use HyperparameterOptimizationOptions, you can use the (compact) model size instead of the cross-validation loss as the optimization
objective by setting the ConstraintType and ConstraintBounds options.
The values of OptimizeHyperparameters override any values you specify using other name-value arguments. For example, setting OptimizeHyperparameters to "auto" causes fitckernel to optimize
hyperparameters corresponding to the "auto" option and to ignore any specified values for the hyperparameters.
The eligible parameters for fitckernel are:
Set nondefault parameters by passing a vector of optimizableVariable objects that have nondefault values. For example:
load fisheriris
params = hyperparameters('fitckernel',meas,species);
params(2).Range = [1e-4,1e6];
Pass params as the value of 'OptimizeHyperparameters'.
By default, the iterative display appears at the command line, and plots appear according to the number of hyperparameters in the optimization. For the optimization and plots, the objective function
is the misclassification rate. To control the iterative display, set the Verbose option of the HyperparameterOptimizationOptions name-value argument. To control the plots, set the ShowPlots field of
the HyperparameterOptimizationOptions name-value argument.
For an example, see Optimize Kernel Classifier.
Example: 'OptimizeHyperparameters','auto'
Output Arguments
Mdl — Trained kernel classification model
ClassificationKernel model object | ClassificationPartitionedKernel cross-validated model object
Trained kernel classification model, returned as a ClassificationKernel model object or ClassificationPartitionedKernel cross-validated model object.
If you set any of the name-value pair arguments CrossVal, CVPartition, Holdout, KFold, or Leaveout, then Mdl is a ClassificationPartitionedKernel cross-validated classifier. Otherwise, Mdl is a
ClassificationKernel classifier.
To reference properties of Mdl, use dot notation. For example, enter Mdl.NumExpansionDimensions in the Command Window to display the number of dimensions of the expanded space.
If you specify OptimizeHyperparameters and set the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions, then Mdl is an N-by-1 cell array of model objects, where N is
equal to the number of rows in ConstraintBounds. If none of the optimization problems yields a feasible model, then each cell array value is [].
Unlike other classification models, and for economical memory usage, a ClassificationKernel model object does not store the training data or training process details (for example, convergence
AggregateOptimizationResults — Aggregate optimization results
AggregateBayesianOptimization object
Aggregate optimization results for multiple optimization problems, returned as an AggregateBayesianOptimization object. To return AggregateOptimizationResults, you must specify
OptimizeHyperparameters and HyperparameterOptimizationOptions. You must also specify the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions. For an example that shows
how to produce this output, see Hyperparameter Optimization with Multiple Constraint Bounds.
FitInfo — Optimization details
structure array
Optimization details, returned as a structure array including fields described in this table. The fields contain final values or name-value pair argument specifications.
Field Description
Solver Objective function minimization technique: 'LBFGS-fast', 'LBFGS-blockwise', or 'LBFGS-tall'. For details, see Algorithms.
LossFunction Loss function. Either 'hinge' or 'logit' depending on the type of linear classification model. See Learner.
Lambda Regularization term strength. See Lambda.
BetaTolerance Relative tolerance on the linear coefficients and the bias term. See BetaTolerance.
GradientTolerance Absolute gradient tolerance. See GradientTolerance.
ObjectiveValue Value of the objective function when optimization terminates. The classification loss plus the regularization term compose the objective function.
GradientMagnitude Infinite norm of the gradient vector of the objective function when optimization terminates. See GradientTolerance.
RelativeChangeInBeta Relative changes in the linear coefficients and the bias term when optimization terminates. See BetaTolerance.
FitTime Elapsed, wall-clock time (in seconds) required to fit the model to the data.
History History of optimization information. This field is empty ([]) if you specify 'Verbose',0. For details, see Verbose and Algorithms.
To access fields, use dot notation. For example, to access the vector of objective function values for each iteration, enter FitInfo.ObjectiveValue in the Command Window.
If you specify OptimizeHyperparameters and set the ConstraintType and ConstraintBounds options of HyperparameterOptimizationOptions, then Fitinfo is an N-by-1 cell array of structure arrays, where N
is equal to the number of rows in ConstraintBounds.
A good practice is to examine FitInfo to assess whether convergence is satisfactory.
HyperparameterOptimizationResults — Cross-validation optimization of hyperparameters
BayesianOptimization object | AggregateBayesianOptimization object | table of hyperparameters and associated values
Cross-validation optimization of hyperparameters, returned as a BayesianOptimization object, an AggregateBayesianOptimization object, or a table of hyperparameters and associated values. The output
is nonempty when OptimizeHyperparameters has a value other than "none".
If you set the ConstraintType and ConstraintBounds options in HyperparameterOptimizationOptions, then HyperparameterOptimizationResults is an AggregateBayesianOptimization object. Otherwise, the
value of HyperparameterOptimizationResults depends on the value of the Optimizer option in HyperparameterOptimizationOptions.
Value of Optimizer Option Value of HyperparameterOptimizationResults
"bayesopt" (default) Object of class BayesianOptimization
"gridsearch" or "randomsearch" Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)
More About
Random Feature Expansion
Box Constraint
• Standardizing predictors before training a model can be helpful.
□ You can standardize training data and scale test data to have the same scale as the training data by using the normalize function.
□ Alternatively, use the Standardize name-value argument to standardize the numeric predictors before training. The returned model includes the predictor means and standard deviations in its Mu
and Sigma properties, respectively. (since R2023b)
• After training a model, you can generate C/C++ code that predicts labels for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.
• fitckernel minimizes the regularized objective function using a Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) solver with ridge (L[2]) regularization. To find the type of LBFGS solver
used for training, type FitInfo.Solver in the Command Window.
□ 'LBFGS-fast' — LBFGS solver.
□ 'LBFGS-blockwise' — LBFGS solver with a block-wise strategy. If fitckernel requires more memory than the value of BlockSize to hold the transformed predictor data, then the function uses a
block-wise strategy.
□ 'LBFGS-tall' — LBFGS solver with a block-wise strategy for tall arrays.
When fitckernel uses a block-wise strategy, it implements LBFGS by distributing the calculation of the loss and gradient among different parts of the data at each iteration. Also, fitckernel
refines the initial estimates of the linear coefficients and the bias term by fitting the model locally to parts of the data and combining the coefficients by averaging. If you specify
'Verbose',1, then fitckernel displays diagnostic information for each data pass and stores the information in the History field of FitInfo.
When fitckernel does not use a block-wise strategy, the initial estimates are zeros. If you specify 'Verbose',1, then fitckernel displays diagnostic information for each iteration and stores the
information in the History field of FitInfo.
• If you specify the Cost, Prior, and Weights name-value arguments, the output model object stores the specified values in the Cost, Prior, and W properties, respectively. The Cost property stores
the user-specified cost matrix (C) without modification. The Prior and W properties store the prior probabilities and observation weights, respectively, after normalization. For model training,
the software updates the prior probabilities and observation weights to incorporate the penalties described in the cost matrix. For details, see Misclassification Cost Matrix, Prior
Probabilities, and Observation Weights.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
The fitckernel function supports tall arrays with the following usage notes and limitations:
• fitckernel does not support tall table data.
• Some name-value pair arguments have different defaults compared to the default values for the in-memory fitckernel function. Supported name-value pair arguments, and any differences, are:
□ 'Learner'
□ 'NumExpansionDimensions'
□ 'KernelScale'
□ 'BoxConstraint'
□ 'Lambda'
□ 'BetaTolerance' — Default value is relaxed to 1e–3.
□ 'GradientTolerance' — Default value is relaxed to 1e–5.
□ 'IterationLimit' — Default value is relaxed to 20.
□ 'BlockSize'
□ 'RandomStream'
□ 'HessianHistorySize'
□ 'Verbose' — Default value is 1.
□ 'ClassNames'
□ 'Cost'
□ 'Prior'
□ 'ScoreTransform'
□ 'Weights' — Value must be a tall array.
□ 'OptimizeHyperparameters'
□ 'HyperparameterOptimizationOptions' — For cross-validation, tall optimization supports only 'Holdout' validation. By default, the software selects and reserves 20% of the data as holdout
validation data, and trains the model using the rest of the data. You can specify a different value for the holdout fraction by using this argument. For example, specify
'HyperparameterOptimizationOptions',struct('Holdout',0.3) to reserve 30% of the data as validation data.
• If 'KernelScale' is 'auto', then fitckernel uses the random stream controlled by tallrng for subsampling. For reproducibility, you must set a random number seed for both the global stream and the
random stream controlled by tallrng.
• If 'Lambda' is 'auto', then fitckernel might take an extra pass through the data to calculate the number of observations in X.
• fitckernel uses a block-wise strategy. For details, see Algorithms.
For more information, see Tall Arrays.
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To perform parallel hyperparameter optimization, use the UseParallel=true option in the HyperparameterOptimizationOptions name-value argument in the call to the fitckernel function.
For more information on parallel hyperparameter optimization, see Parallel Bayesian Optimization.
For general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
Version History
Introduced in R2017b
R2023b: Kernel models support standardization of predictors
Starting in R2023b, fitckernel supports the standardization of numeric predictors. That is, you can specify the Standardize value as true to center and scale each numeric predictor variable by the
corresponding column mean and standard deviation. The software does not standardize the categorical predictors.
You can also optimize the Standardize hyperparameter by using the OptimizeHyperparameters name-value argument. Unlike in previous releases, when you specify "auto" as the OptimizeHyperparameters
value, fitckernel includes Standardize as an optimizable hyperparameter. | {"url":"https://nl.mathworks.com/help/stats/fitckernel.html","timestamp":"2024-11-14T08:05:34Z","content_type":"text/html","content_length":"293158","record_id":"<urn:uuid:5b87394e-609f-4615-809c-28bb2eadedcc>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00458.warc.gz"} |
Georgy Dunaev (Motivation Letter)
To Whom It May Concern,
I am a mathematician and programmer, who moved to Germany several years ago.
For many years i studied studied foundations of mathematics and proof assistants (HOL4, Coq, Isabelle, Metamath).
Currently I am writing a book about foundations of mathematics. (~90pages now)
My activities/achievements related to foundations of mathematics:
1) formulated super-small presentation of foundations of mathematics via Morse-Kelley class theory, for implementing in my proof assistant.
2) As a hobby, I have implemented my LCF proof assistant with tactics, etc. (1st order logic, classtheory Morse-Kelley).
Currently, playing Isabelle's metatheory, for formally proving soundness of higher-order statements of Isabelle, including those about the first order logic.
3) invented some really nice class-theoretic notions and proved theorems about them. (unpublished, but very reliable, some constructions and theorems are formally verified in The Coq Proof Assistant)
4) invented new semantics for certain extension of the 1st order logic, natural & useful: it has nice properties for working with "greek-letter-operators", and nicely treats different flavors of
choice. Proved all necessary lemmas and theorems for its soundness. (unpublished, but very reliable)
5) have an idea of a new "natural" semantics for a metatheory of Isabelle. (just experimenting: It seems to be a bit exotic, but ok.)
My goal is to work in TUM(or in other legal entity in Germany) on any project related to Isabelle or to foundations of mathematics and/or verification in general.
I'd like and I have ability either to participate in existing Isabelle's project, or to do solo projects, and I can propose severak directions of work.
I am very open to new ideas/advises. Let's talk about it!
Kind regards,
Georgy Dunaev.
Last updated: Nov 11 2024 at 01:24 UTC | {"url":"http://isabelle.systems/zulip-archive/stream/202967-New-Members-.26-Projects/topic/Georgy.20Dunaev.20.28Motivation.20Letter.29.html","timestamp":"2024-11-11T04:13:39Z","content_type":"text/html","content_length":"3910","record_id":"<urn:uuid:deb6321f-cd68-4e88-aa4f-0c018da7d54e>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00076.warc.gz"} |
Posterior Probability - (Intro to Probabilistic Methods) - Vocab, Definition, Explanations | Fiveable
Posterior Probability
from class:
Intro to Probabilistic Methods
Posterior probability is the probability of an event or hypothesis after considering new evidence or information. It plays a critical role in Bayesian inference, as it updates the prior probability
in light of observed data, allowing for refined predictions and decisions. This concept connects deeply with the total probability theorem and Bayes' theorem, as well as in constructing and analyzing
Bayesian networks.
congrats on reading the definition of Posterior Probability. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Posterior probability is calculated using Bayes' theorem, which mathematically expresses how to update beliefs based on new information.
2. In Bayesian networks, posterior probabilities allow for the inference of unknown variables based on the known states of other related variables.
3. The calculation of posterior probabilities is crucial for decision-making processes, especially in areas like medical diagnosis and machine learning.
4. The posterior probability can be seen as a new 'updated' belief, which incorporates both prior beliefs and the strength of the evidence observed.
5. Understanding posterior probabilities helps in assessing risks and uncertainties in various fields such as finance, engineering, and social sciences.
Review Questions
• How does posterior probability relate to prior probability and likelihood in Bayesian inference?
□ Posterior probability is derived from prior probability and likelihood through Bayes' theorem. The prior probability represents our initial belief about a hypothesis before any new data is
available. When we obtain new evidence, we calculate the likelihood, which measures how probable that evidence is under different hypotheses. By applying Bayes' theorem, we can update our
prior belief to get the posterior probability, reflecting our revised understanding after considering the new information.
• Discuss the importance of posterior probabilities in decision-making within Bayesian networks.
□ In Bayesian networks, posterior probabilities are crucial because they allow us to make informed decisions based on the relationships between various variables. Once we observe evidence in
the network, we can use it to compute the posterior probabilities of other variables that are dependent on this evidence. This enables us to assess risks and make predictions more accurately
by integrating various sources of information, ultimately leading to better decision-making outcomes in complex scenarios.
• Evaluate how posterior probabilities can influence predictive modeling and risk assessment in real-world applications.
□ Posterior probabilities significantly enhance predictive modeling and risk assessment by providing updated insights based on observed data. In fields such as healthcare, finance, or machine
learning, accurately calculating posterior probabilities allows for dynamic models that adapt as new data comes in. This adaptability is essential for managing uncertainties and making
informed decisions that can lead to improved outcomes. By continuously refining predictions based on new evidence, organizations can optimize their strategies and reduce potential risks
© 2024 Fiveable Inc. All rights reserved.
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ESP Biography
OMKAR DESHPANDE, Founder at The Young Socratics
Major: Computer Science
College/Employer: Self
Year of Graduation: 2008
Brief Biographical Sketch:
Learning and teaching about the big ideas in math, science and philosophy from a historical perspective to K-12 students is my primary interest. For my academic bio, please look at my Stanford
webpage: http://ai.stanford.edu/~omkard
Past Classes
(Clicking a class title will bring you to the course's section of the corresponding course catalog)
P4853: The Galileo Affair in Splash Spring 2016 (Apr. 09 - 10, 2016)
"Future centuries may find it strange that after he retracted an opinion which had not yet been absolutely prohibited in public... so much rigor should be used against a pitiable old septuagenarian
as to keep him in a (private if not public) prison: he is not allowed to return to his city and his house, or to receive visits and comfort from friends. He has infirmities that are almost
inseparable from old age and require almost continuous assistance.... This shook my heart and drove me to tears, as I considered the vicissitudes of human affairs and the fact that he had had so many
uncommon honors and accomplishments, whose memory will last for centuries." (December 5, 1634, Peiresc's plea for a pardon) In 1633, the Roman inquisition condemned Galileo as a suspected heretic for
defending Copernicus's hypothesis of the earth's motion, and denying the scientific authority of Scripture. This incident is the most cited one in the history of science-religion interactions. Does
the trial of Galileo epitomize the conflict between enlightened science and dogmatic religion? We will go beyond the simple narratives of that story to look deeper into the scientific, religious,
philosophical and personal issues at stake in this controversy, based on what the original documents and letters say. Note: This is a history of science class, not a regular science class.
P4854: The Shape of the Earth in Splash Spring 2016 (Apr. 09 - 10, 2016)
"Is there anyone so senseless as to believe that (on the other side of the earth) there are men whose footsteps are higher than their heads?... That crops and trees grow downwards? That the rains and
snow and hail fall upwards to the earth?" (Lactantius, 4th century CE) There was a time when people across different cultures believed that the earth was more or less flat, not because they were
stupid in any way, but because reasoning by common sense led them to it. Even many of the best philosophers in ancient Greece adhered to some version of a flat earth viewpoint. How did some of the
philosophers (or 'scientists' of the time) begin to claim that the earth was spherical? What observations and arguments led them to it? In this course, we will imagine travelling back in time to the
ancient world, two thousand years before the advent of modern science, and understand how educated people convinced themselves and the others that the earth was really spherical, without satellite
pictures and the ability to circumnavigate the world. We will look at the mixture of right and wrong arguments offered by the key philosophers and astronomers of the time, including Aristotle and
Ptolemy. Along the way, we will also understand the physics of Aristotle (which was the dominant physics for 2000 years), and the ancient astronomical explanations of eclipses and the phases of the
moon. Note: This is a history of science class, not a regular science class.
P4855: Great Scientists-Kepler in Splash Spring 2016 (Apr. 09 - 10, 2016)
Want to know how we moved from a geocentric model to the heliocentric model of the universe? Did you know the person after whom NASA named one of their space shuttle? Learn about the key scientific
figure, Kepler, who was instrumental in putting the heliocentric model of Copernicus on firm ground. In this course, we will see how his tenacity and perseverance led to development of his three laws
of planetary motion that inspired Newton’s universal of gravitation. Along the way, we will look at his legendary tussle with Tycho Brahe without whose observations Kepler could not have made much
headway in his theories. We add to this, his work in explaining human vision and behavior of telescopes and his work on logarithms. And we will look at Kepler and his times, how his religious beliefs
influenced his writings, how he devoted as much time to astrology as to astronomy and much more.
P4856: Great Scientists-Newton in Splash Spring 2016 (Apr. 09 - 10, 2016)
Considered by some to be the greatest “scientist” (natural philosopher would be the word used back then!) of all time, Newton shaped the modern world in ways too numerous to mention. In this course,
we will go through his most important contributions: the laws of optics, the binomial theorem, integral and differential calculus, laws of motion and above all his crowning glory, the universal law
of gravitation. But we won’t stop there! We will also discuss his other “outlandish” interests like alchemy and iconic fights with contemporary scientists like Leibniz and Hooke that divided
continental Europe and England. Let us embark on the journey to understand the complete man that Newton was: a loner, a genius, a celibate, a heretic and a bitter critic of his rivals and the first
person in England to get state funeral whose attainment lay in the realm of the mind.
P4570: The Shape of the Earth in Splash Fall 2015 (Nov. 07 - 08, 2015)
"Is there anyone so senseless as to believe that (on the other side of the earth) there are men whose footsteps are higher than their heads?... That crops and trees grow downwards? That the rains and
snow and hail fall upwards to the earth?" (Lactantius, 4th century CE) There was a time when people across different cultures believed that the earth was more or less flat, not because they were
stupid in any way, but because reasoning by common sense led them to it. Even many of the best philosophers in ancient Greece adhered to some version of a flat earth viewpoint. How did some of the
philosophers (or 'scientists' of the time) begin to claim that the earth was spherical? What observations and arguments led them to it? In this course, we will imagine travelling back in time to the
ancient world, two thousand years before the advent of modern science, and understand how educated people convinced themselves and the others that the earth was really spherical, without satellite
pictures and the ability to circumnavigate the world. We will look at the mixture of right and wrong arguments offered by the key philosophers and astronomers of the time, including Aristotle and
Ptolemy. Along the way, we will also understand the physics of Aristotle (which was the dominant physics for 2000 years), and the ancient astronomical explanations of eclipses and the phases of the
moon. Note: This is a history of science class, not a regular science class.
P4571: The Galileo Affair in Splash Fall 2015 (Nov. 07 - 08, 2015)
"Future centuries may find it strange that after he retracted an opinion which had not yet been absolutely prohibited in public... so much rigor should be used against a pitiable old septuagenarian
as to keep him in a (private if not public) prison: he is not allowed to return to his city and his house, or to receive visits and comfort from friends. He has infirmities that are almost
inseparable from old age and require almost continuous assistance.... This shook my heart and drove me to tears, as I considered the vicissitudes of human affairs and the fact that he had had so many
uncommon honors and accomplishments, whose memory will last for centuries." (December 5, 1634, Peiresc's plea for a pardon) In 1633, the Roman inquisition condemned Galileo as a suspected heretic for
defending Copernicus's hypothesis of the earth's motion, and denying the scientific authority of Scripture. This incident is the most cited one in the history of science-religion interactions. Does
the trial of Galileo epitomize the conflict between enlightened science and dogmatic religion? We will go beyond the simple narratives of that story to look deeper into the scientific, religious,
philosophical and personal issues at stake in this controversy, based on what the original documents and letters say. Note: This is a history of science class, not a regular science class.
P4584: Great Scientists: Kepler in Splash Fall 2015 (Nov. 07 - 08, 2015)
Want to know how we moved from a geocentric model to the heliocentric model of the universe? Did you know the person after whom NASA named one of their space shuttle? Learn about the key scientific
figure, Kepler, who was instrumental in putting the heliocentric model of Copernicus on firm ground. In this course, we will see how his tenacity and perseverance led to development of his three laws
of planetary motion that inspired Newton’s universal of gravitation. Along the way, we will look at his legendary tussle with Tycho Brahe without whose observations Kepler could not have made much
headway in his theories. We add to this, his work in explaining human vision and behavior of telescopes and his work on logarithms. And we will look at Kepler and his times, how his religious beliefs
influenced his writings, how he devoted as much time to astrology as to astronomy and much more.
P4585: Great Scientists:Newton in Splash Fall 2015 (Nov. 07 - 08, 2015)
Considered by some to be the greatest “scientist” (natural philosopher would be the word used back then!) of all time, Newton shaped the modern world in ways too numerous to mention. In this course,
we will go through his most important contributions: the laws of optics, the binomial theorem, integral and differential calculus, laws of motion and above all his crowning glory, the universal law
of gravitation. But we won’t stop there! We will also discuss his other “outlandish” interests like alchemy and iconic fights with contemporary scientists like Leibniz and Hooke that divided
continental Europe and England. Let us embark on the journey to understand the complete man that Newton was: a loner, a genius, a celibate, a heretic and a bitter critic of his rivals and the first
person in England to get state funeral whose attainment lay in the realm of the mind.
P4327: The Shape of the Earth in Splash Spring 2015 (Apr. 11 - 12, 2015)
There was a time when people across different cultures believed that the earth was more or less flat, not because they were stupid in any way, but because reasoning by common sense led them to it.
Even many of the best philosophers in ancient Greece adhered to some version of a flat earth viewpoint. How did some of the philosophers (or 'scientists' of the time) begin to claim that the earth
was spherical? What observations and arguments led them to it? In this course, we will imagine travelling back in time to the ancient world, two thousand years before the advent of modern science,
and understand how educated people convinced themselves and the others that the earth was really spherical, without satellite pictures and the ability to circumnavigate the world. We will look at the
arguments offered by the key philosophers and astronomers of the time, including Aristotle and Ptolemy. Along the way, we will also understand the phases of the moon, and how the ancients inferred
that eclipses are caused by the alignment of the sun, moon and the earth. Note: This is a history of science class, not a regular science class.
M4332: The Art of Summation in Splash Spring 2015 (Apr. 11 - 12, 2015)
Summing the numbers from 1 to 5 can be done quickly. Summing the numbers from 1 up to 100 would take a lot more time. Or is there a quick way to do it? How about the general problem of summing the
first N numbers? Drawing inspiration from Pythagoras and his followers, and a precocious elementary school kid who grew up to become the "Prince of Mathematicians", we will discover a number of
different approaches to the problem. We will generalize those approaches to compute the sum of an arithmetic series and geometric series. We will also play with other summations: summing the first N
squares, or the first N cubes, and try to discover connections between these different series. We will follow bacteria as they grow and divide, and ask why they don't conquer the world. We will trace
back a children's nursery rhyme in English all the way back to a mathematical papyrus roll from ancient Egypt. We will compute the number of squares and rectangles on a chessboard, and we will learn
the legend of a wise man who used a chessboard and the power of geometric growth to fool a king into promising something that was impossible for any earthly king to fulfill. And in the process, we
will learn the art of summation.
M4334: The Story of e in Splash Spring 2015 (Apr. 11 - 12, 2015)
Starting from the evolution of interest rates in the Greek, Roman and Christian worlds, students will learn how Euler’s number e emerged in the context of calculating compound interest. The
relationship between natural logarithms and e will also be looked at.
M4338: The story of pi in Splash Spring 2015 (Apr. 11 - 12, 2015)
How was the ratio of the circumference and diameter of a circle understood in Egypt, Mesopotamia, India and China? Students will learn about the approximation of pi derived by Archimedes. Modern
developments will include the examination of the infinite series of Leibnitz, Gregory and Madhava.
P3911: The Flat Earth in Splash Fall 2014 (Nov. 08 - 09, 2014)
There was a time when people across different cultures believed that the earth was more or less flat, not because they were stupid in any way, but because reasoning by common sense led them to it.
Even many of the best philosophers in ancient Greece adhered to some version of a flat earth viewpoint. How did some of the philosophers (or 'scientists' of the time) begin to claim that the earth
was spherical? What observations and arguments led them to it? In this course, we will imagine travelling back in time to the ancient world, two thousand years before the advent of modern science,
and understand how educated people convinced themselves and the others that the earth was really spherical, without satellite pictures and the ability to circumnavigate the world. We will look at the
arguments offered by the key philosophers and astronomers of the time, including Aristotle and Ptolemy. Along the way, we will also understand the phases of the moon, and how the ancients inferred
that eclipses are caused by the alignment of the sun, moon and the earth. Note: This is a history of science class, not a regular science class.
M3925: Great Mathematicians: Euclid in Splash Fall 2014 (Nov. 08 - 09, 2014)
"At the age of 11, I began Euclid, with my brother as tutor. This was one of the great events in my life, as dazzling as first love. I had not imagined there was anything so delicious in the world...
From that moment until I was 38, mathematics was my chief interest and my chief source of happiness." This is how Bertrand Russell, one of the most famous 20th century mathematicians (and also a
Nobel Laureate) described his childhood encounter with Euclid's Elements. It is reputed to be one of the most widely read texts in the Western tradition, next only to the Bible. It has influenced
scientists like Albert Einstein, politicians and lawyers like Abraham Lincoln, philosophers and mathematicians like Rene Descartes, political philosophers like Thomas Hobbes and even the ethics of
Spinoza. After reviewing the state of mathematics prior to Euclid, and the importance of the Museum of Alexandria, where Euclid was the first professor of mathematics, we will look at the chief
contribution of Euclid for which he has been so influential - the axiomatic method, and the way he employs it to construct all of Greek mathematics in the Elements. We will devote most of our
attention to his two-dimensional geometry, which has come to be known as 'Euclidean geometry'. The Elements also has portions on number theory, and we will look at an elegant proof of the fact that
there are an infinite number of primes. Note: Though not required, we would recommend that you also enroll for Great Mathematicians: Thales and Great Mathematicians: Pythagoras if you're enrolling
for this course.
M3929: The Story of Numbers in Splash Fall 2014 (Nov. 08 - 09, 2014)
How did we come up with the modern number systems? What are the important characteristics of the modern numbers that make them different from pre-historic ideas about numbers? Join us for a
fascinating tour of the modern number system beginning from the pre-historic tally counting to the modern decimal systems. We will look at Mesopotamian, Egyptian,Greek, Roman, Chinese, Indian, Arabic
and Mayan number systems and the emergence of the current system that we use.
M3930: Interest Rates and the emergence of the number 'e' in Splash Fall 2014 (Nov. 08 - 09, 2014)
From student loans and mortgages to the financial system, we see the use of interest rates everywhere. Wonder how the idea of charging interest started in the ancient world? Beginning from
Hammurabi's code, we trace the evolution of interest rates in Greek, Roman and Christian world. How is the mysterious number e connected to something as practical as interest rates? You will learn
about all this in this "interest"-ing class.
M3935: Great Mathematicians: Thales in Splash Fall 2014 (Nov. 08 - 09, 2014)
Living in a time when mathematics, science and philosophy were not separate disciplines, Thales of Miletus is reputed to be the earliest philosopher in the Western intellectual tradition. From
creatively determining the distance of a ship from the seashore, to calculating the height of the Great Pyramid using a simple stick; from predicting a solar eclipse to arguing that water is the
origin of everything; from being an absent-minded stargazer who fell into a well, to being a practical-minded merchant who made a good profit, we will look at the limited and uncertain historical
sources that we have about this earliest philosopher who is reputed to have introduced the notion of 'proof' in mathematics.
M3936: Great Mathematicians: Pythagoras in Splash Fall 2014 (Nov. 08 - 09, 2014)
We have all heard of the Pythagorean Theorem, named after the Greek philosopher Pythagoras. But the Pythagorean Theorem was not discovered by Pythagoras. And it would be more accurate to look at
Pythagoras as the founder of a religious community than as a mathematician. In this class, we will look at the life and times of Pythagoras, one of the most enigmatic figures in history. We will look
at his travels, his struggle to find students, and his subsequent rise to fame as the divine founder of a secretive religious community, that embraced mathematics as a core part of its religion,
along with music, reincarnation and abstention from meat-eating. We will look at the relationship between music and mathematics that Pythagoras supposedly uncovered, his creation myth in which
everything arises from number, and his influence on the scientific revolution much later. We will see how the discovery of irrationals shattered the worldview of the Pythagoreans, and we'll look at a
possible proof of the Pythagorean Theorem that Pythagoras (or one of his followers) might have originally come up with. Note: Though not required, we would recommend that you also enroll for Great
Mathematicians: Thales if you're taking this course.
M3692: A Gentle Introduction to Calculus in Splash! Spring 2014 (Apr. 12 - 13, 2014)
It is considered to be one of the greatest achievements of the human mind. But Calculus is also the greatest source of terror and anxiety for students. In this course, we will build your intuition
for this important area of mathematics, preempting any reason to be fearful of it. We will go into a brief history of its development from precursors in ancient Greece and India, to the modern
calculus developed by Newton and Leibnitz in the 17th century. Through problems involving simple geometric concepts that you will participate in solving, you will come away with an appreciation for
the two ideas underlying all of calculus -- integration and differentiation.
M3696: Games of Chance: An Introduction to Probability in Splash! Spring 2014 (Apr. 12 - 13, 2014)
Our lives are significantly influenced by random events -- weather forecasts, stock market fluctuations, the traits we inherit from our parents, car accidents, how well you do in your exams, etc --
these are all examples where chance plays a role. Nowhere is that role more prominent than in games of chance. In tossing a coin, or rolling dice, or flipping cards, we are blatantly playing games of
chance. In fact, it was in the context of gambling that the basic ideas of probability were first invented. We will look at those basics: what is probability, and how do we assign a numerical value
to the likelihood of an event when we have no idea what will happen? Believe it or not, even basic questions like these about probability are still a source of debate among mathematicians (and we
will see why). But that also makes probability one of the most interesting topics to learn about in mathematics. So we invite you to join in -- you will enjoy the experience (with a high
M3697: The Art of Summation in Splash! Spring 2014 (Apr. 12 - 13, 2014)
Summing the numbers from 1 to 5 can be done quickly. Summing the numbers from 1 up to 100 would take more time. Or is there a quick way to do it? How about the general problem of summing the numbers
from 1 to N? We will look at a few different ways to approach problems of this kind, including a trick Gauss used when he was supposedly nine years old! We will generalize the solution to compute the
sum of an arithmetic series. $$\textit{As I was going to Saint Ives,}$$ $$\textit{I met a man with seven wives.}$$ $$\textit{Each wife had seven sacks;}$$ $$\textit{every sack had seven cats;}$$ $$\
textit{Every cat had seven kits;}$$ $$\textit{Kits, cats, sacks and wives,}$$ $$\textit{How many were going to Saint Ives?}$$ This children's rhyme will initiate us into summation of a different kind
of series -- a geometric series. We will also play with other summations: summing the first n squares, or the first n cubes, and try to discover connections between these different series. To sum up,
it'll be fun, so come and join us!
M3698: Great Mathematicians- Pythagoras in Splash! Spring 2014 (Apr. 12 - 13, 2014)
The first of the great Greek mathematicians, Pythagoras had a fascination for numbers and geometric shapes that raised their status to attributes of God! By studying his work, we will see the
beginning of pure mathematics in Greece and contrast it with the more applied mathematics that was prevalent in ancient India and Egypt. Any mention of Pythagoras is incomplete without his theorem,
probably the most known mathematical theorem in the world. Do you know people have claimed over 350 proofs of the theorem? You will rediscover some of these proofs on your own in an interactive
M3699: Great Mathematicians-Euclid in Splash! Spring 2014 (Apr. 12 - 13, 2014)
The first person in Greece to actually systematically publish mathematics books covering both arithmetic and geometry, Euclid was instrumental in bringing ideas of great Greek mathematicians
together. Believe it or not Euclid’s book “the Elements” is the most printed book after the Bible in the western world. We will cover the basic principles of geometry he used to derive many results
in planar geometry. And we will also discuss a fascinating method Euclid found to get the greatest common divisor of two numbers.
M3700: Great Mathematicians-Archimedes in Splash! Spring 2014 (Apr. 12 - 13, 2014)
Considered one of the greatest mathematicians and physicists among the ancients, we will see Archimedes’s contribution specifically in computing areas and volumes of different shapes and his
fascination with the number infinity. Want to know how the value of the famous number pi comes from? To appreciate Archimedes better, you will interactively learn his approximation to pi. | {"url":"https://stanfordesp.learningu.org/teach/teachers/omkard/bio.html","timestamp":"2024-11-11T06:33:49Z","content_type":"application/xhtml+xml","content_length":"44828","record_id":"<urn:uuid:4e02f4c7-d8fc-4bf3-bd2d-ea2c2b9606d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00685.warc.gz"} |
RD Sharma Class 11 Maths Chapter wise Solutions - Free PDF Download
RD Sharma Solutions for Class 11 Maths - Free PDF Download
RD Sharma (Ravi Dutt Sharma) is a math reference book for students who are in grades 6 through 12. It is one of the best books since it teaches you everything you need to know about each chapter. It
is created by following CBSE and NCERT requirements. RD Sharma Solutions for Class 6-12 provided by Vedantu are the best books available on the internet. The solutions are arranged chapter-by-chapter
and topic-by-topic. As a result, students may easily look for the preparation of a required topic that interests them. These solutions include detailed explanations, short tricks, improved drawings,
and particular key points which by practicing will improve a student's problem solving ability.
RD Sharma's Class 11 book includes numerous solved problems. Each chapter's exercises have been updated with new illustrated examples and difficulty levels. All ideas and terminology have been
explored in detail clearly and concisely in each chapter, as well as explained with relevant examples.
The CBSE's most recent syllabus is used in the RD Sharma Class 11 Textbooks. The examples and exercises at the end of each chapter's section/subsection have been grouped in increasing order of
difficulty levels. An exercise consisting of Multiple Choice Questions (MCQs), a summary of rapid revision of ideas and equations has been provided at the end of each chapter.
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FAQs on RD Sharma Class 11 Mathematics Solutions
1. Is RD Sharma a good book for competitive exams like JEE Mains and NTSE?
RD Sharma can make you qualify the JEE Mains exam, but for scoring well, you need to solve other books too. The priority should be reading and understanding NCERT books. For practising all the topics
chapter-wise, you should solve RD Sharma questions, and for their solutions, RD Sharma Solutions of Vedantu is the irreplaceable source. In JEE exam, MCQs are asked, and for a better score, you also
need to solve more MCQs from other books too. The advice is to solve more and more variety of questions for all the concepts.
2. Is solving RD Sharma good enough for the best preparation of Class 11th Maths?
Yes, out of all the reference books available in the market, RD Sharma is above all of them. It has many questions on a single concept which ensure complete practice for a particular topic. It means
you can completely solve the problems and understand the ideas in the solution with internalising the key concept. For Class 11th Maths, you will rarely find any question out of its content. It helps
students to assess their understanding. Students, by going through it, can get the idea of the difficulty level of the upcoming examinations. So have faith and start preparing with Vedantu and
download the PDF of RD Sharma Solutions for Class 11 Maths.
3. How many chapters are in RD Sharma Class 11 Maths Chapter-wise Solutions - Free PDF Download?
In the RD Sharma Textbook for Class 11, there are 33 chapters. Sets, Relations, Functions, Angle Measurement, Trigonometric Functions, Graphs of Trigonometric Functions, Trigonometric Ratios Of
Compound Angles, Transformations Formulae, Trigonometric Ratios Of Multiple And Sub Multiple Angles, Sine And Cosine Formulae And Their Applications, Sine And Cosine Sine And Cosine Formulae And
Their Applications Complex Numbers, Quadratic Equations, Linear Inequalities, Permutations, Combinations, Binomial Theorem, Arithmetic Progression, Geometric Progression, Some Special Series,
Trigonometric Equation, Mathematical Induction, Complex Numbers, Quadratic Equations, and Linear Inequalities. You can download all these chapter solutions in pdf format in Vedantu website or Vedantu
mobile app.
4. How does RD Sharma Class 11 Maths Chapter-wise Solutions PDF help in scoring good marks?
RD Sharma is a math reference book for students in grades 6 through 12. It is one of the best Maths books since it covers almost every concept you need to know about each chapter which is created by
following updated CBSE and NCERT requirements. The greatest book available on the internet is Vedantu's RD Sharma Solutions for Class 6-12. Its solutions are arranged in a chapter-by-chapter and
topic-by-topic format. As a result, students may easily look for the preparation of a certain topic that interests them and that they require. Get RD Sharma Class 11 Maths Chapter-wise Solutions -
Free PDF Download from Vedantu, this will help you to understand concepts in a clear way. Practicing these step-by-step solutions will help you to score good marks in final exams.
5. How to practice with RD Sharma Class 11 Maths Chapter-wise Solutions - Free PDF Download?
Practicing questions from the RD Sharma book guarantees that students learn about the different types of questions that can be asked in an exam and prepares them for various competitive tests.
Students may encounter a difficult question while solving questions from the RD Sharma textbook. Downloading the free pdfs of RD Sharma class 11 Maths solutions present in Vedantu helps you to
practice and understand all chapters. These solutions were created by best subject matter experts at Vedantu, and referring to them would provide them with new ways of learning approaches to solve a
6. What all is given in RD Sharma Class 11 Maths Chapter-wise Solutions - Free PDF Download
The CBSE's updated syllabus is used in the RD Sharma Class 11 Textbooks. The examples and exercises at the end of each chapter's section/subsection have been provided at various difficulty levels.
Exercise consisting of Multiple Choice Questions (MCQs), a summary of rapid revision of ideas and equations has been provided at the end of each chapter. | {"url":"http://eukaryote.org/rd-sharma-class-11-solutions.html","timestamp":"2024-11-06T20:33:23Z","content_type":"text/html","content_length":"744593","record_id":"<urn:uuid:2bbfb4f9-13dc-4395-a08b-5b7b3d295003>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00544.warc.gz"} |
The Mechanics of a Spiral Water Pump
Alex Garcia, Ann Lin, Michael Yep
Creative Commons CC BY 4.0
The fundamental necessity for water is a widespread issue affecting many communities across the globe. In this project, our team sought to provide an innovative solution to this problem for a small
community with access to a relatively close source of flowing water. Resultant flow rate of water was calculated to be 0.498 L/min. Stress and strain analysis on individual subsections of the system
were as follows: -44.2 MPa for bending moment in the rod; -7.04 MPa for shear stresses in the rod of the axis; -8.88E-5 for shear strain in the rod ; -1.11 MPa and .429 MPa of shear stresses in the L
brackets; -0.123 MPa for radial stress of spiral, -0.422 MPa for hoop stress of spiral, -2.10 MPa for bending moment of spiral, and -0.176 MPa for the maximum shear stress of the spiral pump on the
rotating wheel. The focus of the design remained fixated on water acquisition; however, further additions can be made for water purification. | {"url":"https://tr.overleaf.com/articles/the-mechanics-of-a-spiral-water-pump/cfwrtqddfygq","timestamp":"2024-11-04T02:53:21Z","content_type":"text/html","content_length":"60763","record_id":"<urn:uuid:9289f2e4-062c-493d-903d-30530c7a2c66>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00445.warc.gz"} |
What is the ph of hydronium ion online programs that convert
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Free Vedic Maths Books: PDF Download
PDF Drive is your search engine for PDF files. As of today we have 75,627,621 eBooks for you to download for free. No annoying ads, no download limits, enjoy it and don't forget to bookmark and share
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Vedic Maths Books
• The Power of Vedic Maths
170 Pages·2005·4.22 MB·New!
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• Vedic Mathematics Made Easy
169 Pages·2014·3.41 MB
Mathematics: An Introduction.
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• Abacus & Vedic Mathematics
66 Pages·2016·5.18 MB
Institute of
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• Math - Vedic Math Genius.pdf
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• Vedic mathematics & FastMaths
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“ You're not going to master the rest of your life in one day. Just relax. Master the day. Than just keep doing that every day. ” ― Anonymous
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How shall I live, knowing I will die? | {"url":"https://www.pdfdrive.com/vedic-maths-books.html","timestamp":"2024-11-04T02:13:49Z","content_type":"application/xhtml+xml","content_length":"52871","record_id":"<urn:uuid:510ba07a-430b-4c8a-9275-2929bafcb9d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00664.warc.gz"} |
Large numbers
"The greatest thrill I remember from my girlhood -- better than my first kiss, first airplane flight, first taste of mango, first circuit around the ice rink without clinging to a grown-up's sleeve
-- was the heart-lifting moment when I first understood Georg Cantor's Diagonal Proof of the nondenumerability of the real numbers. This proof, the Mona Lisa of set theory (to my mind, the most
satisfying branch of mathematics), changed the way mathematicians thought about infinity."
- What's bigger than a kazillion?.
Could stuff like this be why I'm subscribed to Salon?
Add comment
To avoid spam many websites make you fill out a CAPTCHA, or log in via an account at a corporation such as Twitter, Facebook, Google or even Microsoft GitHub.
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Everyday maths 1 (Wales)
7 Maps
A map gives you a detailed drawing of a place. They are used to find out how to get from one place to another. They use a scale that lets you calculate the actual distance from one place to the
If you look in a holiday brochure you will see lots of maps. They are often used to show how a resort is laid out. They show where a few important places are, such as local shops, hotels, the beach,
swimming pools and restaurants.
It is important to understand how to read a map so that you do not end up too far from the places you want to be near – or too close to the places you want to avoid!
Example: Holiday map
Here is a typical example of a map you find in a holiday brochure.
How far apart is everything on this map? Each square measures 1 cm on the map.
As with scale drawings, the thing you need to know before you can understand the map is the scale and how to read it.
This means that for every 1 cm square on the map there are 10 metres (10 m) in real life.
Using the scale, you can interpret the data on the map and work out how far different places are from one another.
To do this you need to measure the distances on the map and then multiply the distances in centimetres by 10 to get the actual distance in metres.
So on this map the Grooves Nightclub is 1 cm from Hotel Party. In real life that’s 10 m – not very far at all. Knowing this could affect whether you choose to stay at Hotel Party, depending on
whether you like nightclubs or not.
Now try the following activity. Remember to check your answers once you have completed the questions.
Activity 13: Using a map to find distances
Let’s stay with the map of the holiday resort.
Hint: The entrances to the buildings are marked with crosses on the map. You need to measure from these crosses.
1. What is the distance in real life between the pub and Hotel Sun in metres?
2. How far is it in real life from the Super Shop to the Beach Bistro in metres?
3. What is the distance in real life from Grooves Nightclub to the beach in metres?
Now try these:
4. A map has a scale of 1 cm to 5 km. On the map, the distance between two towns measures 6 cm. What is the actual distance between the two towns? Remember to show the units in your answer.
5. A scale is given as 1 cm to 2 km. When measured on a map, the distance from the college to the bus station is 4.5 cm. What is the actual distance?
1. The distance on the map between the pub and Hotel Sun is 4 cm, and the scale is 1 cm : 10 m. Because you need to work out the real measurement, you need to multiply the map measurement by 10:
The actual distance in real life between the pub and Hotel Sun is 40 m.
2. The distance on the map is 2 cm. Using the same calculation, the actual distance in real life between the Super Shop and the Beach Bistro is 20 m.
3. The distance on the map is 6 cm. Using the same calculation, the actual distance in real life between Grooves nightclub and the beach is 60 m.
4. The scale is 1 cm to 5 km. The distance on the map is 6 cm, so multiply 6 × 5 km to give an answer of 30 km.
5. The scale is 1 cm to 2 km. The distance on the map is 4.5 cm, so multiply 4.5 × 2 km to give an answer of 9 km.
In this section you have learned how to use maps. | {"url":"https://www.open.edu/openlearn/mod/oucontent/view.php?id=84739§ion=7","timestamp":"2024-11-13T08:23:49Z","content_type":"text/html","content_length":"131005","record_id":"<urn:uuid:31082888-8a56-48e6-b0ee-faaaa707cf87>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00656.warc.gz"} |
The Milwaukee Public Museum | Smartmeasurement Inc.
Things to Know About The Milwaukee Public Museum
The Milwaukee Public Museum is a natural and human history museum in downtown Milwaukee, Wisconsin. The museum was chartered in 1882 and opened to the public in 1884; it is a not-for-profit
organization operated by the Milwaukee Public Museum, Inc. MPM has three floors of exhibits and the first Dome Theater in Wisconsin. Wikipedia
Are you located near the The Milwaukee Public Museum and looking for the Professional Flow Meter Manufacturer in Milwaukee, WI? Go to the Smartmeasurement Inc. – Flow Meter Manufacturer. it’s only
11 min (8.3 miles) via I-94 W. You can just use this driving direction below:
What are some types of calculators?
There are many different types of calculators available, each of which is designed for specific purposes and applications. Some common types of calculators include:
Basic calculators: These are the most common type of calculators, they are designed for simple arithmetic operations such as addition, subtraction, multiplication, and division.
Scientific calculators: These calculators are designed for more advanced math and scientific applications, and they typically have a wider range of functions such as trigonometry, logarithms, and
statistical analysis.
Graphing calculators: These calculators are designed for graphing and plotting mathematical equations. They typically have a larger screen and a variety of functions for creating and manipulating
graphs. Visit this link for more information.
Financial calculators: These calculators are designed for financial applications such as calculating loan payments, interest, and investment returns.
Programmable calculators: These calculators can be programmed with custom functions and algorithms. They are often used by engineers and scientists for complex calculations.
Handheld calculators: These are portable calculators that are small enough to fit in your pocket or purse. They are typically battery-powered and can be used anywhere.
Desktop calculators: These are larger calculators that are typically used on a desk or tabletop. They may have a larger screen and more advanced features than handheld calculators.
Overall, there are many different types of calculators available, each of which is suited to specific applications and needs. Information about Hare Visit Milwaukee Public Market and Enjoy Your
Smartmeasurement Inc. – Flow Meter Manufacturer
Address: 10437 W Innovation Dr, Milwaukee, WI 53226, United States
Phone: +14142993896
Website: https://smartmeasurement.com/ | {"url":"https://smartmeasurement.com/things-to-know-about-the-milwaukee-public-museum/","timestamp":"2024-11-04T23:08:22Z","content_type":"text/html","content_length":"124633","record_id":"<urn:uuid:4d29cf1d-f78e-4211-831a-b8ae8953d918>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00541.warc.gz"} |
セミナー RIKEN Quantumセミナー
Quantum Fine-Grained Complexity
2024年4月18日(木) 10:30 - 12:00
Harry Buhrman (Chief Scientist for Algorithms and Innovation, Quantinuum, UK)
(The speaker is also a professor at University of Amsterdam & QuSoft. This is a joint seminar with the iTHEMS Quantum Computation Study Group.) One of the major challenges in computer science is to
establish lower bounds on the resources, typically time, that are needed to solve computational problems, especially those encountered in practice. A promising approach to this challenge is the study
of fine-grained complexity, which employs special reductions to prove time lower bounds for many diverse problems based on the conjectured hardness of key problems. For instance, the problem of
computing the edit distance between two strings, which is of practical interest for determining the genetic distance between species based on their DNA, has an algorithm that takes O(n^2) time.
Through a fine-grained reduction, it can be demonstrated that a faster algorithm for edit distance would imply a faster algorithm for the Boolean Satisfiability (SAT) problem. Since faster algorithms
for SAT are generally considered unlikely to exist, this implies that faster algorithms for the edit distance problem are also unlikely to exist. Other problems used for such reductions include the
3SUM problem and the All Pairs Shortest Path (APSP) problem. The quantum regime presents similar challenges; almost all known lower bounds for quantum algorithms are defined in terms of query
complexity, which offers limited insight for problems where the best-known algorithms take super-linear time. Employing fine-grained reductions in the quantum setting, therefore, represents a natural
progression. However, directly translating classical fine-grained reductions to the quantum regime poses various challenges. In this talk, I will present recent results in which we overcome these
challenges and prove quantum time lower bounds for certain problems in BQP, conditioned on the conjectured quantum hardness of, for example, SAT (and its variants), the 3SUM problem, and the APSP
problem. This presentation is based on joint works with Andris Ambainis, Bruno Loff, Florian Speelman, and Subhasree Patro.
会場: セミナー室 (359号室) (メイン会場) / via Zoom
イベント公式言語: 英語 | {"url":"https://ithems.riken.jp/ja/events/seminar?page=4","timestamp":"2024-11-05T02:41:41Z","content_type":"text/html","content_length":"90371","record_id":"<urn:uuid:47387f5e-ff2c-48fa-a537-8191c469017e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00526.warc.gz"} |
A particle executes simple harmonic motion with a frequency $f$. The frequency with which its kinetic energy oscillates is?
Hint: Equation of position of a particle in simple harmonic motion is noted. Velocity of the particle in simple harmonic motion is calculated. Finally, the equation of kinetic energy of the particle
in simple harmonic motion is derived, from which, frequency of oscillation is easily determined.
Formula used:
$1)x=A\sin ft$
$x$ is the position of the particle at time $t$
$A$ is the amplitude of oscillation
$f$ is the frequency of oscillation
$v$ is the velocity of the particle
$K$ is the kinetic energy of the particle
$m$ is the mass of the particle.
Complete step by step answer:
The basic idea is to derive the equation of kinetic energy of a particle undergoing simple harmonic motion. Suppose that a particle is moving in simple harmonic motion. It is given that the frequency
of oscillation of the particle is $f$. Let the position of a particle at a time $t$ be $x$. The position of the particle is given by
$x=A\sin ft$
where $A$ is the amplitude of oscillation.
Now, if $v$ is the velocity of the particle, it is given by
where $dx$ is the change in position and $dt$is the change in time.
Let us substitute the value of $x$ in this formula. We get
$v=\dfrac{dx}{dt}=\dfrac{d(A\sin ft)}{dt}=Af\cos ft$
Now, let us derive the equation of kinetic energy. Kinetic energy is given by
where $m$is the mass of the particle.
Substituting the value of$v$in the above equation, we have
$K=\dfrac{1}{2}m{{v}^{2}}=\dfrac{1}{2}m{{(Af\cos ft)}^{2}}=\dfrac{1}{2}m{{A}^{2}}{{f}^{2}}{{\cos }^{2}}ft$
We know that ${{\cos }^{2}}ft=\dfrac{1+\cos 2ft}{2}$
$\dfrac{1}{2}m{{A}^{2}}{{f}^{2}}{{\cos }^{2}}ft=\dfrac{1}{2}m{{A}^{2}}{{f}^{2}}\left( \dfrac{1+\cos 2ft}{2} \right)=\dfrac{1}{4}m{{A}^{2}}{{f}^{2}}(1+\cos 2ft)$
Therefore, kinetic energy of a particle undergoing simple harmonic motion with frequency $f$ is given by
$K=\dfrac{1}{4}m{{A}^{2}}{{f}^{2}}(1+\cos 2ft)=\dfrac{1}{4}m{{A}^{2}}{{f}^{2}}(1+\cos {{f}_{k}}t)$
From this equation, it is clear that the frequency of oscillation of kinetic energy is $2f$. It can be represented as
Therefore, the kinetic energy of the particle oscillates with a frequency double the frequency with which the particle oscillates.
It can easily be noted that velocity of the particle oscillates with the same frequency with which the particle oscillates.
$v=Af\cos ft=Af\cos {{f}_{v}}t$
${{f}_{v}}$ is the frequency with which the velocity oscillates. Clearly, ${{f}_{v}}=f$. | {"url":"https://www.vedantu.com/question-answer/a-particle-executes-simple-harmonic-motion-with-class-11-physics-cbse-5f5d3ae09427543f910117cb","timestamp":"2024-11-02T14:47:23Z","content_type":"text/html","content_length":"166424","record_id":"<urn:uuid:6be79a2d-462b-4d8d-a837-057a95e6bd0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00162.warc.gz"} |
Algebraic combinatorics
There are many open questions concerning group actions on graphs, designs and other combinatorial systems. For example, in [3] we have classified all flag-transitive Steiner systems (a Steiner system
is a block design in which every pair of points lies in a unique block). One can ask for a similar classification (or just to get as far as one can) with many other classes of flag-transitive | {"url":"https://www.ma.imperial.ac.uk/~mwl/algebr.htm","timestamp":"2024-11-14T06:53:41Z","content_type":"text/html","content_length":"1066","record_id":"<urn:uuid:7201c390-18ea-4319-9c7e-56216f1bfade>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00838.warc.gz"} |
How Do We Use Electromagnetic Waves to Cook Food?
As one moves along the electromagnetic (EM) spectrum from longer wavelengths, such as radio waves, to shorter ones, such as gamma rays, both frequency and energy increase. But why is this? All EM
waves travel at the speed of light, which is about 3 x 10^8 m/s in vacuum. To calculate the speed of a wave, Equation 1 is used.
c = νλ Equation 1,
where c is the speed of light in the unit of m/s, ν is frequency in hertz, and λ is the wavelength in meters. The unit hertz (Hz) is equal to the number of complete waves passing a given point per
second. For any wavelength on the EM spectrum, the wavelength multiplied by its frequency is equal to the speed of light. Light or any other EM wave can be considered as photons which carry a
discrete amount of energy. It is possible to calculate the energy of a photon based on the EM waves’ frequency and Planck's constant using the following equation.
E = hν Equation 2,
where E is equal to energy, h is Planck’s constant, and ν is frequency. Planck’s constant is 6.626 x 10^-34 J▪s. Energy in Joules is equal to Planck's constant in the unit of J▪s times the
frequency in Hz (Equation 2). With this equation, it is possible to observe that as frequency increases, so does the energy. Because Planck’s constant is a very small number, the amount of energy
carried by a single photon is also very small. | {"url":"https://teachersinstitute.yale.edu/curriculum/units/2020/2/20.02.04/4","timestamp":"2024-11-14T01:24:31Z","content_type":"text/html","content_length":"39344","record_id":"<urn:uuid:7dc23eb5-a3de-4071-8197-6d78969be99f>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00413.warc.gz"} |
Gold-Currency Strategy II | Logical Invest
What do these metrics mean?
'Total return is the amount of value an investor earns from a security over a specific period, typically one year, when all distributions are reinvested. Total return is expressed as a percentage of
the amount invested. For example, a total return of 20% means the security increased by 20% of its original value due to a price increase, distribution of dividends (if a stock), coupons (if a bond)
or capital gains (if a fund). Total return is a strong measure of an investment’s overall performance.'
Which means for our asset as example:
• Looking at the total return, or increase in value of 48.2% in the last 5 years of Gold-Currency Strategy II, we see it is relatively lower, thus worse in comparison to the benchmark GLD (77.1%)
• During the last 3 years, the total return is 21.3%, which is smaller, thus worse than the value of 50.7% from the benchmark.
'Compound annual growth rate (CAGR) is a business and investing specific term for the geometric progression ratio that provides a constant rate of return over the time period. CAGR is not an
accounting term, but it is often used to describe some element of the business, for example revenue, units delivered, registered users, etc. CAGR dampens the effect of volatility of periodic returns
that can render arithmetic means irrelevant. It is particularly useful to compare growth rates from various data sets of common domain such as revenue growth of companies in the same industry.'
Which means for our asset as example:
• Looking at the annual return (CAGR) of 8.2% in the last 5 years of Gold-Currency Strategy II, we see it is relatively smaller, thus worse in comparison to the benchmark GLD (12.1%)
• Looking at annual return (CAGR) in of 6.7% in the period of the last 3 years, we see it is relatively lower, thus worse in comparison to GLD (14.7%).
'In finance, volatility (symbol σ) is the degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns. Historic volatility measures a time
series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). Commonly, the higher the
volatility, the riskier the security.'
Which means for our asset as example:
• The 30 days standard deviation over 5 years of Gold-Currency Strategy II is 9.6%, which is lower, thus better compared to the benchmark GLD (15.3%) in the same period.
• During the last 3 years, the volatility is 9.3%, which is lower, thus better than the value of 14.2% from the benchmark.
'Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference.
Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk in
our definition is the semi-deviation, that is the standard deviation of all negative returns.'
Which means for our asset as example:
• Looking at the downside deviation of 7.1% in the last 5 years of Gold-Currency Strategy II, we see it is relatively smaller, thus better in comparison to the benchmark GLD (10.7%)
• Compared with GLD (9.5%) in the period of the last 3 years, the downside risk of 6.5% is smaller, thus better.
'The Sharpe ratio is the measure of risk-adjusted return of a financial portfolio. Sharpe ratio is a measure of excess portfolio return over the risk-free rate relative to its standard deviation.
Normally, the 90-day Treasury bill rate is taken as the proxy for risk-free rate. A portfolio with a higher Sharpe ratio is considered superior relative to its peers. The measure was named after
William F Sharpe, a Nobel laureate and professor of finance, emeritus at Stanford University.'
Using this definition on our asset we see for example:
• Looking at the ratio of return and volatility (Sharpe) of 0.59 in the last 5 years of Gold-Currency Strategy II, we see it is relatively lower, thus worse in comparison to the benchmark GLD
• Looking at Sharpe Ratio in of 0.45 in the period of the last 3 years, we see it is relatively lower, thus worse in comparison to GLD (0.86).
'The Sortino ratio, a variation of the Sharpe ratio only factors in the downside, or negative volatility, rather than the total volatility used in calculating the Sharpe ratio. The theory behind the
Sortino variation is that upside volatility is a plus for the investment, and it, therefore, should not be included in the risk calculation. Therefore, the Sortino ratio takes upside volatility out
of the equation and uses only the downside standard deviation in its calculation instead of the total standard deviation that is used in calculating the Sharpe ratio.'
Which means for our asset as example:
• Looking at the ratio of annual return and downside deviation of 0.81 in the last 5 years of Gold-Currency Strategy II, we see it is relatively lower, thus worse in comparison to the benchmark GLD
• Compared with GLD (1.28) in the period of the last 3 years, the downside risk / excess return profile of 0.63 is lower, thus worse.
'The ulcer index is a stock market risk measure or technical analysis indicator devised by Peter Martin in 1987, and published by him and Byron McCann in their 1989 book The Investors Guide to
Fidelity Funds. It's designed as a measure of volatility, but only volatility in the downward direction, i.e. the amount of drawdown or retracement occurring over a period. Other volatility measures
like standard deviation treat up and down movement equally, but a trader doesn't mind upward movement, it's the downside that causes stress and stomach ulcers that the index's name suggests.'
Which means for our asset as example:
• Compared with the benchmark GLD (9.74 ) in the period of the last 5 years, the Downside risk index of 7.14 of Gold-Currency Strategy II is lower, thus better.
• Looking at Ulcer Index in of 7.99 in the period of the last 3 years, we see it is relatively lower, thus better in comparison to GLD (8.23 ).
'A maximum drawdown is the maximum loss from a peak to a trough of a portfolio, before a new peak is attained. Maximum Drawdown is an indicator of downside risk over a specified time period. It can
be used both as a stand-alone measure or as an input into other metrics such as 'Return over Maximum Drawdown' and the Calmar Ratio. Maximum Drawdown is expressed in percentage terms.'
Using this definition on our asset we see for example:
• Looking at the maximum DrawDown of -13.8 days in the last 5 years of Gold-Currency Strategy II, we see it is relatively higher, thus better in comparison to the benchmark GLD (-22 days)
• During the last 3 years, the maximum drop from peak to valley is -13.8 days, which is higher, thus better than the value of -21 days from the benchmark.
'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Max Drawdown Duration is the worst (the maximum/longest) amount of time an investment has
seen between peaks (equity highs) in days.'
Using this definition on our asset we see for example:
• The maximum days below previous high over 5 years of Gold-Currency Strategy II is 590 days, which is smaller, thus better compared to the benchmark GLD (897 days) in the same period.
• Compared with GLD (436 days) in the period of the last 3 years, the maximum days below previous high of 590 days is higher, thus worse.
'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Avg Drawdown Duration is the average amount of time an investment has seen between peaks
(equity highs), or in other terms the average of time under water of all drawdowns. So in contrast to the Maximum duration it does not measure only one drawdown event but calculates the average of
Using this definition on our asset we see for example:
• Compared with the benchmark GLD (346 days) in the period of the last 5 years, the average days below previous high of 215 days of Gold-Currency Strategy II is smaller, thus better.
• During the last 3 years, the average days under water is 246 days, which is higher, thus worse than the value of 143 days from the benchmark. | {"url":"https://logical-invest.com/app/strategy/gld-uup/gold-currency-strategy-ii","timestamp":"2024-11-04T10:37:28Z","content_type":"text/html","content_length":"60839","record_id":"<urn:uuid:e1fa35e2-72ed-4003-b66b-ea1f9e3c1e9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00591.warc.gz"} |
Absolute modulus : A better understanding
File comment: |x+2|=3-2x
docu1.png [ 43.31 KiB | Viewed 153507 times ]
I had a PM and a profile comment asking about the absolute modulus, its concept and in particular a Question discussed on various occassion "
How many solutions does the equation |x+3|-|4-x|=|8+x| have?
Just thought to write down few concepts I have gathered. I have not gone through various Topics on Absolute Modulus in this Forum, so maybe few points are repetition.
Although difficult for a topic like this, I'll try to follow KISS- Keep It Short and Simple. So, let me touch the concepts now..
what is absolute modulus?
Absolute modulus is the numeric value of any number without any sign or in other words ' the distance from the origin'. It will always be positive.
What kind of Qs can one see in GMAT?
The Q will ask either the values of x or how many values can x take?
most often what one can encounter is a linear Equation with...
a) only one Mod
eg.. |x+2| + 2x= 3..
b) two mods..
c) three mods.. very rare
|x+3|-|4-x|=|8+x| ..
What are the methods
three methods..
1) As the property suggests, Open each modulus in both +ive and -ive ....
2) Critical value
3) Graphical method..
Opening each modulus
It is a time consuming process where you open each mod in both positive and negative and the number of Equations thus formed will increase as we increase the no of mods..
a) only one Mod
eg.. |x+2| + 2x= 3..
i) (x+2) + 2x=3..
x=1/3.. valid value
ii) -(x+2)+2x=3..
but if we substitute x=5 in |x+2| + 2x= 3..... (x+2) will turn out to be a positive value, while we took x=2 to be negative so discard
so one value of x..
b) two mods..
here you will have four equations..
i)(x+2)=(x-3)+1.. both positive
ii)-(x+2)=-(x-3)+1.. both negative
iii)-(x+2)=(x-3)+1..one positive and other negative
iv)(x+2)=-(x-3)+1.. opposite of the one on topc) three mods.. very rare
|x+3|-|4-x|=|8+x| ..
it will further increase the number of equations..
.. time consuming and susceptible to errors in opening of brackets and at times requires more to negate the values found as in first example.
Critical method
ets find what happens in this before trying the Qs as this was the main query..Step 1 :-
for each mod, there is a value of x that will make that mod to 0..
Step 2 :-
the minimum value of a mod will be 0 and at this value of x, the mod has the lowest value...
Once we know this critical value, we work on the mod for values lesser than(<) that or more than(>)that and including the critical value in either of them,
we assign a sign, + or -, depending on what will happen to the value inside the mod in each scenario(in one scenario it will be positive and in other, negative)
Step 3 :-
after assigning the sign, we solve for x and see if the value of x that we get is possible depending on which side of critical value we are working on..
So what are we doing hereWe are assuming a certain region for value of x and then solving for x.. If the value found matches the initial assumption, we take that as a solution or discard that value,
which would mean that there is no value of x in that assumed region
lets see the three examples
a) only one Mod
eg.. |x+2| + 2x= 3..
here x+2 will be 0 at x=-2..
so Critical value =-2..
so two regions are <-2 and >= -2i)when x<-2, |x+2|will be given negative sign..
for this assign any value in that region say -3 in this case x+2 will become -3+2=-1 hence a negative sign..
x-2=3.. x=5, which is not in the region <-2.. so not a valid value..
ii)when x>=-2, |x+2|will be given positive sign..
for this assign any value in that region say 3 in this case x+2 will become 3+2= 5 hence a positive sign..
3x+2=3.. x=1/3, which is in the region >=-2.. so a valid value..
b) two mods..
critical values -2 and 3...
so regions are <-2, -2<=x<3, x>=3..i) x<-2...
x+2 will be -ive and x-3 will be negative ..
eq becomes -(x+2)=-(x-3)+1.. both negative
-x-2=-x+3+1..... no values..
ii) \(-2<=x<3\)..
x+2 will be positive and x-3 will be negative ..
eq becomes (x+2)=-(x-3)+1..
x=1.. valid value
x+2 will be positive and x-3 will be positive ..
eq becomes (x+2)=(x-3)+1..
no valid value..
so the solution is x=1c) three mods.. very rare
|x+3|-|4-x|=|8+x| ..
its time consuming and can be solved similarly..
Graphical method
for graphical method we will have to again use the critical point..
at critical point, it is the lowest value of mod and on either side it increases with a negative slope on one side and positive slope on other side
so it forms a 'V' shape in linear equation and a 'U ' curve for Quadratic Equation..
If the mod has a negative sign in front, -|x+3|, it will have an "inverted V" shape with max value at critical value..
lets see the three examples..
a) only one Mod
eg.. |x+2| + 2x= 3..
critical value at -2 and equation can be written as
|x+2| = 3-2x..
we take y=|x+2| and draw a graph and then take y=3-2x and again draw graph..
the point of intersection is our value..
b) two mods..
here we will take y=|x+2| and y=|x-3|+1
again the point of intersection of two sides will give us the value of x..
c) three mods.. very rare
|x+3|-|4-x|=|8+x| ..
Here we have three critical values, but the graph will still be only two, one for LHS and one for RHS..
It will not be three for three mod as someone has drawn it in one of the discussions on this Q..
again we see the intersection of two graph..
there are no points of intersection , so no solution
1) Opening modulus is time consuming, susceptible to error, and the answer found can still be wrong and has to checked by putting the values in mod again..
should be least priority and should be used by someone has not been able to grasp finer points of other two methods..
2) "Critical method" should be the one used in most circumstances although it requires a good understanding of signs given to the mod when opened within a region.
It has to be the method, when you are looking for values of X..
3) "Graphical method" is useful in finding the number of values of x, as getting accurate values of x may be difficult while estimating from free hand graphs..
but if understood much faster and easier to find sol for Q like
How many solutions does the equation |x+3|-|4-x|=|8+x| have?
Hope it helps atleast a few of you..
File comment: |x+3|-|4-x|=|8+x| ..
docu3.png [ 32.95 KiB | Viewed 150536 times ]
File comment: |x+2|=|x-3|+1..
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On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation (Conference Proceeding) | NSF PAGES
We consider the problem of finding stationary points in Bilevel optimization when the lower-level problem is unconstrained and strongly convex. The problem has been extensively studied in recent
years; the main technical challenge is to keep track of lower-level solutions $y^*(x)$ in response to the changes in the upper-level variables $x$. Subsequently, all existing approaches tie their
analyses to a genie algorithm that knows lower-level solutions and, therefore, need not query any points far from them. We consider a dual question to such approaches: suppose we have an oracle,
which we call $y^*$-aware, that returns an $O(\epsilon)$-estimate of the lower-level solution, in addition to first-order gradient estimators {\it locally unbiased} within the $\Theta(\epsilon)$-ball
around $y^*(x)$. We study the complexity of finding stationary points with such an $y^*$-aware oracle: we propose a simple first-order method that converges to an $\epsilon$ stationary point using $O
(\epsilon^{-6}), O(\epsilon^{-4})$ access to first-order $y^*$-aware oracles. Our upper bounds also apply to standard unbiased first-order oracles, improving the best-known complexity of first-order
methods by $O(\epsilon)$ with minimal assumptions. We then provide the matching $\Omega(\epsilon^{-6})$, $\Omega(\epsilon^{-4})$ lower bounds without and with an additional smoothness assumption on
$y^*$-aware oracles, respectively. Our results imply that any approach that simulates an algorithm with an $y^*$-aware oracle must suffer the same lower bounds.
more » « less | {"url":"https://par.nsf.gov/biblio/10533142-penalty-methods-nonconvex-bilevel-optimization-first-order-stochastic-approximation","timestamp":"2024-11-02T20:31:41Z","content_type":"text/html","content_length":"243766","record_id":"<urn:uuid:b0b3a293-c4ad-46f5-be38-124decad53dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00696.warc.gz"} |
August 21, 2023
The Law of Sines is an important mathematical tool used to solve a variety of problems related to the lengths of sides and angles of triangles. It is an equation that relates the side lengths and
angles of any triangle, allowing us to calculate unknown values when two sides and an angle, or two angles and a side, are known. The Law of Sines is an essential tool for engineers, architects, and
other professionals who must calculate the dimensions of structures and other objects.
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides. This means that if we know the length of two sides and
the measure of their opposite angles, we can calculate the length of the third side and the measure of its opposite angle. This equation can be written in the following form:
a/sin A = b/sin B = c/sin C
Where a, b, and c are the lengths of the three sides of the triangle, and A, B, and C are the angles opposite those sides.
The Law of Sines can be used to solve a variety of problems. For example, it can be used to calculate the lengths of the sides of a triangle when two angles and a side are known. It can also be used
to calculate the angles of a triangle when two sides and an angle are known. This makes it an invaluable tool for engineers and architects who need to calculate the dimensions of structures and other
The Law of Sines can also be used to solve problems involving oblique triangles. An oblique triangle is a triangle with one angle greater than 90 degrees. In this case, the Law of Sines can be used
to calculate the lengths of the sides when two angles and the length of the third side are known.
The Law of Sines is an important mathematical tool that can be used to solve a variety of problems related to the lengths of sides and angles of triangles. It is a simple equation that can be used to
calculate the unknown values when two sides and an angle, or two angles and a side, are known. This makes it an invaluable tool for engineers, architects, and other professionals who must calculate
the dimensions of structures and other objects.… | {"url":"https://caymanlawsociety.com/2023/08/21/","timestamp":"2024-11-08T08:10:43Z","content_type":"text/html","content_length":"38130","record_id":"<urn:uuid:8889e009-8010-467d-a0a4-faa769250d3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00282.warc.gz"} |
Ramanujam's Mathematics Magic of Mathematics Summary and Analysis (like SparkNotes) | Free Book Notes
Ramanujam's Mathematics Magic of Mathematics Summary and Analysis
FreeBookNotes found 1 site with book summaries or analysis of Ramanujam's Mathematics Magic of Mathematics. If there is a Ramanujam's Mathematics Magic of Mathematics SparkNotes, Shmoop guide, or
Cliff Notes, you can find a link to each study guide below.
Among the summaries and analysis available for Ramanujam's Mathematics Magic of Mathematics, there is 1 Short Summary.
Depending on the study guide provider (SparkNotes, Shmoop, etc.), the resources below will generally offer Ramanujam's Mathematics Magic of Mathematics chapter summaries, quotes, and analysis of
themes, characters, and symbols. | {"url":"http://www.freebooknotes.com/summaries-analysis/ramanujams-mathematics-magic-of-mathematics/","timestamp":"2024-11-02T23:53:07Z","content_type":"text/html","content_length":"39161","record_id":"<urn:uuid:441e7cdd-8266-4952-8db0-33286c48e184>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00440.warc.gz"} |
Right-Angled Triangles
A right-angled triangle (also called a right triangle)
is a triangle with a right angle (90°) in it.
The little square in the corner tells us it is a right angled triangle
(I also put 90°, but you don't need to!)
The right angled triangle is one of the most useful shapes in all of mathematics!
(It is used in the Pythagoras Theorem and Sine, Cosine and Tangent for example).
Try it yourself (drag the points):
Two Types
There are two types of right angled triangle:
Isosceles right-angled triangle
One right angle
Two other equal angles always of 45°
Two equal sides
Scalene right-angled triangle
One right angle
Two other unequal angles
No equal sides
Example: The 3,4,5 Triangle
The "3,4,5 Triangle" has a right angle in it.
(Draw one if you ever need a right angle!)
It has no equal sides so it is a scalene right-angled triangle
And, like all triangles, the three angles always add up to 180°.
6701, 6707, 761, 1800, 762, 1801, 3228, 3229, 8997, 8998 | {"url":"http://wegotthenumbers.org/right_angle_triangle.html","timestamp":"2024-11-03T17:15:29Z","content_type":"text/html","content_length":"5628","record_id":"<urn:uuid:93fc96a8-59b4-42c2-8ec6-ec886b042520>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00191.warc.gz"} |
Asymmetrical Drawing & Lettering with the Golden Ratio | Chris Heath | Skillshare
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Asymmetrical Drawing & Lettering with the Golden Ratio
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Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more
Asymmetrical Drawing & Lettering with the Golden Ratio
Introduction to Asymmetry
Abstract Rectangles
Swiss Style Numerals
Roman Style Letters
Dividing a Line by the Golden Ratio
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About This Class
Have you ever wondered how to use the golden ratio?
Let's face it; it's not the easiest thing to wrap your head around. The math may seem mind-boggling, but here's the thing — it's actually quite simple for artists, designers and architects
to use.
In this class we explore, through a series of simple drawing exercises, one of the golden ratio's special properties. This property can best be described as balanced asymmetry. The golden ratio is
one of a number of useful proportions that can be used to introduce asymmetry into our work, and to do this in a harmonious way.
Symmetry is easy. Asymmetry is just as easy if you know where to begin.
All you need for this class is:
• to print the grids that you can download from this class
• some pens and pencils
• an eraser, and
• a straight edge, e.g., a ruler
So lets get started...
Meet Your Teacher
Check out my profile page to discover more classes for artists and designers.
See full profile
Level: Beginner
Hands-on Class Project
To follow the lessons in this class, download the following grids:
• Grid-Golden-Ratio-03.pdf, and
• Grid-Golden-Ratio-05.pdf.
For your project, you may work with any of the grids. Download them all and let your imagination soar. As a hint, try using the grid to draw something you have drawn before or enjoy drawing.
If you find that looking at the grid leaves you feeling blank, have a look around the room for objects you can draw. Don't just look at an object, look at the space around it and its relationship
to other other objects in the room.
Ideas to Explore
• Stripes
• Geometric Patterns
• Repeat Patterns
• Alternating Colours
• Letter and Number Shapes
• Abstract Shapes and Patterns
• and the list goes on..
Please post your experiments in the project area of this class.
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1. Asymmetrical Drawing & Lettering with the Golden Ratio: it's class, I will introduce you to a small range of abstract drawing exercises that are designed to help you incorporate balanced asymmetry
into your own work. With just two grades, we will be using the golden ratio to determine the size and distribution, off lines, big tangles and Kurds I compress. I'm a writer and designer. In the late
20th century, I completed a design thesis for my master's degree on the geometric laws and principles on ornament. I've always been fascinated with how geometry can be used to handle simplicity,
penned complexity and design. With that being abstract design a simple Lego and I call or a complex decorative surface pattern, The golden ratio is one of a number of useful proportions on design.
The other proportions that I commonly use. Other related group two ratio and the Route three ratio. The golden ratio, with its related fivefold symmetry and its fractal property of self similarity at
all scales as incredibly useful when it comes to representing natural forms and now work. If you want to incorporate the golden ratio into your work understanding, it's asymmetrical yet harmonious
division of space. It's perhaps the first flailing hurdle toe become. That is a complex Syria study, and it would be difficult to cover everything about the golden ratio on one class. So in this
class we're just going to focus on how to use the golden ratio Toe Corp rape asymmetry into your work. 2. Introduction to Asymmetry: asymmetry is widespread throughout the natural world. I like to
think of a symmetry. Is the unequal yet harmonious division off space? Asymmetry is also property off continuous proportions. For example, the golden ratio. This means we can make use of the golden
ratio to incorporate asymmetry and where work, for example, to recreate the asymmetrical wings on one side of a butterfly. What we're working with here is the asymmetrical properties of a continuous
proportion to size and arranged the elements of our design for artwork. This is one property that we can use to introduce a sense of harmony and balance to our work. 3. Drawing Tools: For this class,
you will need a pencil, some pins, the printed grids, which you can download from the your project page of this class and a razor and straight ege headers. A ruler or sit square. You don't need to be
good at drawing Freehand. Just explore the downloadable grids. Give the exercises ago thin. Try using the grids to draw whatever it is you normally enjoy drawing. 4. Abstract Lines: on this first or
against a size, we will be using this grid grid number five to sit the worth of her lines in an abstract design. It's easy to focus on making things symmetrical, and it's easy to ignore asymmetry
altogether, putting it in the too hard basket. I encourage you to complete this exercise as a means of pushing yourself outside of your comfort zone. I can promise you it's not much of a push, and I
think you'll actually enjoy it. Just print off a few copies of the grid and discover a range of different rhythms by blocking out the spaces between the grid lines. You can see in my example that
each time I change direction, I started new pattern to see what rhythm unfolds. And I'm really just making this up as I go and relying on the grid to introduce sense of harmony between each pattern.
Well, I have some idea of my hitters toe how I hope this turns out. The resulting rhythm of each pattern is very much a pleasant surprise. If you are a surface pattern designer, you may find these
grids useful for creating striped patterns, perhaps even his background patterns to your floral designs When coloring between the grid lines. I'm also seeing what happens of high color and adjacent
spaces, and conversely, seeing what happens if I leave two or more adjacent spaces white, the range of rhythmic patterns that you can create are endless. 5. Abstract Rectangles: Leslie said. We will
be using the screws. Grid number three. This listen is similar to the first, but I tend to think of these shapes is rectangles. You may think of them as lines with stripes. In this case, we're
working with two months off Rick Tangle that are related by the golden ratio, and we are introducing 1/3 element. And that is the background, which is white, that is, our rectangles and the space
between rectangles are all determined by the golden ratio. I'm essentially sitting up blocks off repeated Rick tangles. Each block exhibits a variation of the same rhythm, and to make this a bit more
interesting, I'm changing direction, creating the illusion of overlapping rectangles and making use of the white space to create a balanced, asymmetric design. In some cases, the wide lines will be
colored, and then some. They will be black. Conversely, some of the thin lines will be colored and some will be black. I guess you could say that I'm introducing counter change to the design and
asymmetric way , and we will also incorporate the white background to enhance asymmetry 6. Swiss Style Numerals: When you think about it, the shapes of letters and numerals are pretty abstract. I
have always liked the typeface is Helvetica and universe. So sitting down with migrant and pencil, I thought had tickle some numerals with thesis was typefaces in mind? The idea here is to
incorporate asymmetry by making use of the grid lines to determine the worth of the numerals. At various points, for example, the thickness off the numeral varies from the horizontal to the vertical
by the golden ratio. When I was a kid, I used to help my dad out by scaling up in hand, drawing these letters and numerals for silage. Next up is a three. I really am making this up as I go, and I'm
not looking at any type faces as a reference. So if you see me stall and waved my pencil, I'm just thinking of what to do next. To start with, I am reproducing the same curves as on the to and
working at the center off the three. You'll also notice that the width of the center portion of the three is also related by the golden ratio to the other dimensions. It's not uncommon to reading
some of the kids or readjust the position off some of the points I prefer to draw lightly before darkening at the curves. Also, I'm aiming for long, deliberate strokes. I'm not building my curves by
sketching. After scanning the numeral and to a victor drawing EP, I've added a few refinements. First, up the squares to the right relate to the cred each squares. Dimensions are related to the
adjacent square by the golden ratio. I like to use thes squares as a unit of measure, much like you would use a ruler with an chisel millimeters on it. The difference is the units differ and scale by
the golden ratio. I ended up putting the center curves that defined the counter space to some circles. Obviously, it's not going to be a perfect matched with circles. I'm just using them as a guide.
I used the pink square toe and set the curve on the rights and used the Green Square toe office it punch point. The reason for these changes is to give the lower half of three more weight. By doing
so, the three doesn't look so top heavy. Is it would if it was perfectly symmetrical. There is more than I would like to do to this numeral, for example, to pull the top and bottom curves just over
the base and median line, but only fit for another day. 7. Roman Style Letters: for our last drawing exercise. Let's tackle a Roman style letter in this case, the letter A. Again. I'm not using a
reference for this. I'm just making it up. It's a go. I sit the apex of the letter using the grid, then set the baseline. The epic sits on the kept height line, so I'm drawing the sun as well, mixed
up of the feet. And these were drawn symmetrically either side of the APICS, I said, a point to draw the diagonal of thick line on the right. Thin established the thickness off the thick line using
the grid. And then I drew the brackets or curves that connect the stems to the syrup. Similarly, a point was sit for the here line on the lift. The worth of the hairline was also sit using the grid.
Most of all, the crossbar was headed and again, the thickness and location of this cross bar is determined using the grid. The thickness off the ethical line was okay, but I decided it might look
better if it was thicker, so I'm redrawing it again, using the grid to determine the thickness. Apart from making it up, it's pretty much done with these drawing exercises. I hope you can see the
value in using a grid on particular. Agreed, that incorporates the golden ratio to build asymmetry into your design and to do it in such a way as to create a harmonious design. Because I have done,
you can use the grid to guide the size and location of each line and curve on your design, and here we have the finished leader. 8. Dividing a Line by the Golden Ratio: hopefully bone there you've
had a go at the exercises. Enough not, Doesn't really matter what I thought I would do here is going Teoh the golden ratio on a little bit more detail. Perhaps this might explain the golden ratio
bitter. So in this bonus, listen, we will be dividing a line using geometry to reveal the golden ratio. Then, in the following listen, we will use this ratio to create the simple, asymmetric grids
used in this class. So let's get started. Step on. Draw a double square rectangle that is a rectangle made of to squeeze side by side. I'm using graph paper to make this easier if you have ever heard
of Dynamic cemetery. The stubble square rectangle is known as the root for rectangle because the length of its long side, when divided by the length of its short side, equals two. Stick to draw a
diagonal from the lower lift of the rectangle to the opposite corner of the rectangle. Step three. Drawing AK with the radius it to the height of the rectangle centric on the end of the diagonal and
draw the air controller and six The diagonal Step four drawer A second ac sent to the sack on the start off the diagonal and sit the radius to the length of the diagonal list the height off the
rectangle. Describe this arc until under six. The long side of the rectangle while I continue to market the strolling, let's have a look at what we've just done. So now we have the length of the
route for rectangle divided and two unequal segments. There is something special about the length off these segments and relation to the length of the rectangle. The length of the rectangle of
divided by the length of the longer segments results in a number, and that number has 1.618 It's a number which just continues forever. Likewise, dividing the length of the longer segment by the
length of the short segment also equals on 0.618 and this is the golden ratio. So if I was to label these lengths A, B and C, I think I over Bay equals B over see what she calls 1.618 This number has
called an irrational number because it goes on forever without repeating and the fact that a divided by B equals be divided by C makes this ratio a continuous proportion. And it's this perfect ratio
that pops up in geometry time and time again. And I'm perfectly on natural systems throughout the observable universe. 9. Grid Setup: What were you doing in this? Listen is creating a Siri's off
related grids. These are the same grids used in this class. I'm using affinity designer. You can follow along with the Adobe Illustrator or any other victor drawing toll. If you don't want to create
these grids, you can download the finished grids. End this affinity designer file from your project page off this class, I've set up a square that is 200 millimeters by 200 millimeters. The sizes
arbitrary only chose the size to fit the grid nicely on a piece of a four paper starting in the lower left corner drawer, a square incidents dimensions to 10 millimeters by 10 millimeters. I chose 10
millimeters so that divides equally into 200. I'm also using the lift lift corner off My objects has a datum to assume into the square end. Duplicate er's three with both dimensions connected,
multiply one of the dimensions off the duplicated square by 0.618 This is the invoice of 1.618 which is the golden ratio. This will sit the rectangle nicely and the lower left corner off the larger
square select your pin toll. Sit the strike toe around 0.5 point. Draw a horizontal line from the lower left corner across to the right that has 200 millimeters long. Duplicate this line and snap it
to the top right corner off the smaller square. Note that this line is not going to snip to the nearest millimetre on the Y Xs because the golden ratio is an irrational number, and it is the golden
ratio that we used to determine the dimension off the smaller square. Sit the worth of this line toe approximately 2.5 point. Duplicate the two lines and drake the copies until the lower line snips
to the lower lift off the larger square. We can now continue duplicating thes two lines using Command J until we reach the top of the bounding square. Group these lines and label them horizontal 01 I
Mr Line. So I'm going to add one more line to the group, duplicate and rotate the group anti clockwise 90 degrees. Renamed the stew Placated group vertical 01 We now have a grid where the asymmetry
is obvious and so this is a grid that we can use later on duplicate the group horizontal 01 Henry Name it horizontal Syria to hide all groups other than horizontal zero to ignore the bottom and top
lines and select the third line up from the bottom and delete it. Continue deleting every second line. This is so that when we rotate the group, we're not duplicating lines. Select the group, rotate
it 180 degrees, duplicate the group horizontal 02 and rename it Physical zero to hide the horizontal groups and show both vertical groups select and rotates the group 90 degrees. So now we have a
range of grids that we can explore, and H Grid unit is full of golden rectangles and squares. 10. What's Next?: I really hope you enjoyed this class and you got a lot out of it. Now, this class is
the first class in the Siris on the golden ratio, End related proportions. So if you would like to know Mawr and to see future classes, please follow me on skill share. Also, drop your questions into
the discussion area and I may be able to incorporate what you're looking for in the next class or two. Remember to download the exercises and give them ago. They are really good exercises in terms
off introducing a symmetry to your work. And what I recommend is after you've done these exercises, try and apply what you normally draw. What you enjoy. Drawing and incorporate those with the grid
and see what happens. But I look forward to seeing you in the mixed class. | {"url":"https://www.skillshare.com/en/classes/asymmetrical-drawing-and-lettering-with-the-golden-ratio/399174620?via=similar-classes","timestamp":"2024-11-09T18:59:24Z","content_type":"application/xhtml+xml","content_length":"370711","record_id":"<urn:uuid:d7befcd7-7cd4-473b-ae7e-7889fb853c4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00188.warc.gz"} |
nd c
template<typename IndexType_, typename ValueType_, typename UnionFunction_>
UnionFind class
An STL-like data structure for the union-find algorithm.
Operations such as the watershed, connected component labeling, and the area opening, as implemented in DIPlib, use the union-find algorithm. It provides an efficient method to set equivalences in
labels. That is, one can re-assign one label to be equivalent to another in O(1) time. Typically, one pass through the image assigns a label to each pixel (Create), and determines which labels should
be equivalent (Union); a second pass changes the label for each pixel to that of the representative label for the corresponding set of equivalent labels (FindRoot).
To stream-line the second pass, we provide here a Relabel method that assigns a unique, consecutive label to each of the correspondence sets.
Each tree element has a value associated to it. This must be a type that is copy-constructible and default-initializable. Ideally, it’s small. The value associated to any tree element that is not a
root is ignored. The unionFunction that the constructor takes is used to compute the value associated to the merged tree when two trees are merged. It should have the following signature:
ValueType_ unionFunction( ValueType_ const& value1, ValueType_ const& value2 );
The IndexType_ template parameter should be an integer, and probably unsigned.
See the code to any of the algorithms that use this class for an example usage.
Derived classes
template<typename IndexType_>
class dip::SimpleUnionFind
A simplified version of dip::UnionFind that doesn’t store any information about the regions, only equivalences.
Constructors, destructors, assignment and conversion operators
UnionFind(UnionFunction_ const& unionFunction) explicit
Default constructor, creates an empty structure
UnionFind(dip::uint n, dip::UnionFind::ValueType value, UnionFunction_ const& unionFunction)
Alternate constructor, creates n trees initialized to value.
using IndexType = IndexType_
The type of the index (or label) that identifies each tree element
using ValueType = ValueType_
The type of the additional data stored for each tree element
Function documentation
template<typename IndexType_, typename ValueType_, typename UnionFunction_>
template<typename Function>
void Iterate(Function function)
Calls function for each tree root.
function is a function or function object that takes the index (IndexType) and ValueType associated to a tree.
template<typename IndexType_, typename ValueType_, typename UnionFunction_>
dip::uint Relabel()
Assigns a new label to each of the trees.
Returns the number of unique labels.
template<typename IndexType_, typename ValueType_, typename UnionFunction_>
template<typename Constraint>
dip::uint Relabel(Constraint constraint)
Assigns a new label to the trees that satisfy constraint, and 0 to the remainder.
constraint is a function or function object that takes the ValueType associated to a tree, and returns true if the tree is to be kept.
Returns the number of unique labels. | {"url":"https://diplib.org/diplib-docs/dip-UnionFind-T.html","timestamp":"2024-11-13T08:41:29Z","content_type":"text/html","content_length":"21633","record_id":"<urn:uuid:ff2f672f-4a39-473d-9fcf-567550b3b854>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00832.warc.gz"} |
Maharashtra State Board Class 7th Maths Chapter 4 Angles and Pairs of Angles Set 4.1 Solution | Maharashtra Board Book Solution
Maharashtra State Board Class 7th Maths Chapter 4 Angles and Pairs of Angles Set 4.1 Solution
Question 1.
Observe the figure and complete the table for ∠AWB.
Points in the interior
Points in the exterior
Points on the arms of the angles
Points in the interior point C, point R, point N, point X
Points in the exterior point T, point U, point Q, point V, point Y
Points on the arms of the angles point A, point W, point G, point B
Question 2.
Name the pairs of adjacent angles in the figures below.
i. ∠ANB and ∠ANC
∠BNA and ∠BNC
∠ANC and ∠BNC
ii. ∠PQR and ∠PQT
Question 3.
Are the following pairs adjacent angles? If not, state the reason.
1. ∠PMQ and ∠RMQ
2. ∠RMQ and ∠SMR
3. ∠RMS and ∠RMT
4. ∠SMT and ∠RMS
1. ∠PMQ and ∠RMQ are adjacent angles.
2. ∠RMQ and ∠SMR not adjacent angles since they do not have separate interiors.
3. ∠RMS and ∠RMT not adjacent angles since they do not have separate interiors.
4. ∠SMT and ∠RMS are adjacent angles.
Intext Questions and Activities
Question 1.
Observe the figure alongside and write the answers. (Textbook pg. no. 24)
1. Write the name of the angle shown alongside___.
2. Write the name of its vertex___.
3. Write the names of its arms___.
4. Write the names of the points marked on its arms___.
1. ∠ABC
2. Point B
3. Ray BA, ray BC
4. Points A, B, C | {"url":"https://maharashtraboardbookolution.in/maharashtra-state-board-class-7th-maths-chapter-4-angles-and-pairs-of-angles-set-4-1-solution/","timestamp":"2024-11-03T21:53:13Z","content_type":"text/html","content_length":"182302","record_id":"<urn:uuid:6a790bb4-d268-4944-8e2b-4bc739b6cdae>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00339.warc.gz"} |
How do I
How do I write divided in Word?
To type a division symbol using the keyboard, press Alt-0247 on the numeric keypad with Num Lock turned on.
Can you do calculations in Word?
Word lets you perform calculations on numerical table data and display the results in the table. For example, you can add a row or column of numbers, or you can add, subtract, multiply, or divide the
contents of individual cells.
How do I insert equation numbers in Word 2007?
To insert an equation in a Word 2007 document, click on the “Insert” menu/tab to see the “Insert” ribbon. In the “Symbols” section, choose “Equation”. You can also press “Alt+=” on your keyboard.
How do I insert an equation in Word 2010?
Insert an equation with Equation Editor
1. On the Insert tab, in the Text group, click Object.
2. In the Object dialog box, click the Create New tab.
3. In the Object type box, click Microsoft Equation 3.0, and then click OK.
4. Use the symbols, templates, or frameworks on the Equation toolbar to edit the equation.
How do I insert equation numbers in Word 365?
To add cations for equations in your document, do the following:
1. When you type an equation (see how to create different equations for more details), at the end of it, type the symbol Hash (#) and then the number in a format that you prefer (usually in the
round brackets, in parentheses).
2. Press Enter.
How do I write math in Word?
In Word, you can insert mathematical symbols into equations or text by using the equation tools. On the Insert tab, in the Symbols group, click the arrow under Equation, and then click Insert New
How do you fix equations in Word?
Here is my fix method:
1. Go to the EQUATION TOOLS tab.
2. Click the small arrow button under Tools to open the Equation Options dialog.
3. Open the popup menu the Default font for math regions, and choose Cambria Math (the only).
4. Press OK button. It should restore the equations.
How do you insert an equation in Word on a Mac?
Using MathType for Microsoft Word for Mac To insert an equation using MathType go to insert tab and select Math icon to open MathType Window. You can then type the equation or handwrite it.
What is the formula for subtraction in Word?
Subtract numbers in a cell To do simple subtraction, use the – (minus sign) arithmetic operator. For example, if you enter the formula =10-5 into a cell, the cell will display 5 as the result.
How do I multiply in Word?
To create a formula, click inside the cell where you want the product to appear and go to the “Layout” tab of the Word Ribbon. Click the “Formula” icon and enter “=PRODUCT” in the “Formula” field.
You must also tell Word with cells to multiply together.
Can you change equation font in Word?
Setting font styles & sizes in an equation is a simple process. In Word 2016, you can change font sizes, styles, or even paragraph style in every equation like a regular text.
How do you make an equation bigger in Word?
You can set this adjustment by following these steps:
1. Choose Spacing from the Format menu. The Equation Editor displays the Spacing dialog box. (See Figure 1.)
2. Click on the Limit height box. The Equation Editor changes the Spacing dialog box.
3. Enter a limit height spacing that is a percentage of normal.
4. Click on OK.
Why can’t I change font in Word?
Cannot change Default Font in Microsoft Word If you like to change the default font, you need to press Ctrl+D and then have to click on Set As Default after picking a font. Finally, you have to
select All documents based on this Normal template option to make the changes available for new documents.
How do I add line numbers to a Word document?
Add line numbers to a section or to multiple sections
1. Click in a section or select multiple sections.
2. On the Layout tab, in the Page Setup group, click Line Numbers.
3. Click Line Numbering Options, and then click the Layout tab.
4. In the Apply to list, click Selected sections.
5. Click Line Numbers.
Can you insert a formula in Word?
Insert a formula in a table cell. Select the table cell where you want your result. On the Table Tools, Layout tab, in the Data group, click Formula. Use the Formula dialog box to create your
How do I number my Word document?
Insert page numbers
1. Select Insert > Page Number, and then choose the location and style you want.
2. If you don’t want a page number to appear on the first page, select Different First Page.
3. If you want numbering to start with 1 on the second page, go to Page Number > Format Page Numbers, and set Start at to 0.
How do I change the equation font in Word 2020?
On the ribbon, go to Insert>Equation. Type in an equation. Once you’re done, select it and on the ‘Design’ tab, click the ‘Normal Text’ button on the Tools box. Next, go to the Home tab, and from the
Font dropdown, select any font you like.
How do I use MathType in Word 2007?
MathType tab in Word
1. Insert Inline Equation Ctrl + Alt + Q (Windows), Ctrl + Q (Mac)
2. Insert Display Equation Alt + Q (Windows), ⌥ + Q (Mac)
3. Insert Right-Numbered Display Equation Alt + Shift + Q (Windows), ⌥ + Shift + Q (Mac)
4. Insert Left-Numbered Display Equation Ctrl + Alt + Shift + Q (Windows), Ctrl + Shift + Q (Mac)
Why is my equation in word not working?
Why is the equation editor selection grayed out? You may have saved your document in a format that does not support the Equation Editor. Try selecting “File” > “Save As…” and save the document as a
“. docx” file or “File” > “Convert” to update the document to the latest format.
How do I use formulas in Word 2007?
Inserting Formulas
1. Place your insertion point in the cell where you want to place the formula.
2. From the Layout tab, in the Data group, click Formula.
3. In the Formula text box, type the desired formula.
4. If necessary, from the Number format pull-down list, select the desired format for the result.
5. Click OK.
How do you use the SUM formula in Word 2007?
Sum a column or row of numbers in a table
1. Click the table cell where you want your result to appear.
2. On the Layout tab (under Table Tools), click Formula.
3. In the Formula box, check the text between the parentheses to make sure Word includes the cells you want to sum, and click OK. =SUM(ABOVE) adds the numbers in the column above the cell you’re in.
How do I get automatic numbering in Word?
Turn on or off automatic bullets or numbering
1. Go to File > Options > Proofing.
2. Select AutoCorrect Options, and then select the AutoFormat As You Type tab.
3. Select or clear Automatic bulleted lists or Automatic numbered lists.
4. Select OK.
How do I change the default math font in Word?
Open Word and create a new equation. Then select the little “additional settings” corner. In the menu, change the “Default font” to “XITS Math”. In order for the changes to take effect, you will have
to create a new equation environment (the current one will not be changed).
How do you insert equation numbers in Word?
Click on the Commands tab if it is not already selected. Select Insert on the left and then Equation Editor on the right. Click and drag the button beside Equation Editor (a square root symbol with
an alpha in it) to the toolbar. From now on, clicking on that button will insert an equation.
How do you move equations in Word?
Do this:
1. Right-click (Win) or ctrl+click (Mac) and choose Format object.
2. In the Layout tab, click In Front of text.
3. Click the equation and move it close to where you want it.
4. It should still be selected, so now hold down the Shift key and click the other objects — the triangle and the equations for the other 2 sides. | {"url":"https://thisisbeep.com/how-do-i-write-divided-in-word/","timestamp":"2024-11-13T15:11:52Z","content_type":"text/html","content_length":"55279","record_id":"<urn:uuid:df89c7c1-6d38-477f-bae9-52b701456712>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00832.warc.gz"} |
# to the nth power
Herman Rubin cik at l.cc.purdue.edu
Fri Nov 9 00:36:28 AEST 1990
In article <1990Nov3.220554.19404 at zoo.toronto.edu>, henry at zoo.toronto.edu (Henry Spencer) writes:
> In article <3809 at idunno.Princeton.EDU> pfalstad at phoenix.Princeton.EDU (Paul John Falstad) writes:
> >Just out of curiosity, is there an efficient algorithm for finding
> >square root of an integer (or the integral part of the square root, if
> >the number is not a perfect square) using only integer arithmetic?
> It's not difficult using Newton's Method, i.e. successive approximations,
> if you have good integer division. Like this:
> sqrtx = ( sqrtx + x/sqrtx ) / 2
> although you'll have to watch roundoff effects to avoid the possibility
> of an infinite loop when the square root isn't exact. You also need an
> initial approximation, although just x/2 is a good start.
Assuming this is done in integer arithmetic, there is only one case of an
infinite loop. Using real arithmetic (of course impossible on a computer),
the iteration is always too large. Thus it should be rounded down. Now
it is possible that this makes it so small that the next iteration is
larger. If one starts with something too large, this can only happen
if x = n(n+2), and this can be detected by the iterate being larger than
the previous value, so that a stopping rule when the iterate does not
decrease, and taking the previous value, is always right, if the initial
value is too large.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907
Phone: (317)494-6054
hrubin at l.cc.purdue.edu (Internet, bitnet) {purdue,pur-ee}!l.cc!hrubin(UUCP)
More information about the Comp.lang.c mailing list | {"url":"http://tuhs.vtda.org/Unix_Usenet/comp.lang.c/1990-November/011685.html","timestamp":"2024-11-11T17:32:05Z","content_type":"text/html","content_length":"4588","record_id":"<urn:uuid:bec6e1e3-feee-4eeb-b00b-6cad6d1245a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00635.warc.gz"} |
Ratio Call BackSpread Options Profit & Loss Calculations
Details about Profit & Loss Calculations for Ratio Call BackSpread Options Trading
Continuing further from our previous post
Ratio Call BackSpread Options Trading Explained: Example & Payoff Charts
, lets see the details about Profit & Loss Calculations for Ratio Call BackSpread.
What are the breakeven points for the Ratio Call BackSpread Options Trading
Break even points are the points or spots when there is no profit - no loss for the option trader. i.e. they are the points on either side of which there is a profit area and a loss area.
As can be seen from the payoff function, the Ratio Call BackSpread has 2 break even points, one on the upper side and one on the lower side - where the final BROWN colored graph crosses the
horizontal axis (x-axis) at
Downward Breakeven Point for Ratio Call BackSpread Options = [Lower strike + net credit amount]
In the above case, it will be = $40 + $7 = $47
Upward Breakeven Point for Ratio Call BackSpread Options = [Higher OTM strike price] + [difference between strikes * number of short contracts] / [number of long calls - number of short calls ] -
[net credit received]
In the above case, it will be = $60 + [($60-$40)*2]/[2-1] - $7 = $60 + $20 -$7 = $73
As seen from the final BROWN colored payoff function chart, they are indicated in the final payoff function as the brown color graph crosses the horizontal axis or x-axis.
What is the maximum profit for Ratio Call BackSpread Options Trading
The profit region of Ratio Call BackSpread is in 2 parts - on the upside and on the downside.
On the upside, the profit can go to unlimited as the stock price continues to rise further and further
On the downside, the profit will be limited and capped to the maximum net credit amount.
Maximum Profit of Ratio Call BackSpread Options = Net Options Premium Received - Brokerage & Commissions Paid (on the downside)
Maximum Profit of Ratio Call BackSpread Options = Unlimited - Brokerage & Commissions Paid (on the upside)
What is the maximum loss for Ratio Call BackSpread Options Trading
The loss is limited and occurs in an inverted pyramid shape - i.e. it goes linearly as the underlying stock price moves (within the loss region between 2 break even points)
Maximum Loss of Call Ratio BackSpread Options = Difference in strike price - Net Options Premium Received - Brokerage & Commissions Paid
= $20 - $7 - Brokerage & Commissions Paid
= $13 - Brokerage & Commissions Paid
What are the risks in trading Ratio Call BackSpread Options Trading
Trading Ratio Call BackSpread Options is not that risky as compared to any other high risk option strategies like naked short calls or short puts. You are usually receiving (net credit) for a an
unlimited profit potential on the upside and a limited profit potential on the downside while your loss regions are restricted to a small inverted pyramid based region.
Any big move in the stock price will directly mean profit to you - limited on the downside, but unlimited on the upside.
Any increase in volatility will mean a profit to the options trader, as increase in volatility will lead to increase in option premiums.
However, Any stability in underlying stock price movements or no movement at all, or decrease in volatility or time decay will be bad for this Call Ratio BackSpread Option position.
- Trader needs to take atleast 3 option positions at the entry and close all 3 at the exit. That will mean high cost of brokerage which may eat into your profits.
- The option trader needs to have a bullish or bearish outlook for trading Ratio Call BackSpread
- Time decay is harmful to the trader as there are net long positions (in terms of 2 long calls and one short).
- This being a net buyer (long) position, its advisable to trade this with longer time to expiry, so that the benefit from Time Decay is minimum. see (
Options Time Decay: Explained with Examples
Let's continue further to the
Greeks for Ratio Call BackSpread Option: Delta, Gamma, Rho, Vega, Theta
0 Comments: Post your Comments
Wish you all profitable derivatives trading and investing activities with safety! = = Post a Comment | {"url":"http://futuresoptionsetc.com/2012/02/ratio-call-backspread-options-profit.html?widgetType=BlogArchive&widgetId=BlogArchive1&action=toggle&dir=open&toggle=YEARLY-1262332800000&toggleopen=MONTHLY-1328083200000","timestamp":"2024-11-15T04:14:39Z","content_type":"application/xhtml+xml","content_length":"59602","record_id":"<urn:uuid:f8208040-2bc1-4a94-aaf7-529099548a92>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00525.warc.gz"} |
om Online Grader
Fito is doing Math homework again. As usual, he would like to write a program that computes the answer automatically. He knows that the teacher might ask the students to solve thousands of them.
Since Fito is new to programming, he needs your help!
Given a positive integer $N$, find the minimum positive integer divisible by both $2$ and $N$. | {"url":"https://matcomgrader.com/problem/9690/an-easy-task-ii/","timestamp":"2024-11-03T03:19:48Z","content_type":"text/html","content_length":"23523","record_id":"<urn:uuid:8ada4587-08d0-4a70-a90b-4e24af4e69a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00205.warc.gz"} |
A Study On The Relationship Between The AM, GM, And HM Tamildada
It is common in mathematics to encounter the AM, GM, and HM connections while studying sequences of numbers. The averages or means of the three series in question are shown here. The Arithmetic Mean,
Geometric Mean, and Harmonic Mean are denoted by AM, GM, and HM Relation. A mean of Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP) is denoted by the letters
AM, GM and HM relation.
Anyone with a basic understanding of mathematical progressions or sequences should derive from the AM, GM and HM relation. When it comes to mathematics, a Mathematical Sequence is an array or
collection of elements that follow a predetermined pattern. A series is referred to as a progression in another context. The three most prevalent types of sequences are arithmetic, geometric, and
harmonic sequences, among others. In mathematics, an arithmetic sequence is a pattern of numbers in which the difference between consecutive terms remains constant throughout the series. A geometric
progression is a sequence of numbers in which each pair of following terms has the same ratio as the previous pair of terms. When the reciprocal of each term is taken in order, harmonic progression
is the sequence that results in the creation of an arithmetic series of terms.
It is vital to get acquainted with these three procedures and the linked equations before looking into their relationship.
Relation between AM, GM, HM
The AM, GM, and HM standards are perhaps the most extensively used and most straightforward. Multiplying the difference between two numerical values by the range’s total value is a simple example
(s). Scientists and researchers quantify a broad range of variables using the arithmetic mean, while people estimate the mean using the national average. Arithmetic means popularity has grown in
recent years, making it the most often used statistic.
Several acronyms are used to denote different types of means, such as the Arithmetic Mean (AM), Geometric (GM), and Harmonic (HM).
• AM – Also known as Arithmetic Mean, is the mean or average of a set of numerical values. When all of the terms in the collection are added together, the result is divided by the total number of
terms in the collection. To compute arithmetic mean, take a number n and divide it by n to get its arithmetic mean and the arithmetic means of all the other numbers in the range. This produces
the central tendency, which may then be compared to all other numbers bigger than it. The central tendency is also known as the arithmetic mean since it is the average of all arithmetic outcomes
that seem to be roughly normal. The usual means that you will see are closest to the average of all numbers greater or equal. If the results of your computation deviate considerably from the
mean, you should change the input to achieve an accurate result.
• GM – The middle term or the mean value is a collection of numbers in the geometric progression, also known as the geometric mean or the geometric mean value. When considering a geometric sequence
with n terms, the geometric mean is defined as the sequence with n terms. The nth root of the product of all of the terms in the sequence is denoted by the letter n.
• HM – Known as the Harmonic Mean, this function calculates the average. It is possible to compute the harmonic mean of a series by dividing the number of values in the sequence by the sum of the
reciprocals. The harmonic mean, abbreviated as HM, is one such technique.
Relation Between AM, GM and HM
Assuming that the numbers a and b represent two distinct real numbers that are positive and unequal in magnitude. The numbers A, G, and H represent their arithmetic, geometric, and harmonic means.
Check below concerning the AM, GM and HM relation.
• The following is the relationship between the letters A, G, and H: A > (greater than) G > (greater than) H.
• The following is valid for the overall relationship between A, G, and H: AH (equals) G2.
• The relationship of AM, GM and HM is as follows: AM multiplied by HM (equals)GM2.
AM GM HM in the study of mathematics
• AM GM HM statistics play a crucial role in complicated calculations.
• As a result of its straightforwardness and simplicity in mathematics, the Arithmetic Mean is one of the most often used measures of central tendency in collecting data, whether grouped or
• In the computation of stock indexes, the geometric mean is used as a formula. In addition, the geometric mean is used to calculate the annual returns on the portfolio’s investments. In addition,
the geometric mean is used to analyse biological processes such as cell division and bacterial growth, among other things.
• Price-earnings ratios and other common multiples are calculated with the use of the harmonic mean in finance. Furthermore, it is used in the computation of the Fibonacci sequence.
• Statistical professionals that use AM, GM, and HM in their work infer and demonstrate that the value of AM is greater than the values of GM and HM. AM is the most significant of the three
variables. The value of GM is greater than the value of HM and lower than the value of AM. The letter HM has a lesser value than the letters AM and GM.
• Tutflix is an ideal stage for everyone. It is an ideal response for understudies since they can watch the workshops on Tutflix, and it is also an amazing instrument for adults who need to find
some new data
Most people use the arithmetic mean to compute the mean or average of any statistical data. The Geometric Mean (GM) is a constant that represents the nth root of the product of terms in any
mathematical sequence of ‘n’ terms. The geometric mean is used to compute growth, investment, surface area, and volume. The harmonic mean is the sum of the reciprocals of all the words in a sequence
divided by their number. It is used to calculate speed, output, and cost. It is all about the AM, GM and HM relations. Bear this in mind if you want to get a high score.
Plagiarism Report | {"url":"https://tamildada.info/a-study-on-the-relationship-between-the-am-gm-and-hm/","timestamp":"2024-11-04T23:29:40Z","content_type":"text/html","content_length":"91056","record_id":"<urn:uuid:f4cd7ba0-5b31-4e6f-9ada-7a6569aa7b6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00558.warc.gz"} |
Access zero-pole-gain data
[z,p,k] = zpkdata(sys) returns the zeros z, poles p, and gain(s) k of the zero-pole-gain model sys.
[z,p,k,Ts] = zpkdata(sys) also returns the sample time Ts.
[z,p,k,Ts,covz,covp,covk] = zpkdata(sys) also returns the covariances of the zeros, poles and gain of the identified model sys.
[z,p,k] = zpkdata(sys,'v') returns the zeros and poles directly as column vectors for SISO zero-pole-gain models.
Example 1
Given a zero-pole-gain model with two outputs and one input
H = zpk({[0];[-0.5]},{[0.3];[0.1+i 0.1-i]},[1;2],-1)
Zero/pole/gain from input to output...
#1: -------
2 (z+0.5)
#2: -------------------
(z^2 - 0.2z + 1.01)
Sample time: unspecified
you can extract the zero/pole/gain data embedded in H with
z =
[ 0]
p =
[ 0.3000]
[2x1 double]
k =
To access the zeros and poles of the second output channel of H, get the content of the second cell in z and p by typing
ans =
0.1000+ 1.0000i
0.1000- 1.0000i
Example 2
Extract the ZPK matrices and their standard deviations for a 2-input, 1 output identified transfer function.
transfer function model
sys1 = tfest(z7, 2, 1, 'InputDelay',[1 0]);
an equivalent process model
sys2 = procest(z7, {'P2UZ', 'P2UZ'}, 'InputDelay',[1 0]);
[z1, p1, k1, ~, dz1, dp1, dk1] = zpkdata(sys1);
[z2, p2, k2, ~, dz2, dp2, dk2] = zpkdata(sys2);
Use iopzplot to visualize the pole-zero locations and their covariances
h = iopzplot(sys1, sys2);
Input Arguments
sys — Zero-pole-gain model
zpk model object
Zero-pole-gain model, specified as a zpk (Control System Toolbox) model object.
Output Arguments
z — Zeros
cell array (default) | column vector
Zeros of the zero-pole-gain model, returned as a cell array with as many rows as outputs and as many columns as inputs. The (i,j) entry z{i,j} is the (column) vector of zeros of the transfer function
from input j to output i.
p — Poles
cell array (default) | column vector
Poles of the zero-pole-gain model, returned as a cell array with as many rows as outputs and as many columns as inputs. The (i,j) entry p{i,j} is the (column) vector of zeros of the transfer function
from input j to output i.
k — Gain
Gain of the zero-pole-gain model, returned as a matrix with as many rows as outputs and as many columns as inputs such that k(i,j) is the gain of the transfer function from input j to output i. If
sys is a transfer function or state-space model, it is first converted to zero-pole-gain form using zpk.
Ts — Sample time
Sample time, specified as a scalar.
covz — Covariance of zeros
cell array
Covariance of zeros, returned as a cell array such that covz{ky,ku} contains the covariance information about the zeros in the vector z{ky,ku}. covz{ky,ku} is a 3-D array of dimension 2-by-2-by-Nz,
where Nz is the length of z{ky,ku}, so that the (1,1) element is the variance of the real part, the (2,2) element is the variance of the imaginary part, and the (1,2) and (2,1) elements contain the
covariance between the real and imaginary parts.
covp — Covariance of poles
cell array
Covariance of poles, returned as a cell array such that covp{ky,ku} contains the covariance information about the poles in the vector p{ky,ku}. covp{ky,ku} is a 3-D array of dimension 2-by-2-by-Np,
where Np is the length of p{ky,ku}, so that the (1,1) element is the variance of the real part, the (2,2) element is the variance of the imaginary part, and the (1,2) and (2,1) elements contain the
covariance between the real and imaginary parts.
covk — Covariance of gain
cell array
Covariance of zeros, returned as a cell array such that covk{ky,ku} contains the covariance information about the zeros in the vector k{ky,ku}. covk{ky,ku} is a 3-D array of dimension 2-by-2-by-Nk,
where Nk is the length of k{ky,ku}, so that the (1,1) element is the variance of the real part, the (2,2) element is the variance of the imaginary part, and the (1,2) and (2,1) elements contain the
covariance between the real and imaginary parts.
Version History
Introduced before R2006a | {"url":"https://kr.mathworks.com/help/ident/ref/dynamicsystem.zpkdata.html","timestamp":"2024-11-08T17:41:49Z","content_type":"text/html","content_length":"95512","record_id":"<urn:uuid:de94c5fe-a942-4a6c-a1ce-6f8125612f29>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00454.warc.gz"} |
Highly optimized 5-error-correcting BCH codes
Keywords: coding theory, root finding, decoding, fast algorithms
Prerequisites: 01405: Algebraic Coding Theory
In hardware implementations of BCH decoders, error location is bottleneck, i.e. finding the error-locator polynomial and afterwards, the roots of the error locator polynomial. In high-density optical
transmission, one would like to be able to correct up to 5 errors as fast as possible. In this project we will consider the following question:
How can we perform decoding for a 5-error-correcting BCH code as fast as possible?
A strategy to answer this question is the following: To speed up finding the error-locator polynomial, one can precompute a so-called generic error-locator polynomial, i.e. one can find an expression
containing the syndromes as additional variables, such that for each actual value of the syndromes one obtains the correct error-locator polynomial from the generic error-locator polynomial by
substituting the syndrome values.
Such expressions are too lengthy in general, but if the number of errors one needs to correct is modest this is more feasible.. To find the roots of the polynomial of degree 5, the strategy is to use
suitably chosen transformations to bring any given polynomial in a standard form. Tricks similar (but algebraically more interesting) to completing the square are generalized and used for this
purpose. Afterwards a strategy will be developed to solve the remaining standard cases, among others using table lookup for special cases. | {"url":"http://algebra.compute.dtu.dk/highly-optimized-5-error-correcting-bch-codes/","timestamp":"2024-11-08T21:01:23Z","content_type":"text/html","content_length":"25592","record_id":"<urn:uuid:64236cd0-1d9c-4914-adb1-76bff2e3beaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00856.warc.gz"} |
First-Principles Calculation of Physical Tensors of α-Diisopropylammonium Bromide (α-DIPAB) Molecular Ferroelectric Crystal
We report accurate calculations of tonsorial elements of α-Diisopropylammonium bromide (α-DIPAB) molecular ferroelectric crystal. In particular, elastic, piezoelectric and dielectric tensors were
computed using density functional theory (DFT)-based Vienna ab initio simulation package (VASP). The determination of above parameters allows an accurate description of the energy landscape for
modeling of realistic devices at finite temperatures. We determine the major physical tensors in energy expansion of total energy per volume of un-deformed crystal to provide experimentalists with
valuable information for designing and fabrication of pyroelectric detectors, capacitors, piezoelectric devices based on α-DIPAB. The spontaneous polarization P[s] was calculated using Berry phase
approach and found to be 22.64 μC/cm^2 in agreement with reported theoretical value. Furthermore, we calculate dynamical Born effective charge tensor to get a deeper insight into the bonding network
and lattice dynamic of α-DIPAB crystal. The neighboring layers of DIPA molecules were found to be strongly crenelated due to the strong short-ranged electrostatic repulsion between Br sites in the
DIPAB crystal structure. The organization of species in DIPA molecular layer as well as in the bromine “stitching” layer is essential for accurate calculation of DIPAB elastic properties. Having
understood the actual network bonding in α-DIPAB, we calculated the components of the elastic moduli tensor. Our results indicate that a Young's modulus of 50–150 GPa and a shear modulus of 4–26 GPa
were found. Thus, α-DIPAB phase has a great potential to be a terrific candidate for flexible electronic device applications. The value of the principle component of electronic contribution to the
static dielectric tensor of α-DIPAB is found to be ≈2.5, i.e., 50% smaller than that in typical perovskite-based ferroelectrics. Therefore, α-DIPAB is anticipated to exhibit creative materials'
innovations. It could be potential candidate as insulating layer of polymer thick films. Its mechanical, insulating and elastic properties make it eligible for switch keys and flex-circuit
applications. Furthermore, clamped-ion piezoelectric tensor is calculated. Our results indicate a reasonable piezoelectric response of this polar crystal making it a low cost attractive candidate for
piezoelectric applications.
• elastic and dielectric properties
• molecular ferroelectrics
• piezoelectric and dielectric tensors
• Vienna ab initio simulation package (VASP)
• α-diisopropylammonium bromide
Dive into the research topics of 'First-Principles Calculation of Physical Tensors of α-Diisopropylammonium Bromide (α-DIPAB) Molecular Ferroelectric Crystal'. Together they form a unique | {"url":"https://khazna.ku.ac.ae/en/publications/first-principles-calculation-of-physical-tensors-of-%CE%B1-diisopropyl","timestamp":"2024-11-09T07:59:48Z","content_type":"text/html","content_length":"63491","record_id":"<urn:uuid:0818efb1-b911-438f-9cff-097792df5a06>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00528.warc.gz"} |
What is the distance between M(7,−1,5) and the origin? | Socratic
What is the distance between #M(7,−1,5)# and the origin?
1 Answer
The answer is $5 \sqrt{3}$.
We simply use the Pythagorean Theorem for this calculation. We can draw a right triangle on the $x y$ plane and extend this for another right triangle to the $z$ plane. Let $d$ be the length of the
diagonal on the $x y$ plane. Then
$l = \sqrt{{d}^{2} + {z}^{2}}$
${d}^{2} = {x}^{2} + {y}^{2}$
$l = \sqrt{{x}^{2} + {y}^{2} + {z}^{2}}$
and substituting the values from your question:
$l = \sqrt{{7}^{2} + {\left(- 1\right)}^{2} + {5}^{2}} = \sqrt{49 + 1 + 25} = \sqrt{75} = 5 \sqrt{3}$
We can generalize the distance between any 2 - 3D points as:
$l = \sqrt{{\left({P}_{{2}_{x}} - {P}_{{1}_{x}}\right)}^{2} + {\left({P}_{{2}_{y}} - {P}_{{1}_{y}}\right)}^{2} + {\left({P}_{{2}_{z}} - {P}_{{1}_{z}}\right)}^{2}}$
The order of ${P}_{1}$ and ${P}_{2}$ doesn't matter because the difference is squared which always results in a positive value.
Impact of this question
1963 views around the world | {"url":"https://socratic.org/questions/what-is-the-distance-between-m-7-1-5-and-the-origin#107881","timestamp":"2024-11-08T05:53:34Z","content_type":"text/html","content_length":"34034","record_id":"<urn:uuid:4747b5a4-e41a-4c7e-bfbc-93ea10f90172>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00130.warc.gz"} |
Catalan Numbers
Now what exactly are these numbers? Here are the first few:
N Nth Catalan Number
For the general case, I could for example give you one of these definitions (where C(N) denotes the N-th Catalan number):
1. C(N) := choose(2N,N) / (n+1)
2. C(0) := 1, and C(N+1) := Sum(i=0 to n) C(i)*C(n-i)
3. C(N) := The number of binary trees that can be built with N nodes.
4. C(N) := The number of possible triangulations (with N triangles) of a convex polygon with N+2 sides.
It turns out that all these definitions are equivalent. But there are big differences in quality:
• 1. and 2. give a definition without any meaning. Without meaning, they're senseless. These might be helpful when we need to calculate the values, but they're bad definitions in a deeper sense.
• 3. and 4. are great, and the binary tree problem is probably the problem where most of us first saw the Catalan Numbers. Unfortunately, these don't come with mathematical formulas to calculate
the values. But it's fairly easy to prove that the number of binary trees is equivalent to the recursive 2. definition.
On this page, I will therefore use the 3. definition (the one with binary trees). | {"url":"https://algods.fandom.com/wiki/Catalan_Numbers","timestamp":"2024-11-06T02:17:17Z","content_type":"text/html","content_length":"147936","record_id":"<urn:uuid:ce3aff0a-3f45-4f93-afa6-18b905d2cf67>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00415.warc.gz"} |
Precision Agriculture (PA) uses technologies with the aim of increasing productivity and reducing the environmental impact by means of site-specific application of agricultural inputs. In order to
make it economically feasible, it is essential to improve the current methodologies as well as proposing new ones, in which data regarding productivity, soil, and compound indicators are used to
determine Management Areas (MAs). These units are heterogeneous areas within the same region. With these methodologies, data mining (DM) techniques and algorithms may be used. In order to integrate
DM techniques to PA, the aim of this study was to associate MAs created for soy productivity using the Fuzzy C-means algorithm by SDUM software over a 9.9-ha plot as the reference method. It was in
opposition to the grouping of 2, 3, and 4 clusters obtained by the K-means classification algorithms, with and without the Principal Component Analysis (PCA), and the EM algorithm using chemical and
physical data of the soil samples collected in the same area during the same period. The EM algorithm with PCA modeling had a superior performance than K-means based on hit rates. It is noteworthy
that the greater the number of analyzed MAs, the lower the percentage of hits, in agreement with the result shown by SDUM, which shows that two MAs compose the best configuration for this studied
algorithms; EM; KDD; K-means; Weka
Associating technology to agriculture has been increasingly relevant because of the need for increased productivity and profitability, less use of pesticides, and reducing the environmental impact on
several rural areas (MOLIN et al., 2015MOLIN, J. P.; AMARAL, L. R.; COLAÇO, A. Agricultura de Precisão. São Paulo: Editora Oficina de Textos, 2015.). This later approach is the basis of Precision
Agriculture (PA).
PA implementation and maintenance costs are generally a problem for smallholder farmers. Hence, the division of agricultural areas into smaller homogeneous units, known as Management Areas (MAs), is
considered as an alternative for the application of PA (DOERGE, 2000DOERGE, T. A. Management zone concepts. Site-Specific Management Guidelines, 2000.) since it allows the use of conventional
equipment as well as decreasing the number of soil analyses required to setting input recommendations.
MAs can be defined in several manners. JOHANNSEN et al. (2000)JOHANNSEN, C. J.; CARTER, P. J.; ERICKSON, B. J.; MORRIS, D. K.; WILLIS, P. R. A cornucopia of agricultural applications. Space Imaging,
Thornton, p.22-23, jan./feb., 2000. show an approach using remote sensing to obtain vegetation indexes and associate them to soil sampling grids. Other approaches consider the farmer sensitivity from
empirical knowledge, although the most disseminated method in the literature for the setting-up of MAs consists in grouping chemical and physical soil parameters as well as relief data, taken from
strategic georeferenced areas (MOLIN; FAULIN, 2013MOLIN, J. P.; FAULIN, G. C. Spatial and temporal variability of soil electrical conductivity related to soil moisture. Scientia Agricola, Piracicaba,
v.70, n.1, p.1-5, 2013.). Several farmers own a large volume of data related to their property, from which new information and standards can be retrieved to support the decision-making process (
CARVALHO; MILANI, 2013CARVALHO, D. R.; MILANI, C. S. Pós-Processamento em KDD. Revista de Engenharia e Tecnologia, Ponta Grossa, v.5, n.1, p.151-162, 2013.; LUNARDELLI et al., 2014LUNARDELLI, R. S.
A.; TONELLO, I. M. S.; MOLINA, L. G. A constituição da memória dos procedimentos em saúde no contexto do prontuário eletrônico do paciente. Informação & Informação, Londrina, v.19, n.3, p.107-124,
set./dez. 2014.).
RODRIGUES & CORÁ (2015)RODRIGUES, M.; CORÁ, J. Management Zones Using Fuzzy Clustering Based On Spatialtemporal Variability Of Soil And Corn Yield. Engenharia Agrícola, Jaboticabal, v.35, n.3,
p.470-483, 2015. reported a lack of consensus on how many MAs should be created for this to be feasible. To resolve this, the implementation of clusters may indicate how many MAs need to be created
based on statistical criteria thereof (ODEH et al., 1992ODEH, I. O. A.; CHITTLEBOROUGH, D. J.; MCBRATNEY, A. B. Soil pattern recognition with fuzzy-c-means: Application to classification and
soil-landform interrelationships. Soil Science Society of America Journal, Madison, v.56, n.2, p.505-516, 1992.).
To find out the number of clusters (in this case MAs) should be generated, it is considered that the lower the count of clusters, the easier it will be for farmers to apply inputs at variable rates
on their crops (PEDROSO, 2010PEDROSO, M.; TAYLOR, J.; TISSEYRE, B.; CHARNOMORDIC, B.; GUILLAUME, S. A segmentation algorithm for the delineation of agricultural management zones. Computers and
Eletronics in Agriculture, Netherlands, v.70, n.1, p.199-208, 2010.; BAZZI et al., 2013BAZZI, C.; SOUZA, E.; URIBE-OPAZO, M.; NÓBREGA, L.; ROCHA, D. Management Zones Definition Using Soil Chemical
And Physical Attributes In A Soybean Area. Engenharia Agrícola, Jaboticabal, v.34, n.5, p.952-964, 2013.). When using empirical methods, the ideal number is estimated to be between three or four MAs
(SUSZEK, et al., 2012SUSZEK, G.; SOUZA, E. G.; URIBE-OPAZO, M. A.; NÓBREGA, L. H. P. Determination of management zones from normalized and standardized equivalent productivity maps in the soybean
culture. Engenharia Agrícola, Jaboticabal, v.32, n.5, p.895-905, 2012.) without disregarding the use of two MAs.
Computational tools stand out among the technologies associated with PA, within which data mining (DM) is inserted. DM is the process of discovering new information (whether known or unknown) from
large volumes of data (TAN et al, 2012TAN, P.; STEINBACH, M.; KUMAR, V. Introdução ao datamining: mineração de dados, Rio de Janeiro: Editora Ciência Moderna, 2012.). FAYYAD (1996)FAYYAD, U.;
PIATETSKY-SHAPIRO, G.; SMYTH, P. From Data Mining to Knowledge Discovery in Databases. Palo Alto: American Association for Artificial Intelligence, 1996. presented a definition from the perspective
of machine learning, stating that DM is a step towards the knowledge of the discovery process, which consists of analyzing data and applying discovery algorithms which produce a set of data
standards. These stages constitute the KDD process (Knowledge Discovery in Databases).
The KDD process is a non-trivial, interactive, and iterative method of identifying comprehensible, valid, new, and potentially useful standards from large data sets. Moreover, it is constituted by
selection, process, transformation, DM, and assessment stages (FAYYAD, 1996FAYYAD, U.; PIATETSKY-SHAPIRO, G.; SMYTH, P. From Data Mining to Knowledge Discovery in Databases. Palo Alto: American
Association for Artificial Intelligence, 1996.), involving a series of areas related in a multidisciplinary manner (CARVALHO & DALLAGASSA, 2014CARVALHO, D. R.; DALLAGASSA, M. R. Mineração de dados:
aplicações, ferramentas, tipos de aprendizado e outros subtemas. AtoZ: novas práticas em informação e conhecimento, Curitiba, v.3, n.2, p.82-86, 2014.).
Among the algorithms associated with DM are those for data grouping. Therefore, data is partitioned into homogeneous clusters, maximizing the similarity of objects within the same cluster, thus
minimizing that of objects of different clusters (TAN et al, 2012TAN, P.; STEINBACH, M.; KUMAR, V. Introdução ao datamining: mineração de dados, Rio de Janeiro: Editora Ciência Moderna, 2012.). The
algorithms may be standardized to create clusters from 2 to n (n is the total number of data).
These algorithms use similarity measures to generate clusters for the decision-making process, determining the data distribution on the respective groups. There are several measures to calculate
similarity, including distance, correlation, and association (TAN et al., 2012TAN, P.; STEINBACH, M.; KUMAR, V. Introdução ao datamining: mineração de dados, Rio de Janeiro: Editora Ciência Moderna,
2012.). When the data set is constituted by quantitative attributes, distance metrics may be applied to calculate the similarity among the data (METZ & MORNARD, 2006METZ, J.; MONARD, M. C. Projeto e
implementação do módulo de clustering hierárquico do discover. São Carlos: ICMC, USP, 2006. Tech. Rep.).
Using such DM techniques and inherent concepts from PA, the needs (excess or lack of nutrients) on the georeferenced collection point could be detected. Each point is categorized into a cluster,
precisely orienting the actions for each sampled point to be normalized according to recommendations from relevant entities.
Although several computational tools offer assistance with DM algorithms, all the results generated must be analyzed and interpreted by specialists in the area, creating a useful knowledge (TAN et
al., 2012TAN, P.; STEINBACH, M.; KUMAR, V. Introdução ao datamining: mineração de dados, Rio de Janeiro: Editora Ciência Moderna, 2012.). It is important to highlight that DM algorithms are
independent of cropped area or of the number of collected samples for analysis, which confirms their potential in PA.
The overall aim of this study was to use data mining techniques to assess management areas in Precision Agriculture. Thus, the generated groupings with EM (Expectation–maximization) and K-means
algorithms (with Euclidean Distance, with and without Principal Component Analysis) were overlapped with soil physical and chemical data from one farming area split into two, three, and four MAs,
with the Fuzzy C-means algorithm by SDUM Software with productivity data from this area.
Study Area and Parameters Used
Soil chemical and physical data were gathered from a 9.9-ha plot of a rural property located in the Serranópolis do Iguaçu, Paraná state - Brazil (Figure 1).
Data were gathered in 2014 from 42 georeferenced samples (points highlighted in Figure 1) which compose the database panel. The records consisted of altitude; contents of sand, clay, silt and organic
matter; pH, being the soil classified as type 3 according to the normative instruction number 2 (October 9, 2008, Ministry of Agriculture, Livestock and Supply); arithmetic average of the
standardized soybean yield for the years of 2012, 2013 and 2014; soil resistance to penetration measured by an automatic soil compaction meter (SoloTrack - PLG5200) for the depth layers of 0 to 10
cm, 10 to 20 cm and 20 to 30 cm.
Computing Tools
Software Weka^1 1 http://www.cs.waikato.ac.nz/ml/weka/ , Surfer^2 2 http://www.goldensoftware.com/products/surfer , and SDUM^3 3 http://200.201.88.199/portalpos/index.php/livros were used.
Weka was used for data pre-processing, cleaning, and normalization, as well as for conducting the K-means and EM algorithms, assisting in the constitution of the Principal Component Analysis (PCA) to
re-implement K-means.
Surfer was used to interpolating the productivity data as well as generating the contour maps and showing the clusters, which are the visual forms of MAs.
SDUM was used to execute the Fuzzy C-means algorithm on average productivity data from 2012, 2013, and 2014, considering these results as the reference method for data overlapping and visual
comparison with data grouped on Weka, both for two, three, and four MAs. In addition, this tool highlights the best configuration of management areas further generated, facilitating thus a
comparative analysis.
Pre-Processing of Data
During this phase, data was prepared to reduce discrepancies and possible inconsistencies introduced by failures or noises. At this time, the data were modified according to proper formats suitable
for DM by aggregation, generalization, normalization, building and selection of attributes or even data reduction (TAN et al, 2012TAN, P.; STEINBACH, M.; KUMAR, V. Introdução ao datamining: mineração
de dados, Rio de Janeiro: Editora Ciência Moderna, 2012.). The collected data showed no missing values, and outliers and inconsistencies were not found.
The data were normalized to a single scale, between zero and one, achieving parametric uniformity for algorithms, which must be conducted to establish a data standard, excluding the hypothesis of
algorithm influence on data. It has no effect on data representation in the field since the relationship between them still has the same original ratio; however, they are suitable for the application
of necessary algorithms in the Weka software.
Neither dimensionality nor data discretization had to be made since the number and format were already adequate to execute the algorithms.
Data Processing
Principal Component Analysis (PCA) was carried out in order to indicate possible exclusions of attributes in the database, with the aim of attaining better results of the grouping algorithms.
These data were subjected to the K-means, Fuzzy C-means, and EM algorithms with the purpose of identifying possible clusters. Since the groups to which each object belongs are unknown beforehand,
non-supervised learning techniques were used.
Euclidean Distance was performed for all executions of the K-means and Fuzzy C-means algorithms, and the variations of two, three, and four clusters for all algorithms. Algorithm EM has no distance
parameter for execution since it is based on probabilistic models.
The inverse squared distance was used to interpolate data, being applied to the pre-established clusters by the grouping algorithms, showing the generated MAs.
After normalizing the data, processing was carried out in six steps, as follows:
1. Execution of the K-means algorithm on the data set for two, three, and four clusters;
2. Principal Component Analysis following criteria described by Jolliffe (JOLLIFFE, 1972JOLLIFE, I. T. Discarding variables in a principal component analysis. Journal of Applied Statistics,
Abingdon, v.21, p.160-173, 1972.).
3. Execution of the K-means algorithm on the data set after PCA conversion, removing attributes suggested in step 2 for two, three, and four clusters;
4. Execution of the EM algorithm on the data set after PCA conversion, removing attributes suggested in step 2 for two, three, and four clusters;
5. Creation of dot maps containing the generated clusters;
6. Execution of the Fuzzy C-means algorithm on the same data used for the other algorithms, by means of SDUM software.
Steps 1 to 5 were carried out with the support from Weka tool and Surfer was used for georeferencing of clusters.
Step 6 consisted of executing the results assumed as the reference method, in addition to performing the Fuzzy C-means algorithm using the same data from the other algorithms. These analyses were
performed through SDUM software, which implements interpolator and applies ANOVA at a 5% significance comparing the average productivity of each area and, thereby, identifying which of management
area created is the best, among two, three or four clusters.
The first sequence of the K-means algorithm executions was conducted only with normalized data without PCA. Figure 2 illustrates the generated groupings for two, three, and four clusters,
respectively in (a), (b) and (c).
FIGURE 2
Groupings generated by the K-means algorithm - (a) 2 clusters, (b) 3 clusters and (c) 4 clusters.
The classification error rates without conducting PCA were 16%, 14%, and 13% for groupings in two, three, and four clusters, respectively (Table 1). It was observed that the points were categorized
into mixed clusters since the geographical location of each point was not considered. Hence, each point was exclusively classified according to the result of the algorithm for input parameters.
TABLE 1
K-means - Points per cluster and error rate.
By applying the PCA technique and the criteria by JOLLIFFE (1972)JOLLIFE, I. T. Discarding variables in a principal component analysis. Journal of Applied Statistics, Abingdon, v.21, p.160-173, 1972.
, which suggested the exclusion of two attributes (organic matter and sand), the K-means algorithm was newly executed to generate two, three, and four clusters, shown respectively in (a), (b) and
(c), in Figure 3.
FIGURE 3
Groupings generated by the K-means algorithm + PCA – (a) 2 clusters, (b) 3 clusters and (c) 4 clusters.
For the K-means algorithm execution with PCA, the data classification errors were reduced by 5%, on average, for the three cluster options. Therefore, the excluded attributes by PCA and JOLLIFFE
(1972)JOLLIFE, I. T. Discarding variables in a principal component analysis. Journal of Applied Statistics, Abingdon, v.21, p.160-173, 1972. criteria positively contributed to improving the generated
clusters. The distribution of points was homogeneous among the clusters, except for the case of four clusters, in which the number of points (2 points) was well below the other clusters (Table 2).
TABLE 2
K-means + PCA – Points per cluster and error rate.
The results of EM algorithm run after applying PCA with the criteria by JOLLIFFE (1972)JOLLIFE, I. T. Discarding variables in a principal component analysis. Journal of Applied Statistics, Abingdon,
v.21, p.160-173, 1972. for two, three, and four clusters are shown in (a), (b) and (c) of Figure 4, respectively. If compared to the algorithm K-means, EM algorithm results had greater differences
for MAs with 3 (Figures 2b and 3b) and 4 (Figures 2c and 3c) clusters. This fact is which is justified by the heuristics used by each grouping algorithm (K-means and EM) to generate the clusters,
which had already been observed by Johann et al. (2013)JOHANN, J. A.; ROCHA, J. V.; OLIVEIRA, S. R. M.; RODRIGUES, L. H. A.; LAMPARELLI, R. A. C. Data mining techniques for identification of
spectrally homogeneous areas using NDVI temporal profiles of soybean crop. Engenharia Agrícola, Jaboticabal, v.33, n.3, p.511-524, 2013.. Although Weka does not offer the error rate of the groupings
conducted with EM, it enables visualizing the distribution of the points among the clusters (Table 3).
FIGURE 4
Groupings generated by the EM + PCA algorithm – (a) 2 clusters, (b) 3 clusters and (c) 4 clusters.
TABLE 3
Error Rate of points by cluster for the EM + PCA algorithm.
In SDUM, the interpolation by the Inverse Square Distance and generation of the maps for two, three, and four MAs was conducted using the Fuzzy C-means algorithm, and it was defined as the reference
method for this study.
Once the reference method was obtained, it was compared to the groupings obtained by algorithms K-means and EM for the soil chemical and physical data. Such comparison was performed by overlapping
the georeferenced maps as shown in Figure 5, in which (a) shows the K-means algorithm for two, three, and four MAs, (b) presents the clusters by K-means with PCA, whereas in (c) are the clusters by
EM algorithm with PCA.
FIGURE 5
Overlap of groupings for 2, 3, and 4 clusters and the map of MAs where (2a), (2b) and (2c) K-means, (3a), (3b) and (3c) PCA + K-means and (4a), (4b) and (4c) PCA + EM.
Based on this overlap (Figure 5), the hits may be quantified for the clusters generated by each grouping algorithm (Table 4) when overlapped with the maps of the reference model.
TABLE 4
Hit rate of K-means and EM groupings on the Reference Method.
EM algorithm with PCA stood out across all executions if compared to the performance of K-means, and it had the highest hit rate (Table 4). The greater the number of analyzed MAs, the lower the
percentage of hits across all executions, confirming the evaluation created by SDUM through ANOVA, which was conducted with productivity data before executing DM algorithms, which shows the division
into two MAs as the most appropriate one for this study area.
The K-means algorithm also had a lower performance than Algorithm EM if compared to the reference method, and its highest hit rate was 38% (K-means with PCA for three MAs).
It was also observed that the application of PCA increased the hit margin with algorithm K-means in 17% at its best performance (Table 4), indicating that, excluding the sand and organic matter
attributes, in fact, improved the generation of the clusters.
After generating the MAs with the Fuzzy C-means algorithm for two, three, and four MAs, they were evaluated with the assistance of the SDUM software, in which ANOVA indicated homogeneity between MAs
with 3 and 4 clusters for the analyzed data. When considering the Tukey's test, two MAs made up the best configuration for the study area, being the only non-homogeneous set across the generated
classes, indicating that, in fact, there are two different MAs. Therefore, with 5% of significance, the generated MAs with three and four clusters showed at least one pair of classes considered
homogeneous in relation to each other.
Using Data Mining techniques to evaluate Management Areas has proven to be a method providing innovative and conclusive results. In addition, the Principal Component Analysis technique maximized the
generation of clusters, showing an increase in hit rate across all executions conducted with the K-means algorithm.
Once a point-point distance analysis was not carried out in EM algorithm structuring, there was an unbalanced distribution of the MAs since in all executions, at least one of the MAs had a number of
points near or superior to 50%.
The soil attributes sand and organic matter had no direct influence on the creation of MAs in the study area when considering average productivity, used by the reference method.
The grouping obtained with the EM algorithm and the application of PCA for two MAs were more assertive, while the K-means algorithm without PCA for four MAs had the worst overlapping if compared to
the reference method from SDUM.
• 1
Article developed in the discipline “Data Mining and Knowledge Discovery” in the postgraduate program in Agricultural Engineering – PGEAGRI, Western Paraná State University - Cascavel, in 2015.
• 1
• 2
• 3
• BAZZI, C.; SOUZA, E.; URIBE-OPAZO, M.; NÓBREGA, L.; ROCHA, D. Management Zones Definition Using Soil Chemical And Physical Attributes In A Soybean Area. Engenharia Agrícola, Jaboticabal, v.34,
n.5, p.952-964, 2013.
• CARVALHO, D. R.; MILANI, C. S. Pós-Processamento em KDD. Revista de Engenharia e Tecnologia, Ponta Grossa, v.5, n.1, p.151-162, 2013.
• CARVALHO, D. R.; DALLAGASSA, M. R. Mineração de dados: aplicações, ferramentas, tipos de aprendizado e outros subtemas. AtoZ: novas práticas em informação e conhecimento, Curitiba, v.3, n.2,
p.82-86, 2014.
• DOERGE, T. A. Management zone concepts Site-Specific Management Guidelines, 2000.
• FAYYAD, U.; PIATETSKY-SHAPIRO, G.; SMYTH, P. From Data Mining to Knowledge Discovery in Databases. Palo Alto: American Association for Artificial Intelligence, 1996.
• JOHANN, J. A.; ROCHA, J. V.; OLIVEIRA, S. R. M.; RODRIGUES, L. H. A.; LAMPARELLI, R. A. C. Data mining techniques for identification of spectrally homogeneous areas using NDVI temporal profiles
of soybean crop. Engenharia Agrícola, Jaboticabal, v.33, n.3, p.511-524, 2013.
• JOHANNSEN, C. J.; CARTER, P. J.; ERICKSON, B. J.; MORRIS, D. K.; WILLIS, P. R. A cornucopia of agricultural applications. Space Imaging, Thornton, p.22-23, jan./feb., 2000.
• JOLLIFE, I. T. Discarding variables in a principal component analysis Journal of Applied Statistics, Abingdon, v.21, p.160-173, 1972.
• LUNARDELLI, R. S. A.; TONELLO, I. M. S.; MOLINA, L. G. A constituição da memória dos procedimentos em saúde no contexto do prontuário eletrônico do paciente. Informação & Informação, Londrina,
v.19, n.3, p.107-124, set./dez. 2014.
• METZ, J.; MONARD, M. C. Projeto e implementação do módulo de clustering hierárquico do discover. São Carlos: ICMC, USP, 2006. Tech. Rep.
• MOLIN, J. P.; AMARAL, L. R.; COLAÇO, A. Agricultura de Precisão São Paulo: Editora Oficina de Textos, 2015.
• MOLIN, J. P.; FAULIN, G. C. Spatial and temporal variability of soil electrical conductivity related to soil moisture. Scientia Agricola, Piracicaba, v.70, n.1, p.1-5, 2013.
• ODEH, I. O. A.; CHITTLEBOROUGH, D. J.; MCBRATNEY, A. B. Soil pattern recognition with fuzzy-c-means: Application to classification and soil-landform interrelationships. Soil Science Society of
America Journal, Madison, v.56, n.2, p.505-516, 1992.
• PEDROSO, M.; TAYLOR, J.; TISSEYRE, B.; CHARNOMORDIC, B.; GUILLAUME, S. A segmentation algorithm for the delineation of agricultural management zones. Computers and Eletronics in Agriculture,
Netherlands, v.70, n.1, p.199-208, 2010.
• RODRIGUES, M.; CORÁ, J. Management Zones Using Fuzzy Clustering Based On Spatialtemporal Variability Of Soil And Corn Yield. Engenharia Agrícola, Jaboticabal, v.35, n.3, p.470-483, 2015.
• SUSZEK, G.; SOUZA, E. G.; URIBE-OPAZO, M. A.; NÓBREGA, L. H. P. Determination of management zones from normalized and standardized equivalent productivity maps in the soybean culture. Engenharia
Agrícola, Jaboticabal, v.32, n.5, p.895-905, 2012.
• TAN, P.; STEINBACH, M.; KUMAR, V. Introdução ao datamining: mineração de dados, Rio de Janeiro: Editora Ciência Moderna, 2012.
Publication Dates
• Publication in this collection
Jan-Feb 2017
• Received
28 Apr 2016
• Accepted
16 Aug 2016 | {"url":"https://www.scielo.br/j/eagri/a/b7KF435xzDnxVWMnC7YtQ8F/?lang=en","timestamp":"2024-11-04T15:18:56Z","content_type":"text/html","content_length":"120984","record_id":"<urn:uuid:8e47c03d-f236-43f9-bd89-67e500e006a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00343.warc.gz"} |
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Quant thread for CAT 2011
Member • Jul 18, 2011
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Last change on this file since 5599 was 5599, checked in by sexton, 16 years ago
(1) update to validation paper - references and loose text and (2) update to urs2sts testing
File size: 27.1 KB
1 %Anuga validation publication
2 %
3 %Geoscience Australia and others 2007-2008
5 % Use the Elsevier LaTeX document class
6 %\documentclass{elsart3p} % Two column
7 %\documentclass{elsart1p} % One column
8 %\documentclass[draft]{elsart} % Basic
9 \documentclass{elsart} % Basic
11 % Useful packages
12 \usepackage{graphicx} % avoid epsfig or earlier such packages
13 \usepackage{url} % for URLs and DOIs
14 \usepackage{amsmath} % many want amsmath extensions
15 \usepackage{amsfonts}
16 \usepackage{underscore}
17 \usepackage{natbib} % Suggested by the Elsevier style
18 % Use \citep and \citet instead of \cite
21 % Local LaTeX commands
22 %\newcommand{\Python}{\textsc{Python}}
23 %\newcommand{\VPython}{\textsc{VPython}}
24 \newcommand{\pypar}{\textsc{mpi}}
25 \newcommand{\Metis}{\textsc{Metis}}
26 \newcommand{\mpi}{\textsc{mpi}}
28 \newcommand{\UU}{\mathbf{U}}
29 \newcommand{\VV}{\mathbf{V}}
30 \newcommand{\EE}{\mathbf{E}}
31 \newcommand{\GG}{\mathbf{G}}
32 \newcommand{\FF}{\mathbf{F}}
33 \newcommand{\HH}{\mathbf{H}}
34 \newcommand{\SSS}{\mathbf{S}}
35 \newcommand{\nn}{\mathbf{n}}
37 \newcommand{\code}[1]{\texttt{#1}}
42 \begin{document}
45 \begin{frontmatter}
46 \title{On The Validation of A Hydrodynamic Model}
49 \author[GA]{D.~S.~Gray}
50 \ead{Duncan.Gray@ga.gov.au}
51 \author[GA]{O.~M.~Nielsen}
52 \ead{Ole.Nielsen@ga.gov.au}
53 \author[GA]{M.~J.~Sexton}
54 \ead{Jane.Sexton@ga.gov.au}
55 \author[GA]{L.~Fountain}
56 \author[GA]{K.~VanPutten}
57 \author[ANU]{S.~G.~Roberts}
58 \ead{Stephen.Roberts@anu.edu.au}
59 \author[UQ]{T.~Baldock}
60 \ead{Tom.Baldock@uq.edu.au}
61 \author[UQ]{M.~Barnes}
62 \ead{Matthew.Barnes@uq.edu.au}
64 \address[GA]{Georisk Project,
65 Geospatial and Earh Monitoring Division,
66 Geoscience Australia, Canberra, Australia}
68 \address[ANU]{Department of Mathematics,
69 Australian National University, Canberra, Australia}
71 \address[UQ]{University of Queensland, Brisbane, Australia}
74 % Use the \verb|abstract| environment.
75 \begin{abstract}
76 Modelling the effects on the built environment of natural hazards such
77 as riverine flooding, storm surges and tsunami is critical for
78 understanding their economic and social impact on our urban
79 communities. Geoscience Australia and the Australian National
80 University have developed a hydrodynamic inundation modelling tool
81 called ANUGA to help simulate the impact of these hazards.
82 The core of ANUGA is a Python implementation of a finite-volume method
83 for solving the conservative form of the Shallow Water Wave equation.
85 In this paper, a number of tests are performed to validate ANUGA. These tests
86 range from benchmark problems to wave and flume tank examples.
87 ANUGA is available as Open Source to enable
88 free access to the software and allow the scientific community to
89 use, validate and contribute to the software in the future.
91 %This method allows the study area to be represented by an unstructured
92 %mesh with variable resolution to suit the particular problem. The
93 %conserved quantities are water level (stage) and horizontal momentum.
94 %An important capability of ANUGA is that it can robustly model the
95 %process of wetting and drying as water enters and leaves an area. This
96 %means that it is suitable for simulating water flow onto a beach or
97 %dry land and around structures such as buildings.
99 \end{abstract}
102 \begin{keyword}
103 % keywords here, in the form: keyword \sep keyword
104 % PACS codes here, in the form: \PACS code \sep code
106 Hydrodynamic Modelling \sep Model validation \sep
107 Finite-volumes \sep Shallow water wave equation
109 \end{keyword}
111 \date{\today()}
112 \end{frontmatter}
117 % Begin document in earnest
118 \section{Introduction}
119 \label{sec:intro}
121 Hydrodynamic modelling allows impacts from flooding, storm-surge and
122 tsunami to be better understood, their impacts to be anticipated and,
123 with appropriate planning, their effects to be mitigated. A significant
124 proportion of the Australian population reside in the coastal
125 corridors, thus the potential of significant disruption and loss
126 is real. The extent of
127 inundation is critically linked to the event, tidal conditions,
128 bathymetry and topography and it not feasible to make impact
129 predictions using heuristics alone.
130 Geoscience
131 Australia in collaboration with the Mathematical Sciences Institute,
132 Australian National University, is developing a software application
133 called ANUGA to model the hydrodynamics of floods, storm surges and
134 tsunami. These hazards are modelled using the conservative shallow
135 water equations which are described in section~\ref{sec:model}. In
136 ANUGA these equations are solved using a finite volume method as
137 described in section~\ref{sec:model}. A more complete discussion of the
138 method can be found in \citet{Nielsen2005} where the model and solution
139 technique is validated on a standard tsunami benchmark data set
140 or in \citet{Roberts2007} where the numerical method and parallelisation
141 of ANUGA is discussed.
142 This modelling capability is part of
143 Geoscience Australia's ongoing research effort to model and
144 understand the potential impact from natural hazards in order to
145 reduce their impact on Australian communities \citep{Nielsen2006}.
146 ANUGA is currently being trialled for flood
147 modelling \citep{Rigby2008}.
149 The validity of other hydrodynamic models have been reported
150 elsewhere, with \citet{Hubbard02} providing an
151 excellent review of 1D and 2D models and associated validation
152 tests. They described the evolution of these models from fixed, nested
153 to adaptive grids and the ability of the solvers to cope with the
154 moving shoreline. They highlighted the difficulty in verifying the
155 nonlinear shallow water equations themselves as the only standard
156 analytical solution is that of \citet{Carrier58} that is strictly for
157 non-breaking waves. Further,
158 whilst there is a 2D analytic solution from \citet{Thacker81}, it appears
159 that the circular island wave tank example of Briggs et al will become
160 the standard data set to verify the equations.
162 This paper will describe the validation outputs in a similar way to
163 \citet{Hubbard02} to
164 present an exhaustive validation of the numerical model.
165 Further to these tests, we will
166 incorporate a test to verify friction values. The tests reported in
167 this paper are:
168 \begin{itemize}
169 \item Verification against the 1D analytical solution of Carrier and
170 Greenspan (p~\pageref{sec:carrier})
171 \item Testing against 1D (flume) data sets to verify wave height and
172 velocity (p~\pageref{sec:stage and velocity})
173 \item Determining friction values from 1D flume data sets
174 (p~\pageref{sec:friction})
175 \item Validation against a genuinely 2D analytical
176 solution of the model equations (p~\ref{sec:XXX})
177 \item Testing against the 2D Okushiri benchmark problem
178 (p~\pageref{sec:okushiri})
179 \item Testing against the 2D data sets modelling wave run-up around a circular island by Briggs et al.
180 (p~\pageref{sec:circular island})
181 \end{itemize}
184 Throughout the paper, qualitative comparisons will be drawn against
185 other models. Moreover, all source code necessary to reproduce the
186 results reported in this paper is available as part of the ANUGA
187 distribution in the form of a test suite. It is thus possible for
188 anyone to readily verify that the implementation meets the
189 requirements set out by these benchmarks.
192 %Hubbard and Dodd's model, OTT-2D, has some similarities to ANUGA, and
193 %whilst the mesh can be refined, it is based on rectangular mesh.
195 %The ANUGA model and numerical scheme is briefly described in
196 %section~\ref{sec:model}. A more detailed description of the numerical
197 %scheme and software implementation can be found in \citet{Nielsen2005} and
198 %\citet{Roberts2007}.
199 The six case studies to validation and verify ANUGA
200 will be presented in section~\ref{sec:validation}, with the
201 conclusions outlined in section~\ref{sec:conclusions}.
203 NOTE: This is just a brain dump at the moment and needs to be incorporated properly
204 in the text somewhere.
206 Need some discussion on Bousssinesq type models - Boussinesq equations get the
207 nonlinearity and dispersive effects to a high degree of accuracy
209 moving wet-dry boundary algorithms - applicability to coastal engineering
211 Fuhrman and Madesn 2008 \cite{Fuhrman2008}do validation - they have a Boussinesq type
212 model, finite
213 difference (therefore needing a supercomputer), 4th order, four stage RK time stepping
214 scheme.
216 their tests are (1) nonlinear run-up on periodic and transient waves on a sloping
217 beach with excellent comparison to analytic solutions (2) 2d parabolic basin
218 (3) solitary wave evolution through 2d triangular channel (4) solitary wave evolution on
219 conical island (we need to compare to their computation time and note they use a
220 vertical exaggeration for their images)
222 excellent accuracy mentioned - but what is it - what does it mean?
224 of interest is that they mention mass conservation and calculate it throughout the simulations
226 Kim et al \cite{DaiHong2007} use Riemann solver - talk about improved accuracy by using 2nd order upwind
227 scheme. Use finite volume on a structured mesh. Do parabolic basic and circular island. Needed?
229 Delis et all 2008 \cite{Delis2008}- finite volume, Godunov-type explicit scheme coupled with Roe's
230 approximate Riemann solver. It accurately describes breaking waves as bores or hydraulic jumps
231 and conserves volume across flow discontinuties - is this just a result of finite volume?
233 They also show mass conservation for most of the simulations
235 similar range of validation tests that compare well - our job to compare to these as well
237 \section{Mathematical model, numerical scheme and implementation}
238 \label{sec:model}
240 The ANUGA model is based on the shallow water wave equations which are
241 widely regarded as suitable for modelling 2D flows subject to the
242 assumptions that horizontal scales (e.g. wave lengths) greatly exceed
243 the depth, vertical velocities are negligible and the fluid is treated
244 as inviscid and incompressible. See e.g. the classical texts
245 \citet{Stoker57} and \citet{Peregrine67} for the background or
246 \citet{Roberts1999} for more details on the mathematical model
247 used by ANUGA.
249 The conservation form of the shallow water wave
250 equations used in ANUGA are:
251 \[
252 \frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial
253 x}+\frac{\partial \GG}{\partial y}=\SSS
254 \]
255 where $\UU=\left[ {{\begin{array}{*{20}c}
256 h & {uh} & {vh} \\
257 \end{array} }} \right]^T$ is the vector of conserved quantities; water depth
258 $h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities
259 entering the system are bed elevation $z$ and stage (absolute water
260 level above a reference datum such as Mean Sea Level) $w$,
261 where the relation $w = z + h$ holds true at all times.
262 The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given
263 by
264 \[
265 \EE=\left[ {{\begin{array}{*{20}c}
266 {uh} \hfill \\
267 {u^2h+gh^2/2} \hfill \\
268 {uvh} \hfill \\
269 \end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c}
270 {vh} \hfill \\
271 {vuh} \hfill \\
272 {v^2h+gh^2/2} \hfill \\
273 \end{array} }} \right]
274 \]
275 and the source term (which includes gravity and friction) is given
276 by
277 \[
278 \SSS=\left[ {{\begin{array}{*{20}c}
279 0 \hfill \\
280 -{gh(z_{x} + S_{fx} )} \hfill \\
281 -{gh(z_{y} + S_{fy} )} \hfill \\
282 \end{array} }} \right]
283 \]
284 where $S_f$ is the bed friction. The friction term is modelled using
285 Manning's resistance law
286 \[
287 S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy}
288 =\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}
289 \]
290 in which $\eta$ is the Manning resistance coefficient.
292 %%As demonstrated in our papers, \cite{modsim2005,Roberts1999} these
293 %%equations provide an excellent model of flows associated with
294 %%inundation such as dam breaks and tsunamis. Question - how do we
295 %%know it is excellent?
297 ANUGA uses a finite-volume method as
298 described in \citet{Roberts2007} where the study area is represented by an
299 unstructured triangular mesh in which the vector of conserved quantities
300 $\UU$ is maintained and updated over time. The flexibility afforded by
301 allowing unstructed meshes rather than fixed resolution grids
302 is the ability for the user to refine the mesh in areas of interest
303 while leaving other areas coarse and thereby conserving computational
304 resources.
307 The approach used in ANUGA are distinguished from many
308 other implementations (e.g. \citet{Hubbard02} or \citet{Zhang07}) by the
309 following features:
310 \begin{itemize}
311 \item The fluxes across each edge are computed using the semi-discrete
312 central-upwind scheme for approximating the Riemann problem
313 proposed by \citet{KurNP2001}. This scheme deals with different
314 flow regimes such as shocks, rarefactions and sub to super
315 critical flow transitions using one general approach. We have
316 found this scheme to be pleasingly simple, robust and efficient.
317 \item ANUGA does not employ a shoreline detection algorithm as the
318 central-upwind scheme is capable of resolving fluxes arising between
319 wet and dry cells. ANUGA does optionally bypass unnecessary
320 computations for dry-dry cell boundaries purely to improve performance.
321 \item ANUGA employs a second order spatial reconstruction of triangles
322 to produce a piece-wise linear function construction of the conserved
323 quantities. This function is allowed to be discontinuous across the
324 edges of the cells, but the slope of this function is limited to avoid
325 artificially introduced oscillations. This approach provides good
326 approximation of steep gradients in the solution. However,
327 where the depths are very small compared to the bed-slope a linear
328 combination between second order and first order reconstructions is
329 employed to guarantee numerical stability that may arise form very
330 small depths.
331 \end{itemize}
333 In the computations presented in this paper we use an explicit Euler
334 time stepping method with variable timestepping subject to the
335 CFL condition:
336 \[
337 \delta t = \min_k \frac{r_k}{v_k}
338 \]
339 where $r_k$ refers to the radius of the inscribed circle of triangle
340 $k$, $v_k$ refers to the maximal velocity calculated from fluxes
341 passing in or out of triangle $k$ and $\delta t$ is the resulting
342 'safe' timestep to be used for the next iteration.
345 ANUGA utilises a general velocity limiter described in the
346 manual which guarantees a gradual compression of computed velocities
347 in the presence of very shallow depths:
348 \begin{equation}
349 \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h},
350 \end{equation}
351 where $h_0$ is a regularisation parameter that controls the minimal
352 magnitude of the denominator. The default value is $h_0 = 10^{-6}$.
355 ANUGA is mostly written in the object-oriented programming
356 language Python with computationally intensive parts implemented
357 as highly optimised shared objects written in C.
359 Python is known for its clarity, elegance, efficiency and
360 reliability. Complex software can be built in Python without undue
361 distractions arising from idiosyncrasies of the underlying software
362 language syntax. In addition, Python's automatic memory management,
363 dynamic typing, object model and vast number of libraries means that
364 ANUGA scripts can be produced quickly and can be adapted fairly easily to
365 changing requirements.
369 \section{Validation}
370 \label{sec:validation} Validation is an ongoing process and the purpose of this paper
371 is to describe a range of tests that validate ANUGA as a hydrodynamic model.
372 This section will describe the six tests outlined in section~\ref{sec:intro}.
373 Run times where specified measure the model time only and exclude model setup,
374 data conversions etc. All examples were timed on a a 2GHz 64-bit
375 Dual-Core AMD Opteron(tm) series 2212 Linux server. %This is a tornado compute node (cat /proc/cpuinfo).
378 \subsection{1D analytical validation}
380 Tom Baldock has done something here for that NSW report
382 \subsection{Stage and Velocity Validation in a Flume}
383 \label{sec:stage and velocity}
384 This section will describe tilting flume tank experiments that were
385 conducted at the Gordon McKay Hydraulics Laboratory at the University of
386 Queensland that confirm ANUGA's ability to estimate wave height
387 and velocity. The same flume tank simulations were also used
388 to explore Manning's friction and this will be described in the next section.
390 The flume was set up for dam-break experiments, having a
391 water reservior at one end. The flume was glass-sided, 3m long, 0.4m
392 in wide, and 0.4m deep, with a PVC bottom. The reservoir in the flume
393 was 0.75m long. For this experiment the reservoir water was 0.2m
394 deep. At time zero the reservoir gate is manually opened and the water flows
395 into the other side of the flume. The water ran up a flume slope of
396 0.03 m/m. To accurately model the bed surface a Manning's friction
397 value of 0.01, representing PVC was used.
399 % Neale, L.C. and R.E. Price. Flow characteristics of PVC sewer pipe.
400 % Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE.
401 % pp. 109-129. 1964.
403 Acoustic displacement sensors that produced a voltage that changed
404 with the water depth was positioned 0.4m from the reservoir gate. The
405 water velocity was measured with an Acoustic Doppler Velocimeter 0.45m
406 from the reservoir gate. This sensor only produced reliable results 4
407 seconds after the reservoir gate opened, due to limitations of the sensor.
410 % Validation UQ flume
411 % at X:\anuga_validation\uq_sloped_flume_2008
412 % run run_dam.py to create sww file and .csv files
413 % run plot.py to create graphs heere automatically
414 % The Coasts and Ports '2007 paper is in TRIM d2007-17186
415 \begin{figure}[htbp]
416 \centerline{\includegraphics[width=4in]{uq-flume-depth}}
417 \caption{Comparison of wave tank and ANUGA water height at .4 m
418 from the gate}\label{fig:uq-flume-depth}
419 \end{figure}
421 \begin{figure}[htbp]
422 \centerline{\includegraphics[width=4in]{uq-flume-velocity}}
423 \caption{Comparison of wave tank and ANUGA water velocity at .45 m
424 from the gate}\label{fig:uq-flume-velocity}
425 \end{figure}
427 Figure~\ref{fig:uq-flume-depth} shows that ANUGA predicts the actual
428 water depth very well, although there is an initial drop in water depth
429 within the first second that is not simulated by ANUGA.
430 Water depth and velocity are coupled as described by the nonlinear
431 shallow water equations, thus if one of these quantities accurately
432 estimates the measured values, we would expect the same for the other
433 quantity. This is demonstrated in Figure~\ref{fig:uq-flume-velocity}
434 where the water velocity is also predicted accurately. Sediment
435 transport studies rely on water velocity estimates in the region where
436 the sensors cannot provide this data. With water velocity being
437 accurately predicted, studies such as sediment transport can now use
438 reliable estimates.
441 \subsection{1D flume tank to verify friction}
442 \label{sec:friction}
443 The same tilting flume tank was used to validate stage and velocity
444 was used to validate the ANUGA friction model. A ground slope of 1:20,
445 reservior lenght of 0.85m and damn depth of 0.4 m was used to verify
446 the friction. The PVC bottom of the tank is equivalent to a friction
447 value of 0.01. {\bf Add ref } Depth sensors were placed 0.2, 0.3,
448 0.4, 0.5 and 0.6 m from the bed gate.
451 As described in the model equations in ~\ref{sec:model}, the bed
452 friction is modelled using the Manning's model. {\bf Add the formula}
453 Validation of this model was carried out by comparing results
454 from ANUGA against experimental results from flume wave tanks.
456 This experiment was simulated twice by ANUGA: without using the
457 friction model {\bf Duncan says: It really used the friction model, with a
458 value of 0.0, representing no friction model. Is it ok to say
459 'without using the model?'} and using the friction model with a
460 Manning's friction value of 0.01. The results from both of these
461 simulations were compared against the experimental flume tank results
462 using the Root Mean Square Relative Error (RMSRE). The RMSRE was
463 summed over all of the depth sensors, for the first 30 seconds of the
464 experiment. This resulted in one number which represents the error
465 between tow data sets, with a lower number representing less
466 differences. The RMSRE for no friction model was 0.380, the RMSRE for
467 the friction model with a Manning's friction value of 0.01 was
468 0.358. So for this experiment using a friction value given from a
469 standard fricition table improved the accuracy of the ANUGA
470 simulation. {\bf Add ref to table}
472 % Validation UQ friction
473 % at X:\anuga_validation\uq_friction_2007
474 % run run_dam.py to create sww file and .csv files
475 % run plot.py to create graphs, and move them here
476 \begin{figure}[htbp]
477 \centerline{\includegraphics[width=4in]{uq-friction-depth}}
478 \caption{Comparison of wave tank and ANUGA water height at .4 m
479 from the gate, simulated using a Mannings friction of 0.0 and
480 0.1.}\label{fig:uq-friction-depth}
481 \end{figure}
483 \subsection{Okushiri Wavetank Validation}
484 \label{sec:okushiri}
485 As part of the Third International Workshop on Long-wave Runup
486 Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four
487 benchmark problems were specified to allow the comparison of
488 numerical, analytical and physical models with laboratory and field
489 data. One of these problems describes a wave tank simulation of the
490 1993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A
491 significant feature of this tsunami was a maximum run-up of 32~m
492 observed at the head of the Monai Valley. This run-up was not
493 uniform along the coast and is thought to have resulted from a
494 particular topographic effect. Among other features, simulations of
495 the Hokkaido tsunami should capture this run-up phenomenon.
497 This dataset has been used by to validate tsunami models by
498 a number of tsunami scientists. Examples include Titov ... lit review
499 here on who has used this example for verification (Leharne?)
501 \begin{figure}[htbp]
502 %\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}}
503 \centerline{\includegraphics[width=4in]{ch5.png}}
504 \centerline{\includegraphics[width=4in]{ch7.png}}
505 \centerline{\includegraphics[width=4in]{ch9.png}}
506 \caption{Comparison of wave tank and ANUGA water stages at gauge
507 5,7 and 9.}\label{fig:val}
508 \end{figure}
511 \begin{figure}[htbp]
512 \centerline{\includegraphics[width=4in]{okushiri-model.jpg}}
513 \caption{Complex reflection patterns and run-up into Monai Valley
514 simulated by ANUGA and visualised using our netcdf OSG
515 viewer.}\label{fig:run}
516 \end{figure}
518 The wave tank simulation of the Hokkaido tsunami was used as the
519 first scenario for validating ANUGA. The dataset provided
520 bathymetry and topography along with initial water depth and the
521 wave specifications. The dataset also contained water depth time
522 series from three wave gauges situated offshore from the simulated
523 inundation area. The ANUGA model comprised $41404$ triangles
524 and took about $1330$ s to run on the test platform described in
525 Section~\ref{sec:validation}.
527 The script to run this example is available in the ANUGA distribution in the subdirectory
528 \code{anuga_validation/automated_validation_tests/okushiri_tank_validation}.
531 Figure~\ref{fig:val} compares the observed wave tank and modelled
532 ANUGA water depth (stage height) at one of the gauges. The plots
533 show good agreement between the two time series, with ANUGA
534 closely modelling the initial draw down, the wave shoulder and the
535 subsequent reflections. The discrepancy between modelled and
536 simulated data in the first 10 seconds is due to the initial
537 condition in the physical tank not being uniformly zero. Similarly
538 good comparisons are evident with data from the other two gauges.
539 Additionally, ANUGA replicates exceptionally well the 32~m Monai
540 Valley run-up, and demonstrates its occurrence to be due to the
541 interaction of the tsunami wave with two juxtaposed valleys above
542 the coastline. The run-up is depicted in Figure~\ref{fig:run}.
544 This successful replication of the tsunami wave tank simulation on a
545 complex 3D beach is a positive first step in validating the ANUGA
546 modelling capability.
548 \subsection{Runup of solitary wave on circular island wavetank validation}
549 \label{sec:circular island}
550 This section will describe the ANUGA results for the experiments
551 conducted by Briggs et al (1995). Here, a 30x25m basin with a conical
552 island is situated near the centre and a directional wavemaker is used
553 to produce planar solitary waves of specified crest lenghts and
554 heights. A series of gauges were distributed within the experimental
555 setup. As described by Hubbard and Dodd \cite{Hubbard02}, a number of
556 researchers have used this benchmark problem to test their numerical
557 models. {\bf Jane: check whether these results are now avilable as
558 they were not in 2002}. Hubbard and Dodd \cite{Hubbard02} note that a
559 particular 3D model appears to obtain slightly better results than the
560 2D ones reported but that 3D models are unlikely to be competitive in
561 terms of computing power for applications in coastal engineering at
562 least. Choi et al \cite{Choi07} use a 3D RANS model (based on the
563 Navier-Stokes equations) for the same problem and find a very good
564 comparison with laboratory and 2D numerical results. An obvious
565 advantage of the 3D model is its ability to investigate the velocity
566 field and Choi et al also report on the limitation of depth-averaged
567 2D models for run-up simulations of this type.
569 Once results are availble, need to compare to Hubbard and Dodd and draw any conclusions
570 from nested rectangular grid vs unstructured gird.
571 Figure \ref{fig:circular screenshots} shows a sequence of screenshots depicting the evolution of the solitary wave as it hits the circular island.
573 \begin{figure}[htbp]
574 \centerline{
575 \includegraphics[width=5cm]{circular1.png}
576 \includegraphics[width=5cm]{circular2.png}}
577 \centerline{
578 \includegraphics[width=5cm]{circular3.png}
579 \includegraphics[width=5cm]{circular4.png}}
580 \centerline{
581 \includegraphics[width=5cm]{circular5.png}
582 \includegraphics[width=5cm]{circular6.png}}
583 \centerline{
584 \includegraphics[width=5cm]{circular7.png}
585 \includegraphics[width=5cm]{circular8.png}}
586 \centerline{
587 \includegraphics[width=5cm]{circular9.png}
588 \includegraphics[width=5cm]{circular10.png}}
589 \caption{Screenshots of the evolution of solitary wave around circular island.}
590 \label{fig:circular screenshots}
591 \end{figure}
594 \clearpage
596 \section{Conclusions}
597 \label{sec:conclusions}
598 ANUGA is a flexible and robust modelling system
599 that simulates hydrodynamics by solving the shallow water wave
600 equation in a triangular mesh. It can model the process of wetting
601 and drying as water enters and leaves an area and is capable of
602 capturing hydraulic shocks due to the ability of the finite-volume
603 method to accommodate discontinuities in the solution.
604 ANUGA can take as input bathymetric and topographic datasets and
605 simulate the behaviour of riverine flooding, storm surge,
606 tsunami or even dam breaks.
607 Initial validation using wave tank data supports ANUGA's
608 ability to model complex scenarios. Further validation will be
609 pursued as additional datasets become available.
610 The ANUGA source code and validation case studies reported here are available
611 at \url{http://sourceforge.net/projects/anuga}.
613 something about use on flood modelling community and their validation initiatives
616 %\bibliographystyle{plainnat}
617 \bibliographystyle{elsart-harv}
618 \bibliography{anuga-bibliography}
620 \end{document}
for help on using the repository browser. | {"url":"https://anuga.anu.edu.au/browser/anuga_work/publications/anuga_2007/anuga_validation.tex?rev=5599","timestamp":"2024-11-08T21:30:30Z","content_type":"application/xhtml+xml","content_length":"100464","record_id":"<urn:uuid:baf542b0-5463-414c-ad05-48cb5743b0eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00288.warc.gz"} |
HtaGlossary.net | beta (β)
In hypothesis testing, the probability of a Type II error, i.e. the probability of concluding incorrectly that a null hypothesis is true.
Note: For example, β could be the probability of concluding that an intervention is not effective when it has a true effect. (1-β) is the statistical power of a test allowing for rejection of a null
hypothesis that is truly false (e.g. detecting the effect of an intervention that truly exists).
(Related concepts: hypothesis testing, statistical power) | {"url":"https://htaglossary.net/beta-(%CE%B2)","timestamp":"2024-11-01T20:37:54Z","content_type":"text/html","content_length":"44355","record_id":"<urn:uuid:eeb49fb4-2d65-4f84-a417-3e5e009aaf6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00391.warc.gz"} |
STAB22H3 Lecture 7: Box Plot Diagrams, Density Curves, Normal Distributi... - OneClass
STAB22H3 Lecture 7: Box Plot Diagrams, Density Curves, Normal Distribution, Mean, Standard Deviation, Z-Scores & Standardization
Document Summary
Stab22 - lecture 7 - box plot diagrams, density curves, normal distribution, mean, It"s a 5 number summary which includes the following: a maximum, minimum, 1st quartile (q1), median (q2) and 3rd
quartile (q3). You can find q1,q2 and q3 from the blue box given. The most bottom line of the blue box is q1. The top line of the blue box is q3. You also have an upper-fence and a lower fence: Lower
fence = q1 - 1. 5/qr (*remember: iqr = q3-q1) Maximum and minimum are the highest and lines that come out of the the graph (maximum is 35 while minimum is probably 10). Upper-fence and lower-fence
are the two blue lines (upper-fence is 31. 5 and lower-fence is 3. 5). You can calculate them based on the equations. Values bigger than the upper fence, or lower than the lower fence are considered
outliers (notice the little black dot right above 35 in slide 29) | {"url":"https://oneclass.com/class-notes/ca/utsc/sta-actb/stab-22h3/2334210-stab22h3-lecture-7.en.html","timestamp":"2024-11-04T21:53:31Z","content_type":"text/html","content_length":"644553","record_id":"<urn:uuid:ce30bd46-44d6-4132-894b-11dedd8f25e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00675.warc.gz"} |
Active Prompting
π ¦ Active Prompting
π ¦ This article is rated medium
Reading Time: 7 minutes
Last updated on October 3, 2024
Overview of Active Prompting
Information and Links
What is Active Prompting?
Active Prompting (or Active-Prompt)^ is a technique for improving Chain-of-Thought (CoT) prompting performance by selectively human-annotating exemplars where the model shows the most uncertainty.
This approach helps maximize the efficiency of human annotation efforts by focusing only on the most challenging questions for the model.
Active Prompting consists of four main steps:
1. Uncertainty Estimation:
• It prompts the model several times ($k$) with unlabeled questions using Chain-of-Thought Prompting with a few human-written chains-of-thought or Zero-Shot Chain-of-Thought (CoT) prompting without
human-written chains-of-thought to generate possible answers with intermediate steps for a set of unlabeled questions.
□ It calculates the uncertainty $u$ based on $k$ answers via a selected uncertainty metric.
2. Selection: The questions with the highest uncertainty are selected for human annotation.
3. Annotation: The selected questions from step 2 are manually annotated by humans to provide more accurate answers.
4. Inference: The newly annotated exemplars are used to improve the modelβ s performance in answering future questions.
Active Prompting saves significant human resources by reducing the need to annotate all training data. It outperforms other techniques such as Automatic Chain-of-Thought prompting, Random
Chain-of-Thought prompting, and Self-Consistency on a range of reasoning tasks. Active Prompting research^ is the first to show the benefits of selective question annotation in CoT prompting for
solving complex reasoning tasks.
How to Use Active Prompting?
Letβ s break down the Active Prompting process with an example. Assume you have a pool of $n$ unlabeled questions.
Step 1. Uncertainty Estimation
First, you prompt the model multiple times ($k$) for each unlabeled question using:
• A number of annotated examplars if you want to use CoT option
• "Think step-by-step" if you want to use Zero-Shot CoT option
Letβ s say you choose the CoT option. You provide exemplars (Q1 and Q2), then ask your pool question (Q3). Repeat this process $k$ times for each question.
Prompting k times with CoT option
Q1: Josh and Anna were both born on August 17th, but in different years. To consolidate celebrations they also got married on August 17 when Josh turned 22. If today theyβ re celebrating 30 years of
marriage and their combined age is exactly 5 times what Joshβ s age was when they married, how old was Anna when they got married?
A1: Letβ s think step by step. To calculate how old was Anna when they got married, we have to know their combined age, Joshβ s age after 30 years, and Annaβ s age after 30 years from their
marriage. Since their combined age is 5 times Joshβ s age when he got married, their combined age is 5 * 22 = 110 years. Josh must be 30 years older than his age when they got married, so he is 22 +
30 = 52 years old now. Therefore, Annaβ s current age will be 110 - 52 = 58 years. If they married 30 years ago, Anna must have been 58 - 30 = 28 years old when they married The answer is 28.
Q2: John buys a chair. He then buys a table that is 3 times the price of the chair. Then, he buys a couch that is 5 times the price of the table. If John paid $380 for all these items, what is the
price of the couch?
A2: Letβ s think step by step. To calculate the price of the couch, we need to know the price of the chair, the price of the table, and the relation between the chair, table, couch, and total money
paid. Let x be the price of the chair, 3 * x be the price of the table, and 5 * (3 * x) = 15 * x be the price of the couch. The relationship between the chair, table, couch, and the total price paid
is x + 3 * x + 15 * x = $380, which is 19 * x = 380, and x=20. The price of the couch is 15 * x, which is 15 * 20 = $300. The answer is 300.
Q3: John has 5 apples, and he gives 2 to Mary. How many apples does John have left?
As a result you will get $k$ answers for each of your $n$ questions.
Next, you need to measure the uncertainty of the model for each question based on the $k$ answers it generates for a given question.
To do that, you select the uncertainty metric. An example metric could be disagreement:
You use the disagreement among $k$ generated answers for a given question from the pool. The disagreement calculates the unique answers in the predictions.
• You count the number of unique answers generated for a question, $h$
• Calculate the disagreement by dividing $uncertainty = h/k$
Step 2. Selection
Then, you select the questions with the highest uncertainty based on the metric. For simplicity, let's review just one example question from the set of the most uncertain questions.
Imagine you take disagreement as an useartainty metric. You prompt the model $k$ times and find that the below question consistently yields different LLM's outputs meaning the model is uncertain in
the answer:
John has 5 apples, and he gives 2 to Mary.
How many apples does John have left?
This is just one example while in reality there can be many of them.
Step 3. Annotation
You manually annotate the selected question to provide a clear, correct answer:
Annotated Question
Q: John has 5 apples, and he gives 2 to Mary. How many apples does John have left?
A: John starts with 5 apples. He gives away 2 apples. Therefore, he has 5 - 2 = 3 apples left.
Step 4. Inference
This annotated question becomes an example for the model.
The code for Active Prompting is open-sourced and available for further research and implementation at shizhediao/active-prompt.
What are the Experimental Results for Active Prompting?
Active Prompting has demonstrated superior performance across several benchmarks, including arithmetic, commonsense, and symbolic reasoning tasks. It consistently outperforms traditional CoT and
other baseline techniques, highlighting its effectiveness in enhancing LLM capabilities.
• Self-Consistency: Active-Prompt outperforms self-consistency across most tasks. Active-Prompt shows a 7.2% improvement over SC on arithmetic reasoning tasks. For commonsense and symbolic
reasoning, Active-Prompt consistently outperforms SC across all tasks.
• Chain-of-Thought: Compared to CoT, Active-Prompt demonstrates significant improvements. For instance, Active-Prompt scores 83.4% on GSM8K, compared to 63.1% by CoT. Across other datasets (e.g.,
ASDiv, SVAMP, AQUA), Active-Prompt outperforms CoT by margins ranging from 1.0% to 15.4%, showing the robustness of the active selection method.
• Random Chain-of-Thought: Active-Prompt also shows consistent improvements over Random-CoT, with higher average performance across the datasets.
• Automatic Chain-of-Thought: Though Auto-CoT produces decent results, particularly with code-davinci-002, Active-Prompt surpasses it in every task.
Limitations of Active Prompting
Despite its advantages, Active Prompting has some limitations:
• Human Annotation Required: Some level of human involvement is needed to annotate the most uncertain questions.
• Choosing the right uncertainty metric matters: The way we measure uncertainty can impact performance, so we need to pick the right one based on the task at hand.
Active prompting really enhances how well Large Language Models solve complex reasoning problems. By focusing on the questions the model is most uncertain about, we make the annotation process
efficient and tailor it to boost the model's learning.
Get AI Certified by Learn Prompting | {"url":"https://learnprompting.org/docs/advanced/thought_generation/active_prompting","timestamp":"2024-11-01T20:52:15Z","content_type":"text/html","content_length":"1053425","record_id":"<urn:uuid:bed74a81-a2cc-4ec0-9479-b89759834adf>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00618.warc.gz"} |
Ch. 3 Introduction to Polynomial and Rational Functions - Precalculus | OpenStax
Chapter Outline
Digital photography has dramatically changed the nature of photography. No longer is an image etched in the emulsion on a roll of film. Instead, nearly every aspect of recording and manipulating
images is now governed by mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or
computer interprets the numbers and converts them to a visual image. Photo editing software uses complex polynomials to transform images, allowing us to manipulate the image in order to crop details,
change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how
mathematics can be used in such applications. | {"url":"https://openstax.org/books/precalculus/pages/3-introduction-to-polynomial-and-rational-functions","timestamp":"2024-11-08T14:44:08Z","content_type":"text/html","content_length":"373261","record_id":"<urn:uuid:80719ac4-1d0b-4931-ac05-525f2df651d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00669.warc.gz"} |
Digitized by the Internet Archive
in 2008 with funding from
Microsoft Corporation
BY VIRGIL SNYDER, Ph.D.
JOHN IRWIN HUTCHINSON, Ph.D Of Cornell University
NEW YORK •:• CINCINNATI •:• CHICAGO
THE MODERN MATHEMATICAL SERIES. Lucien Augustus Wait,
{Senior Professor of Mathematics in Cornell University ,)
General Editor.
This series includes the following works : BRIEF ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. ELEMENTARY ANALYTIC GEOMETRY. By J. H. Tanner and Joseph
Allen. DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyder. INTEGRAL CALCULUS. By D. A. Murray. DIFFERENTIAL AND INTEGRAL CALCULUS. By Virgil Snyder and
J. I. Hutchinson. ELEMENTARY TEXTBOOK ON THE CALCULUS. By Virgil Snyder
and J. I. Hutchinson.
HIGH SCHOOL ALGEBRA. By J. H. Tanner. ELEMENTARY ALGEBRA. By J. H. Tanner. ELEMENTARY GEOMETRY. By James McMahon.
COPYRIGttl', 1912, BY
The present volume is the outgrowth of the requirements for students in engineering and science in Cornell University, for whom a somewhat brief but adequate introduction to the Calculus is
The guiding principle in the selection and presentation of the topics in the following pages has been the ever increasing pressure on the present-day curriculum, especially in applied science, to
limit the study of mathematics to a minimum of time and to the topics that are deemed of most immediate use to the professional course for which it is preparatory.
To what extent it is wise and justifiable to yield to this pressure it is not our purpose to discuss. But the constantly accumulating details in every pure and applied science makes this attitude a
very natural one towards mathematics, as well as towards several other subjects which are subsidiary to the main object of the given course.
This desire to curtail mathematical training is strikingly evidenced by the numerous recent books treating of Calculus for engineers, for chemists, or for various other professional students. Such
books have no doubt served a useful purpose in various ways. But we are of the opinion that, in spite of the unquestioned advantages of learning a new method by means of its application to a specific
field, a student would ordinarily acquire too vague and inaccurate a command of the fundamental ideas of the Calculus by this one-sided presenta- tion. While a suitable illustration may clear up the
4/( • ;/f ;'l :*•/;• \ * ., 'breface
of an abstract theory, too constant a dwelling among applica- tions alone, especially from one point of view, is quite as likely to prevent the learner from grasping the real significance of a vital
In recognition of the demand just referred to, we have made special effort to present the Calculus in as simple and direct a form as possible consistent with accuracy and thoroughness. Among the
different features of our treatment, we may single out the following for notice.
The derivative is presented rigorously as a limit. This does not seem to be a difficult idea for the student to grasp, espe- cially when introduced by its geometrical interpretation as the slope of
the line tangent to the graph of the given func- tion. For the student has already become familiar with this notion in Analytic Geometry, and will easily see that the analytic method is virtually
equivalent to a particular case of the process of differentiation employed in the Calculus.
In order to stimulate the student's interest, easy applications of the Differential Calculus to maxima and minima, tangents and normals, inflexions, asymptotes, and curve tracing have been introduced
as soon as the formal processes of differentia- tion have been developed. These are followed by a discussion of functions of two or more independent variables, before the more difficult subject of
infinite series is introduced.
In the chapter on expansion, no previous knowledge of series is assumed, but conditions for convergence are discussed, and the criteria for determining the interval of convergence of those series
that are usually met with in practice are derived.
A chapter on the evaluation of indeterminate forms and three chapters on geometric applications furnish ample illus-
PREFACE 5
tration of the uses of infinite series in a wide range of problems.
By reason of its significance in applications, it does not seem advisable to omit the important principle of rates. Arising out of the familiar notion of velocity, it affords an early glimpse into
applications of the Calculus to Mechanics and Physics. We do not propose to make the Calculus a treatise on Mechanics, as seems to be the tendency with some writers; but a final chapter on
applications to such topics of Mechanics as are easy to comprehend at this stage is thought advisable and sufficient. Especially in treating of center of gravity, the formulas have been derived in
detail, first for n particles, and then, by a limit- ing process, for a continuous mass. This was considered the more desirable, as textbooks in applied mathematics frequently lack in rigor in
discussing the transition from discrete particles to a continuous mass. Besides, the derivation of these formu- las affords a very good application of the idea of the definite integral as the limit
of a sum. This idea has been freely and consistently used in the derivation of all applied formulas in the Integral Calculus. However, as the formula for the length of arc in polar coordinates is
especially difficult of derivation by this method, we have deduced it from the corresponding formula for rectangular coordinates by a transformation of the variable of integration.
In-order to make the number of new ideas as few as possible, the notions of infinitesimals and orders of infinitesimals have been postponed to the last article on Duhamel's principle. This principle
seems to flow naturally and easily from the need of completing the proof of the formulas for center of gravity. The teacher may omit this article, but its presence should at
6 PREFACE
least serve the important end of calling the attention of the student to the fact that there is something yet to be done in order to make the derivations complete.
Some teachers will undoubtedly prefer to do a minimum amount of work in formal integration and use integral tables in the chapters on the applications. For such the first chapter of the Integral
Calculus might suffice for drill in pure integration. The problems in this chapter are numerous, and, for the most part, quite easy, and should furnish the student a ready insight into the essential
principles of integration.
The characteristic features of the books on the Calculus previously published in this series have been retained. The extensive use of these books by others, and a searching yearly test in our own
classroom experience convince us that any far- reaching change could not be undertaken without endangering the merits of the book. The changes that have been made are either in the nature of a slight
rearrangement, or of the addi- tion of new illustrative material, particularly in the applications.
We wish to acknowledge our indebtedness to our colleagues, who have added many helpful suggestions ; to Professor I. P. Church, of the College of Civil Engineering, for a number of very useful
problems in applications of integration (See Exs. 14-18, pp. 318-320, and Exs. 6-7, pp. 323-324), and particu- larly to Professor James McMahon, who has carefully read all the manuscript, assisted
throughout in the proof reading, and made many improvements in the text.
Fundamental Principles
1. Elementary definitions 15
2. Illustration : Slope of a tangent to a curve . . . .16
3. Fundamental theorems concerning limits 17
4. Continuity of functions 19
5. Comparison of simultaneous increments of two related variables 20
6. Definition of a derivative 21
7. Process of differentiation 22
8. Differentiation of a function of a function .... 23
CHAPTER II Differentiation of the Elementary Forms
9. Differentiation of the product of a constant and a variable 25
10. Differentiation of a sum . 26
11. Differentiation of a product 27
12. Differentiation of a quotient 28
13. Differentiation of a commensurable power of a function . 29
14. Differentiation of implicit functions 33
15. Elementary transcendental functions .... 34
16. Differentiation of loga x and loga u 34
17. Differentiation of the simple exponential function 36
18. Differentiation of an incommensurable power . 37
19. Limit of ^— as 6 approaches 0 38
20. Differentiation of sin u 39
21. Differentiation of cos u 40
22. Differentiation of tan u 40
23. Differentiation of sin-1 u 42
24. Table of fundamental forms 44
Successive Differentiation
25. Definition of the nth derivative 47
26. Expression for the nth derivative in certain cases ... 49
CHAPTER IV Maxima and Minima
27. Increasing and decreasing functions 51
28. Test for determining intervals of increasing and decreasing . 51
29. Turning values of a function 53
30. Critical values of the variable 55
31. Method of determining whether <f>'(x) changes its sign in pass-
ing through zero or infinity ....... 55
32. Second method of determining whether <p'(x) changes its sign
in passing through zero ... ..... 57
33. The maxima and minima of any continuous function occur
alternately .59
34. Simplifications that do not alter critical values .... 59
35. Geometric problems in maxima and minima .... 60
CHAPTER V Rates and Differentials
36. Rates. Time as independent variable 68
37. Abbreviated notation for rates 72
38. Differentials often substituted for rates 74
39. Theorem of mean value 74
Differential of an Area, Arc, Volume, and Surface of Revolution
40. Differential of an area 78
41. Differential of an arc 79
42. Trigonometric meaning of — , — .80
dx dy
43. Differential of the volume of a surface of revolution ... 81
44. Differential of a surface of revolution
45. Differential of arc in polar coordinates
46. Differential of area in polar coordinates
CHAPTER VII Applications to Curve Tracing
47. Equation of tangent and normal ....
48. Length of tangent, normal, subtangent, and subnormal
49. Concavity upward and downward 60. Algebraic test for positive and negative bending
51. Concavity and convexity toward the axis .
52. Hyperbolic and parabolic branches .
53. Definition of a rectilinear asymptote .
Determination of Asymptotes
54. Method of limiting intercepts 96
55. Method of inspection. Infinite ordinates, asymptotes parallel
to axes 97
56. Method of substitution. Oblique asymptotes .... 99
57. Number of asymptotes 102
Polar Coordinates
58. Meaning of p — 104
59. Relation between ^ and p— 105
dx dp
60. Length of tangent, normal, polar subtangent, and polar sub-
normal 105
Differentiation of Function
61. Definition of continuity
62. Partial differentiation
63. Total differential
64. Total derivative . •
65. Differentiation of implicit functions
66. Geometric interpretation .
67. Successive partial differentiation
68. Order of differentiation indifferent
10 CONTENTS
Change of Variable
article page
69. Interchange of dependent and independent variables . . 124
70. Change of the dependent variable 125
71. Change of the independent variable 126
72. Simultaneous changes of dependent and of independent variables 126
CHAPTER X Expansion of Functions
73. Convergence and divergence of series 132
74. General test for convergence , 133
75. Interval of convergence 138
70. Remainder after n terms , . 140
77. Maclaurin's expansion of a function in a power series . .141
78. Taylor's series 148
79. Rolle's theorem 150
80. Form of remainder in Maclaurin's series 150
81. Another expression for the remainder 153
Indeterminate Forms
82. Definition of an indeterminate form » 157
83. Indeterminate forms may have determinate values . . . 158
84. Evaluation by development 160
85. Evaluation by differentiation ....... 161
86. Evaluation of the indeterminate form g- 165
CHAPTER XII Contact and Curvature
87. Order of contact 167
88. Number of conditions implied by contact 168
89. Contact of odd and of even order 169
90. Circle of curvature 172
91 . Length of radius of curvature ; coordinates of center of curvature 1 72
92. Limiting intersection of normals 174
93. Direction of radius of curvature . . . ' • • 175
94. Total curvature of a given arc ; average curvature . . . 176
95. Measure of curvature at a given point 177
. 96. Curvature of an arc of a circle 178
97. Curvature of osculating circle 178
98. Direct derivation of the expressions for k and B in polar co-
ordinates 180
99. Definition of an evolute 182
100. Properties of the evolute 184
CHAPTER XIII Singular Points
101. Definition of a singular point
102. Determination of singular points of algebraic curves
103. Multiple points
104. Cusps .........
105. Conjugate points
106. Family of curves
107. Envelope of a family of curves
108. The envelope touches every curve of the family
109. Envelope of normals of a given curve
110. Two parameters, one equation of condition
12 CONTENTS
General Principled oe Integration
111. The fundamental problem 209
112. Integration by inspection 210
113. The fundamental formulas of integration .... 211
114. Certain general principles 212
115. Integration by parts 216
116. Integration by substitution 219
117. Additional standard forms 222
118. Integrals of the forms C(^x + B)dx and C (Ax + B)dx # »M
J ax2 + bx + c J ^/ax-2 + bx + c
119. Integrals of the forms f — and
J (Ax + B) Vax2 + bx + c
121. Decomposition of rational fractions
122. Case I. Factors of the first degree, none repeated
123. Case II. Factors of the first degree, some repeated
124. Case II J. Occurrence of quadratic factors, none repeated
125. Case IV. Occurrence of quadratic factors, some repeated
126. General theorem
CHAPTER II 120. Reduction Formulas 229
Integration of Rational Fractions
Integration by Rationalization
127. Integration of functions containing the irrationality y/ax + b 248
128. Integration of expressions containing Vax2 + bx + c . . 249
129. The substitution V± t1 ± k2 -z 253
Integration of Trigonometric Functions
130. Integration by substitution
181. Integration of ( sec2na; dx, \ csc2»x dx
132. Integration of I secTOx tan'2n+1x dx, j cscmx coV^+h; dx
133. Integration of i tan"x dx, \ cotnx dx
134. Integration of ( sin'"x cosnx dx
135. Integration of f — , f-
J a + b cos nx J a
+ b sin nx -f c cos nx 136. Integration of I eax sin wxcJx, je^cos nxdx
Integration as a Summation. Areas
137. Areas 268
138. Expression of area as a definite integral 270
139. Generalization of the area formula 273
140. Certain properties of definite integrals 274
141. Maclaurin's formula 276
142. Remarks on the area formula - 277
143. Precautions to be observed in evaluating definite integrals . 283
144. Calculation of area when x and y are expressible in terms of a
third variable 289
145. Areas in polar coordinates 291
146. Approximate integration. The trapezoidal rule . . . 292
147. Simpson's rule 294
148. The limit of error in approximate integration .... 295
Geometrical Applications
149. Volumes by single integration 298
150. Volume of solid of revolution 302
151. Lengths of curves. Rectangular coordinates .... 306
152. Lengths of curves. Polar coordinates 309
153. Measurement of arcs by the aid of parametric representation . 310
154. Area of surface of revolution 312
155. Various geometrical problems leading to integration . 315
CHAPTER VIII Successive Integration
156. Functions of a single variable .
157. Integration of functions of several variables
158. Integration of a total differential
159. Multiple integrals . . .
160. Definite multiple integrals
161. Plane areas by double integration .
162. Volumes
Some Applications of Integral Calculus to Problems of Mechanics
163. Liquid pressure on a plane vertical wall . . . . 338
164. Center of gravity . 340
165. Moment of inertia . . 346
166. Duhamel's theorem . . 348
Trigonometric Formulas . 352
Logarithmic Formulas . 353
1. Elementary definitions. A constant number is one that retains the same value throughout an investigation in which it occurs. A variable number is one that changes from one value to another during
an investigation. If the variation of a number can be assigned at will, the variable is called independent; if the value of one number is determined when that of another is known, the former is
called a dependent variable. The depend- ent variable is called also a function of the independent variable.
E.g., 3 x2, 4vx — 1, cos x, are all functions of x.
Functions of one variable x will be denoted by the symbols /(#), <f>(x), • ••, which are read as "/of x" " <f> of x" etc. ; simi- larly, functions of two variables, x, y, will be denoted by such
expressions as
f(?,y),F(x,y), ••••
When a variable approaches a constant in such a way that the difference between the variable and the constant may be- come and remain smaller than any fixed number, previously assigned, the constant
is called the limit of the variable.
2. Illustration : Slope of a tangent to a curve. To obtain the slope of the tangent to a curve at a point P upon it, first take the slope of the line joining P = (xly yx) to another point (x2, y2)
upon the curve, then determine the limiting value of the slope
as the second point approaches to coincidence with the first, always remaining on the curve.
Ex. 1. Determine the slope of the tangent to the curve
at the point (2, 4) upon it.
Here, x\ = 2, y\ = 4. Let x2 = 2 + h, yi = 4 + k, where h, k are so related that the point (x2, y*) lies on the curve.
Thus 4 + k = (2 + h)\
or h = 4 A + A2- (1)
m = y* -
The slope -Xi becomes
x2 -
4+ Tc 2 + h k
_ 2
which from (1) may be written in the form
The ratio k : h measures the slope of the line joining (xh yx) to (ar2, ys) • When the second point approaches the first as a limiting position, the first member of equation (2) assumes the
indeterminate form -, but the second member approaches the definite limit 4. When
the two points approach coincidence, a definite slope 4 is obtained, which is that of the tangent to the curve y = x2 at the point (2, 4).
It may happen that h, k appear in both members of the equation which defines the slope, as in the next example.
Ex. 2. If x2 + y1 — «2? find the slope of the tangent at the point Oi> yd- Since
Xl* + i/!2 = a2, (asi + ny+ (t/1 + ky = a\
hence 2 hx1 + A2 + 2 /fr/i + fc2 = 0,
from which - = — - — — — h 2 ?/i + k
k To obtain the limit of -, put h, k h
each equal to zevo in tlie second member.
lim * = _*!.
h±o h ?/i
This step is more fully justified in the next article. This result agrees with that obtained by elementary geome- try. The slope of the radius to the circle a2 + y2 = a2 through
the point (xlf yx) is — , and the slope of the tangent is the nega-
tive reciprocal of that of the radius to the point of tangency, since the two lines are perpendicular.
3. Fundamental theorems concerning limits. The following theorems are useful in the processes of the Calculus.
Theorem 1. If a variable a approaches zero as a limit, then lea will also approach zero, k being any finite constant.
That is, if a = 0,
then Jca = 0.
For, let c be any assigned number. By hypothesis, a can be- come less than -, hence ka can become less than c, the arbi-
k ■ '
* For convenience, the symbol = will be used to indicate that a variable approaches a constant as a limit; thus the symbolic form x = a is to be read " the variable x approaches the constant a as a
limit." el. calc — 2
trary, assigned number, hence ka approaches zero as a limit. (Definition of a limit.)
Theorem 2. Given any finite number n of variables a, (3, y, •••, each of which approaches zero as a limit, then their sum will approach zero as a limit. For the sum of the n variables does not at any
stage numerically exceed n times the largest of them, which by Theorem 1 approaches zero.
Theorem 3. If each of two variables approaches zero as a limit, their product will approach zero as a limit. More gen- erally, if one variable approaches zero as a limit, then its product with any
other variable having a finite limit will have the limit zero, by Theorem 1.
Theorem 4. If the sum of a finite number of variables is variable, then the limit of their sum is equal to the sum of their limits ; i.e.,
lim (x + y + • • •) = lim x + lim y + For, if x = a, y = b, • • •,
then x = a + a, y = b -\- (3, •-•,
wherein a = 0, fi = 0, •• • ; (Def . of limit)
hence x + y+ ••• = (o+ &+ •••) + (« + fi+ •••)>
but a + p+->- =0, (Th. 2)
hence, from the definition of a limit,
lim (x + y + •••) = a-\-b-\- ••• = lim x -f- lim y + •••.
Theorem 5. If the product of a finite number of variables is .variable, then the limit of their product is equal to the product of their limits.
For, let x = a + a, y = b+($,
wherein a = 0, (3 = 0,
so that lim x = a, lim y = b.
Form the product
xy = (a + a)(b + fi) = ab + «6 -f /3a + «0. Then lim sc?/ = lim (ab + ab + (5a + a(5)
= ab + lim ab + lim 0a + lim a/3 (Th. 2)
= ab. (Th. 1)
Hence lim xy = lim a; • lim y.
In the case of a product of three variables x, y, z, we have lim xyz = lim xy • lim z (Th. 5)
= lim x lim y lim 3, and so on, for any finite number of variables.
Theorem 6. If the quotient of two variables as, y is vari- able, then the limit of their quotient is equal to the quotient of their limits, provided these limits are not both infinite or not both
y lim y
4. Continuity of functions. When an independent variable x, in passing from a to b, passes through every intermediate value, it is called a continuous variable. A function f(x) of an independent
variable x is said to be continuous at any value xl} or in the vicinity of xXi when f(x^) is real, finite, and determi- nate, and such that in whatever way x approaches a^,
From the definition of a limit it follows that corresponding to a small increment of the variable, the increment of the
For, since x = -y,
lim x = lim - lim yy
and hence ,. x lim x lim - =
function is also small, and that corresponding to any number c, previously assigned, another number 8 can be determined, such that when h remains numerically less than 8, the differ-
is numerically less than c.
Thus, the function of Fig. 3 is continuous between the values xx and xY -f- 8, while the functions of Fig. 4 and Fig. 5 are dis- continuous. In the former of these two the function becomes infinite
at x = c, while in the latter the difference between the value of the function at c + h and c — h does not approach zero with h, but approaches the finite value AB as h ap- proaches zero.
When a function is continuous for every value of x between a and b, it is said to be continuous within the interval from a to b.
5. Comparison of simultaneous increments of two related vari- ables. The illustrations of Art. 2 suggest the following general procedure for comparing the changes of two related variables. Starting
from any fixed pair of values x1} y^ represented graph- ically by the abscissa and ordinate of a chosen point P on a given curve whose equation is given, we change the values of
x and y by the addition of small amounts h and k respectively, so chosen that the new values xL + h and yl + k shall be the coordinates of a point P2 on the curve. The amount h added to xx is called
the increment of x and is entirely arbitrary. Likewise, k is called the increment of y ; it is not arbitrary but depends upon the value of h ; its value can be calcu- lated when the equation of the
curve is given, as is shown by equation (1). These increments are not necessarily positive. In the case of continuous functions, h may always be taken positive. The sign of k will then depend upon
the function under consideration. The slope of the line
PjP2 is then - and the slope of the tangent line at Pj is the
limit of - as h and consequently k approach zero.
The determination of the limit of the ratio of k to h as h and k approach zero is the fundamental problem of the Differential Calculus. The process is systematized in the following ar- ticles. While
the related variables are here represented by ordinate and abscissa of a curve, they may be any two related magnitudes, such as space and time, or volume and pressure of a gas, etc.
6. Definition of a derivative. If to a variable a small incre- ment is given, and if the corresponding increment of a contin- uous function of the variable is determined, then the limit of the ratio
of the increment of the function to the increment of the variable, when the latter increment approaches the limit zero, is called the derivative of the function as to the variable.
k That is, the derivative is the limit of - as h approaches zero,
For the purpose of obtaining a derivative in a given case it is convenient to express the process in terms of the following steps:
1. Give a small increment to the variable.
2. Compute the resulting increment of the function.
3. Divide the increment of the function by the increment of the variable.
4. Obtain the limit of this quotient as the increment of the variable approaches zero.
7. Process of differentiation. In the preceding illustrations, the fixed values of x and of y have been written with sub- scripts to show that only the increments h, k vary during the algebraic
process of finding the derivative, also to emphasize the fact that the limit of the ratio of the simultaneous incre- ments h, k depends upon the particular values which the variables x, y have, when
they are supposed to take these in- crements. With this understanding the subscripts will hence- forth be omitted. Moreover, the increments h, k will, for greater distinctness, be denoted by the
symbols Ax, Ay, read " increment of x," " increment of y."
If the four steps of Art. 6 are applied to the function y = <£ (x), the results become y + £fy=<f>(x + \x),
Ay = <j>(x + Ax) — <f>(x) = A<f> (x),
Ay _<f>(x + Ax) — <j> (x) _ A<£ (x)
Ax Ax Ax
. lira -AJ = Km {+(* + **)- »(*) | =Um Aj>^
A# Ax Ax
' FUNDAMENTAL PRINCIPLES 23
The operation here indicated is for brevity denoted by the
symbol — , and the resulting derivative function by <f>'(x); thus dx
dy _d<f>(x) _ lim f <f>(x + Ax)-<j>(x)'
The new symbol -^ is not (at the present stage at least) to ax
be looked upon as a quotient of two numbers dy, dx, but rather as a single symbol used for the sake of brevity in place of the expression " derivative of y with regard to x."
The process of performing this indicated operation is called the differentiation of <f> (x) with regard to x.
EXERCISES Find the derivatives of the following functions with regard to x.
5. I.
6. xn, n being a positive integer.
1. x2- 2x-, 2x\ 3; x.
2. 3x*-4:x + 3.
3. 1 4*'
**-2 + i.
9. y = Vx.
10. y = x~$.
f 1
[Put #2 = x, and apply the rules.]
8. Differentiation of a function of a function. Suppose that y, instead of being given directly as a function of x, is expressed as a function of another variable u, which is itself expressed as a
function of x. Let it be required to find the derivative of y with regard to the independent variable x.
Let y =f(u), in which u is a function of x. When x changes to the value au + Aaj, let u and y, under the given relations,
change to the values u + Aw, y + A?/. Then
A?/ _ Aw Aw ,
Aa; — Aw Ax
hence, equating limits (Th. 5, Art. 3),
dy _dy da _ df(u) du dx ~~ du dx~ du dx
This result may be stated as follows :
The derivative of a function of u with regard to x is equal to the product of the derivative of the function with regard to w, and the derivative of u with regard to x.
1. Given v = 3u2-l, M = 3x2 + 4; find ^-
dy du
du dx
dx du dx K '
2. Given ?/ =3m2-4u+ 5,« = 2x3-5; find ^ •
3. Given y = -,w = 5a;2-2x + 4; find ^ •
1 -r3 3 _ , rfv
3 m2 3 a:3 aar
dv In recent articles, the meaning of the symbol -f was ex-
plained and illustrated ; and a method of expressing its value, as a function of x, was exemplified, in cases in which y was a simple algebraic function of x, by direct use of the definition. This
method is not always the most convenient one in the dif- ferentiation of more complicated functions.
The present chapter will be devoted to the establishment of some general rules of differentiation which will, in many cases, save the trouble of going back to the definition.
The next five articles treat of the differentiation of algebraic functions and of algebraic combinations of other differentiable functions.
9. Differentiation of the product of a constant and a variable.
Let y = cu,
Then y + Ay = c(u + Au),
A?/ = c(m + Au) — cu = cAu,
Ay Au
Ax Ax'
therefore dy du dx~~ dx
Thus d(cu) _ du dx dx
The derivative of the product of a coyistant and a variable is equal to the constant multiplied by the derivative of the variable.
10. Differentiation of a sum.
Let 2/==M_[_<y_W;_j_ ...
in which u. v, w, ••> are functions of x.
Then y + Ay = u + Au + v + Av — w — Aw + • • •,
Ay = Au + A?; — Aiv + • • •,
Ay _ Au ,Av_ Aw
Ax Ax Ax Ax '
dy _du dv dw dx dx dx dx
Hence -f(u + v- w+ . ..)=f^+^-^+ .. (2)
doc doc doc doc
The derivative of the sum of a finite number of fractions is equal to the sum of their derivatives.
Cor. If y = u + c, c being a constant, then y + Ay = u + Au + c, hence Ay = Au,
and dy = du
dx dx
This last equation asserts that all functions which differ from each other only by an additive constant have the same derivative.
Geometrically, the addition of a constant has the effect of moving the curve y = u(x) parallel to the y-axis ; this opera- tion will obviously not change the slope at points that have
the same x.
-c /rtN dy du , dc
From (2), -f- = — + — ;
dx dx dx
but from the fourth equation above,
dy _du% dx dx'
dc hence, it follows that — = 0. dx
The derivative of a constant is zero.
If the number of functions is infinite, Theorem 4 of Art. 3 may not apply; that is, the limit of the sum may not be equal to the sum of the limits, and hence the derivative of the sum may not be
equal to the sum of the derivatives. Thus the derivative of an infinite series cannot always be found by differentiating it term by term.
11. Differentiation of a product.
Let y = uv, wherein u, v are both functions of x.
Then ^=(U + *UW + *V)-UV = u^ + v^ + ^ . to,. Ax Ax Ax Ax Ax
Now let Aa; approach zero, using Art. 3, Theorems 4, 5, and
noting that if — has a finite limit, then the limit of Avf—)
Ax \AxJ
is zero. '
The result may be written in the form
d(uv) = udv + vdu (3)
doc dx doc
The derivative of the product of two functions is equal to the sum of the products of the first factor by the derivative of the sec- ond, and the second factor by the derivative of the first.
This rule for differentiating a product of two functions may be stated thus : Differentiate the product, treating the first factor as constant, then treating the second factor as constant, and add
the two results.
Cor. To find the derivative of the product of three functions uvw.
Let y = uvw.
By (3), *y = w±(uv)+uv^
dx dx dx
The result may be written in the form
d(uvw)=uvdw + vwdu + wudv (4
doc dx dm dx
By induction the following rule is at once derived :
The derivative of the product of any finite number of factors is equal to the sum of the products obtained by midtiplying the de- rivative of each factor by cdl the other factors.
12. Differentiation of a quotient.
Let y = - , u, v both being functions of x.
u -f Au u Au Av
! V U
A?/ v -\- Av v _ Ax Ax Ax ~~ Ax v(y + Av)
Passing to the limit, we obtain the result
d (u\- dx dx (5)
dx\v J v1
TJie derivative of a fraction, the quotient of two functions, is equal to the denominator multiplied by the derivative of the nu- merator minus the numerator multiplied by the derivative of the
denominator, divided by the square of the denominator.
13. Differentiation of a commensurable power of a function. Let y = un, in which it is a function of x. Then there are three cases to consider :
1. n a positive integer.
2. n a negative integer.
3. n a commensurable fraction.
1. n a positive integer.
This is a particular case of (4), the factors u, v, w, ••• all
being equal. Thus
dy n_xdu
dx dx
2. n a negative integer.
Let n = — m, in which m is a positive integer.
Then y = un = u~m = — ,
«* dl = ^-dl by (5), and Case (1)
dx u2m dx
— mu~m~ idu, dx'
dy _ n-l dtt
dx wit" — • dx
3. 7i a commensurable fraction.
Let n=*-, where p, g are both integers, which may be either
• q
positive or nega
Then y = un = u9 ;
hence if = fir,
i.e. dec cte
Solving for the required derivative, we obtain
dx V dxJ
hence -*-Un = nun-14". (6)
dx, doc
The derivative of any commensurable power of a function is equal to the exponent of the power multiplied by the power | {"url":"https://ia804606.us.archive.org/10/items/elementarytextbo00snydrich/elementarytextbo00snydrich_hocr.html","timestamp":"2024-11-11T10:56:38Z","content_type":"application/xhtml+xml","content_length":"1048984","record_id":"<urn:uuid:7b14be5d-23e1-4c88-98b0-b93305200c4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00853.warc.gz"} |
Liquid Drop Model of Nucleus | Definition, Facts & Uses | nuclear-power.com
Liquid Drop Model of Nucleus
What is liquid drop model of nucleus?
In nuclear physics, the liquid drop model of the nucleus describes forces in atomic nuclei as if a tiny liquid drop formed the atomic nucleus. But in this nuclear scale, the fluid is made of nucleons
(protons and neutrons). The liquid drop model considers that the forces on the nucleons on the surface are different from those on nucleons on the interior, where other attracting nucleons completely
surround them. This is similar to taking into account surface tension as a contributor to the energy of a tiny liquid drop.
Key Facts
• Scattering experiments suggest that nuclei have approximately constant density.
• Nuclei have their volume and surface, where forces act differently.
• In the ground state, the nucleus is spherical.
• If sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a dumbbell shape and then maybe split into two fragments.
• The Weizsaecker formula is an empirically refined form of the liquid drop model for the binding energy of nuclei. It has the following terms:
□ Volume term
□ Surface term
□ Asymmetry term
□ Pairing term
• Using the Weizsaecker formula, the binding energies and also masses of atomic nuclei can be derived. Therefore, we can also derive the energy released per fission.
Liquid Drop Model of Nucleus
One of the first models that could describe the behavior of the nuclear binding energies and therefore of nuclear masses was the mass formula of von Weizsaecker (also called the semi-empirical mass
formula – SEMF) published in 1935 by German physicist Carl Friedrich von Weizsäcker. This theory is based on the liquid drop model proposed by George Gamow.
According to this model, the atomic nucleus behaves like the molecules in a drop of liquid. But in this nuclear scale, the fluid is made of nucleons (protons and neutrons), held together by the
strong nuclear force. The liquid drop model of the nucleus considers that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. Other
attractive nucleons completely surround the interior nucleons. Here is the analogy with the forces that form a drop of liquid.
In the ground state, the nucleus is spherical. If sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a dumbbell shape and then maybe split into two fragments.
Since these fragments are more stable, the splitting of such heavy nuclei must be accompanied by energy release. This model does not explain all the properties of the atomic nucleus but does explain
the predicted nuclear binding energies.
The nuclear binding energy as a function of the mass number A and the number of protons Z based on the liquid drop model can be written as: This formula is called the Weizsaecker Formula (or the
semi-empirical mass formula). The physical meaning of this equation can be discussed term by term.
Nuclear binding energy curve.
Source: hyperphysics.phy-astr.gsu.edu
With the aid of the Weizsaecker formula, the binding energy can be calculated very well for nearly all isotopes. This formula provides a good fit for heavier nuclei. For light nuclei, especially for
^4He, it provides a poor fit. The main reason is the formula does not consider the internal shell structure of the nucleus.
The coefficients a[V], a[S], a[C], a[A,] and a[P] must be known to calculate the binding energy. The coefficients have units of megaelectronvolts (MeV) and are calculated by fitting to experimentally
measured masses of nuclei. They usually vary depending on the fitting methodology. According to ROHLF, J. W., Modern Physics from α to Z0, Wiley, 1994., the coefficients in the equation are the
following: Using the Weizsaecker formula. Also, the mass of an atomic nucleus can be derived and is given by:
m = Z.m[p] +N.m[n] -E[b]/c^2
where m[p] and m[n] are the rest mass of a proton and a neutron, respectively, and E[b] is the nuclear binding energy of the nucleus. From the nuclear binding energy curve and the table, it can be
seen that, in the case of splitting a ^235U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ^235U nucleus.
According to the Weizsaecker formula, the total energy released for such reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.
Table of binding energies fo some nuclides. Calculated according to the semi-empirical mass formula.
The minimum excitation energy required for fission to occur is known as the
critical energy (E[crit])
threshold energy
This table shows critical energies compared to binding energies of the last neutron of a number of nuclei. | {"url":"https://www.nuclear-power.com/nuclear-power/fission/liquid-drop-model/","timestamp":"2024-11-07T22:48:26Z","content_type":"text/html","content_length":"107168","record_id":"<urn:uuid:f30fd905-6f1e-4c0d-8e56-959d98ff080a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00335.warc.gz"} |
The effect of ageostrophy on the stability of thin oceanic vortices
This paper examines the stability of vortices in a two-layer ocean on the f-plane. The mean depth h̄[1] of the upper layer is assumed to be much smaller than the depth h̄[2] of the lower layer. Using
the primitive equations, we derive an asymptotic criterion for baroclinic instability of compensated (i.e. confined to the upper layer) vortices. Surprisingly, it coincides exactly with a similar
criterion derived from the quasigeostrophic equations [Benilov, E.S., 2003. Instability of quasigeostrophic vortices in a two-layer ocean with thin upper layer. J. Fluid Mech. 475, 303-331]. Thus, to
leading order in h̄[1]/h̄[2], ageostrophy does not affect the stability properties of thin compensated vortices. As a result, whether a vortex is stable or not, depends on its shape, not amplitude
(although the growth rate of an unstable vortex does depend on its amplitude).
• Ageostrophy
• Baroclinic instability
• Ocean vortices
Dive into the research topics of 'The effect of ageostrophy on the stability of thin oceanic vortices'. Together they form a unique fingerprint. | {"url":"https://pure.ul.ie/en/publications/the-effect-of-ageostrophy-on-the-stability-of-thin-oceanic-vortic","timestamp":"2024-11-02T05:40:59Z","content_type":"text/html","content_length":"48153","record_id":"<urn:uuid:7d495c6f-194b-42cc-b894-af7128eeda52>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00786.warc.gz"} |
Parameters Measured
This section of the manual lists each of the acoustic measurements VoiceSauce is capable of making, and describes how they are made.
One of the critical measurements made by VoiceSauce is of the fundamental frequency, f0. VoiceSauce uses this measurement to estimate the location of harmonics. VoiceSauce can make measurements of F0
using four different algorithms - Straight (Kawahara et al. 1998), the Snack Sound Toolkit (Sjölander 2004), Praat, or Sun's Subharmonic-to-Harmonic Ratio method. (In VoiceSauce, these are
abbreviated strF0, sF0, pF0, shrF0.) VoiceSauce defaults to the Straight measurements of F0 to locate and measure harmonic amplitudes. Straight is used to find F0 at one millisecond intervals. As of
version 1.28 (December 23, 2016), the Straight algorithm in VoiceSauce is different from in all previous versions - it is now Kawahara's "XSX", taken from his new TANDEM-Straight package. The new
algorithm is implemented here in a way that makes its output very similar, but not identical, to results of the old algorithm (Muliticue/"NDF"). For technical discussion and comparisons of these and
other F0 estimators, see Kawahara et al. (2016); Tsanas et al. (2014).
An occasional console message, at least in older versions of VoiceSauce, is "Multicue failed: switching to exstraightsource". This somewhat cryptic message is related to the pitch estimators in the
Straight package. The "Multicue" pitch estimator is quite complicated and uses cues from various calculations. However, from experience, we have found that it sometimes crashes on certain signals.
When this happens VoiceSauce will switch to the simpler "extrastraightsource" algorithm which is also part of the Straight package. Since the newest Straight does not use the Multicue estimator, we
expect fewer crashes.
The Snack Sound Toolkit (Sjölander 2004) is used by default to find the frequencies and bandwidths of the first four formants. By default, VoiceSauce uses the covariance method, a pre-emphasis of
0.96, and a window length of 25 ms with a frame shift of 1 ms. This frame shift is used to match the f0 estimation by the Straight algorithm.
The formant values calculated by the console version of Snack (which is used when VoiceSauce is run under Windows) vs. the values calculated when Snack is called via the Tcl shell (default for OSX)
can differ noticeably (see https://github.com/voicesauce/opensauce-python/issues/27#issuecomment-316565993 for Terri Yu’s demonstration of some observed differences). This is because the options
implemented in the console version of Snack appear to be a subset of those implemented in the full Snack executable, so the settings are likely to be different. As always, inspection of obtained
values for obvious errors is recommended.
Praat can also be used to estimate formant frequencies and bandwidths. By default, Praat is configured to run with the number of formants set to 4 and the maximum formant frequency set to 6000Hz -
these settings can be changed in the Settings window. As of July 2015, VoiceSauce allows Praat's fractional (x.5) values.
Harmonic and Formant Amplitude
By default, the pitch track obtained from the Straight algorithm (Kawahara et al. 1998) is used to locate the harmonics; any of the other options may be chosen instead in the Settings window. By
default, formant frequencies are obtained from the Snack Sound Toolkit (Sjölander 2004), but Praat can be specified instead in the Settings window.
In traditional FFT analysis, changing the cutting window can change the features of the extracted spectrum. VoiceSauce computes harmonic magnitudes pitch-synchronously over a three pitch period
window (by default; this value can be changed under Settings). This eliminates much of the variability obtained in spectra computed over a fixed time window. The harmonics are located using standard
optimization techniques which locate the maximum of the spectrum around peak locations, as estimated by F0. This is equivalent to using a very long FFT window.
VoiceSauce measures the amplitudes of various harmonics: H1, H2, H4, A1 (the harmonic nearest F1), A2, A3, H2K (the harmonic nearest 2000 Hz), H5K, H1-H2, H2-H4, H1-A1, H1-A2, H1-A3, H4-H2K and
H2K-H5K, as well as corrected versions of all these measures (except for H5k). The individual harmonic amplitudes are not normalized, and will vary with, e.g., overall loudness. For this reason, the
harmonic difference measures like H1-H2 are more commonly used, as they provide a kind of within-token normalization.
Note that the reliability of these measures depends on the successful estimation of their component parameters. If the F0 is not well-tracked, then all the measures that include H1 (or any other
harmonic) will be problematic. Similarly, if one or more formants are not well-tracked, then the corresponding measures will be problematic. Thus if the estimate of F1 is wrong, then A1 and H1-A1
will be wrong too, even for the uncorrected measures. Errors in F1 estimation are especially likely for breathy, nasal, or high-pitched vowels. Obviously, all the amplitude corrections described in
the next section also crucially depend on accurate formant estimation. Therefore it is recommended that the F0 and formant estimates be checked to verify the integrity of the voice measures derived
from them.
Note that there are no formant-corrected version of the 5K measure. This was due to the observation that formant estimation inaccuracies increase significantly with the higher formants. Also, the
sample rate of the input file needs to be higher than 10KHz in order to return a meaningful 5K measure.
Amplitude Corrections
All the harmonic-amplitude voice measurements can be corrected for the effect of formant frequencies, using an algorithm developed by Iseli & Alwan (2004, 2006, 2007). This is done so that voice
parameters can be compared across segments with different formant frequencies, e.g. different vowel qualities. A formant boosts the amplitude of any nearby harmonic(s), so uncorrected values of
harmonic amplitudes reflect both the source and the filter. Uncorrected outputs should be used only when comparing matched speech samples for which the filter functions will be essentially the same.
In some studies, only segments with a very high F1 (e.g. low vowels) and relatively low F0 are used, so that the H1 and H2 frequencies will be well below the F1 frequency and uncorrected H1-H2 should
be unaffected by the formants. Under this method, uncorrected parameters based on harmonics above H2 cannot be used, and most vowel qualities cannot be studied. The alternative of applying the
formant corrections allows any mix of segments (or at least, any mix of voiced oral sonorants), with any formant frequencies, to be combined together for comparisons across the full range of harmonic
amplitude measures. Formant-corrected harmonic-amplitude measures are thus especially important when using natural speech samples in which segment sets cannot be controlled.
Amplitude measures are corrected every frame using the formant frequencies obtained by default from the Snack toolkit, or from Praat if specified by the user (under Settings). Formant bandwidths are
by default calculated by the formula from Hawks & Miller (1995). That is, the formant bandwidths estimated by Snack and Praat and included in VoiceSauce's output are not used by default in the
corrections. Under Settings, the "Bandwidth" setting gives the option of switching to the estimated values of the formant bandwidths - from either Snack or Praat, whichever is used for the formant
frequencies. Here is how the various amplitude measures are corrected:
H1* - uses F1, F2 and B1, B2 (from formula, or calculated)
H2* - uses F1, F2 and B1, B2 (from formula, or calculated)
H4* - uses F1, F2 and B1, B2 (from formula, or calculated)
H2k* - uses F1, F2, F3 and B1, B2, B3 (from formula, or calculated)
H5k - only uncorrected is available
A1* - uses F1, F2 and B1, B2 (from formula, or calculated)
A2* - uses F1, F2 and B1, B2 (from formula, or calculated)
A3* - uses F1, F2, F3 and B1, B2, B3 (from formula, or calculated)
The measures are then smoothed with a moving average filter with a default length of 20 milliseconds. As noted in the previous section, if the estimates of the formant frequencies are not accurate,
these corrections will be problematic.
It has been noted that formant bandwidths calculated by formula, while overall more reliable than using token-specific measurements, can cause large errors in corrections of H4 and H2k when these
happen to be at a formant frequency: the harmonic amplitudes are reduced to too-low values. Using measured bandwidths, if those are reasonable, eliminates this problem. VoiceSauce was modified in
June 2015 to allow such use of measured bandwidths, as described above.
In the literature, corrected measures are indicated with an asterisk. For example, H1*-H2* is the standard way of indicating the corrected form of H1-H2. However, because of limitations imposed by
Matlab, asterisks cannot be used in VoiceSauce's output. Instead, a "c" indicates a corrected measure. Thus, "H1c" is H1*. For clarity, a "u" is used for uncorrected measures. Thus, "H1u" is
uncorrected H1.
Energy refers to the Root Mean Square (RMS) energy, calculated at every frame over a variable window equal to five pitch pulses by default. The variable window effectively normalizes the energy
measure with F0 to reduce the correlation between them.
Ceptral peak prominence (CPP) calculations are based on the algorithm described in Hillenbrand et al. (1994). A variable window length equal to 5 pitch periods is used by default for the
calculations. After multiplying the data with a Hamming window, the data is then transformed into the real cepstral domain. The CPP is found by performing a maximum search around the quefrency of the
pitch period. This peak is normalized to the linear regression line which is calculated between 1 ms and the maximum quefrency.
Harmonic to Noise Ratio
Harmonic-to-noise ratio (HNR) measures are derived from the algorithm in de Krom (1993). Using a variable window length equal to 5 pitch periods by default, the HNR measurements are found by
liftering the pitch component of the cepstrum and comparing the energy of the harmonics with the noise floor. HNR05 measures the HNR between 0-500Hz, HNR15 measures the HNR between 0-1500Hz and HNR25
measures the HNR between 0-2500Hz.
Subharmonic to Harmonic Ratio
The Subharmonic-to-harmonic ratio (SHR) measure is derived from the algorithm in Sun (2002), quantifying the amplitude ratio between subharmonics and harmonics. It therefore characterizes speech with
alternating pulse cycles (period-doubling).
Strength of Excitation
The Strength of Excitation (SoE) measure is derived from the algorithm in Murty and Yegnanaraya (2008), "Epoch extraction from speech signals", also described in detail by Mittal et al. (2014),
"Study of the effect of vocal tract constriction on glottal vibration" in JASA. Measured at "the instant of significant excitation of the vocal-tract system during production of speech", it
represents "the relative amplitude of impulse-like excitation" then. SoE values depend on the signal energy, and so should generally be normalized by forming a ratio between 2 sounds within an
utterance (the sound of interest vs a reference). The SoE measure is accompanied by the "Epoch" parameter, which indicates where each SoE value comes from in the file (to the closest frame, by
default 1 msec - not to the sample point as in the original algorithm). It is important for these parameters that VoiceSauce's frame shift be much shorter than the length of a glottal cycle. The
default value of 1 msec will usually be fine, but for very high F0s the frame shift should be decreased. And, if the frame shift has been changed to a higher value, the epochs will not be recorded
accurately. Note that the Epoch parameter is informative only when the output is shown for every frame - here a "1" indicates an epoch, and only epochs have SoE values. When the value for Epoch is 0,
then the value for SoE is also 0. In contrast, if the output is shown averaged within one or more "sub-segments", the value(s) for Epoch will be meaningless (the mean of some number of 1s and 0s).
However, mean values for the SoE parameter are taken only over non-zero values, so these means are informative. | {"url":"https://phonetics.ucla.edu/voicesauce/documentation/parameters.html","timestamp":"2024-11-08T07:08:55Z","content_type":"text/html","content_length":"18391","record_id":"<urn:uuid:dfb28974-2bdc-4976-a63d-883a6979c58f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00719.warc.gz"} |
Lowering the cost of anonymization
4.3.3 Improving coverage and utility
In Section 4.2, we presented a framework for computing differentially private statistics, and we explained how it could be implemented as an extension of a general-purpose SQL engine. SQL has major
usability advantages: the ability to write SQL queries is more common than the ability to write code in a traditional programming language. The declarative aspect of SQL also gives us a large freedom
to rewrite or optimize the way queries are run, and semantic analysis is much easier to perform on SQL queries rather than in a Turing-complete programming language.
However, more powerful programming languages and computation models also come with a number of distinct advantages. In this section, we quickly present an alternative implementation of our framework
on Apache Beam [17], then we illustrate the possible benefits that can be gained by such an alternative.
Privacy on Beam
SQL is the tool of choice of many data analysts, but many engineers who build pipelines for large-scale systems opt for more powerful computation frameworks, like Apache Beam [17], an evolution of
prior technologies like MapReduce [95]. Apache Beam is a programming model in which a client can describe a data processing pipeline, and run it in a massively parallel fashion without having to
implement parallelization themselves, or worry about technical details like redundancy.
Apache Beam has client libraries written in Java, Go and Python. They allow clients to specify complex data transformations that would be difficult to express in SQL, either because of syntactic
inelegance (e.g. chaining a large number of search & replace operations on a string) or lack of language support (e.g. computations that require loops, or calling an external service). Such examples
are very common for complex data pipelines, which explains the popularity of frameworks like Apache Beam in organizations running large-scale computational tasks. Furthermore, even though frameworks
like Apache Beam have a declarative aspect (high-level API functions do not provide interface guarantees on their execution plan), it is often easier to understand which actual computation tasks are
executed when running a pipeline, allowing engineers to finely optimize their performance if needed.
Migration costs are one of the main obstacles to the wide-scale adoption of privacy technologies: nobody likes to hear that they will need to change their practices and infrastructure entirely to add
a layer of privacy protection, and asking people to do so is usually a losing battle. This is especially the case if their use case requires using a more powerful framework than the one we are trying
to push them towards. Thus, it is crucial to implement differentially privacy features in the tools that people are already using. This is a core reason why we invested in adapting the model we
presented in Section 4.2 to frameworks like Apache Beam. We have done so and published an open-source implementation in Go, available at [320]. In this section, we discuss the advantages of using a
more powerful framework, and the challenges that came with this project.
Let us give an example, freely inspired from the public Codelab for Privacy on Beam [83]. The following Apache Beam pipeline, using the Go SDK, assumes that the input data is a collection of structs
of type Visit, which encapsulate information about the visit of individuals to a given shop.
Suppose we want to compute the total euros spent by weekday. A traditional Apache Beam pipeline would look like the following, assuming that the initial collection is stored in a variable named input
func extractDayAndSpend(v Visit) (time.Date, int) {
return v.Day, v.EurosSpent
// Initialize the pipeline and its scope
p := beam.NewPipeline()
s := p.Root()
input := readInput(s, "path/to/file")
// Extract day and spend from each Visit
extracted := beam.ParDo(s, extractDayAndSpend, input)
// Sum the spend of each user per day
totalSpent := stats.SumPerKey(s, extracted)
// Determine where to results and execute the pipeline
writeOutput(s, output)
runner.Execute(context.Background(), p)
In the code above, input is a PCollection<Visit>: it is conceptually similar to an array, except elements are not directly accessible, as a PCollection can represent a very large collection stored
across multiple machines. Then, extracted and totalSpent are PCollections with (key, value) type <time.Date,int>.
Now, to make this pipeline differentially private, we need to first encapsulate the input into a PrivatePCollection, which will implicitly track user identifiers.
// Encapsulate the input
epsilon, delta := 1.0, 1e-10
privacySpec := pbeam.NewPrivacySpect(epsilon, delta)
pcol := pbeam.MakePrivateFromStruct(
s, input, privacySpec, "VisitorID"
// Extract day and spend from each Visit
extracted := pbeam.ParDo(s, extractDayAndSpend, pcol)
// Sum the spend of each user per day
sumParams := pbeam.SumParams{
MaxPartitionsContributed: 5,
MinValue: 0,
MaxValue: 100,
totalSpent := pbeam.SumPerKey(s, extracted, sumParams)
The MaxPartitionsContributed, MinValue, and MaxValue parameters are conceptually the same as the , L and U introduced in Section 4.2: how many different partitions a single user can contribute to,
and how many euros can they contribute at most in a single partition.
A SQL query is conceptually easy to understand: there is a clear input and output, and it is relatively easy to check that it satisfies all the constraints of our framework (see Section 4.2.3.0). By
contrast, an Apache Beam pipeline can have multiple outputs, and contains transformations with arbitrarily complex logic implemented in a Turing-complete language. Thus, it is much more difficult to
automatically analyze its semantics, or to use rewriting operations like in SQL. This makes the adaptation of our framework difficult.
We solve this problem by introducing PrivatePCollection. The high-level intuition is that a PrivatePCollectionencapsulates a PCollection, and provides a simple yet powerful interface guarantee: all
PCollections that are output from this PrivatePCollection are differentially private, with the parameters and user identifier given during initialization. Not only does this cleanly delineates
between the private and the non-private part, but it also prevents library users from mistakenly copying data that has not been made differentially private yet.
This interface is also easy to understand for programmers already familiar with Apache Beam: PrivatePCollection can be used just like PCollection, with additional constraints to ensure privacy
guarantees. Ideally, an operation is allowed by the interface if and only if it is compatible with our privacy model.
Behind the scenes, a PrivatePCollection<Visit> is actually a PCollection of type <userID,Visit>: we keep track of the user identifier associated with each record, and this user identifier is then
used for all privacy-sensitive operations. This plays the same role as the subquery constraints from Section 4.2.3.0, which also guarantee that each record belongs to a single user. Importantly,
per-user transformations are still allowed: using pbeam.ParDo, the equivalent of beam.ParDo, a client can use arbitrary logic to transform a record, including filtering records out and creating
multiple records from a single one: the outputs of a pbeam.ParDo transformation on a record owned by a given user ID will also be owned by the same user ID.
The power of a programming language like Go is convenient to clients of the library, as it enables them to implement complex transformations. It also presents an opportunity for us to surface more
options to the client, so they can fine-tune the behavior of their differential privacy pipeline and optimize the utility of the data they generate.
We briefly touched on an example in Section 4.3.1.0: in the SQL implementation of our framework, we compute averages as the average of per-user averages, for usability reasons: many aggregations have
the same two options (lower and upper bound), so having a third option for averages would be confusing and error-prone, as SQL options are specified as unnamed arguments to the corresponding
function. When using Apache Beam, this problem is somewhat mitigated: we can use structs with named fields as options for each aggregation, which makes the code easier to read.
The possibilities offered by a powerful framework like Apache Beam do not stop there. In the SQL version of our framework, a single value of was used within all aggregations of a query, and the
privacy budget was shared equally across aggregations. Again, usability is a main reason for these technical choices: surfacing too many options is awkward and confusing. But these choices are not
optimal in general, and Apache Beam offers more customization options so pipeline owners can tweak these values to optimize the utility of their data more finely.
Finally, in our SQL system, the partition selection primitive was silently added next to the other aggregations present in a query. By contrast, in Privacy on Beam, generating a list of partitions
can be a standalone primitive, and differentially private aggregations can use a fixed list of partitions. This is not only useful to manually set the parameters used within this primitive (exact
mechanism, privacy budget, number of partitions each user can contribute to), but also allows an analyst to manually specify a list of partitions that is not data-dependent, and skip thresholding
Gaussian noise
The Laplace mechanism is, by far, the most common way of building differentially private algorithms. Its simplicity makes it particularly easy to reason about and use in a framework like ours: it
provides perfect -DP (with ) for a single metric, it scales linearly with the sensitivity of each aggregation, and with the maximum number of partitions that a single user can contribute to. This
last fact is a direct application from the basic composition theorem (Proposition 5, Section 2.1.6.0), which is known not to be tight [133, 213]. In particular, in [213], authors prove the following
composition theorem, which is tight in general.
Theorem 15 (Theorem 3.3 in [213]). For any and , the sequential composition of independent -DP mechanisms satisfies -DP, with:
for all integers , where:
Thus, a natural option for improving utility is to use this result to split the privacy budget between different aggregations, and between the different contributions of a single user across
partitions in each aggregation. However, this first idea is less attractive than it initially appears. First, it only brings significant utility gains for large values of (larger than 20), which
limits its usefulness in practice, as the maximum number of contributions a single user contributes to is often lower. Second, it is very tricky to implement: one needs to reverse the formula above
to split a given privacy budget, and we very quickly encounter floating-point issues when doing so.
Note that this tight composition theorem is generic: it holds for any -DP mechanism. But in our framework, we really only care about the noise mechanism used in practice. If we can find a better
composition result for Laplace noise, or even change the type of noise we are using, we might get stronger results and pay a smaller complexity cost.
The Gaussian mechanism turns out to be a strong candidate for such an optimization: instead of using noise drawn from a Laplace distribution, the Gaussian mechanism adds Gaussian noise to the result
of the original query. We first introduce this mechanism, then discuss its implementation, and finish by pointing out a few natural open questions.
The Gaussian mechanism was first proposed in [131], and is a fundamental building block to differentially private machine learning [1, 39]. Let us first introduce the concepts necessary to define
this mechanism formally.
Definition 85 (Gaussian distribution). The Gaussian distribution of mean and of variance is the probability distribution with probability density function:
Using the Gaussian distribution provides benefits besides the utility gains we will discuss shortly. This distribution plays a central in many scientific theories and practices, so most data analysts
and engineers are already familiar with it and its basic properties. It also has a much more “predictable” behavior, with tails decaying extremely fast: the probability that a random variable sampled
from a Gaussian distribution of mean 0 and variance ends up outside is less than one in a million.
Crucially, the Gaussian mechanism depends on a different definition of sensitivity than the Laplace mechanism: the sensitivity.
Definition 86 ( sensitivity). Let be a deterministic function. The sensitivity of is the smallest number such that for all datasets and differing only in one record:
where is the norm.
We can now introduce a tight analytical result from [28], that quantifies the variance of the Gaussian mechanism necessary and sufficient to obtain differential privacy based on the -sensitivity of a
Theorem 16 (Theorem 8 in [28]). Let be a deterministic function with finite sensitivity . The mechanism defined by , where is a random vector of where each coordinate is independently sampled from a
Gaussian distribution of mean and with variance , is -differentially private iff:
where is the cumulative distribution function of a Gaussian distribution of mean 0 and variance 1.
This formula looks complex, but it is straightforward to implement a binary-search-based algorithm to find out the lowest possible that gives -DP given the sensitivity .
Importantly, this formula is invariant if and are scaled by the same number: the standard deviation of the noise required for -DP is a linear function of . This insight is the main reason why
Gaussian noise is particularly well-suited to contexts where each user can contribute to many partitions. Indeed, recall that the norm of a vector is defined by:
Each partition corresponds to a particular dimension of : to compute , if a user can contribute to at most partitions, all but of these coordinates are going to be . If the maximum change to each
partition is , then : the magnitude of the noise scales with the square root of the number of partitions contributed to. With typical values of , this makes Gaussian noise a better choice for values
of on the order of or larger.
Implementing Gaussian noise properly presents a few challenges. First, naively implementing this continuous probability distribution on finite computers will also introduce vulnerabilities. This fact
is well-known for Laplace noise [283], but a quick analysis of common random number generation libraries used in various languages shows that the problem would arise for Gaussian noise as well if
implemented naively: to generate 64-bit floating-point numbers, Java uses the Marsaglia polar method [269] with only 53 bits of randomness [299, 300], Python uses the same method [306, 325], Go uses
the Ziggurat method, which only uses 32 bits of randomness 99% of the time [305], etc. Thus, the noisy values will necessarily be sparse within the space of all possible 64-bit floating-point values,
which will create the same vulnerability than the one described in [283]. It is therefore crucial to focus some attention to this problem, and come up with robust implementations that are proven to
uphold -DP guarantees. A handful of solutions have recently been proposed [61, 321, 361], so we assume that this problem can be solved.
Another challenge is that we do not want to have Gaussian noise as the only option in our framework: some applications require pure -DP, and for low values of , using Laplace noise is much better for
utility, even for relatively large values of . So we need to design a system in which we can easily switch one type of noise for another. However, Laplace noise and Gaussian noise use different
sensitivities, which makes this non-trivial. With Laplace noise, we could simply use the standard composition theorem to split the privacy budget across different partitions, and consider each
partition independently. With Gaussian noise however, we have to have a high-level overview of what each aggregation is doing in each partition, we cannot simply rely on splitting the value of in
equal parts.
A natural way to solve this design question is to pass the number of partitions that each user contributes to as a parameter to each aggregator, and pass the value of associated to the entire
aggregation. This way, each aggregator can compute the appropriate value of the noise. Note that yet another subtlety appears here: Theorem 16 assumes that there is a uniform bound on the
per-partition contribution of individual users, but this is not always the case in practice, for example if we use approximate bounds determination (Section 4.2.5.0) per-partition. Luckily, it is
relatively straightforward to extend this result in the case where a user can contribute more in some partitions than others.
Proposition 31 (Generalized analytical Gaussian mechanism). Let be a deterministic function such that for all databases and differing in a single record, and for all :
where denotes the -th coordinate of . The mechanism defined by , where is a random vector of where the -th coordinate is independently sampled from a Gaussian distribution of mean and with variance ,
is -differentially private if for all :
Proof. Note that this mechanism is equivalent to computing , dividing each coordinate by , adding i.i.d. Gaussian noise according to Theorem 16 with , then multiplying each coordinate by .
Once reformulated this way, the result is almost immediate: the first rescaling gives a mechanism of sensitivity , the noise addition step guarantees -differential privacy according to Theorem 16,
and the result follows directly by post-processing. □
Finally, note that the core insight behind using Gaussian noise to improve the utility of differentially private queries is to consider the release of multiple metrics together, as if they were a
single query outputting a single output in , instead of considering them separately. Doing this for a single histogram is natural. An additional possible extension is to make the same reasoning
across entirely different queries. Consider, for example, the following queries.
COUNT(DISTINCT uid)
FROM access_logs
GROUP BY browser_agent
Listing 4.13: Query counting users, grouped by browser agent
COUNT(DISTINCT uid)
FROM access_logs
GROUP BY hour_of_day
Listing 4.14: Query counting users, grouped by hour of day
Both queries have a per-partition sensitivity of , and we might choose different cross-partition contribution bounds for the two queries (say, and ). To make the result of both queries
-differentially private using Gaussian noise, it is natural to split the budget between the two. But for the same reason as before, it is more optimal to consider them as a single query, where each
user can contribute to partitions: this allows us to scale the noise by instead of . This leads to a natural trade-off between utility and design best practices: such optimizations require a
simultaneous view of all queries and the way each of them is implemented. This goes against typical good practices in design, which require compartmentalization of a system into simple parts. | {"url":"https://desfontain.es/thesis/ImprovingCoverageAndUtility.html","timestamp":"2024-11-11T11:07:25Z","content_type":"text/html","content_length":"181964","record_id":"<urn:uuid:74243f26-0719-4747-98db-af8f731c8893>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00190.warc.gz"} |
Divide by 3 [9 different ways]
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In this article, we will go through different ways to divide by 3 in C++ Standard Template Library (STL) and general software techniques. Methods used to divide number by 3 are:
1. Solution using a basic method
2. Solution using fma() library function, works for any positive number
3. A logarithmic solution
4. Solution using iterative way
5. Using Bitwise operator
6. Using File
7. Using div() function from standard library
8. Using counters is a basic solution
9. Using itoa() function
We will dive into each method in depth.
1. Solution using a basic method:
Here we use a simple way of finding divisbility by 3. What we doing here is decrementing the number by 3 and then checking whether it is greater than 3 or not simultaneously after every loop we are
increasing the quotient (i.e. result, here). Hence when the loop reaches the number less than or equal to 3, we will finally get the final increment in result and which wil be out required output.
int div3(int x)
int reminder = abs(x);
int result = 0;
while(reminder >= 3)
return result;
int main()
cout<<"Provide the no.";
return 0;
2. Solution using fma() library function, works for any positive number:
The fma() function takes three arguments a, b and c, and returns (ab)+c without losing precision. The fma() function is defined in the cmath header file. Here we use a method in which we multiply 3
with 1 and add it by 3 until we reach the provided number. Simultaneously we increment the result after every loop. Hence when loop reaches equal or upto number we get our final quotient as our
#include <bits/stdc++.h>
#include <stdio.h>
#include <math.h>
using namespace std;
int main()
int number = 8;//Any +ve no.
int temp = 3, result = 0;
while(temp <= number){
temp = fma(temp, 1, 3); //fma(a, b, c) is a library function and returns (a*b) + c.
result = fma(result, 1, 1);
cout<<"\n\n"<<number<<"divided by 3 =\n"<<result;
3. A logarithmic solution to this would be:
In mathematics a way to find quotient of number divide by 3 is using logarithmic formula i.e. log(pow(exp(numerator),pow(denominator,-1))).
So here we simply apply the logaritmic formula log(pow(exp(number),0.33333333333333333333)) by including <math.h> header file.
#include <bits/stdc++.h>
#include <math.h>
using namespace std;
int main()
cout<<"Provide number";
int result=log(pow(exp(number),0.33333333333333333333))
4. Solution using iterative way:
Using for loop iteration, We will iterate the variable till 3 then will check with the provided number whether it is less than the provided number, we will increment gResult each time. Once we reach
the number we will get our quotient as gResult as our required output.
#include <bits/stdc++.h>
#include <stdio.h>
#include <math.h>
using namespace std;
int main()
int aNumber = 500;
int gResult = 0;
int aLoop = 0;
int i = 0;
for(i = 0; i < aNumber; i++)
if(aLoop == 3)
aLoop = 0;
cout<<"Result of" aNumber/ 3 <<"="<< gResult;
return 0;
5. Using Bitwise operator:
Here the function performs the same sum function, but it requires the + operator, so all you have left to do is to add the values with bit-operators. The left-shift and right-shift operators are
equivalent to multiplication and division by 2 respectively.
// replaces the + operator
int add(int x, int y)
while (x) {
int t = (x & y) << 1;
y ^= x;
x = t;
return y;
int divideby3(int num)
int sum = 0;
while (num > 3) {
sum = add(num >> 2, sum);
num = add(num >> 2, num & 3);
if (num == 3)
sum = add(sum, 1);
return sum;
6. Using File:
In this method we applied mathematical formulae of divison by creating file.
The 'fwrite' writes number bytes (number being 123456 in the example above).
The 'rewind' resets the file pointer to the front of the file.
The 'fread' reads a maximum of number "records" that are divisor in length from the file, and returns the number of elements it read.
If you write 30 bytes then read back the file in units of 3, you get 10 "units". i.e. 30 / 3 = 10.
#include <bits/stdc++.h>
#include <stdio.h>
#include <stdlib.h>
using namespace std;
int main()
FILE * fp=fopen("temp.dat","w+b");
int number=12346;
int divisor=3;
char * buf = calloc(number,1);
int result=fread(buf,divisor,number,fp);
printf("%d / %d = %d", number, divisor, result);
return 0;
7. Using div() function from standard library:
Here we used standard div() function from standard library. It returns the integral quotient and remainder of the division of number by denom ( number/denom ) as a structure of type div_t, ldiv_t or
lldiv_t, which has two members: quot and rem.
#include <bits/stdc++.h>
#include <stdio.h>
#include <stdlib.h>
using namespace std;
int main(int argc, char *argv[])
int num = 1234567;
int den = 3;
div_t r = div(num,den); // div() is a standard C++ function.
return 0;
8. Using counters is a basic solution:
Here first the number is equated after every increment . Simultaneoulsy, after every third check we increment result till we reach the number. Hence we will get result as our quotient and also our
required output.
int DivBy3(int num) {
int result = 0;
int counter = 0;
while (1) {
if (num == counter) //Modulus 0
return result;
counter = abs(~counter); //++counter
if (num == counter) //Modulus 1
return result;
counter = abs(~counter); //++counter
if (num == counter) //Modulus 2
return result;
counter = abs(~counter); //++counter
result = abs(~result); //++result
9.Using itoa() function:
itoa() is a standard C++ function. itoa function converts integer into null-terminated string. Here, using itoa to convert to a base 3 string. Drop the last trit and convert back to base 10.
int div3(int i) {
char str[42];
sprintf(str, "%d", INT_MIN); // Put minus sign at str[0]
if (i>0) // Remove sign if positive
str[0] = ' ';
itoa(abs(i), &str[1], 3); // Put ternary absolute value starting at str[1]
str[strlen(&str[1])] = '\0'; // Drop last digit
return strtol(str, NULL, 3); // Read back result
All the ways discussed above implements different ways to divide by 3 in C++ Standard Template Library. | {"url":"https://iq.opengenus.org/divide-by-3/","timestamp":"2024-11-02T21:51:22Z","content_type":"text/html","content_length":"65585","record_id":"<urn:uuid:d4f18f22-44e7-437c-a07a-c438612d687f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00309.warc.gz"} |
WeBWorK Standalone Renderer
Using a graph, it can be shown that the approximate $x$-coordinates of the points of intersection of the two curves $y=x^4$ and $y=3x-x^3$ are $x = 0$ and $x \approx 1.17.$ Use this information to
approximate the area of the region bounded by these curves.
Make sure your answer (based on the estimates given above) is correct to two decimal places.
Area = | {"url":"https://wwrenderer.libretexts.org/render-api?sourceFilePath=Library/UCSB/Stewart5_6_1/Stewart5_6_1_34.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&answersSubmitted=0&showSummary=1&displayMode=MathJax&language=en&outputFormat=nosubmit","timestamp":"2024-11-11T15:04:41Z","content_type":"text/html","content_length":"5173","record_id":"<urn:uuid:8a9e9626-1105-4d84-9124-21ad90d23f8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00448.warc.gz"} |
Where to Find Dividend Growth RatesWhere to Find Dividend Growth Rates - The Dividend Guy Blog
On this blog, and other dividend investing blogs, we talk a lot about the dividend growth rate of the various companies we track. As part of good dividend analysis determining what the dividend
growth rate has been is important, primarily because we need to see if the growth rate is accelerating or slowing. I have some suggestions for you to help determine these growth rates – the first one
more difficult than the second but depends on the data available for the stock you are examining.
However, before we get into finding the data here is a definition of the dividend growth rate from Investopedia:
The annualized percentage rate of growth that a particular stock’s dividend undergoes over a period of time. The time period included in the analysis can be of any interval desired, and is calculated
by using the least squares method, or by simply taking a simple annualized figure over the time period.
Method #1: Manually Calculating the Dividend Growth Rate
The first method I am going to present to determine the dividend growth rate is manual. It requires some sleuthing to get the dividends paid per year for a company. The best place to find this date
that I have found is on the company website for each company. If you get the most recent annual report, you should be able to find the dividends paid per year for at least the past 5-years. Each
company presents this data differently (i.e. number of years) however most companies that have strong dividend growth records are proud to highlight their dividends and present as much data as
Once this data has been obtained, I then head over to Investopedia to use their handy Compound Annual Growth Calculator (CAGR). This tool requires three data inputs: the ending value or the most
recent dividends per year number, the beginning value or the earliest dividends per year number, and finally the number of periods you have the data for. For example, let’s say in 2004 Coca-Cola’s
dividends per year was $0.88 and in 2007 it was $1.36. Using the calculator to input the data for the three year period, we get a dividend growth rate of 15.62%.
The benefit of doing the dividend growth rate calculations using this manual method is that you can run the numbers for various periods. For example, you can calculate the CAGR for 10-years, 5-years,
and 2-years. With this data you will know if the dividend growth is accelerating or decelerating. This is a crucial piece of analytical data to fully understand the health of the company. Obviously,
as investors we prefer to see that growth rate accelerating.
Method #2: Using Website to Obtain the Dividend Growth Rate
The second method is quicker but not as informative, because it does not given the investor the ability to obtain the dividend growth rate for various periods of time. This method involves simply
visiting websites to obtain a static number for the dividend growth rate. One good website for this is Morningstar. In their stock analysis section, they have a page called ‘Dividend Interpreter’
that provides a 5-year dividend growth rate. Again, this is a quick and dirty way to get a number, but if you are weeding out stocks with the highest growth rates, then this is a good place to start.
I hope this helps you in your dividend analysis of the dividend stocks you are interested.
if you are looking for more dividend stock information & analysis, you can subscribe to my free newsletter:
1. Hi,
Reuters.com also provides the 5 year dividend growth rate under their “ratios” section. That is where I get information and comparisons between stocks and their respective industry averages.
Hope this helps.
2. Cool – thanks Tyler. I have never seen that section of their site before.
3. thanks for that investopedia compounding calculator…great tool. Happy Father’s Day!
4. […]The Dividend Guy showed Where to Find Dividend Growth Rates.[…]
5. If anyone is interested in the math behind calculating CAGR…Using the example above of Coca-Cola: “2004 Coca-Cola’s dividends per year was $0.88 and in 2007 it was $1.36”
1. 1.36/.88 = 1.54 (Dividends grew 154% over the 3 year span)
2. (1.54 ^.33)-1 = .15615 = 3 Year CAGR (.33 1/#years, in this case 1/3)
This is useful for calculating growth rates for returns, eps, revenue, etc.
The Investopedia definition mentions using the least squares method, which is more complicated.
Leave a Reply Cancel reply | {"url":"https://thedividendguyblog.com/where-to-find-dividend-growth-rates/","timestamp":"2024-11-11T02:50:04Z","content_type":"text/html","content_length":"109226","record_id":"<urn:uuid:c5b12e50-f22f-4509-b224-d155cd12c2b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00483.warc.gz"} |
Swirling flow field reconstruction and cooling performance analysis based on experimental observations using physics-informed neural networks
Swirling flow could always be found in the combustor of gas turbine or aircraft engine (Figure 1). Generated by the swirler, swirling flow is capable of improving the atomization condition, and
stabilizing the combustion. Inevitably, the high thermal loads associated with the swirling jet often cause ablation of the combustor endwall, necessitating the design of effective thermal protection
modules (Wurm et al., 2009, 2013). While the effective thermal protection modules are highly determined based on the swirling flow field. The accurate measurement of the swirling flow field is
therefore crucial for the diagnosis of combustion stabilization and thermal protection. Furthermore, an accurate flow field is also important for optimizing combustion processes, improving
efficiency, and reducing emissions.
Numerical investigations have been widely conducted from various perspectives (Xia et al., 1998; Spencer et al., 2008; Xiao and Zheng, 2014; Andrei et al., 2015). Although the numerical results
revealed valuable three-dimensional mechanism of swirling flow to some degree, it was found that the numerical results weren’t always satisfactory. The recirculation zone, and swirling jet were
always grossly predicted due to their inherent anisotropy (Jones et al., 2005) and the isotropic nature of the Boussinesq hypothesis used in numerical studies. In addition, another challenge for
numerical simulation was that it was always difficult to obtain the expected results when the boundary conditions were unknown or deviated from the reality, which would always be applied when the
boundary conditions were highly complex. Therefore, experimental investigations such as hot wire (Vu and Gouldin, 1982), laser Doppler velocimetry (Brum and Samuelsen, 1987), laser anemometry (Syred
et al., 1994), and planar particle image velocimetry (PIV) (Andreini et al., 2014; Lenzi et al., 2022) could provide more accurate and reliable reference data, without knowing the boundary
conditions. Besides for the flow field diagnosis, the cooling performance diagnosis of the common cooling design in the combustor, such as effusion cooling, slot cooling, was also widely conducted
because of its most intuitive display of the cooling performance affected by the swirling flow (Andreini et al., 2014, 2015, 2017). In recent years, there have been researches investigating the
effect of swirling flow on the cooling performance of cooling design based on the experimental observations, which have unveiled valuable flow and heat transfer characteristics (Jiang et al., 2022;
Lenzi et al., 2023). However, the observations are always sparse or planar patterns due to the methodological limitations, resulting in the decreased three-dimensional analyticity. Therefore,
leveraging experimental observations of swirling flow field would be beneficial for both scientific and industrial purposes. Inferring three-dimensional turbulent flow field through two-dimensional
observations is a highly nonlinear mapping issue. Artificial neural networks, currently, are widely adopted fitting nonlinear functions for its outperformed capability to represent nonlinear mapping.
To address this issue, Physics-informed neural networks (PINNs) have been proposed as a powerful approach to reconstruct three-dimensional flow fields from sparse or planar observations.
Designed by Raissi et al. (2018), PINNs have been widely developed in resolving inverse problem for various scenarios, such as material sciences (Lu et al., 2020; Shukla et al., 2020), chemistry (
Pfau et al., 2020; Ji et al., 2021), geophysics (Li et al., 2020; Weiqiang et al., 2021), and topology optimization (Lu et al., 2021). PINNs also bring attentions to the field of flow and heat
transfer for various scenarios. By inserting the prior knowledge of fluid mechanics such as continuity equation and Navier-Stokes (N-S) equations into the PINNs, the quantities in the domain field
could be regressed. PINNs have shown the capabilities of leveraging the limited observations to reconstruct the laminar flow field. (Raissi et al., 2020; Cai et al., 2021a,c). For the turbulence
problem, the Reynolds-averaged Navier–Stokes (RANS) method is adopted generally because of its higher efficiency and better convergence properties in numerical simulations. However, the derived term
Reynolds stress, is unclosed yet, leading to the increased nonlinearity when resolving inverse problems for PINNs. Currently, the researches on RANS method are limited. Eivazi et al. (2022) evaluated
the PINNs’ performance on the two-dimensional RANS equations without importing any turbulent viscous models. Cases of airfoil, periodic hill, etc. were applied, which validated that PINNs were
capable of mining the latent mechanics of the turbulent model. Von Saldern et al. (2022) adopted the axis-symmetric swirling jet data and imported simplified Boussinesq hypothesis to evaluate PINNs’
regression on axis-symmetric RANS equations. They found that the velocity components were matched with the experimental results while the Reynolds stress components were poorly estimated. More
complex cases of jet in crossflow were investigated by Huang et al. (2023), they designed the 2-rank tensor-basis eddy viscosity (t-EV) model to better fit PINNs’ structure and represent the
anisotropy, effectively improving PINNs’ regression. It’s found from the above evidence that the investigated case is relatively simple and most of the training datasets are extracted from the clean
numerical simulation. The performance of PINNs on predicting complex turbulent flow field haven’t been deeply evaluated, especially based on the experimental observations. Meanwhile, the observations
from above researches only contain flow information. For the swirling flow in the combustor, scalar field is equally important as flow field. It’s believed that leveraging both scalar and flow field
could further enhance the reconstruction performance, which also hasn’t been investigated yet.
The present work aims to reconstruct the complex swirling flow field based on limited experimental observations using PINNs. The effects of datasets on reconstruction are discussed preliminarily so
that an effective sampling strategy is designed. Furthermore, the multi-source strategy would be adopted, where the scalar observation is introduced to improve PINNs’ prediction. Finally, the
reconstructed vortex distributions would be shown to provide the evidences of the cooling performance destruction under swirling flow. The investigations in this study would provide the alternative
to obtain the three-dimensional swirling flow field leveraging experimental observations with deep learning and deepen the understanding between the film cooling performance and swirling flow from a
novel method, which is potentially beneficial for the cooling structure improvement in the future.
Multilayer perception (MLP)
MLP is one of the essential parts of PINNs, which is a fully connected neural network. When MLP is being trained, the known input and output would be fed as the labels so that the parameters in
neural network would be optimized to minimize the loss functions. Activation functions are added to increase the nonlinearity of the mapping. In this study, the mapping is expressed as:
where uι¯, p, and uι′uȷ′ represent the mean velocity components, pressure, and Reynolds stress, respectively. Previous study (Huang et al., 2023) has provided the in-depths information about the
determination of hyperparameter and MLP structure on solving RANS problems. In this study, 10 hidden layers, each containing 300neurons is prescribed in MLP structure (Figure 2). Swish function is
selected as the activation function, expressed as follows:
Loss function
The loss functions of PINNs consist of two parts: loss functions of physics constraints and experimental observation. To obtained the mean flow field, the RANS equations are preparedly designed into
the loss function of physics constraints:
The derived term uι′uȷ′¯, traditionally, is closed by semiempirical model based on the Boussinesq eddy viscosity hypothesis:
where S[ij] is the mean rate of strain tensor, νt is the eddy viscosity, and K is the turbulent kinetic energy (TKE). The assumption of isotropic νt is considered as the causes that lead to the
unsatisfactory numerical results. Meanwhile, the application of Eq. 5 in PINNs couldn’t perform well in previous work. Huang et al. (2023) designed the t-EV model in consideration of the anisotropy.
This model has been validated its capability of well-predicting jet in crossflow field. Therefore, the modified symmetric 2-rank anisotropic eddy viscosity νt(ij) is imported in this study so that
Eq. 5 could be rewritten as
where νt(ij) is symmetrically expressed as:
Thus, another six formulations would be designed into the loss function of physics constraints:
Note that the derivatives in these PDEs are obtained by automatic differentiation (AD) (Baydin et al., 2018). Therefore, the derivatives of these PDEs can be calculated by applying the chain rule
under backpropagation. To satisfy these physical equations, the loss functions of physics constraints should approach zeros, which is formulated as
where N[e] is the number of physical equations. N[d] is the domain points in the selected regions.
For the observation loss functions, the two-dimensional and two-component (2D2C) results from PIV measurements are chosen as the labels, containing 2D2C mean velocities and their corresponding
Reynolds stress. Furthermore, the non-slip boundary condition is added as known labels. Therefore, the loss function of the observations is reported as
The total loss then written by summing the weighted loss of above loss functions:
where ωe and ωobs are the weighting coefficients. In this study, both of the weighting coefficients are set as the same because of the equal importance. Tensorflow (Abadi et al., 2016) is adopted as
the deep-learning package to construct PINNs model. Adam optimizer (Kingma and Ba, 2014), with default hyper-parameters is chosen to minimize the loss function due to its efficient optimization
algorithm. The batch sizes for both loss functions of physical equations and experimental observations are set as 20,000. A single NVIDIA Tesla V100s GPU is chosen for PINNs’ training. The losses
during training are depicted in Figure 3, where three samples preliminarily serve as training datasets. When the loss was found to be stable, the PINNs’ regression would be regarded as the
convergence. Generally, it’s observed that the losses tend to be stable after 30,000.
Multi-sources strategies
For the swirling flow in the combustor, scalar field is equally important as flow field. As the key variable to evaluate the film cooling performance, film cooling effectiveness was relatively
measurable in the scalar field. Therefore, besides for the source from flow observation, the source from scalar field would also be imported to further improve PINNs’ regression. In this study, the
scalar (i.e., film cooling effectiveness) observation of effusion plate is supplemented. The scalar distribution, as shown in Figure 4, is experimentally obtained by pressure-sensitive paint (PSP)
measurement (Jiang et al., 2022). The PSP molecules could return to the ground state via the oxygen quenching process thus the cooling effectiveness could be acquired according to the calibrated
luminosity-pressure correlation. The adopted 1,000 pairs of snapshots were captured for each case with a frequency of 1,800Hz. The uncertainty of the PSP measurement was calculated to be around
The scalar equation is thus added into the loss function of physical equations:
where Pr is the Prandtl number, set as 0.71 in this study. uι′c′¯ is the turbulent scalar flux. Similarly, the derived term uι′c′¯ is also unclosed. In this study, the simple gradient diffusion
hypothesis (SGDH) is adopted:
where Prt is the turbulent Prandtl number, set as 0.85 in this study. To better characterize the anisotropy, the isotropic νt is substituted as νt(ij). Therefore, other three physical equations are
Dataset establishment
The dataset is established employing the PIV measurement (Jiang et al., 2022), where 1,000 pairs of snapshots were captured for each case. The spatial resolution of each snapshot was 0.023mm/pixel
with a frequency of 1Hz. The uncertainty of the PIV measurement was calculated to be around 3%. As shown in Figure 5, seven observations were accessed during the measurement, containing two planes
at XoZ direction (Y=5 and 70mm) and 5 planes (Z=0, ±25.25, and ±50.5mm) at XoY direction. In subsequent study, the coordinate is normalized, with a measured length L=25.25mm. To find out
PINNs’ performance based on various number of observations, several training datasets are designed, as listed in Table 1. The two planes at XoZ direction are adopted throughout all datasets while the
planes at XoY direction are increasingly sampled.
Results and discussion
Effect of sampling strategy on field reconstruction
Number of observations
In this section, we focus on investigating the effect of the number of observations on reconstructing swirling flow, with the streamwise velocity as the main variable of interest. Following the
sampling strategy employed in previous work (Cai et al., 2021b), sample 1 consists of four planes surrounding the regions of interest. However, unlike the well-reconstructed fields obtained in
previous work, the performance of PINNs in predicting swirling flow fields is poor. As shown in Figure 6, (a) depicts experimental observations from PIV measurement, while (b) represents predictions
made by PINNs using sample 1. It’s observed that the test dataset fails to capture the characteristics of the swirling flow, especially at the exit of the swirler, where the swirling jet should have
emerged. Owing to the supplement of plane 1, the region where jet hit on the effusion plate could be predicted, but presented with shrunk area. This suggests that the vanilla sampling strategy is
insufficient in leveraging the limited information provided by sample 1 to reconstruct complex swirling flow fields. It is therefore suggested that increasing the number of observations in the
training dataset could improve the performance of PINNs. Subsequently, sample 2 and 3, which include additional information on the swirling flow field, will be adopted in the subsequent study.
Figure 7 presents the predictions of PINNs using sample 2, where planes 4 and 5 serve as the test dataset. Compared to the predictions from sample 1, the overall performance shows improvement. Figure
6(a) shows that the jet at the exit of the swirler is captured for plane 4, although the velocity magnitude is weakened. However, a failed prediction is obtained for plane 5 in Figure 7b. The
magnitudes of the recirculation zone and swirling jet at the swirler’s exit are much weaker than those of the experiments, as marked with red lines. In contrast, a better prediction is achieved when
PINNs are trained with sample 3, as shown in Figure 8. The predicted streamwise velocity distribution of plane 3 shows satisfactory agreement with that of the experiment, including the recirculation
zone, the jet at the swirler’s exit, and near the effusion plate. Furthermore, an improved denoise on the distribution could also be observed. Thus, the increasingly improved predictions suggest that
compared to simple turbulent flow, swirling flow requires a higher number of fed observations to better capture the three-dimensional and complex swirling flow characteristics using PINNs.
Effective sampling strategy
From the results above, it can be concluded that the jet at the swirler’s exit is the most sensitive region to sampling strategies. In addition, Karniadakis et al. (2021) revealed that PINNs models
often struggled on penalizing the PDEs residuals when tackling the solution with steep target gradients. Therefore, the regions with steep velocity gradients, such as the swirling jets, brought by
the complex vortex structures and intense shearing would similarly increase the difficulty in PINNs’ accurate regression. Therefore, supplementing partial information at these sensitive regions can
improve the prediction of PINNs. To improve the performance of PINNs, sample 1, which performed the poorest, was selected for investigation. In the updated sampling strategy, partial information
containing the jet at the exit of the swirler was imported for planes 6 and 7 in the XoY direction, as shown in Figure 9a. The updated sample was named as sample 4. The predictions of PINNs with
sample 4, shown in Figure 9b, indicate a significant improvement in the prediction of the unknown zones, such as the jet near the effusion plate and recirculation zones, compared with the poorly
performing sample 1. However, it is also observed that the prediction of the jet highly close to the effusion plate is still unsatisfactory, where the streamwise velocity impinging onto the effusion
plate is weakened. Statistically listed in the Table 1, compared with the poorly performed sample 1, sample 4 had the training volume of 106,308, where the additional observation of sample 4 reduced
the numbers by 50% and 75%, respectively, compared with sample 2 and sample 3. To quantitatively analyse the PINNs’ performances in the test dataset of, relative L[2] error is imported as the
metrics, reported as:
Sample 1 and 4 were chosen as the comparison due to the same test dataset. It was found that the relative L[2] error of the test dataset has been effectively dropped from 0.84 to 0.24, with an
reduction of 71.4%. Therefore, it can be concluded that the performance of PINNs for the prediction of complex swirling flow can be improved by increasing the number of observations, particularly in
sensitive regions, to better capture the three-dimensional and complex characteristics of swirling flows.
Multi-source strategy
The scalar distribution of the effusion plate (shown in Figure 4) highlights the complex interaction between the effusion film and swirling flow. The low FCE zone is attributed to the mixing of
swirling jets with the film, resulting in film coverage destruction. Supplementing observations of scalar sources prompts the PINNs to extract latent flow information and improve their performance.
The investigation focuses on samples 2 and 4. In sample 2, the results in Figure 10 reveal improvements in the predicted flow below Y/L=1. The regions marked by the red lines show more agreement
with experimental observations, indicating a better distribution compared to the single-source strategy. For Plane 4 (Figure 10a), the marked zone is corrected as a higher velocity zone. While for
plane 5 (Figure 10b), although the high streamwise velocity at the swirler outlet is not predicted accurately, the jet’s majority at the exit of the swirler agrees well with the experiments,
indicating a better distribution than the single-source strategy. Note that the improvements propagating vertically are clearly reduced after Y/L>1, where only limited correction could be observed,
and the streamwise velocities of recirculation zone are still ill-matched with that of experimental observations. For sample 4, as shown in Figures 11a and 11b, the improvement is also observed,
mainly for the marked regions near the effusion plate. To quantitatively analyse the amelioration provided by the multi-source strategy, the streamwise velocity profiles at the ill-performed marked
region are shown in Figure 11c. It’s found that the more upstream zone shows the more noticeable improvement. For plane 5 (Z/L=1), whose ill-prediction is concentrated in the downstream zone (
Figure 9), the mean error of streamwise velocity weakly reduces from 18.7% to 13.5% after adopting multi-source strategy. While for planes 3 (Z/L=0) and 4 (Z/L=−1), the improvements are obvious,
where the mean errors are reduced from 18.1% to 6.9% and 6.8% to 1.6%, respectively. Note that the improvements for planes 5 and 3 near the jet impingement zone (Y/L<0.5) are limited to some
extents due to the insufficient downstream observations, suggesting that a satisfactory reconstruction of swirling flow field requires both designed multi-source and sampling strategies.
To show the three-dimensional analysability of the reconstructed flow field, PINNs regression based on sample 4 and the multi-source strategy is selected as the investigation. As shown in Figure 12,
the vortex structures are extracted from the predicted velocity field based on the Q criterion with a normalized value of 0.113. Note that only half of the vortex can be predicted because the PIV
experiments only captured the half of the swirling flow field. The shown vortexes are found as a ring-like pattern similar with the previous literature (Chen and Driscoll, 1989; Oberleithner et al.,
2011), named as recirculation vortex or recirculation bubble. The contours of vortexes in Figures 12a and 12b are marked by vertical velocity of Y direction and lateral velocity of Z direction,
respectively. The black streamlines indicate the general swirling direction of swirling flow. The adiabatic cooling effectiveness is also shown. It’s found from the vortex structure distributions
that the vortexes are mainly captured near the exit of the swirler (around 0<X/L<2.5), corresponding to the direct impingement region of the swirling jet in Figure 11b. From the contour in Figure
12a, it’s observed that the lateral velocity near the effusion plate is shown as negative value, indicating that the swirling flow would induce the effusion film deviating towards the opposite
direction of the Z-axis. Therefore, the cooling effectiveness distribution presents the swept pattern. The region marked with the dashed red rectangle shows the obvious deviation of the cooling
effectiveness distribution because of the strongest sweeping effect brought by the closest interaction of swirling flow. As the swirling flow develops downstream (2.5<X/L<4.5), there are still
vortexes be recognized and marked as the negative lateral velocity, meaning that the swirling flow could continuously affect the effusion cooling and deviate the cooling effectiveness distribution.
It’s then found from Figure 12b that both positive and negative values of vertical velocity are found as balanced distributions. The region with positive and negative value indicates that the
swirling flow go through the upwash and downwash motions, respectively. It’s found near the exit of the swirler that the vertical velocity distributed as the negative value, revealing that the
swirling flow downwards impacts and damages the surface of the effusion film, reducing the cooling effectiveness. Similarly, the region marked with the dashed red rectangle still shows the most
severe deterioration of cooling performance than the surroundings due to the most direct impingement of swirling flow. Due to the occlusion of the vortex structures, it could be observed combining
with Figure 4 that the effusion films at Z/L<−1 are less destructed due to the upwash motion of swirling flow. As the swirling flow develops downstream, the recognized vortexes are marked as weakly
positive values. It is revealed that the swirling flow is slightly lifted off, meaning that the destructions on the cooling performance are gradually attenuated, thus leading to the recovery in
cooling effectiveness.
In this study, the reconstructions of swirling flow field were conducted using PINNs fed by limited observations, where the observations of velocity and scalar were experimentally obtained. The
sampling and ameliorated strategies were mainly investigated. Effects of the number of observations was firstly evaluated, it was found that the reconstruction improved as the number of observations
increased. Based above, zones of swirling jet show the high sensitivities towards number of observations. To optimize the dataset, the effective sample 4, containing the partial swirling jets with
complex vortex structures and intense shearing (i.e., steep velocity gradients region) was designed by reducing the additional observation by 50% and 75% compared to sample 2 and 3, respectively,
still resulting in a well-reconstructed flow field. Compared with sample 1, the relative L[2] error of sample 4 showed an effective reduction of 71.4%.
To further improve the PINNs’ prediction, the multi-source strategy was employed by supplementing the observations from the scalar field into PINNs. Additional prior knowledge of scalar
transportation was also supplemented into the loss functions. The sample 2 and 4 were selected as the investigations. It was found that the prediction of swirling jet for sample 2 was obviously
improved. For sample 4, statistically compared with the previous, the mean errors of streamwise velocity at the marked region were reduced by 61.9%, 76.5%, and 27.8% for plane 3, 4, and 5,
respectively. The supplementary scalar distribution was assumed to strengthen the latent information of the complex interactions between swirling flow and the effusion film.
Finally, the three-dimensional analysis of the reconstructed flow field unveiled that swirling flow vortex structures presented a crucial role in cooling effectiveness. It was found that these
vortexes, especially near the swirler exit, significantly impacted the cooling performance, inducing a swept and deteriorated pattern in the cooling effectiveness distribution. As the swirling flow
developed downstream, the vortexes’ influence on cooling effectiveness gradually attenuated, leading to a recovery in cooling performance. From the engineering perspective, it is advisable to
allocate a greater proportion of coolants or design a more effective cooling structure to the region near the swirler exit, especially near the central side following the swirling flow direction.
The investigation in this study explored the method on reconstructing complex turbulent flow field using deep learning based on the experimental results, which provided the alternative for diagnosing
the complex flow and ameliorating flow field. This reconstructed turbulent flow and vortex field highlighted the intricate relationship between vortex structures and cooling effectiveness, which
intended to deepen the understanding of effusion cooling under swirling flow condition from a novel method. Considering that the investigated region was the swirling flow field, the more complex
phenomena haven’t been captured accurately, such as the chemical reactions and the interactions near the wall between swirling flow and effusion film. In addition, the current study still required a
large volume of data for a satisfactory flow field reconstruction. Therefore, the methodology proposed in this study was not intended to substitute the numerical simulation or other three-dimensional
experimental diagnosis. It was aimed to propose one solution to leverage the existing but limited observations to potentially improve the flow field analysability for both scientific and engineering
purposes. Future studies would focus on finer reconstruction of the complex turbulent wall-bounded flow. Furthermore, due to the PINNs regression is a case-by-case process. Future studies would also
focus on the acceleration for PINNs with more efficient designed deep learning structure. | {"url":"https://journal.gpps.global/Swirling-flow-field-reconstruction-and-cooling-performance-analysis-based-on-experimental,185745,0,2.html","timestamp":"2024-11-11T00:56:16Z","content_type":"application/xhtml+xml","content_length":"169935","record_id":"<urn:uuid:4037d44e-39f1-4bed-8f7f-b1fa66fc9aea>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00371.warc.gz"} |
Introduction to Thermal Boundary Layer in Circular - Convection Heat Transfer - Heat Transfer | Video Summary and Q&A | Glasp
Introduction to Thermal Boundary Layer in Circular - Convection Heat Transfer - Heat Transfer | Summary and Q&A
6.6K views
May 23, 2023
Introduction to Thermal Boundary Layer in Circular - Convection Heat Transfer - Heat Transfer
This video explains the two cases of thermal boundary layer in a circular pipe, where the surface temperature is either higher or lower than the fluid temperature.
Key Insights
• 😘 The thermal boundary layer in a circular pipe varies depending on whether the surface temperature is higher or lower than the fluid temperature.
• ✋ In both cases, the temperature gradients are higher at the boundary and decrease towards the center of the pipe.
• 💁 A fully developed thermal boundary layer is formed at a certain distance from the entry of the pipe.
• 💠 The temperature profile follows a parabolic shape when the surface temperature is lower and a decreasing exponential shape when the surface temperature is higher.
click the bell icon to get latest videos from equator hello friends we have seen in conviction that there are velocity boundary layer and thermal boundary layer now let us consider the thermal
boundary layer in case of a internal flow that is pipe the thermal boundary layer in circular pipe there are two cases that one must consider first case wher... Read More
Questions & Answers
Q: How does the temperature profile look like in a circular pipe when the surface temperature is lower than the fluid temperature?
When the surface temperature is lower, the temperature gradients are higher at the boundary and decrease towards the center of the pipe. The temperature profile follows a parabolic shape, with the
maximum temperature at a certain point.
Q: What happens to the temperature profile in a circular pipe when the surface temperature is higher than the fluid temperature?
In this case, the fully developed flow has higher temperature at the wall of the pipe and decreases towards the center. The temperature profile forms a decreasing exponential shape, with a minimum
temperature reached at a certain point.
Q: What is the thermal entry length in a circular pipe for laminar flow?
The length of the thermal entry region in laminar flow can be determined using the empirical relationship le = 0.0575 RdT * PRd, where Rd is the Reynolds number and PRd is the Prandtl number.
Q: How is the thermal entry length calculated in turbulent flow?
For turbulent flow, the length of the thermal entry region is given by le = 0.0575 Re D * D, where Re is the Reynolds number and D is the diameter of the pipe.
Summary & Key Takeaways
• The video discusses the thermal boundary layer in circular pipes, focusing on two cases: surface temperature higher than fluid temperature and surface temperature lower than fluid temperature.
• In the first case, before the entry of the pipe, the velocity profile remains constant while the temperature varies. As we move towards the thermal entry region, the temperature gradients
increase at the boundary and decrease towards the center of the pipe. At a certain distance, a fully developed thermal boundary layer is formed with a parabolic temperature profile.
• In the second case, the temperature profile initially has lower temperature gradients at the boundary, but as the flow becomes fully developed, the temperature at the wall of the pipe increases
while it decreases towards the center. A minimum temperature is reached at a certain point.
Explore More Summaries from Ekeeda 📚 | {"url":"https://glasp.co/youtube/p/introduction-to-thermal-boundary-layer-in-circular-convection-heat-transfer-heat-transfer","timestamp":"2024-11-04T00:59:32Z","content_type":"text/html","content_length":"356975","record_id":"<urn:uuid:3c396def-cd67-4b85-9dc7-5db35eaf08e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00267.warc.gz"} |
Why Does The Division Get rounded To An Integer? - Python4U
Why Does The Division Get rounded To An Integer?
In this tutorial, we will learn Why does the division gets rounded to an integer in python we have different ways of dividing two digits like by ‘/’ or ‘//’ ‘/’ is used for getting the exact value
after dividing whereas in case of ‘//’ it is not the same case it will simply get floor value.
And here in this post, we will get to know the exact differences between both the ways to divide the numbers and when we need which one or we can say when we should use which operator to divide the
As we know When we divide a number there is a prob that we might get the output in the form of decimal numbers so at some times we might want to avoid such cases when we do not want to get the
decimal values in that case we have a number of ways to avoid the decimal values which could be round off, floor values and we can simply truncate the decimal values.
The use of all three methods depends upon the requirements of the output as at some times we simply need the floor value and at the time we need to round off the digits that we got after calculations
as the floor values get changed in case of round off.
Here will take the example of all three cases and try to find out the difference between them and how they all differ from each other.
Let’s understand the difference between all of these.
import math
# Truncating a float to an integer
num = 3.14159
truncated_num = int( num)
print( truncated_ num) # Output: 3
# Truncating a decimal to a certain number of decimal places
dec = 3.14159
truncated_ dec = math.trunc( dec * 100) / 100
print( truncated_ dec) # Output: 3.14
import math
# Rounding a float to the nearest integer
num = 3.6
rounded_ num = round( num)
print(rounded_num) # Output: 4
# Rounding a decimal to a certain number of decimal places
dec = 3.14159
rounded_ dec = round( dec, 2)
print (rounded_ dec) # Output: 3.14
import math
# Flooring a float to the nearest integer
num = 3.6
floored_ num = math. floor( num)
print( floored_ num) # Output: 3
# Flooring a decimal to a certain number of decimal places
dec = 3.14159
floored_ dec = math. floor( dec * 100) / 100
print( floored_ dec) # Output: 3.14
To learn more about Why does the division gets rounded to an integer visit: Rounding off of decimal numbers.
To learn more about python solutions of problems and concepts visit: Python Tutorials And Problems.
To learn more solutions and concepts of different other programming languages visit: beta python programming languages Solutions.
Leave a Comment | {"url":"https://betapython.com/why-does-the-division-get-rounded-to-an-integer/","timestamp":"2024-11-11T06:11:54Z","content_type":"text/html","content_length":"91800","record_id":"<urn:uuid:3429bd80-dd10-4863-a750-271001b83a7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00455.warc.gz"} |
Exploring Number Theory and Digit Sums
Elon Tusk ๐
Exploring Number Theory and Digit Sum: Unveiling New Possibilities in Exploreing Large Numbers ๐ ข๐
Number theory, the study of integers and their properties, has captivated mathematicians for centuries. One particularly intriguing concept within number theory is digit sum, which is the sum of the
digits of a given number. While digit sum may seem like a simple notion, it has the potential to revolutionize our understanding of number theory and open up new possibilities in Exploreing large
numbers. ๐ ๐
Understanding Digit Sum ๐ งฎ
The digit sum of a number is calculated by adding up all of its individual digits. For example, the digit sum of 1234 is 1 + 2 + 3 + 4 = 10. Digit sum has several interesting properties and
applications in number theory, such as:
• Divisibility tests: The digit sum can be used to quickly check if a number is divisible by certain divisors, such as 3 or 9. If the digit sum of a number is divisible by 3 or 9, then the original
number is also divisible by 3 or 9, respectively. ๐ ขโ
• Recursive digit sum: The process of calculating the digit sum can be repeated on the resulting sum until a single-digit number is obtained. This is known as the recursive digit sum or digital
root. For example, the recursive digit sum of 1234 is 1, as 1 + 2 + 3 + 4 = 10, and 1 + 0 = 1. ๐ ๐ ฟ
Digit Sum and Exploreing Large Numbers ๐ ๐
Exploreing large numbers is a crucial problem in cryptography and computer science, as many encryption systems rely on the difficulty of Exploreing large composite numbers. Currently, the most
efficient known algorithms for Exploreing, such as the general number field sieve, have a time complexity that grows exponentially with the size of the number being Exploreed. ๐ ป๐
However, exploring the relationship between digit sum and the Explores of a number could potentially lead to new insights and more efficient Exploreing methods. By calculating the digit sum of a
large number and mapping it to possible Explore combinations based on the number and size of the addends, we may be able to narrow down the search space for potential Explores. ๐ โ
For example, consider the number 1234567890. Its digit sum is 45, which can be obtained by adding various combinations of numbers, such as:
• 9 + 9 + 9 + 9 + 9 = 45
• 18 + 18 + 9 = 45
• 22 + 23 = 45
By analyzing these addend combinations and their relationships to the Explores of the original number, we may uncover new patterns and insights that could help us develop more efficient Exploreing
algorithms. ๐ ก๐ ข
Other Applications of Digit Sum ๐ ๐
Digit sum has numerous applications beyond number theory and Exploreing, including:
1. Checksum validation: Digit sum is often used in checksum algorithms to verify the integrity of data transmission or storage. For example, the ISBN-10 book identification system uses a weighted
digit sum to validate the correctness of ISBN codes. ๐ โ ๏ธ
2. Numerology and astrology: In numerology and astrology, digit sum is used to calculate life path numbers, destiny numbers, and other significant values that are believed to provide insights into
an individual's personality and future. ๐ ฎโ จ
3. Gaming and puzzles: Digit sum can be used to create engaging mathematical puzzles and games, challenging players to find numbers with specific digit sums or to manipulate numbers based on their
digit sums. ๐ ฎ๐ งฉ
Conclusion ๐
As we continue to explore the fascinating world of number theory and digit sum, we may uncover new ways to approach complex problems like Exploreing large numbers. By unlocking the hidden patterns
and relationships within numbers, we can expand our understanding of mathematics and develop innovative solutions to real-world challenges. ๐ ๐ ก
The potential applications of digit sum extend far beyond number theory, impacting fields such as data validation, numerology, and gaming. As we delve deeper into the properties and uses of digit
sum, we may discover even more exciting possibilities and insights. ๐ ๐ ซ
So, let us embrace the power of number theory and digit sum, and embark on a journey of mathematical exploration and discovery. Who knows what groundbreaking ideas and solutions await us as we
unravel the mysteries of numbers? ๐ ๐ ญ | {"url":"https://www.elontusk.org/blog/Math/DigitSum","timestamp":"2024-11-10T03:15:13Z","content_type":"text/html","content_length":"33806","record_id":"<urn:uuid:8c8d4379-d1a1-4ef4-81c0-32aeb578c422>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00659.warc.gz"} |
AIEEE 2006 | Waves Question 123 | Physics | JEE Main - ExamSIDE.com
AIEEE 2006
MCQ (Single Correct Answer)
A string is stretched between fixed points separated by $$75.0$$ $$cm.$$ It is observed to have resonant frequencies of $$420$$ $$Hz$$ and $$315$$ $$Hz$$. There are no other resonant frequencies
between these two. Then, the lowest resonant frequency for this string is
AIEEE 2005
MCQ (Single Correct Answer)
When two tuning forks (fork $$1$$ and fork $$2$$) are sounded simultaneously, $$4$$ beats per second are heated. Now, some tape is attached on the prong of the fork $$2.$$ When the tuning forks are
sounded again, $$6$$ beats per second are heard. If the frequency of fork $$1$$ is $$200$$ $$Hz$$, then what was the original frequency of fork $$2$$ ?
AIEEE 2005
MCQ (Single Correct Answer)
Out of Syllabus
An observer moves towards a stationary source of sound, with a velocity one-fifth of the velocity of sound. What is the percentage increase in the apparent frequency ?
AIEEE 2004
MCQ (Single Correct Answer)
The displacement $$y$$ of a particle in a medium can be expressed as, $$y = {10^{ - 6}}\,\sin $$ $$\left( {100t + 20x + {\pi \over 4}} \right)$$ $$m$$ where $$t$$ is in second and $$x$$ in meter. The
speed of the wave is | {"url":"https://questions.examside.com/past-years/jee/question/a-string-is-stretched-between-fixed-points-separated-by-75-2006-marks-4-boo8t7fzf1xcgko3.htm","timestamp":"2024-11-11T10:43:26Z","content_type":"text/html","content_length":"304848","record_id":"<urn:uuid:5ac57460-9b27-45ec-a3e4-9c54a7c6396e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00163.warc.gz"} |
Method for calculating a density of stem cells in a cell image, electronic device, and storage medium
Method for calculating a density of stem cells in a cell image, electronic device, and storage medium
A method for calculating a density of stem cells in a cell image and an electronic device are provided. A plurality of preset ratios and a plurality of density calculation models can be used to
perform hierarchical density calculations on the cell image. Starting from the largest preset ratio (the first preset ratio) reduction of the cell image to no reduction, the density calculation is
performed on the cell image using a model starting with a highest density calculation (the first density calculation model) to a model with the smallest density calculation (the third density
calculation model), which can quickly detect densities of various stem cells. Using different preset ratios and corresponding density calculation models for calculation, it is not necessary to
calculate the number of stem cells to obtain the density of stem cells, which improves a calculation efficiency of the density of stem cells.
The present disclosure relates to a technical field of imaging, specifically a method for calculating a density of stem cells in a cell image, an electronic device, and a storage medium.
By calculating the number and volume of stem cells shown in an image, the actual density of stem cells can be calculated or estimated. However, known methods of calculating the number and volume of
stem cells in the image may have lower efficiencies.
Therefore, a rapid estimation or calculation of the density of stem cells shown in an image is desirable.
FIG. 1 shows a flowchart of a method for calculating a density of stem cells in a cell image provided in an embodiment of the present disclosure.
FIG. 2 shows a schematic structural diagram of a device for calculating a density of stem cells in a cell image provided in an embodiment of the present disclosure.
FIG. 3 shows a schematic structural diagram of an electronic device provided in an embodiment of the present disclosure.
The accompanying drawings combined with the detailed description illustrate the embodiments of the present disclosure hereinafter. It is noted that embodiments of the present disclosure and features
of the embodiments can be combined when there is no conflict.
Various details are described in the following descriptions for a better understanding of the present disclosure, however, the present disclosure may also be implemented in other ways other than
those described herein. The scope of the present disclosure is not to be limited by the specific embodiments disclosed below.
Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the present disclosure belongs. The
terms used herein in the present disclosure are only for the purpose of describing specific embodiments and are not intended to limit the present disclosure.
Optionally, the method for calculating a density of stem cells in a cell image of the present disclosure is applied to one or more electronic devices. The electronic device includes hardware such as,
but not limited to, a microprocessor and an Application Specific Integrated Circuit (ASIC), Field-Programmable Gate Array (FPGA), Digital Signal Processor (DSP), embedded devices, etc.
The electronic device may be a device such as a desktop computer, a notebook, a palmtop computer, or a cloud server. The electronic device can interact with users by using a keyboard, a mouse, a
remote control, a touch panel, or a voice control device.
FIG. 1 is a flowchart of a method for calculating a density of stem cells in a cell image in an embodiment of the present disclosure. The method for calculating a density of stem cells in a cell
image is applied to electronic devices. According to different needs, the order of the steps in the flowchart can be changed, and some can be omitted.
In block S11, acquiring a cell image.
The cell image refers to an image of cells that needs to be analyzed regarding a density of the stem cells shown in the cell image. The cells shown in the cell image may include, but is not limited
to, stem cells, other cells, and impurities.
In some embodiments, before acquiring the cell image, the method further includes: acquiring a plurality of first training images; selecting, from the plurality of first training images, images with
a density of stem cells greater than or equal to a preset first density as first positive sample images, and images with a density of stem cells less than the preset first density as first negative
sample images; reducing image sizes of the first positive sample images and image sizes of the first negative sample images according to a preset first ratio; training and obtaining a first density
calculation model with the reduced first positive sample images and the reduced first negative sample images.
The plurality of first training images are pre-collected images for training the first density calculation model. These first training images include cell images with different densities of stem
cells. The images with a density of stem cells greater than or equal to the preset first density can be selected as the first positive sample images, and the images with a density of stem cells less
than the preset first density can be selected as the first negative sample images. The preset first density may be 80%. Assuming that the preset first ratio is 60%, image sizes of the reduced first
positive sample images and image sizes of the reduced first negative sample images are the first training images reduced by 60%. By training the reduced first positive sample images and the reduced
first negative sample images, the first density calculation model is obtained. The first density calculation model can perform a density calculation on the cell image reduced by 60%.
In some embodiments, before acquiring the cell image, the method further includes: acquiring a plurality of second training images; for a target second ratio of the plurality of preset second ratios,
selecting, from the plurality of second training images, images with a density of stem cells greater than or equal to a preset second density corresponding to the target second ratio as second
positive sample images, and images with a density of stem cells less than the preset second density corresponding to the target second ratio as second negative sample images; reducing image sizes of
the second positive sample images and image sizes of the second negative sample images according to the target second ratio; training and obtaining a second density calculation model corresponding to
the target second ratio with the reduced second positive sample images and the reduced second negative sample images.
The plurality of second training images are pre-collected images for training the second density calculation models. Each second density calculation model corresponds to each preset second ratio.
These second training images include cell images with different densities of stem cells. The images with a density of stem cells greater than or equal to each preset second density can be selected as
the second positive sample images, and the images with a density of stem cells less than the each preset second density can be selected as the second negative sample images. Assuming that one preset
second density is 60% and one preset second ratio corresponding to the one preset second density is 40%, image sizes of the reduced second positive sample images and image sizes of the reduced second
negative sample images are the second training images reduced by 40%. By training the reduced second positive sample images and the reduced second negative sample images, the second density
calculation model is obtained. The second density calculation model can perform a density calculation on the cell image reduced by 40%.
In some embodiments, before acquiring the cell image, the method further includes: acquiring a plurality of third training images; selecting, from the plurality of third training images, images with
a density of stem cells greater than or equal to a preset third density as third positive sample images, and images with a density of stem cells less than the preset third density as third negative
sample images; training and obtaining a third density calculation model with the reduced third positive sample images and the reduced third negative sample images.
The plurality of third training images are pre-collected images for training the third density calculation model. These third training images include cell images with different densities of stem
cells. The images with a density of stem cells greater than or equal to the preset third density can be selected as the third positive sample images, and the images with a density of stem cells less
than the preset third density can be selected as the third negative sample images. The preset third density may be 10%, for example. By training the reduced third positive sample images and the
reduced third negative sample images, the third density calculation model is obtained. The third density calculation model can perform a density calculation on the cell image that has not been
In block S12, reducing an image size of the cell image according to a preset first ratio and obtaining a first reduced image.
In an embodiment of the present disclosure, the image size of the cell images may be reduced according to the preset first ratio to obtain the first reduced image. Assuming that the preset first
ratio is 60%, the first reduced image is the cell images reduced by 60%. The size of the first reduced image is 40% of the size of the cell image.
In block S13, calculating a density of stem cells in the first reduced image by using a pre-trained first density calculation model.
The first density calculation model is used to calculate density of stem cells in the first reduced image. After calculating the density of stem cells in the first reduced image by using the
pre-trained first density calculation model, the electronic device determines whether the density of stem cells in the first reduced image is greater than or equal to the preset first density. The
preset first density may be a greater value, such as 80%. The density of stem cells in the first reduced image is greater than or equal to the preset first density (for example, 80%), or the density
of stem cells in the first reduced image is less than the preset first density (for example, 80%).
In block S14, if the density of stem cells in the first reduced image is greater than or equal to the preset first density, outputting a result that a density of stem cells in the cell image is
greater than or equal to the preset first density.
Assuming that the preset first density is 80%, if the density of stem cells in the first reduced image is greater than or equal to the preset first density, a text prompt or warning giving
information that “a density of stem cells is greater than or equal to 80%” can be output.
In block S15, if the density of stem cells in the first reduced image is less than the preset first density, calculating at least one density of stem cells in the cell image according to a plurality
of preset second ratios and a plurality of pre-trained second density calculation models.
The plurality of preset second ratios are set in advance. The plurality of preset second ratios are in one-to-one correspondence with the plurality of pre-trained second density calculation models.
The preset first ratio is greater than each of the plurality of preset second ratios. For example, the preset first ratio may be 60%, and the plurality of preset second ratios may be 50%, 40%, 30%,
20%, and so on. The density of the plurality of pre-trained second density calculation models corresponding to the plurality of preset second ratios may be 60%, 50%, 40%, 30%, and so on.
In some embodiments, the method of calculating at least one density of stem cells in the cell image according to a plurality of preset second ratios and a plurality of pre-trained second density
calculation models includes: obtaining a largest second ratio among the plurality of preset second ratios; reducing the image size of the cell image according to the largest second ratio and
obtaining a second reduced image; calculating a density of stem cells in the second reduced image by using a pre-trained second density calculation model corresponding to the largest second ratio;
when the density of stem cells in the second reduced image is less than a preset second density corresponding to the largest second ratio, obtaining a second largest second ratio among the plurality
of preset second ratios; reducing the image size of the cell image according to the second largest second ratio and obtaining a third reduced image; calculating a density of stem cells in the third
reduced image by using the pre-trained second density calculation model corresponding to the second largest second ratio.
Assuming that the plurality of preset second ratios are 50%, 40%, 30%, and 20%, a sorting result obtained by sorting the plurality of preset second ratios is 50%, 40%, 30%, and 20%. Among the
plurality of preset second ratios, the largest second ratio is 50%, the second largest second ratio is 40%, the third largest second ratio is 30%, and the smallest second ratio is 20%.
First, the image size of the cell image can be reduced by 50% to obtain the second reduced image, and then the second reduced image can be input into a second density calculation model corresponding
to the preset second ratio of 50%, thus the density of stem cells in the second reduced image can be obtained. The density of stem cells in the second reduced image can be less than a preset second
density corresponding to the preset second ratio 50% or be greater than or equal to the preset second density corresponding to the largest second ratio 50%.
If the density of stem cells in the second reduced image is less the preset second density corresponding to the largest second ratio 50%, the image size of the cell image is reduced by 40% (the
second largest second ratio is 40%) to obtain the third reduced image, and then the third reduced image can be input into a second density calculation model corresponding to the preset second ratio
of 40%, thus the density of stem cells in the third reduced image can be obtained.
If the density of stem cells in the third reduced image is greater than or equal to a preset second density corresponding to the largest second ratio 40%, then according to the sorting result, the
third largest second ratio among the plurality of preset second ratios is 30%, the image size of the cell image is reduced by 30%, and a second density calculation model corresponding to the preset
second ratio of 30% is used to calculate a density of stem cells in the cell images reduced by 30%.
In some embodiments, if the density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to the largest second ratio, the method further
includes: determining that the density of stem cells in the cell image is greater than or equal to the preset second density corresponding to the largest second ratio but less than the preset first
The preset first density is greater than each preset second density. Each preset second density is greater than the preset third density.
In the above embodiment, when the density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to the largest second ratio, because the preset
first density is greater than each preset second density, it indicates that a density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to
the largest second ratio, but less than the preset first density.
When it is determined that the density of stem cells in the cell image is greater than or equal to the preset second density corresponding to the largest second ratio, but less than the preset first
density, the calculation of the density of the stem cell in the cell image ends.
In some embodiments, if the at least one density of stem cells in the cell image is less than the preset second density, the method further includes: calculating a density of stem cells in the cell
image by using the pre-trained third density calculation model.
The preset third density, in comparison with the preset first density and each preset second density, is a relatively small density, such as 10%.
When it is determined that the at least one density of stem cells in the second reduced image is less than the preset second density, the cell image do not need to be reduced at this time, and the
cell image is directly input into the third density calculation model. An output result of the third density calculation model may be that the density of stem cells in the cell image is less than the
preset third density (for example, 10%), or that the density of stem cells in the cell image is greater than or equal to the preset third density (for example, 10%) but less than the preset second
density (for example, 20%).
The preset first ratio, each preset second ratio, and the preset third ratio are all values greater than 0 and less than 1.
In the method provided by the embodiment of the present disclosure, a plurality of preset ratios and a plurality of density calculation models can be used to perform hierarchical density calculations
on the cell image. Starting from the largest preset ratio (the first preset ratio) reduction of the cell image to no reduction, the density calculation is performed on the cell image using a model
starting with a highest density calculation (the first density calculation model) to a model with the smallest density calculation (the third density calculation model), which can quickly detect
densities of various stem cells. For the reduced images, those with a higher density of stem cells are easy to be detected, and those with a lower density of stem cells are not easy to be detected.
Using different preset ratios and corresponding density calculation models for calculation, it is not necessary to calculate the number of stem cells to obtain the density of stem cells, which
improves a calculation efficiency of the density of stem cells.
FIG. 2 shows a schematic structural diagram of a device for calculating a density of stem cells in a cell image provided in the embodiment of the present disclosure.
In some embodiments, the device for calculating a density of stem cells in a cell image 2 runs in an electronic device. The device for calculating a density of stem cells in a cell image 2 can
include a plurality of function modules consisting of program code segments. The program code of each program code segments in the device for calculating a density of stem cells in a cell image 2 can
be stored in a memory and executed by at least one processor to perform image processing (described in detail in FIG. 2).
As shown in FIG. 2, the device for calculating a density of stem cells in a cell image 2 can include: an acquisition module 201, a reduction module 202, a training module 203, a calculation module
204, and an output module 205. A module as referred to in the present disclosure refers to a series of computer-readable instruction segments that can be executed by at least one processor and that
are capable of performing fixed functions, which are stored in a memory. In some embodiment, the functions of each module will be detailed.
The above-mentioned integrated unit implemented in a form of software functional modules can be stored in a non-transitory readable storage medium. The above software function modules are stored in a
storage medium and includes several instructions for causing an electronic device (which can be a personal computer, a dual-screen device, or a network device) or a processor to execute the method
described in various embodiments in the present disclosure.
The acquisition module 201 acquires a cell image.
The cell image refers to an image of cells that needs to be analyzed regarding a density of the stem cells shown in the cell image. The cells shown in the cell image may include, but is not limited
to, stem cells, other cells, and impurities.
In some embodiments, before acquiring the cell image, the acquisition module 201 acquires a plurality of first training images. The acquisition module 201, selects, from the plurality of first
training images, images with a density of stem cells greater than or equal to a preset first density as first positive sample images, and images with a density of stem cells less than the preset
first density as first negative sample images. The reduction module 202 reduces image sizes of the first positive sample images and image sizes of the first negative sample images according to a
preset first ratio. The training module 203 trains and obtains a first density calculation model with the reduced first positive sample images and the reduced first negative sample images.
The plurality of first training images are pre-collected images for training the first density calculation model. These first training images include cell images with different densities of stem
cells. The images with a density of stem cells greater than or equal to the preset first density can be selected as the first positive sample images, and the images with a density of stem cells less
than the preset first density can be selected as the first negative sample images. The preset first density may be 80%. Assuming that the preset first ratio is 60%, image sizes of the reduced first
positive sample images and image sizes of the reduced first negative sample images are the first training images reduced by 60%. By training the reduced first positive sample images and the reduced
first negative sample images, the first density calculation model is obtained. The first density calculation model can perform a density calculation on the cell image reduced by 60%.
In some embodiments, before acquiring the cell image, the acquisition module 201 acquires a plurality of second training images. For a target second ratio of the plurality of preset second ratios,
the acquisition module 201, selects, from the plurality of second training images, images with a density of stem cells greater than or equal to a preset second density corresponding to the target
second ratio as second positive sample images, and images with a density of stem cells less than the preset second density corresponding to the target second ratio as second negative sample images.
The reduction module 202 reduces image sizes of the second positive sample images and image sizes of the second negative sample images according to the target second ratio. The training module 203
trains and obtains a second density calculation model corresponding to the target second ratio with the reduced second positive sample images and the reduced second negative sample images.
The plurality of second training images are pre-collected images for training the second density calculation models. Each second density calculation model corresponds to each preset second ratio.
These second training images include cell images with different densities of stem cells. The images with a density of stem cells greater than or equal to each preset second density can be selected as
the second positive sample images, and the images with a density of stem cells less than the each preset second density can be selected as the second negative sample images. Assuming that one preset
second density is 60% and one preset second ratio corresponding to the one preset second density is 40%, image sizes of the reduced second positive sample images and image sizes of the reduced second
negative sample images are the second training images reduced by 40%. By training the reduced second positive sample images and the reduced second negative sample images, the second density
calculation model is obtained. The second density calculation model can perform a density calculation on the cell image reduced by 40%.
In some embodiments, before acquiring the cell image, the acquisition module 201 acquires a plurality of third training images. The acquisition module 201, selects, from the plurality of third
training images, images with a density of stem cells greater than or equal to a preset third density as third positive sample images, and images with a density of stem cells less than the preset
third density as third negative sample images. The training module 203 trains and obtains a third density calculation model with the reduced third positive sample images and the reduced third
negative sample images.
The plurality of third training images are pre-collected images for training the third density calculation model. These third training images include cell images with different densities of stem
cells. The images with a density of stem cells greater than or equal to the preset third density can be selected as the third positive sample images, and the images with a density of stem cells less
than the preset third density can be selected as the third negative sample images. The preset third density may be 10%, for example. By training the reduced third positive sample images and the
reduced third negative sample images, the third density calculation model is obtained. The third density calculation model can perform a density calculation on the cell image that has not been
The reduction module 202 reduces an image size of the cell image according to a preset first ratio and obtaining a first reduced image.
In an embodiment of the present disclosure, the image size of the cell images may be reduced according to the preset first ratio to obtain the first reduced image. Assuming that the preset first
ratio is 60%, the first reduced image is the cell images reduced by 60%. The size of the first reduced image is 40% of the size of the cell image.
The calculation module 204 calculates a density of stem cells in the first reduced image by using a pre-trained first density calculation model.
The first density calculation model is used to calculate density of stem cells in the first reduced image. After calculating the density of stem cells in the first reduced image by using the
pre-trained first density calculation model, the electronic device determines whether the density of stem cells in the first reduced image is greater than or equal to the preset first density. The
preset first density may be a greater value, such as 80%. The density of stem cells in the first reduced image is greater than or equal to the preset first density (for example, 80%), or the density
of stem cells in the first reduced image is less than the preset first density (for example, 80%).
The output module 205, if the density of stem cells in the first reduced image is greater than or equal to the preset first density, outputs a result that a density of stem cells in the cell image is
greater than or equal to the preset first density.
Assuming that the preset first density is 80%, if the density of stem cells in the first reduced image is greater than or equal to the preset first density, a text prompt or warning giving
information that “a density of stem cells is greater than or equal to 80%” can be output.
If the density of stem cells in the first reduced image is less than the preset first density, the calculation module 204 calculates at least one density of stem cells in the cell image according to
a plurality of preset second ratios and a plurality of pre-trained second density calculation models.
The plurality of preset second ratios are set in advance. The plurality of preset second ratios are in one-to-one correspondence with the plurality of pre-trained second density calculation models.
The preset first ratio is greater than each of the plurality of preset second ratios. For example, the preset first ratio may be 60%, and the plurality of preset second ratios may be 50%, 40%, 30%,
20%, and so on. The density of the plurality of pre-trained second density calculation models corresponding to the plurality of preset second ratios may be 60%, 50%, 40%, 30%, and so on.
In some embodiments, the method of calculating at least one density of stem cells in the cell image according to a plurality of preset second ratios and a plurality of pre-trained second density
calculation models includes: obtaining a largest second ratio among the plurality of preset second ratios; reducing the image size of the cell image according to the largest second ratio and
obtaining a second reduced image; calculating a density of stem cells in the second reduced image by using a pre-trained second density calculation model corresponding to the largest second ratio;
when the density of stem cells in the second reduced image is less than a preset second density corresponding to the largest second ratio, obtaining a second largest second ratio among the plurality
of preset second ratios; reducing the image size of the cell image according to the second largest second ratio and obtaining a third reduced image; calculating a density of stem cells in the third
reduced image by using the pre-trained second density calculation model corresponding to the second largest second ratio.
Assuming that the plurality of preset second ratios are 50%, 40%, 30%, and 20%, a sorting result obtained by sorting the plurality of preset second ratios is 50%, 40%, 30%, and 20%. Among the
plurality of preset second ratios, the largest second ratio is 50%, the second largest second ratio is 40%, the third largest second ratio is 30%, and the smallest second ratio is 20%.
First, the image size of the cell image can be reduced by 50% to obtain the second reduced image, and then the second reduced image can be input into a second density calculation model corresponding
to the preset second ratio of 50%, thus the density of stem cells in the second reduced image can be obtained. The density of stem cells in the second reduced image can be less than a preset second
density corresponding to the preset second ratio 50% or be greater than or equal to the preset second density corresponding to the largest second ratio 50%.
If the density of stem cells in the second reduced image is less the preset second density corresponding to the largest second ratio 50%, the image size of the cell image is reduced by 40% (the
second largest second ratio is 40%) to obtain the third reduced image, and then the third reduced image can be input into a second density calculation model corresponding to the preset second ratio
of 40%, thus the density of stem cells in the third reduced image can be obtained.
If the density of stem cells in the third reduced image is greater than or equal to a preset second density corresponding to the largest second ratio 40%, then according to the sorting result, the
third largest second ratio among the plurality of preset second ratios is 30%, the image size of the cell image is reduced by 30%, and a second density calculation model corresponding to the preset
second ratio of 30% is used to calculate a density of stem cells in the cell images reduced by 30%.
In some embodiments, if the density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to the largest second ratio, the output module 205
determines that the density of stem cells in the cell image is greater than or equal to the preset second density corresponding to the largest second ratio but less than the preset first density.
The preset first density is greater than each preset second density. Each preset second density is greater than the preset third density.
In the above embodiment, when the density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to the largest second ratio, because the preset
first density is greater than each preset second density, it indicates that a density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to
the largest second ratio, but less than the preset first density.
When it is determined that the density of stem cells in the cell image is greater than or equal to the preset second density corresponding to the largest second ratio, but less than the preset first
density, the calculation of the density of the stem cell in the cell image ends.
In some embodiments, if the at least one density of stem cells in the cell image is less than the preset second density, the method further includes: calculating a density of stem cells in the cell
image by using the pre-trained third density calculation model.
The preset third density, in comparison with the preset first density and each preset second density, is a relatively small density, such as 10%.
When it is determined that the at least one density of stem cells in the second reduced image is less than the preset second density, the cell image do not need to be reduced at this time, and the
cell image is directly input into the third density calculation model. An output result of the third density calculation model may be that the density of stem cells in the cell image is less than the
preset third density (for example, 10%), or that the density of stem cells in the cell image is greater than or equal to the preset third density (for example, 10%) but less than the preset second
density (for example, 20%).
The preset first ratio, each preset second ratio, and the preset third ratio are all values greater than 0 and less than 1.
In the device provided by the embodiment of the present disclosure, a plurality of preset ratios and a plurality of density calculation models can be used to perform hierarchical density calculations
on the cell image. Starting from the largest preset ratio (the first preset ratio) reduction of the cell image to no reduction, the density calculation is performed on the cell image using a model
starting with a highest density calculation (the first density calculation model) to a model with the smallest density calculation (the third density calculation model), which can quickly detect
densities of various stem cells. For the reduced images, those with a higher density of stem cells are easy to be detected, and those with a lower density of stem cells are not easy to be detected.
Using different preset ratios and corresponding density calculation models for calculation, it is not necessary to calculate the number of stem cells to obtain the density of stem cells, which
improves a calculation efficiency of the density of stem cells.
The embodiment also provides a non-transitory readable storage medium having computer-readable instructions stored therein. The computer-readable instructions are executed by a processor to implement
the steps in the above-mentioned method for calculating a density of stem cells in a cell image, such as in steps in blocks S11-S15 shown in FIG. 1:
In block S11: acquiring a cell image;
In block S12: reducing an image size of the cell image according to a preset first ratio and obtaining a first reduced image;
In block S13: calculating a density of stem cells in the first reduced image by using a pre-trained first density calculation model;
In block S14, if the density of stem cells in the first reduced image is greater than or equal to the preset first density, outputting a result that a density of stem cells in the cell image is
greater than or equal to the preset first density;
In block S15: if the density of stem cells in the first reduced image is less than the preset first density, calculating at least one density of stem cells in the cell image according to a plurality
of preset second ratios and a plurality of pre-trained second density calculation models.
The computer-readable instructions are executed by the processor to realize the functions of each module/unit in the above-mentioned device embodiments, such as the modules 201-205 in FIG. 2.
FIG. 3 is a schematic structural diagram of an electronic device provided in an embodiment of the present disclosure. The electronic device 3 may include: a memory 31, at least one processor 32, and
computer-readable instructions 33 stored in the memory 31 and executable on the at least one processor 32, for example, image recognition programs. memory 31 and the at least one processor 32
connected by at least one communication bus 34. The processor 32 executes the computer-readable instructions to implement the steps in the embodiment of the method for calculating a density of stem
cells in a cell image, such as in steps in block S11-S15 shown in FIG. 1. Alternatively, the processor 32 executes the computer-readable instructions to implement the functions of the modules/units
in the foregoing device embodiments, such as the modules 201-205 in FIG. 2.
For example, the computer-readable instructions can be divided into one or more modules/units, and the one or more modules/units are stored in the memory 31 and executed by the at least one processor
32. The one or more modules/units can be a series of computer-readable instruction segments capable of performing specific functions, and the instruction segments are used to describe execution
processes of the computer-readable instructions in the electronic device 3. For example, the computer-readable instruction can be divided into the acquisition module 201, the reduction module 202,
the training module 203, the calculation module 204, and the output module 205 as in FIG. 2.
The electronic device 3 can be an electronic device such as a desktop computer, a notebook, a palmtop computer, and a cloud server. Those skilled in the art will understand that the schematic diagram
3 is only an example of the electronic device 3 and does not constitute a limitation on the electronic device 3. Another electronic device 3 may include more or fewer components than shown in the
figures or may combine some components or have different components. For example, the electronic device 3 may further include an input/output device, a network access device, a bus, and the like.
The at least one processor 32 can be a central processing unit (CPU), or can be another general-purpose processor, digital signal processor (DSPs), application-specific integrated circuit (ASIC),
Field-Programmable Gate Array (FPGA), another programmable logic device, discrete gate, transistor logic device, or discrete hardware component, etc. The processor 32 can be a microprocessor or any
conventional processor. The processor 32 is a control center of the electronic device 3 and connects various parts of the entire electronic device 3 by using various interfaces and lines.
The memory 31 can be configured to store the computer-readable instructions and/or modules/units. The processor 32 may run or execute the computer-readable instructions 33 and/or modules/units stored
in the memory 31 and may call up data stored in the memory 31 to implement various functions of the electronic device 3. The memory 31 mainly includes a storage program area and a storage data area.
The storage program area may store an operating system, and an application program required for at least one function (such as a sound playback function, an image playback function, etc.), etc. The
storage data area may store data (such as audio data, phone book data, etc.) created according to the use of the electronic device 3. In addition, the memory 31 may include a high-speed random access
memory, and may also include a non-transitory storage medium, such as a hard disk, an internal memory, a plug-in hard disk, a smart media card (SMC), a secure digital (SD) Card, a flashcard, at least
one disk storage device, a flash memory device, or another non-transitory solid-state storage device.
When the modules/units integrated into the electronic device 3 are implemented in the form of software functional units having been sold or used as independent products, they can be stored in a
non-transitory readable storage medium. Based on this understanding, all or part of the processes in the methods of the above embodiments implemented by the present disclosure can also be completed
by related hardware instructed by computer-readable instructions. The computer-readable instructions can be stored in a non-transitory readable storage medium. The computer-readable instructions,
when executed by the processor, may implement the steps of the foregoing method embodiments. The computer-readable instructions include computer-readable instruction codes, and the computer-readable
instruction codes can be in a source code form, an object code form, an executable file, or some intermediate form. The non-transitory readable storage medium can include any entity or device capable
of carrying the computer-readable instruction code, such as a recording medium, a U disk, a mobile hard disk, a magnetic disk, an optical disk, a computer memory, or a read-only memory (ROM).
In the several embodiments provided in the preset application, the disclosed electronic device and method can be implemented in other ways. For example, the embodiments of the devices described above
are merely illustrative. For example, divisions of the units are only logical function divisions, and there can be other manners of division in actual implementation.
In addition, each functional unit in each embodiment of the present disclosure can be integrated into one processing unit, or can be physically present separately in each unit or two or more units
can be integrated into one unit. The above modules can be implemented in a form of hardware or in a form of a software functional unit.
The present disclosure is not limited to the details of the above-described exemplary embodiments, and the present disclosure can be embodied in other specific forms without departing from the spirit
or essential characteristics of the present disclosure. Therefore, the present embodiments are to be considered as illustrative and not restrictive, and the scope of the present disclosure is defined
by the appended claims. All changes and variations in the meaning and scope of equivalent elements are included in the present disclosure. Any reference sign in the claims should not be construed as
limiting the claim. Furthermore, the word “comprising” does not exclude other units nor does the singular exclude the plural. A plurality of units or devices stated in the system claims may also be
implemented by one unit or device by using software or hardware. Words such as “first” and “second” are used to indicate names, but not in any particular order.
Finally, the above embodiments are only used to illustrate technical solutions of the present disclosure and are not to be taken as restrictions on the technical solutions. Although the present
disclosure has been described in detail with reference to the above embodiments, those skilled in the art should understand that the technical solutions described in one embodiment can be modified,
or some of the technical features can be equivalently substituted, and that these modifications or substitutions are not to detract from the essence of the technical solutions or from the scope of
the technical solutions of the embodiments of the present disclosure.
1. A method for calculating a density of stem cells in a cell image, the method comprising:
acquiring a cell image;
reducing an image size of the cell image according to a preset first ratio and obtaining a first reduced image;
calculating a density of stem cells in the first reduced image by using a pre-trained first density calculation model; and
when the density of stem cells in the first reduced image is less than a preset first density, calculating at least one density of stem cells in the cell image according to a plurality of preset
second ratios and a plurality of pre-trained second density calculation models, wherein the plurality of preset second ratios are in one-to-one correspondence with the plurality of pre-trained
second density calculation models.
2. The method for calculating a density of stem cells in a cell image of claim 1, the method further comprising:
when the at least one density of stem cells in the cell image is less than a preset second density, calculating a density of stem cells in the cell image by using a pre-trained third density
calculation model.
3. The method for calculating a density of stem cells in a cell image of claim 1, wherein calculating at least one density of stem cells in the cell image according to a plurality of preset second
ratios and a plurality of pre-trained second density calculation models comprises:
obtaining a largest second ratio among the plurality of preset second ratios;
reducing the image size of the cell image according to the largest second ratio and obtaining a second reduced image;
calculating a density of stem cells in the second reduced image by using a pre-trained second density calculation model corresponding to the largest second ratio;
when the density of stem cells in the second reduced image is less than a preset second density corresponding to the largest second ratio, obtaining a second largest second ratio among the
plurality of preset second ratios;
reducing the image size of the cell image according to the second largest second ratio and obtaining a third reduced image; and
calculating a density of stem cells in the third reduced image by using a pre-trained second density calculation model corresponding to the second largest second ratio.
4. The method for calculating a density of stem cells in a cell image of claim 3, the method further comprising:
if the density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to the largest second ratio, determining that the density of stem
cells in the cell image is greater than or equal to the preset second density corresponding to the largest second ratio but less than the preset first density.
5. The method for calculating a density of stem cells in a cell image of claim 4, before acquiring the cell image, the method further comprising:
acquiring a plurality of first training images;
selecting, from the plurality of first training images, images with a density of stem cells greater than or equal to the preset first density as first positive sample images, and images with a
density of stem cells less than the preset first density as first negative sample images;
reducing image sizes of the first positive sample images and image sizes of the first negative sample images according to the preset first ratio; and
training and obtaining the first density calculation model with the reduced first positive sample images and the reduced first negative sample images.
6. The method for calculating a density of stem cells in a cell image of claim 5, before acquiring the cell image, the method further comprising:
acquiring a plurality of second training images;
for a target second ratio of the plurality of preset second ratios, selecting, from the plurality of second training images, images with a density of stem cells greater than or equal to a preset
second density corresponding to the target second ratio as second positive sample images, and images with a density of stem cells less than the preset second density corresponding to the target
second ratio as second negative sample images;
reducing image sizes of the second positive sample images and image sizes of the second negative sample images according to the target second ratio; and
training and obtaining a second density calculation model corresponding to the target second ratio with the reduced second positive sample images and the reduced second negative sample images.
7. The method for calculating a density of stem cells in a cell image of claim 6, before acquiring the cell image, the method further comprising:
acquiring a plurality of third training images;
selecting, from the plurality of third training images, images with a density of stem cells greater than or equal to a preset third density as third positive sample images, and images with a
density of stem cells less than the preset third density as third negative sample images;
training and obtaining the third density calculation model with the reduced third positive sample images and the reduced third negative sample images.
8. A electronic device comprising a memory and a processor, the memory stores at least one computer-readable instruction, and the processor executes the at least one computer-readable instruction to:
acquire a cell image;
reduce an image size of the cell image according to a preset first ratio and obtain a first reduced image;
calculate a density of stem cells in the first reduced image by using a pre-trained first density calculation model; and
when the density of stem cells in the first reduced image is less than a preset first density, calculate at least one density of stem cells in the cell image according to a plurality of preset
second ratios and a plurality of pre-trained second density calculation models, wherein the plurality of preset second ratios are in one-to-one correspondence with the plurality of pre-trained
second density calculation models.
9. The electronic device of claim 8, wherein the processor further to:
when the at least one density of stem cells in the cell image is less than a preset second density, calculate a density of stem cells in the cell image by using a pre-trained third density
calculation model.
10. The electronic device of claim 8, wherein calculating at least one density of stem cells in the cell image according to a plurality of preset second ratios and a plurality of pre-trained second
density calculation models comprises:
obtaining a largest second ratio among the plurality of preset second ratios;
reducing the image size of the cell image according to the largest second ratio and obtaining a second reduced image;
calculating a density of stem cells in the second reduced image by using a pre-trained second density calculation model corresponding to the largest second ratio;
when the density of stem cells in the second reduced image is less than a preset second density corresponding to the largest second ratio, obtaining a second largest second ratio among the
plurality of preset second ratios;
reducing the image size of the cell image according to the second largest second ratio and obtaining a third reduced image; and
calculating a density of stem cells in the third reduced image by using a pre-trained second density calculation model corresponding to the second largest second ratio.
11. The electronic device of claim 10, wherein the processor further to:
if the density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to the largest second ratio, determining that the density of stem
cells in the cell image is greater than or equal to the preset second density corresponding to the largest second ratio but less than the preset first density.
12. The electronic device of claim 11, before acquiring the cell image, wherein the processor further to:
acquire a plurality of first training images;
select, from the plurality of first training images, images with a density of stem cells greater than or equal to the preset first density as first positive sample images, and images with a
density of stem cells less than the preset first density as first negative sample images;
reduce image sizes of the first positive sample images and image sizes of the first negative sample images according to the preset first ratio; and
train and obtain the first density calculation model with the reduced first positive sample images and the reduced first negative sample images.
13. The electronic device of claim 12, before acquiring the cell image, the processor further to:
acquire a plurality of second training images;
for a target second ratio of the plurality of preset second ratios, select, from the plurality of second training images, images with a density of stem cells greater than or equal to a preset
second density corresponding to the target second ratio as second positive sample images, and images with a density of stem cells less than the preset second density corresponding to the target
second ratio as second negative sample images;
reduce image sizes of the second positive sample images and image sizes of the second negative sample images according to the target second ratio; and
train and obtain a second density calculation model corresponding to the target second ratio with the reduced second positive sample images and the reduced second negative sample images.
14. The electronic device of claim 13, before acquiring the cell image, wherein the processor further to:
acquire a plurality of third training images;
select, from the plurality of third training images, images with a density of stem cells greater than or equal to a preset third density as third positive sample images, and images with a density
of stem cells less than the preset third density as third negative sample images; and
train and obtain the third density calculation model with the reduced third positive sample images and the reduced third negative sample images.
15. A non-transitory storage medium having stored thereon at least one computer-readable instructions that, when the at least one computer-readable instructions are executed by a processor to
implement the following method:
acquiring a cell image;
reducing an image size of the cell image according to a preset first ratio and obtaining a first reduced image;
calculating a density of stem cells in the first reduced image by using a pre-trained first density calculation model; and
when the density of stem cells in the first reduced image is less than a preset first density, calculating at least one density of stem cells in the cell image according to a plurality of preset
second ratios and a plurality of pre-trained second density calculation models, wherein the plurality of preset second ratios are in one-to-one correspondence with the plurality of pre-trained
second density calculation models.
16. The non-transitory storage medium according to claim 15, the method further comprising:
when the at least one density of stem cells in the cell image is less than a preset second density, calculating a density of stem cells in the cell image by using a pre-trained third density
calculation model.
17. The non-transitory storage medium of claim 15, wherein calculating at least one density of stem cells in the cell image according to a plurality of preset second ratios and a plurality of
pre-trained second density calculation models comprises:
obtaining a largest second ratio among the plurality of preset second ratios;
reducing the image size of the cell image according to the largest second ratio and obtaining a second reduced image;
calculating a density of stem cells in the second reduced image by using a pre-trained second density calculation model corresponding to the largest second ratio;
when the density of stem cells in the second reduced image is less than a preset second density corresponding to the largest second ratio, obtaining a second largest second ratio among the
plurality of preset second ratios;
reducing the image size of the cell image according to the second largest second ratio and obtaining a third reduced image; and
calculating a density of stem cells in the third reduced image by using a pre-trained second density calculation model corresponding to the second largest second ratio.
18. The non-transitory storage medium of claim 17, the method further comprising:
if the density of stem cells in the second reduced image is greater than or equal to the preset second density corresponding to the largest second ratio, determining that the density of stem
cells in the cell image is greater than or equal to the preset second density corresponding to the largest second ratio but less than the preset first density.
19. The non-transitory storage medium of claim 18, before acquiring the cell image, the method further comprising:
acquiring a plurality of first training images;
selecting, from the plurality of first training images, images with a density of stem cells greater than or equal to the preset first density as first positive sample images, and images with a
density of stem cells less than the preset first density as first negative sample images;
reducing image sizes of the first positive sample images and image sizes of the first negative sample images according to the preset first ratio; and
training and obtaining the first density calculation model with the reduced first positive sample images and the reduced first negative sample images.
20. The non-transitory storage medium of claim 19, before acquiring the cell image, the method further comprising:
acquiring a plurality of second training images;
for a target second ratio of the plurality of preset second ratios, selecting, from the plurality of second training images, images with a density of stem cells greater than or equal to a preset
second density corresponding to the target second ratio as second positive sample images, and images with a density of stem cells less than the preset second density corresponding to the target
second ratio as second negative sample images;
reducing image sizes of the second positive sample images and image sizes of the second negative sample images according to the target second ratio; and training and obtaining a second density
calculation model corresponding to the target second ratio with the reduced second positive sample images and the reduced second negative sample images.
Referenced Cited
U.S. Patent Documents
20170350805 December 7, 2017 Murata
20190156947 May 23, 2019 Nakamura
20190228527 July 25, 2019 Ramirez
20200097701 March 26, 2020 Chukka
20210295960 September 23, 2021 Kalkstein
20210303818 September 30, 2021 Randolph
Foreign Patent Documents
109461495 March 2019 CN
111492437 August 2020 CN
Other references
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14. Springer International Publishing, 2016, pp. 274-290. (Year: 2016).
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1160-1166. (Year: 2016).
• Xie, Weidi, J. et al. “Microscopy cell counting and detection with fully convolutional regression networks.” Computer methods in biomechanics and biomedical engineering: Imaging & Visualization
6.3 (2018), pp. 1-10 (Year: 2018).
• Guo, Yue, et al. “Sau-net: A universal deep network for cell counting.” Proceedings of the 10th ACM international conference on bioinformatics, computational biology and health informatics, 2019,
pp. 299-306 (Year: 2019).
• He, Shenghua, et al. “Automatic microscopic cell counting by use of deeply-supervised density regression model.” Medical Imaging 2019: Digital Pathology. vol. 10956. SPIE, 2019, pp. 1-8. (Year:
• Zhang, Dongdong, et al. “Cell counting algorithm based on YOLOv3 and image density estimation.” 2019 IEEE 4th international conference on signal and image processing (ICSIP). IEEE, 2019, pp.
920-924. (Year: 2019).
• Jiang, Ni, et al. “A cell counting framework based on random forest and density map.” Applied sciences (Nov. 24, 2020), pp. 1-18 (Year: 2020).
• Ding, Xin, et al. “Classification Beats Regression: Counting of Cells from Greyscale Microscopic Images based on Annotation-free Training Samples.” arXiv preprint arXiv:2010.14782 (Oct. 29,
2020), pp. 1-9 (Year: 2020).
• He, Shenghua, et al. “Deeply-Supervised Density Regression for Automatic Cell Counting in Microscopy Images.” arXiv preprint arXiv:2011.03683 (Nov. 11, 2020), pp. 1-29. (Year: 2020).
• Shenghua He, Kyaw Thu Minn, Lilianna Solnica-Krezel, Mark Anastasio, and Hua Li “Automatic microscopic cell counting by use of deeply-supervised density regression model”, Proc. SPIE 10956,
Medical Imaging 2019: Digital Pathology, 109560L, Mar. 18, 2019 (Year: 2019).
• Shenghua He, Kyaw Thu Minn, Lilianna Solnica-Krezel, Mark A. Anastasio, Hua Li, Deeply-supervised density regression for automatic cell counting in microscopy images, Medical Image Analysis, vol.
68, 2021, 101892 (Year: 2021).
Patent History
Patent number
: 12111244
: Nov 11, 2021
Date of Patent
: Oct 8, 2024
Patent Publication Number
20220178814 Assignee
HON HAI PRECISION INDUSTRY CO., LTD.
(New Taipei)
Wan-Jhen Lee
(New Taipei),
Chin-Pin Kuo
(New Taipei),
Chih-Te Lu
(New Taipei)
Primary Examiner
Alex Kok S Liew Application Number
: 17/523,989
International Classification: G06K 9/00 (20220101); G01N 15/1433 (20240101); G01N 33/483 (20060101); G06T 7/00 (20170101); | {"url":"https://patents.justia.com/patent/12111244","timestamp":"2024-11-14T03:32:06Z","content_type":"text/html","content_length":"129791","record_id":"<urn:uuid:b675422a-d10c-407d-b6fa-eb22721b17ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00088.warc.gz"} |
Python bin() Function
The popular number system types named “Binary Numbers” are represented by “1” or “0” and are expressed in base 2. In Python, the “0b” is the string format representation of the binary. Sometimes,
while dealing with a number system in Python, we need to convert the different number systems into a binary system. To do this the “bin()” method is utilized in Python.
This Python tutorial will offer you a comprehensive tutorial on the Python “bin()” function via the below contents:
What is the “bin()” Function in Python?
The “bin()” function in Python is a built-in function that converts an integer or a string to its binary representation. The prefix “0b” will always precede the retrieved value.
In this syntax, the “number” parameter represents an integer number that needs to be converted into its binary representation.
Return Value
If an integer is given, the “bin()” method returns a string that represents its binary equivalent. Otherwise, it raises a TypeError for a non-integer argument
Example 1: Retrieving the Binary Representation of Integer
In this code, the integer “num” variable is passed to the “bin()” method to retrieve the binary representation:
The above code retrieves the binary representation:
Example 2: Retrieving the Binary Representation of Non-Integer
In the below code, the “non-integer” value is passed to the “bin()” method to retrieve the TypeError. Take the following code as an example:
num = 'one'
According to the following output, the above code retrieves the TypeError:
Example 3: Retrieving the Binary Representation of Integer Using User-Defined Function
We defined a function that accepts an integer as an argument and returns its binary form without the “0b” prefix that indicates a binary number:
def bin_func(num):
n = bin(num)
return n[2:]
The above code retrieves the binary representation:
Example 4: Retrieving the Binary Representation of Custom Object Using the “bin()” Function and “index()” Method
Here, the class named “num” is defined with an attribute named “num”. Next, the class special method called “__index__” is defined. This method defines how the object can be converted into an integer
and retrieve the attribute value. The bin() function is called on the instance of class num:
class num:
num = 15
def __index__(self):
return self.num
The above output retrieves the below output:
Example 5: Retrieving the Binary Representation of Hexadecimal and Octal Value
In the below-provided code block, the “bin()” function takes the hexadecimal and octal representation value as well as retrieves the binary representation:
hexadecimal = 0xf
octal = 0o17
The binary representation of the input code is shown below:
In Python, the “bin()” function is utilized to convert a decimal, octal, or hexadecimal value into its binary representation. This function retrieves the TypeError when the non-integer is passed as
an argument. We can also use the “bin()” function along with the “index()” method to convert the custom object to binary representation. This tutorial demonstrated a comprehensive tutorial on
Python’s “bin()” function. | {"url":"https://linuxhint.com/python-bin-function/","timestamp":"2024-11-02T05:04:00Z","content_type":"text/html","content_length":"170856","record_id":"<urn:uuid:32b70f7e-026f-429e-bb85-7453897bd441>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00749.warc.gz"} |
In the limit lim x^-2=oo as x->0, how do you find delta>0 such that whenever 0<absx<delta, x^-2>10,000? | HIX Tutor
In the limit #lim x^-2=oo# as #x->0#, how do you find #delta>0# such that whenever #0<absx<delta#, #x^-2>10,000#?
Answer 1
We want #delta# so that #0 < x < delta# implies #x^-2 > 10,000#.
(Note that #0 < x# makes the absolute value superfluous.)
Solve #delta ^-2 > 10,000#
#100 > delta#, so if #0 < x < delta# then
#1/x^2 > 1/ delta^2 > 10,000#
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Answer 2
To find a delta value such that whenever 0 < |x| < delta, x^-2 > 10,000, we can start by rewriting the inequality as x^-2 > 10,000. Taking the reciprocal of both sides, we get x^2 < 1/10,000. Now,
let's solve for x. Taking the square root of both sides, we have |x| < 1/100. This means that whenever 0 < |x| < 1/100, x^-2 > 10,000 holds true. Therefore, we can choose delta = 1/100 as a suitable
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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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what happens in high speed of ball mill
WEBJul 5, 2017 · Effects of Ball Milling Velocity and Ball Volume Fraction. EDEM as a powerful software enables to collect data from the dynamic behavior of the entire ball milling simulation
process. In order to explore the milling efficiency of the models, the average speed of balls, the maximum speed of balls, and the magnitude of torque on . | {"url":"https://www.restaurantsanremo.fr/7434-what_happens_in_high_speed_of_ball_mill.html","timestamp":"2024-11-02T18:51:52Z","content_type":"text/html","content_length":"39673","record_id":"<urn:uuid:53217893-2726-4f9d-960c-3ce62e62bf1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00450.warc.gz"} |
Heat Transfer by Radiation in Combustion Systems in context of Heat Transfer by Radiation
31 Aug 2024
Heat Transfer by Radiation in Combustion Systems: A Crucial Aspect of Thermal Management
In combustion systems, heat transfer plays a vital role in the efficient operation and performance of various industrial processes, such as power generation, chemical synthesis, and propulsion
systems. Among the three primary modes of heat transfer – conduction, convection, and radiation – radiation is particularly significant in high-temperature combustion environments. In this article,
we will delve into the concept of heat transfer by radiation in combustion systems, exploring its importance, mechanisms, and mathematical formulations.
Importance of Radiation Heat Transfer
In combustion systems, temperatures can reach extremely high levels, often exceeding 1000°C (1832°F). At these elevated temperatures, conduction and convection become less effective due to the
reduced thermal conductivity and convective heat transfer coefficients. In contrast, radiation heat transfer remains an efficient mechanism for transferring energy over long distances, even in the
presence of obstacles or complex geometries.
Radiation heat transfer is particularly important in combustion systems because it:
1. Enhances efficiency: By allowing heat to be transferred more efficiently, radiation reduces the temperature gradients and thermal stresses within the system.
2. Reduces emissions: Radiation can help reduce NOx and particulate matter (PM) emissions by minimizing the formation of pollutants at high temperatures.
3. Improves performance: Radiation heat transfer can improve the overall performance of combustion systems by optimizing fuel consumption, reducing energy losses, and increasing power output.
Mechanisms of Radiation Heat Transfer
Radiation heat transfer occurs when electromagnetic radiation is emitted by hot surfaces or particles and absorbed by cooler surfaces or particles. The process involves three main steps:
1. Emission: Hot surfaces or particles emit thermal radiation in the form of photons.
2. Transmission: Photons travel through the medium (gas, solid, or liquid) until they are absorbed or scattered.
3. Absorption: Cooler surfaces or particles absorb the incident radiation, converting it into heat energy.
Mathematical Formulations
To quantify radiation heat transfer, we can use various mathematical formulations, such as:
1. Stefan-Boltzmann Law: The total radiative flux (q) emitted by a surface is proportional to its temperature (T) and the fourth power of its absolute temperature (T^4):
q = εσ(T^4)
where ε is the emissivity, σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W/m²K^4), and T is the surface temperature in Kelvin.
1. Planck’s Law: The spectral radiative flux (q_λ) emitted by a surface at a specific wavelength (λ) can be calculated using Planck’s law:
q_λ = ε(λ) × B(λ, T)
where B(λ, T) is the Planck function and ε(λ) is the spectral emissivity.
1. Radiative Heat Transfer Coefficient: The radiative heat transfer coefficient (h_r) can be calculated using the following formula:
h_r = εσ(T^4) / (1 - ε)
where ε is the emissivity, σ is the Stefan-Boltzmann constant, and T is the surface temperature in Kelvin.
Heat transfer by radiation plays a vital role in combustion systems, enabling efficient energy transfer over long distances. Understanding the mechanisms and mathematical formulations of radiation
heat transfer is crucial for optimizing the performance and efficiency of various industrial processes. By applying these principles, engineers can design more effective thermal management
strategies, reducing emissions and improving overall system performance.
1. Incropera, F. P., & DeWitt, D. P. (2002). Fundamentals of Heat and Mass Transfer. John Wiley & Sons.
2. Holman, J. P. (2010). Heat Transfer. McGraw-Hill Education.
3. Viskanta, R. (1985). Radiation heat transfer. In Advances in Heat Transfer (Vol. 15, pp. 223-310). Academic Press.
About the Author
[Your Name] is a thermal engineer with expertise in heat transfer and combustion systems. With a strong background in mathematical modeling and simulation, [Your Name] has published several papers on
radiation heat transfer and its applications in various industrial processes.
Related articles for ‘Heat Transfer by Radiation’ :
• Reading: Heat Transfer by Radiation in Combustion Systems in context of Heat Transfer by Radiation
Calculators for ‘Heat Transfer by Radiation’ | {"url":"https://blog.truegeometry.com/tutorials/education/9900fb9f323edf78cceef0b1687dfbd6/JSON_TO_ARTCL_Heat_Transfer_by_Radiation_in_Combustion_Systems_in_context_of_Hea.html","timestamp":"2024-11-12T22:40:49Z","content_type":"text/html","content_length":"19785","record_id":"<urn:uuid:165388af-a0d2-487c-bce1-11f17d46b305>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00022.warc.gz"} |
Fraction Calculations Made Easy – Learn How to Calculate Fractions - My Blog
Fraction Calculations Made Easy – Learn How to Calculate Fractions
Are you facing problems in mathematical calculations? If yes, you then are not by myself. Millions of people, mainly college students, face extreme challenges in managing math issues. Fraction
calculation is one of the maximum challenging tasks for maximum college students.
If you need to research the ways of calculating fractions, read this weblog until the give up.
This article will discuss fractions and a number of the first-class methods to calculate them correctly. So, let’s begin without any similarly ado!
What are Fractions?thelasergal.com
In mathematics, fractions are the numerical portions that denote values less than one. A fraction may be a part of any larger quantity out of an entire, wherein the entire may be any variety or a
component. The following instance will help you apprehend this idea easily.
Imagine you have a pizza that is divided into 8 same parts. Now, in case you need to indicate anybody selected part of the pizza, you may define it as 1/eight, which suggests that out of 8 same
components, we are relating to 1 part of pizza.
So, fractional numbers are especially used to degree components of an entire. For example,
One-0.33 (half of)quincyoffers.com
One-fourth (1/4)gemcitybeat.com
Two-thirds (2/three)
Basic Elements of Fractionsandrealchin.com
A fraction has a easy-to-examine structure. There is a numerator and a denominator in a fraction divided by means of a line referred to as the fractional bar.
The integer above the bar which receives divided is the numerator. Similarly, the integer underneath the fractional bar which divides is known as the denominator.
● Numerator
The numerator defines how many fractional elements are decided on. The numerator is placed just above the fractional bar inside the higher segment of the fraction.
For example: In the fraction x/y, the numerator is x.
● Denominatorpasfait.com
The denominator denotes the quantity of identical additives into which a hollow could be divided. The denominator is located below the fractional bar in the lower segment of the fraction. The
denominator specifies what number of portions the whole may be divided into.
For example: In the fraction x/y, the denominator is y.
● Fraction Bar
The fraction bar is the line that separates the numerator from the denominator.
Types of Fractions
Fractions have different types primarily based at the numerator and denominator. Some of them are discussed underneath.
1. Unit Fractionfastsquaring.com
This is the kind of fraction in which the numerator is 1.
For example: half of, 1/4, 1/8, and greater.
2. Proper Fraction
These are the fractions in which the numerator is smaller than its denominator.
For instance: 3/7, five/eight, four/5, and many others.
3. Improper Fractionsupplycommon.com
An improper fraction is a fragment wherein the numerator value is greater than the denominator value.
For Example: 8/5, 18/10, etc.
Four. Mixed fractionslinuxpatent.com
A blended fraction is a fraction that is the mixture of a whole quantity and a right fraction.
For instance, five ¾, where five is the entire variety, and three/4 is the right fraction.
Five. Like fractions
These are the fractions which have the equal denominators.
For example: three/10, 2/10, 7/10, and 1/10, and so forth.
6. Unlike fractions
If fractions have extraordinary denominators, then they will be referred to as unlike fractions.
For instance: five/7, 9-11, 2/15, and 23/36, etc.
7. Equivalent fractions
A fraction with exceptional numerators and denominators but equal whilst decreased to its handiest shape.
To locate equivalent fractions of any shared fraction:
Multiply the numerator and the denominator of the fraction through the equal variety.
Divide the numerator and the denominator of the fraction through the equal range.
Let’s discover the 2 fractions which might be equivalent to a few/5.
Equivalent Fraction 1: Multiply the numerator and the denominator with the equal number 2.
Three/five= (three × 2) / (five × 2) = 6/10
Equivalent Fraction 2: Multiply the numerator and the denominator with the same number three.
This manner, three/five = (3 × three) / (5 × 3) = 9/15
Therefore, 6/10, 9/15, and 3/5 are equivalent fractions.
Rules for Fraction Calculation
Following are a few regulations you need to examine before fixing issues primarily based on fractions.
Rule #1: It is important to make sure that denominators are equal earlier than including or subtracting fractions. Thus, you could use a not unusual denominator to feature or subtract fractions.
Rule #2: The numerators and denominators are improved whenever we multiply two fractions. You want to simplify the fraction after that.
Rule #three: We need to locate the reciprocal of any other fraction and then multiply with the primary one to get the solution to divide the fraction from some other fraction.
How to Calculate Fractions?
Calculating fractions is hard, in particular in case you don’t recognize the calculation strategies. However, there are one-of-a-kind methods to calculate fractions, which includes the subsequent.
1- Adding or Subtracting Fractions
● Identify Fractions with Like Denominatorssuperbglove.com
To add or subtract any fraction, make sure they’ve common denominators before you’re making your calculations. So, look at the denominators of the fractions to ensure they’re the equal.
For example: 1/five + four/five
● Find A Common Denominator If the Denominators Are Unlike
If the denominators within the fractions are not the identical, then it’s miles important in an effort to exchange the fractions to have the same denominators. To search for a commonplace
denominator, multiply every a part of the fraction via the denominator of the alternative fraction.
For example, to test the not unusual denominator for 1/four + 4/5, multiply 1 and 4 by using five, and multiply 4 and five through 4. You will get 5/20 + sixteen/20. Now, you can calculate the
● Add or Subtract the Numerators to Calculate the Fractions
After finding the commonplace denominator and multiplying the numerators if required, it’s time to feature or subtract. Add or subtract the numerators, mention the output over a dividing line, and
place the not unusual denominator below the road.
For instance:
5/20 + sixteen/20 = 21/20.
It is essential to remember the fact that denominators gained’t be introduced or subtracted.
● Simply The Sum
If the fractions don’t have a common denominator, and you have needed to discover it, then you may have a massive fraction that may be simplified.
For example, when you have a resultant cost of 32/40, then you may simplify it to 4/5.
2- Multiplying and Simplifying Fractions
● Turn Mixed Fractions or Whole Numbers into Improper Fractions
To multiply without ambiguity, you need to paintings with right or wrong fractions. If you have got an entire variety or combined fraction that you want to multiply, truly convert it into its
For example, to multiply three/6 by 9, turn 9 into a fragment. Then, you could multiply three/6 by using nine/1.
If you have got a blended fraction like 2. 1/5, convert it into an wrong fraction, 11/five, earlier than you multiply.
● Multiply The Numerators and Denominators
Rather than adding the numerators, multiply each of the numerators and write the result over your dividing line. You are also required to multiply the denominators and mention the end result beneath
the road.
For example, multiply 2/five via 5/6 and multiply 2 by using five to get the numerator. Multiply five by using 6 to get the denominator. Your answer might be 10/30.
● Simplify Your Result
In unique situations, you’ll be required to reduce the end result to a simplified fraction, especially if you begin with incorrect fractions. Find the greatest common component and use it to simplify
the numerator and denominator.
For instance, if your answer is 10/30, then 10 is the greatest commonplace issue. Reduce the fraction by using 10 to get 1/three.
Three- Dividing Fractions
● Invert The Second Fraction
Flipping the second one fraction is the handiest way to divide fractions, despite the fact that they have in contrast to denominators.
For example, with 10/8 ÷ 2/four, you should turn the two/4 fraction to seem 4/2.
● Multiply The Numerators and Denominators
After inverting the second one fraction, multiply the fractions at once in front of the numerators to multiply them. Mention the result over a dividing line and multiply the denominators. Mention the
result beneath the dividing line.
To continue the instance, multiply 10/8 via four/2 to get 40/sixteen.
● Simplify The Results
If the consequent solution is an wrong fraction or can be decreased, simplify the fraction.
Use the greatest commonplace issue to decrease the fraction.
For instance, the finest not unusual element for 10/8 is two, so your simplified solution is 5/four.
Since that is an improper fraction, convert it into a whole range with a fraction. 5/four turns into 1. 1/four.
Solving fractions is a arduous task for lots students. However, calculating fractions is something every student needs to analyze to finish the given assignments. The information shared in this blog
submit enables you find out about fractions, their differing types and methods to calculate fractions accurately. | {"url":"https://foggle.xyz/2024/05/09/fraction-calculations-made-easy-learn-how-to-calculate-fractions/","timestamp":"2024-11-14T18:16:54Z","content_type":"text/html","content_length":"53485","record_id":"<urn:uuid:407459ba-3820-4b9f-ac1d-2b5516a1c198>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00429.warc.gz"} |
: A New Harmony
My book is an in-depth music theory text, outlining a musical notation based on prime-factorization of frequency ratios, and the design of both my lattice diagrams and planetary graphs to
represent those prime factors. The second part of the book is a very extensive examination of different historical just-intonation tuning systems, as well as an exploration of myriad different
tunings used by contemporary microtonal composers. It's been about 14 years in the works.
I should state at the outset that I find it impossible to believe that "atonality" can exist, since there are always proportional relationships between simultaneously-heard notes, whether simple
and clearly perceived, or complex and vague.
My work began originally as an attempt to create what would be for me a more lucid microtonal notation, avoiding the proliferation of large numbers in ratios, and enabling me to display the
pitches on a staff with as much standard music-notation convention as possible.
As I've explored the subject more deeply, and debated it with colleagues on the Tuning Forum, it has grown into something far deeper and more meaningful to me. I have by now become caught up in
the numerological belief that there is something special inherent in the series of prime numbers, as a final reduction or simplification of our understanding of musical harmony or tonality.
The introductory chapters present the background on human perception of pitch and our increasing sophistication in notating musical pitches.
Next are mathematical explanations of ratios, equal-temperaments, and prime-factoring. Then I describe a graph of sonance, Matrix Addition for the calculation of intervals, and my Planetary
The rest of the book shows how my lattice diagrams are designed by making use of this prime factorization, with the diagrams becoming more and more complex as I give an overview thru both
increasing prime limits and a comprehensive historical chronology.
Partch claimed that
"In terms of consonance man's use of musical materials has followed the scale of musical intervals [which] begins with absolute consonance (1 to 1) and progresses into an infinitude of
dissonance, the consonance of the intervals decreasing as the odd numbers of their ratios increase."
Although I find an element of truth in this, I wish to show by my manner of presentation that suprisingly large prime numbers were a part of many older theories.
The historical aspect gives all of the different just-intonation tuning systems I have found in doing my research, presented from one viewpoint--one which, I have found, makes it easier to
understand how different tuning theories have evolved and interacted with each other.
It goes back to ancient Greek and Indian systems, analyzes the work of several medieval European theorists, including in-depth surveys of Boethius's On Music, the Musica Enchiriadis, and
Marchetto of Padua's Lucidarium, and progresses all the way up to present-day composers such as La Monte Young, Ben Johnston, Ezra Sims, and many younger composers, with special emphasis given to
Harry Partch and Arnold Schoenberg. I also present what I believe are the first microtonal analyses of gifted 'popular' composer/performers such as Robert Johnson and Jimi Hendrix.
As can be inferred by my inclusion of Schoenberg, I have also given some reference to some of the more popular equal-temperaments, especially (unavoidably)12-equal, and how some theorists have
"justified" or explained the use of these temperaments by emphasis on their good or bad representation of particular ratios.
I'm still working on the 4th edition of it (October 2000), so it's not really ready to be printed. A text of the 3rd edition is available for download here in a zip file (291 K), in Microsoft Word
format (212 pages). It does not include any musical examples or lattice diagrams. | {"url":"http://www.tonalsoft.com/sonic-arts/monzo/book/book.htm","timestamp":"2024-11-06T08:54:05Z","content_type":"text/html","content_length":"8137","record_id":"<urn:uuid:428eb96b-404f-456b-a626-3106d0f5416a>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00262.warc.gz"} |
Lesson 9
Perimeter Problems
Warm-up: Estimation Exploration: Statue of Liberty (10 minutes)
The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information.
• Groups of 2
• Display the image.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
Student Facing
The Statue of Liberty has two square bases—one larger than the other. The larger base has side lengths of 132 feet each.
Estimate the perimeter of the smaller square base.
Record an estimate that is:
│ too low │ about right │ too high │
│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)│
Activity Synthesis
• “If you wanted to know the perimeter of the star-shaped base, how would you find it?” (We’d need to know the length of each side and add the lengths together. If the lengths were the same we
could count them and multiply the length by number of sides.)
• Consider asking:
□ “Based on this discussion does anyone want to revise their estimate?”
Activity 1: Missing Measurements (15 minutes)
The purpose of this activity is for students to find the length of a missing side of a shape when the perimeter is given, using any strategy that makes sense to them. The synthesis highlights the
variety of methods students used to solve the problem.
• Groups of 2
• “In an earlier lesson, we found the perimeter of shapes when not all the side lengths were labeled. Now, let’s find some missing side lengths when we know the perimeter.”
• 5–7 minutes: partner work time
• Monitor for students who:
□ subtract each side length from the perimeter
□ double a given side length, subtract the result from the perimeter, and divide to find the other two sides
□ divide the perimeter when the side lengths are all equal
Student Facing
1. This pentagon has a perimeter of 32 cm. What is the length of the missing side? Explain or show your work.
2. This rectangle has a perimeter of 56 feet. What are the lengths of the unlabeled sides? Explain or show your work.
3. This pentagon has a perimeter of 65 inches. All the sides are the same length. What is the length of each side? Explain or show your work.
Activity Synthesis
• Select previously identified students to share their strategies. Be sure to share at least one method (more if possible) for each problem.
• Consider asking:
□ “When would this strategy be most useful?”
□ “Did anyone think about it in a different way?”
Activity 2: Can I Use Perimeter? (20 minutes)
The purpose of this activity is for students to solve problems in situations that involve perimeter (MP2). Students may draw diagrams with length labels or simply reason arithmetically. They also
explain how each problem does or does not involve perimeter. The activity synthesis provides an opportunity to begin discussing the difference between area and perimeter, which will be fully explored
in upcoming lessons.
MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any
questions about the context.
Advances: Reading, Representing
Engagement: Provide Access by Recruiting Interest. Synthesis: Invite students to generate a list of additional examples of needing to know the perimeter of an object or space that connect to their
personal backgrounds and interests.
Supports accessibility for: Language, Visual-Spatial Processing
• “Take some time to solve these problems on your own.”
• 5 minutes: independent work time
• “Share with your partner your reasoning on your favorite problem.”
• 2 minutes: partner discussion
• Monitor for a variety of ways students solve these problems, such as by drawing a diagram or writing expressions or equations. Identify one student to share for each problem, with a variety of
ways shown across the problems.
Student Facing
Solve each problem. Explain or show your reasoning.
1. A rectangular park is 70 feet on the shorter side and 120 feet on the longer side. How many feet of fencing is needed to enclose the boundary of the park?
2. Priya drew a picture and is framing it with a ribbon. Her picture is square and one side is 9 inches long. How many inches of ribbon will she need?
3. A rectangular flower bed has a fence that measures 32 feet around. One side of the flower bed measures 12 feet. What are the lengths of the other sides?
4. Kiran took his dog for a walk. Their route is shown. How many blocks did they walk?
5. A room is 10 feet by 8 feet. How many tiles will be needed to cover the floor if each tile is 1 square foot?
Advancing Student Thinking
If students say they aren’t sure how to get started on a problem, consider asking:
• “What is the problem about?”
• “How could you represent the problem?”
Activity Synthesis
• Select previously identified students to share their reasoning for each problem. After each problem, consider asking: “Did anyone solve this problem in a different way?”
• If possible, keep the student work displayed for the lesson synthesis.
Lesson Synthesis
“Look back through the problems you solved in the last activity. Discuss with your partner whether each problem involves perimeter.”
“How do you know if a situation involves perimeter?” (If it’s about finding the distance around something. If answering the question means adding up all the side lengths of a shape.)
“Why was perimeter not useful in the last problem about tiling a floor?” (The perimeter would give the length around the outside of the room, not how many tiles covered the whole room. To know how
many tiles cover the whole room is to find the area of the room.)
“What is the difference between perimeter and area?” (Perimeter is the distance around the outside of a shape. Area is the amount of space a shape covers.)
Cool-down: Sides of a Pool (5 minutes)
Student Facing
In this section, we learned that perimeter is the boundary of a flat shape.
We can find the length of a perimeter by adding the lengths of all the sides, or by using multiplication when there are sides with the same length.
\((2 \times 9) + (2 \times 21)\)
We used our knowledge of shapes to find the perimeter even when some side lengths were missing, and to use the perimeter to find missing side lengths.
For example, if we know the perimeter of this rectangle is 32 feet, we can find the lengths of the three unlabeled sides. | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-3/unit-7/lesson-9/lesson.html","timestamp":"2024-11-03T07:43:31Z","content_type":"text/html","content_length":"101574","record_id":"<urn:uuid:0df3fe85-c043-41f7-a0e7-f45b0e8339cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00518.warc.gz"} |